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We use a second-order rotational invariant Green’s function method (RGM) and the high-temperature expansion (HTE) to calculate the thermodynamic properties, of the kagome-lattice spin-$S$ Heisenberg antiferromagnet with nearest-neighbor exchange $J$. While the HTE yields accurate results down to temperatures of about $T/S(S+1) \sim J$, the RGM provides data for arbitrary $T \ge 0$. For the ground state we use the RGM data to analyze the $S$-dependence of the excitation spectrum, the excitation velocity, the uniform susceptibility, the spin-spin correlation functions, the correlation length, and the structure factor. We found that the so-called $\sqrt{3}\times\sqrt{3}$ ordering is more pronounced than the $q=0$ ordering for all values of $S$. In the extreme quantum case $S=1/2$ the zero-temperature correlation length is only of the order of the nearest-neighbor separation. Then we study the temperature dependence of several physical quantities for spin quantum numbers $S=1/2,1,\dots,7/2$. As increasing $S$ the typical maximum in the specific heat and in the uniform susceptibility are shifted towards lower values of $T/S(S+1)$ and the height of the maximum is growing. The structure factor ${\cal S}(\mathbf{q})$ exhibits two maxima at magnetic wave vectors $\mathbf{q}={\mathbf{Q}_i}, i=0,1,$ corresponding to the $q=0$ and $\sqrt{3}\times\sqrt{3}$ state. We find that the $\sqrt{3}\times \sqrt{3}$ short-range order is more pronounced than the $q=0$ short-range order for all temperatures $T \ge 0$. For the spin-spin correlation functions, the correlation lengths and the structure factors, we find a finite low-temperature region $0 \le T < T^*\approx a/S(S+1)$, $a
\approx 0.2$, where these quantities are almost independent of $T$.
author:
- 'P. Müller'
- 'A. Zander'
- 'J. Richter'
bibliography:
- 'JR\_RGM.bib'
title: 'Thermodynamics of the kagome-lattice Heisenberg antiferromagnet with arbitrary spin $S$.'
---
Introduction\
{#sec:intro}
=============
One of the most prominent and at the same time challenging spin models with a frustration induced highly degenerated classical ground state (GS) manifold is the kagome Heisenberg antiferromagnet (KHAF) [@Elser1990; @Chalker1992; @Harris1992; @Singh1992; @Reimers1993; @elstner1993; @elstner1994; @Henley:1995; @Nakamura1995; @tomczak1996thermodynamical; @Waldtmann1998; @Yu2000; @Lhuillier_thermo_PRL2000; @Huber2001; @Bernhard2002; @Schmalfus2004; @Bernu2005; @Singh2007; @Li2007; @Rigol2007; @Zhitomirsky2008; @DMRG_PRL08; @Laeuchli2009; @Evenbly2010; @Goetze2011; @Nakano2011; @Iqbal2011; @Yan2011; @Laeuchli2011; @Depenbrock2012; @Rousochatzakis2013; @Iqbal2013; @Rousochatzakis2014; @Xie2014; @Lohmann2014; @Munehisha2014; @Kolley2015; @Goetze2015; @Changlani2015; @Liu2015; @Picot2015; @Nishimoto2015; @Liu2016; @Shimokawa2016; @Oitmaa2016; @Goetze2016; @Laeuchli2016; @Pollmann2017; @Xie2017; @Singh2017; @Xi-Chen2017]. This degeneracy is lifted by fluctuations ([*order from disorder*]{} mechanism) [@villain; @shender2; @Henley:1995]. Particular attention has been paid to the extreme quantum spin-half case \[\]. Although, there is consensus on the absence of magnetic long-range order (LRO) the nature of the spin-liquid GS is still under debate. Meanwhile also for spin quantum number $S=1$ there is evidence that the KHAF does not exhibit magnetic LRO [@Goetze2011; @Goetze2015; @Changlani2015; @Liu2015; @Picot2015; @Nishimoto2015]. Recently it has been argued that there is a route to magnetic GS LRO in the KHAF as increasing the spin quantum number to $S\ge3/2$, see [@Goetze2011; @Goetze2015; @Oitmaa2016; @Liu2016].
Except the theoretical work there is also a large activity on the experimental side. Among the $S=1/2$ kagome compounds, Herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$ is a promising candidate for a spin liquid, see [@herbertsmithite2007; @herbertsmithite2007a; @Hiroi2009; @herbertsmithite2009; @herbertsmithite2010; @herbertsmithite2012]. Examples for kagome magnets with higher spin $S$ are deuteronium jarosite (D$_3$O)Fe$_3$(SO$_4$)$_2$(OD)$_6$ with spin $S=5/2$, see [@Fak_2007], and the recently studied Cr-Jarosite KCr$_3$(OH)$_6$(SO$_4$)$_2$ with spin $S=3/2$, see [@Okubo2017].
![ Illustration of the two most relevant classical states. Left: $q=0$ state with magnetic wave vector $\mathbf{Q}_0=(2\pi/\sqrt{3},0)$. Right: $\sqrt{3}\times\sqrt{3}$ state with magnetic wave vector $\mathbf{Q}_1=(0,4\pi/3)$. “$+$” and “$-$” symbols denote plaquettes of different vector spin chirality. The arrows indicate the basis vectors $\mathbf{a}_1$ and $\mathbf{a}_2$. []{data-label="Fig1"}](fig1)
Due to the [*order from disorder*]{} mechanism two different coplanar states may be selected by fluctuations: (i) The so called $q=0$ state with a corresponding magnetic wave vector $\mathbf{Q}_0=(2\pi/\sqrt{3},0)$ (Fig. \[Fig1\], left), which has a magnetic unit cell that is identical to the geometrical one. (ii) The so called $\sqrt{3}\times\sqrt{3}$ state (Fig. \[Fig1\], right) with a corresponding magnetic wave vector $\mathbf{Q}_1=(0,4\pi/3)$ which has a three times larger unit cell, cf. e.g., [@Zhitomirsky2008]. Moreover, both states are characterized by different vector chirality patterns, see Fig. \[Fig1\]. The selection of one of these states is a subtle issue and depends on spin quantum number, anisotropy etc., see, e.g., [@Sachdev_1992; @Chubukov:92; @Henley:1995; @Goetze2011; @Zhito_PRL_XXZ; @Chernyshev2015; @Goetze2015; @Goetze2016]. While for the widely studied GS properties a plethora of many-body methods are available, the tool box for the calculation of finite-temperature properties of highly frustrated quantum magnets is sparse. Here we use two universal approaches suitable to calculate thermodynamic quantities of Heisenberg quantum spin systems of arbitrary lattice geometry, namely the Green-function technique [@Gasser2001; @Nolting2009; @Froebrich2006] and the high-temperature expansion [@elstner1993; @elstner1994; @2DJ1J2; @singh2012; @Kapellasite; @Oitmaa2006; @Bernu2001; @Bernu2005; @Bernu2015; @Lohmann2011; @Lohmann2014; @Richter2015; @Schmidt2017; @Singh2017].
We study the kagome lattice with antiferromagnetic ($J>0$) nearest-neighbor interaction $$\begin{aligned}
\label{model}
\hat{H} = J \hspace{-2mm} \sum_{\langle
m\alpha,n\beta\rangle}\bm{\hat{S}}_{m\alpha}\bm{\hat{S}}_{n\beta}\; , \;
\bm{\hat{S}}_{m\alpha}^2 =S(S+1), \label{eq_ham}\end{aligned}$$ where the Greek indices ($\alpha, \beta=1,2,3$) run over the spins in a geometrical unit cell (that contains three sites) and the latin indices $n$ and $m$ label the unit cells given by the basis vectors $\mathbf{a}_1=(0,2)$ and $\mathbf{a}_2=(\sqrt{3},1)$.
The paper is organized as follows: In Sec. \[methods\] we briefly illustrate the applied methods. In Sec. \[sec:GS\] we describe the properties of the model at zero temperature, followed by the discussion of finite-temperature properties of the model in Sec. \[finite\_T\]. In Sec. \[sec:sum\] we summarize our findings.
Methods
=======
Rotation-invariant Green’s function method (RGM) \[RGM\]
--------------------------------------------------------
A rotation-invariant formalism of the Green’s function method was first introduced by Kondo and Yamaji [@Kondo1972] to describe short-range order (SRO) of the one-dimensional $S=1/2$ Heisenberg ferromagnet at $T>0$. They decoupled the hierarchy of equation of motions in second order, i.e., one step beyond the usual random-phase approximation (RPA) [@Tyablikov1967; @Gasser2001; @Nolting2009] and introduced rotational invariance by setting $\langle \hat{S}^z_{i}\rangle=0$ in the equations of motions. Within this rotation-invariant scheme possible magnetic LRO is described by the long-range part in the two-point spin correlators. Furthermore, the approximation made by the decoupling of higher-order correlators is improved by introducing so-called vertex parameters, see below. In the following decades the rotation-invariant Green’s function method (RGM) was further elaborated to include arbitrary spin $S$, antiferromagnetic spin systems including frustrated ones and also more complex spin-lattices with non-primitive unit cells [@Rhodes1973; @Shimahara1991; @Suzuki1994; @Barabanov94; @Ihle1997; @Ihle1999; @Yu2000; @Ihle2001; @Bernhard2002; @Schmalfus2004; @Schmalfus2006; @Schmalfus2005; @Junger2005; @Juh_sz_Junger_2009; @Haertel2008; @Haertel2010; @Haertel2011; @Haertel2011a; @Haertel2013; @Antsygina2008; @Mikheyenkov2013; @Vladimirov2014; @Mueller2015; @Mikheyenkov2016; @Vladimirov2017; @2017Mullera; @2017Muller]. At the present time the RGM is a well established method and has been successfully used in numerous recent publications on the theory of frustrated spin systems [@Yu2000; @Ihle2001; @Bernhard2002; @Schmalfus2004; @Schmalfus2006; @Schmalfus2005; @Haertel2008; @Haertel2010; @Haertel2011; @Haertel2011a; @Haertel2013; @Mueller2015; @Junger2005; @Juh_sz_Junger_2009; @Mikheyenkov2013; @Mikheyenkov2016; @2017Mullera; @2017Muller].
The early papers using the RGM [@Yu2000; @Bernhard2002; @Schmalfus2004] to study the KHAF were restricted to the spin-$1/2$ case and used a simple minimal version of the RGM, see below. In the present paper we extend the RGM approach to arbitrary values of the spin quantum number $S\ge1$ and improve the previous RGM studies going beyond the minimal version by introducing one more vertex parameter. Moreover, we provide a more comprehensive analysis of the thermodynamic quantities by considering, e.g. the temperature dependence of the structure factor and correlation lengths.
The basic quantity that has to be determined within the RGM is the (retarded) Green’s function $\langle\langle\hat{S}_{\mathbf{q}\alpha}^{+};\hat{S}_{\mathbf{\mathbf{q}\beta}}^{-}\rangle\rangle_{\omega}$, which is related to the dynamic wavelength-dependent susceptibility $\langle\langle\hat{S}_{\mathbf{q}\alpha}^{+};\hat{S}_{\mathbf{\mathbf{q}\beta}}^{-}\rangle\rangle_{\omega}=-\chi^{+-}_{\alpha\beta\mathbf{q}}(\omega)$. To determine $\langle\langle\hat{S}_{\mathbf{q}\alpha}^{+};\hat{S}_{\mathbf{\mathbf{q}\beta}}^{-}\rangle\rangle_{\omega}$ we use the equation of motion (EoM) up to second order, $$\begin{aligned}
\omega\langle\langle\hat{S}_{\mathbf{q}\alpha}^{+};\hat{S}_{\mathbf{\mathbf{q}\beta}}^{-}\rangle\rangle_{\omega}&=&\langle[\hat{S}_{\mathbf{q}\alpha}^{+},\hat{S}_{\mathbf{\mathbf{q}\beta}}^{-}]_{-}\rangle+\langle\langle\textrm{i}\dot{\hat{S}}_{\mathbf{q}\alpha}^{+};\hat{S}_{\mathbf{\mathbf{q}\beta}}^{-}\rangle\rangle_{\omega},\nonumber \\
\omega\langle\langle\textrm{i}\dot{\hat{S}}_{\mathbf{q}\alpha}^{+};\hat{S}_{\mathbf{\mathbf{q}\beta}}^{-}\rangle\rangle_{\omega}&=&\langle[\textrm{i}\dot{\hat{S}}_{\mathbf{q}\alpha}^{+},\hat{S}_{\mathbf{\mathbf{q}\beta}}^{-}]_{-}\rangle-\langle\langle\ddot{\hat{S}}_{\mathbf{q}\alpha}^{+};\hat{S}_{\mathbf{\mathbf{q}\beta}}^{-}\rangle\rangle_{\omega}.
\label{EoM}\end{aligned}$$ Naturally, for an interacting many-body problem more complicated (i.e., higher-order) Green’s functions appear in the EoM. It is in order to mention here that the RPA, that can be obtained by applying the EoM only once (first line in Eq. (\[EoM\])), has the disadvantage that only phases with magnetic LRO can be described properly, since the Green’s function is proportional to magnetic order parameters [@Tyablikov1967; @Gasser2001; @Nolting2009]. In contrast, SRO can be adequately described by the RGM due to including the next order in the EoM, see the second line in Eq. (\[EoM\]). The operator $\ddot{\hat{S}}_{\mathbf{q}\alpha}^{+}$ appearing in second-order contains several combinations of three-spin operators. These products of three-spin operators are simplified by the decoupling scheme along the lines of, e.g., [@Suzuki1994; @Juh_sz_Junger_2009; @Haertel2011; @2017Muller] which can be sketched as follows:
$$\begin{aligned}
\label{decoup_RGM}
\hat{S}_A^-\hat{S}_B^+\hat{S}_C^+ & \rightarrow & \alpha^{}_{AB} c^{+-}_{AB}\hat{S}_C^+ + \alpha^{}_{AC}c^{+-}_{AC}S_B^+,\\ \nonumber
\hat{S}_A^z\hat{S}_B^z\hat{S}_C^+ & \rightarrow & \frac 1 2 \alpha^{}_{AB}c^{+-}_{AB}\hat{S}_C^+,\\ \nonumber
\hat{S}_A^z\hat{S}_A^z\hat{S}_B^+ & \rightarrow & c^{zz}_{AA}\hat{S}_B^+=\frac 1 2 c^{+-}_{AA}\hat{S}_B^+,\\ \nonumber
\hat{S}_A^-\hat{S}_B^+\hat{S}_A^+ & \rightarrow & c^{+-}_{AA}\hat{S}_B^+ + \lambda_{AB}c^{+-}_{AB}\hat{S}_A^+,\\ \nonumber
\hat{S}_A^z\hat{S}_B^z\hat{S}_A^+ & \rightarrow & \frac 1 2 \lambda_{AB}c^{+-}_{AB}\hat{S}_A^+,\\ \nonumber
\hat{S}_A^-\hat{S}_B^+\hat{S}_B^+ & \rightarrow & 2\lambda_{AB}c^{+-}_{AB} \hat{S}_B^+,\end{aligned}$$
where $A\ne B\ne C\ne A$ are sites of the kagome lattice, $c^{+-}_{AB}=\langle\hat{S}_{A}^{+}\hat{S}_{B}^{-}\rangle$ and the conservation of total $S^z$ is implied, i.e., $c^{+z}_{AB}=c^{-z}_{AB}=0$.
In Eq. (\[decoup\_RGM\]) two classes of so-called vertex parameters, $\alpha^{}_{AB}$ and $\lambda^{}_{AB}$, are introduced to improve the approximation made by the decoupling. The parameter $\alpha^{}_{AB}$ enters the decoupling scheme if all sites are different from each other, see lines 1 and 2 in Eq. (\[decoup\_RGM\]). In line 3 of Eq. (\[decoup\_RGM\]) the correlation $\langle \hat{S}_{A}^{+}\hat{S}_{A}^{-}\rangle$ is determined by using the sum rule (operator identity) $\hat{\mathbf{S}}^2= \hat{S}^+\hat{S}^- - \hat{S}^z +
(\hat{S}^z)^2$, i.e., due to $\langle \hat{S}^z\rangle=0$ within the RGM we have $3\langle (\hat{S}^z)^2\rangle =
\langle \hat{\mathbf{S}}^2 \rangle= \langle \hat{S}^+\hat{S}^-\rangle
+\langle (\hat{S}^z)^2 \rangle$ and finally $\langle \hat{S}_{A}^{+}\hat{S}_{A}^{-}\rangle = \frac{2}{3}S(S+1)$. The other class of vertex parameters, $\lambda^{}_{AB}$, present in lines 4, 5 and 6 of Eq. (\[decoup\_RGM\]) appears only for $S>1/2$ if two sites coincide and the remaining correlation function cannot be obtained by an operator identity.
Then the EoM reads $$\begin{aligned}
(\omega^2\mathbb{I}-F_{\mathbf{q}})\chi^{+-}_\mathbf{q}(\omega)&=&-M_{\mathbf{q}}, \label{rgm_green}\end{aligned}$$ where $M_\mathbf{q}$ (moment matrix), $F_{\mathbf{q}}$ (frequency matrix) and $\chi_\mathbf{q}$ (susceptibility matrix) are hermitian 3$\times$3-matrices and $\mathbb{I}$ is the identity matrix. Performing corresponding calculations as described above the components $M^{\alpha
\beta}_\mathbf{q}=\langle[\textrm{i}\dot{\hat{S}}_{\mathbf{q}\alpha}^{+},\hat{S}_{\mathbf{q}\beta}^{-}]\rangle$ of the moment matrix are obtained as $$\begin{aligned}
M_{\mathbf{q}}= & & \\ 4Jc_{1,0} & &\left(\begin{array}{ccc}
-2 & \cos(\frac{\sqrt{3}q_{x}-q_{y}}{2}) & \cos(q_{y})\\ \nonumber
\cos(\frac{\sqrt{3}q_{x}-q_{y}}{2}) & -2 & \cos(\frac{\sqrt{3}q_{x}+q_{y}}{2})\\
\cos(q_{y}) & \cos(\frac{\sqrt{3}q_{x}+q_{y}}{2}) & -2
\end{array}\right). \label{Mq} \\ \nonumber\end{aligned}$$ The elements of frequency matrix of the spin excitations $$\begin{aligned}
F_{\mathbf{q}}= \left(\begin{array}{ccc}
F_{\mathbf{q}}^{1,1} & F_{\mathbf{q}}^{1,2} & F_{\mathbf{q}}^{1,3}\\
F_{\mathbf{q}}^{1,2} & F_{\mathbf{q}}^{2,2} & F_{\mathbf{q}}^{2,3}\\
F_{\mathbf{q}}^{1,3} & F_{\mathbf{q}}^{2,3} & F_{\mathbf{q}}^{3,3}
\end{array}\right), \\ \nonumber\end{aligned}$$ are given by $$\begin{aligned}
\frac{3}{2}J^{-2}F_{\mathbf{q}}^{1,1}=6\tilde{\lambda}_{1,0}+6\tilde{\alpha}_{1,1}+6\tilde{\alpha}_{2,0}+4S(S+1) \\
+3\left(\cos\left(\sqrt{3}q_{x}-q_{y}\right)+\cos(2q_{y})+2\right)\tilde{\alpha}_{1,0}, \nonumber\\
\frac{3}{4}J^{-2}F_{\mathbf{q}}^{2,2} =3\tilde{\lambda}_{1,0}+3\tilde{\alpha}_{1,1}+3\tilde{\alpha}_{2,0}+2S(S+1) \nonumber\\
+3\left(\cos\left(\sqrt{3}q_{x}\right)\cos(q_{y})+1\right)\tilde{\alpha}_{1,0}, \nonumber\\
\frac{3}{2}J^{-2}F_{\mathbf{q}}^{3,3} =6\tilde{\lambda}_{1,0}+6\tilde{\alpha}_{1,1}+6\tilde{\alpha}_{2,0}+4S(S+1) \nonumber\\
+3\left(\cos\left(\sqrt{3}q_{x}+q_{y}\right)+\cos(2q_{y})+2\right)\tilde{\alpha}_{1,0}, \nonumber \end{aligned}$$
$$\begin{aligned}
(\sqrt{2}J)^{-2}F_{\mathbf{q}}^{1,2}&=& \cos\left(\frac{1}{2}\left(\sqrt{3}q_{x}+3q_{y}\right)\right)\tilde{\alpha}_{1,0} \\
&-&\cos\left(\frac{1}{2}\left(\sqrt{3}q_{x}-q_{y}\right)\right)\left(\tilde{\lambda}_{1,0}+3\tilde{\alpha}_{1,0}
+\tilde{\alpha}_{1,1}+\tilde{\alpha}_{2,0}+\frac{2}{3}S(S+1)\right), \nonumber\\
(\sqrt{2}J)^{-2}F_{\mathbf{q}}^{1,3}&=& \cos\left(\sqrt{3}q_{x}\right)\tilde{\alpha}_{1,0}-\cos(q_{y})\left(\tilde{\lambda}_{1,0}
+3\tilde{\alpha}_{1,0}+\tilde{\alpha}_{1,1}+\tilde{\alpha}_{2,0}+\frac{2}{3}S(S+1)\right), \nonumber\\
(\sqrt{2}J)^{-2}F_{\mathbf{q}}^{2,3}&=& \cos\left(\frac{1}{2}\left(\sqrt{3}q_{x}-3q_{y}\right)\right)\tilde{\alpha}_{1,0}\nonumber \\
&-&\cos\left(\frac{1}{2}\left(\sqrt{3}q_{x}+q_{y}\right)\right)\left(\tilde{\lambda}_{1,0}
+3\tilde{\alpha}_{1,0}+\tilde{\alpha}_{1,1}+\tilde{\alpha}_{2,0}+\frac{2}{3}S(S+1)\right), \nonumber \end{aligned}$$
where we have used the abbreviations $$\begin{aligned}
\tilde{\alpha}_{i,j} = \alpha_{i,j}c_{i,j},
\quad
\tilde{\lambda}_{i,j} = \lambda_{i,j}c_{i,j},\end{aligned}$$ and lattice symmetry is used to identify equivalent correlators. The indices $i,j$ indicate lattice sites separated by the vector $\mathbf{R}_{i,j}=\mathbf{r}_i-\mathbf{r}_j=i\mathbf{a_1}/2+j\mathbf{a_2}/2$, i.e., $c_{ij} \equiv \langle \hat{S}^+_{\mathbf{0}}
\hat{S}^-_{\mathbf{R}_{i,j}}\rangle$. Their common eigenvectors $|{\gamma\mathbf{q}}\rangle$ and their eigenvalues ($M_{\mathbf{q}}|{\gamma\mathbf{q}}\rangle=m_{\gamma\mathbf{q}}|{\gamma\mathbf{q}}\rangle, F_{\mathbf{q}}|{\gamma\mathbf{q}}\rangle=\omega^2_{\gamma\mathbf{q}}|{\gamma\mathbf{q}}\rangle$, with $\gamma=1,2,3$) are needed to solve a system of self-consistent equations. The square-root of the eigenvalues of the frequency matrix $F_\mathbf{q}$ can be identified as the branches $\omega_{\gamma\mathbf{q}}$, $\gamma=1,2,3$, of the excitation spectrum.
Finally, the dynamic wavelength-dependent susceptibility reads $$\label{chi_oq}
\chi^{+-}_\mathbf{q\alpha\beta}(\omega)
= -\sum_{\gamma} \frac{m_{\gamma\mathbf{q}}}{\omega^2-\omega^2_{\gamma\mathbf{q}}}\langle\alpha|{\gamma\mathbf{q}}\rangle\langle{\gamma\mathbf{q}}|\beta\rangle$$ and the static $\mathbf{q}$-dependent susceptibility is given by $$\label{chi_q}
\chi_{\mathbf{q}}= \lim_{\omega\rightarrow0}\frac{1}
{2n_{\rm uc}}\sum_{\alpha,\beta}
\chi^{+-}_{\mathbf{q}\alpha\beta}(\omega),$$ where $n_{\rm uc}=3$ is the number of sites in the geometric unit cell. The correlation functions are obtained by applying the spectral theorem $$\begin{aligned}
c_{m\alpha,n\beta} & = & \frac{1}{\mathcal{N}}\sum_{\mathbf{q}}c_{\mathbf{q}\alpha\beta}\cos(\mathbf{q}\mathbf{r}_{m\alpha,n\beta}),\label{eq_correlRGM}\end{aligned}$$ with $$\begin{aligned}
c_{\mathbf{q}\alpha\beta}&=&\sum_{\gamma}\frac{m_{\gamma\mathbf{q}}}{2\omega_{\gamma\mathbf{q}}}(1+2n(\omega_{\gamma\mathbf{q}}))
\langle\alpha|{\gamma\mathbf{q}}\rangle\langle{\gamma\mathbf{q}}|\beta\rangle, \label{eq_cqRGM}\end{aligned}$$ where $\mathcal{N}$ is the number of unit cells and $n(\omega_{\gamma\mathbf{q}})$ is the Bose-Einstein distribution function. At the $\Gamma$ point ($\mathbf{q}=\mathbf{0}$) the eigenvectors have the very simple form $|{1\mathbf{0}}\rangle=(1,0,-1)/\sqrt{2}$, $|{2\mathbf{0}}\rangle=(1,-2,1)/\sqrt{6}$, and $|{3\mathbf{0}}\rangle=(1,1,1)/\sqrt{3}$.
After straightforward calculations we get $$\begin{aligned}
\label{eigsys}
m_{1{\bf q}} & = & -12J c_{1,0},\\
%
\nonumber
m_{2{\bf q}} & = & -2J c_{1,0} (3 + D_{\bf q}),\\
%
\nonumber
m_{3{\bf q}} & = & -2J c_{1,0}(3 -D_{\bf q}),\\
%
\nonumber
\omega_{1{\bf q}}^2 & = & 6J^2(\frac{2}{3}S(S+1) + \tilde{\lambda}_{1,0} + 2\tilde{\alpha}_{1,0} + \tilde{\alpha}_{1,1} + \tilde{\alpha}_{2,0}),\\
\nonumber
\omega_{2{\bf q}}^2 & = & J^2(\frac{2}{3}S(S+1) + \tilde{\lambda}_{1,0} + 2\tilde{\alpha}_{1,0} + \tilde{\alpha}_{1,1} + \tilde{\alpha}_{2,0} \\ \nonumber
& & - \tilde{\alpha}_{1,0} (3 - D_{\bf q}))(3 + D_{\bf q}),\\
\nonumber
%
\omega_{3{\bf q}}^2 & = & J^2(\frac{2}{3}S(S+1) + \tilde{\lambda}_{1,0} + 2\tilde{\alpha}_{1,0} + \tilde{\alpha}_{1,1} + \tilde{\alpha}_{2,0} \\ \nonumber
& & - \tilde{\alpha}_{1,0} (3 + D_{\bf q}))(3-D_{\bf q}),\\
\nonumber
D_{\bf q}^2 & = & 3+2\cos(2q_y)+2\cos(\sqrt{3}q_x-q_y)\\
\nonumber
& &+2\cos(\sqrt{3}q_x+q_y) .\end{aligned}$$ Obviously, we have one flat band, namely $\omega_{1{\bf q}}$, and two dispersive branches $\omega_{2{\bf q}}$ and $\omega_{3{\bf q}}$, where $\omega_{3{\bf q}}$ is the acoustic branch.
The static uniform susceptibility is given by (cf. Eqs. (\[chi\_oq\]) and (\[chi\_q\])) $$\begin{aligned}
\chi_{0}&=&\lim_{\mathbf{q}\rightarrow \mathbf{0}}\chi_{\mathbf{q}}\label{eq_susc0RGM}
=\underset{\mathbf{q}\rightarrow\mathbf{0}}{\textrm{lim}}\frac{m_{3\mathbf{q}}}{2\omega^2_{3\mathbf{q}}} \\
&=&\frac{-c_{1,0}}{J(\frac{2}{3}S(S+1)+\tilde{\lambda}_{1,0}-4\tilde{\alpha}_{1,0}+\tilde{\alpha}_{1,1}+\tilde{\alpha}_{2,0})}. \nonumber\end{aligned}$$ The magnetic correlation length $\xi_{\mathbf{Q}}$ is obtained by expanding the susceptibility $\chi_{\mathbf{Q+q}}=\sum_{\alpha,\beta}\chi^{+-}_{\alpha\beta\mathbf{Q+q}}/(2n_{uc})\approx\chi_{\mathbf{Q}}/(1+\xi_{\mathbf{Q}}^2\mathbf{q}^2)$ in the neighborhood of the corresponding magnetic wave vector $\mathbf{Q}$, see, e.g., [@Schmalfus2004; @Schmalfus2005; @Junger2005; @Haertel2010; @Haertel2011; @Haertel2011a; @Haertel2013; @Mueller2015; @2017Mullera; @2017Muller]. While for the $q=0$ state the expansion is straightforward and yields $\xi_{\mathbf{Q}_0}=\sqrt{J\alpha_{1,0}\chi_{\mathbf{Q}_0}}$, the corresponding susceptibility for the $\sqrt{3}\times \sqrt{3}$ state $\chi_{\mathbf{Q}_1}
=\frac{-c_{1,0}}{J(\frac{2}{3}S(S+1)+\tilde{\lambda}_{1,0}+2\tilde{\alpha}_{1,0}+\tilde{\alpha}_{1,1}+\tilde{\alpha}_{2,0})}=m_{1{\bf q}}/(2\omega_{1{\bf
q}}^2)$ is a quotient of two $\mathbf{q}$-independent quantities, cf. Eq. (\[eigsys\]). Having in mind the above relation between $\xi_{\mathbf{Q}_0}$ and $\chi_{\mathbf{Q}_0}$ and the fact that both quantities would simultaneously diverge at a transition point to magnetic LRO, see, e.g., [@2017Mullera; @2017Muller], we choose $\xi_{\mathbf{Q}_1}=\sqrt{J\alpha_{1,0}\chi_{\mathbf{Q}_1}}$ as a measure of the correlation length related to a possible $\sqrt{3}\times \sqrt{3}$ ordering. In what follows we will use the term ’correlation length’ for $\xi_{\mathbf{Q}_1}$, too. To analyze magnetic ordering we can use the static magnetic structure factor ${\cal S}(\mathbf{q})=(1/N)\sum_{i,j}\langle
\hat{\mathbf{S}}_i\hat{\mathbf{S}}_j\rangle\cos(\mathbf{q}\mathbf{R}_{i,j})$, which is related to $c_{\mathbf{q}\alpha\beta}$, cf. Eq. (\[eq\_cqRGM\]).
The final step in the RGM approach is to find as many equations as there are unknown quantities in the RGM equations, where except the correlation functions entering the EoM also the introduced vertex parameters $\alpha_{i,j}(T)$ and $\lambda_{i,j}(T)$ have to be determined. Then, by numerical solution of the resulting system of coupled self-consistent equations the physical quantities can be determined. Taking into account all possible vertex parameters $\alpha_{i,j}(T)$ and $\lambda_{i,j}(T)$ would noticeably exceed the number of available equations. Within the minimal version of the RGM one takes into account only one vertex parameter in each class, i.e., $\alpha_{i,j}(T)=\alpha(T)$ and $\lambda_{i,j}(T)=\lambda(T)$. Note that this simple version with only one $\alpha$ parameter ($\lambda(T)\equiv0$) was used in the early RGM kagome papers for the spin-half case, see [@Yu2000; @Bernhard2002; @Schmalfus2004]. This approach is particularly appropriate for ferromagnets [@Kondo1972; @Suzuki1994; @Haertel2008; @Junger2005; @Schmalfus2005; @Antsygina2008; @Haertel2010; @Haertel2011; @Haertel2011; @Mueller2015; @2017Mullera; @2017Muller], where all correlation functions have the same sign. However, for antiferromagnets typically the consideration of one additional vertex parameter allowing to distinguish between nearest-neighbor and further-neighbor correlations may yield a significant improvement of the method, see, e.g., [@Ihle1997; @Ihle2001; @Schmalfus2006; @Haertel2013; @Vladimirov2014; @Vladimirov2017]. Thus, we set $\alpha_{i,j}(T)=\alpha_{1}(T)$, if ($i,j$) are nearest neighbors sites, and $\alpha_{i,j}(T)=\alpha_{2}(T)$, if ($i,j$) are not nearest neighbors sites. (For the minimal version $\alpha_{2}=\alpha_{1}$ holds.) Note that in the relevant equations the vertex parameters $\lambda_{i,j}(T)$ only appear for nearest-neighbors sites $i$ and $j$, i.e., we set consistently $\lambda_{i,j}(T)=\lambda(T)$.
The required equations to determine all unknown quantities are as follows: For every unknown correlation function the spectral theorem yields one equation, cf. Eqs. (\[eq\_correlRGM\]) and (\[eq\_cqRGM\]). Another equation is given by the sum rule $\bm{\hat{S}}_{m\alpha}^2 =S(S+1)$, which determines, e.g., one vertex parameter, say $\alpha_1$. For the missing two vertex parameters, $\alpha_2$ and $\lambda$, we follow [@Shimahara1991; @Ihle1997; @Ihle2001; @Junger2005; @Juh_sz_Junger_2009; @Haertel2011; @Vladimirov2014; @Mueller2015; @2017Muller; @Vladimirov2017] and use the ansatzes $r_1(T)=(\alpha_{1}(T)-\alpha_{1}(\infty))/(\lambda(T)-\lambda(\infty))=r_1(0)$ and $r_2(T)=(\alpha_{1}(T)-\alpha_{1}(\infty))/(\alpha_{2}(T)-\alpha_{2}(\infty))=r_2(0)$, where the values $\alpha_1(\infty)=\alpha_2(\infty)=1$ and $\lambda(\infty)=1-3/(4S(S+1))$ are known and can be verified by comparison with the high-temperature expansion, see, e.g., [@Junger2005]. (Note that in the minimal version of the RGM only one of these two equations, namely $r_1(T)$, has to be solved, because $\alpha_{1}=\alpha_{2}$.) For the vertex parameter $\lambda(T)$ at zero temperature we use the well-tested ansatz $\lambda(0)=2-1/S$ [@Juh_sz_Junger_2009; @Haertel2011; @Vladimirov2014; @Mueller2015]. Last but not least, for the extended version we determine the additional vertex parameter $\alpha_{2}(0)$ by adjusting the GS energy to the values obtained by high-order coupled cluster method (CCM) [@Goetze2011; @Goetze2015], which is known to yield precise values for $E_0$, see, e.g., Fig. 7 in [@Xie2014].
High Temperature Expansion (HTE) \[HTE\]
----------------------------------------
In addition to the RGM, we use a general high temperature expansion (HTE) code, see [@Lohmann2011; @Lohmann2014], to discuss the thermodynamics of the KHAF. We compute the series of the susceptibility $\chi_{0}=\sum_nc_n\beta^n$ and the specific heat $C=\sum_nd_n\beta^n$ up to order 11. To extend the region of validity of the power series Padé approximants are a conventional transformation. These approximants are ratios of two polynomials of degree $m$ and $n$: $[m,n]=P_m(x)/Q_n(x)$. Furthermore the series of the correlation functions $\langle
\hat{\mathbf{S}}_i
\hat{\mathbf{S}}_j\rangle$ are analyzed up to 11th order, which we use to consider the static magnetic structure factor ${\cal S}(\mathbf{q})=(1/N)\sum_{i,j}\langle
\hat{\mathbf{S}}_i\hat{\mathbf{S}}_j\rangle\cos(\mathbf{q}\mathbf{R}_{i,j})$, see, e.g., [@Richter2015]. The structure factor is one of the main outcomes of neutron diffraction measurements, where the maxima of the structure factor indicate the favored magnetic ordering.
Results\[results\]
==================
In what follows we set the energy scale of the model (\[model\]) by fixing the exchange constant $J=1$.
Zero-temperature properties\[sec:GS\]
-------------------------------------
![ RGM GS results for the dispersion of the magnetic excitations $\omega_{\gamma\mathbf{q}}/S$ ($\gamma=1,2,3$) for $S=1/2$ (red lines) and $S=3$ (blue lines) compared with data of the LSWT (black lines) along a typical path in the first Brillouin zone (see inset). LSWT formulas for $\omega_{\gamma\mathbf{q}}/S$ can be found, e.g., in [@Chernyshev2015]. []{data-label="Fig4"}](fig4-eps-converted-to.pdf)
![ Main: Normalized RGM GS excitation velocity $v(0)/S$ in dependence on the inverse spin quantum number $1/S$. Inset: Position $E_{\rm flat}/S$ of the flat band in dependence on the inverse spin quantum number $1/S$. []{data-label="Fig5"}](fig5-eps-converted-to.pdf)
We start with the discussion of the RGM results for the GS properties using the minimal as well as the extended (i.e., with CCM input) version of the RGM. In Fig. \[Fig2\] we show the GS energy $E_0/S^2$ as a function of the inverse spin quantum number $1/S$. It is obvious that the minimal version leads to significant higher energy values, where for $S=1/2$ the difference is smallest. It is also obvious, that the minimal version does not yield the correct classical large-$S$ limit, $\lim_{S\to\infty}
E_0/S^2 = -1$. Thus, we conclude that the minimal version is only applicable for small values of $S$. This conclusion is supported by the data for the static uniform susceptibility $\chi_{0}$ shown in Fig. \[Fig3\]. In what follows (i.e., figures subsequent to Fig. \[Fig3\]), we therefore focus on the discussion of the results obtained by the extended version, i.e., unless stated otherwise, all data presented below belong to the extended version. Now we discuss the excitation spectrum shown in Fig. \[Fig4\]. We mention first, that in linear spin-wave theory (LSWT) $\omega_{\gamma\mathbf{q}}/S$ is independent of $S$, the flat band $\omega_1$ is exactly at zero energy and the two dispersive branches, $\omega_{2\mathbf{q}}$ and $ \omega_{3\mathbf{q}}$, are degenerate [@Chernyshev2015]. The RGM provides an improved description of the excitation energies. The flat band is of course also present, but its position $E_{\rm flat}$ depends on $S$, where $E_{\rm flat}/S$ decreases almost linearly with $1/S$ down to $E_{\rm flat}/S = 0$ as $S \to \infty$, see inset of Fig. \[Fig5\]. Moreover, the degeneracy of $\omega_{2\mathbf{q}}$ and $ \omega_{3\mathbf{q}}$ is lifted and there is a noticeable dependence of the dispersive branches on $S$. In particular, in the extreme quantum case $S=1/2$ the dispersion relations deviate strongly from the LSWT. As increasing $S$ the RGM data approach the LSWT result.
The GS excitation velocity $v$ corresponding to the linear expansion of the lowest branch $\omega_{3\mathbf{q}}$ around the $\Gamma$ point is given by $v^2=(\frac{2}{3}S(S+1) + \tilde{\lambda}_{1,0} -4\tilde{\alpha}_{1,0} + \tilde{\alpha}_{1,1} + \tilde{\alpha}_{2,0})$. The LSWT result is $v_{LSWT}=\sqrt{3} S$. Numerical data for $v$ are shown in Fig. \[Fig5\]. While in LSWT $v/S$ is independent of $S$, within the RGM there is a noticeable dependence of $v/S$ on $S$.
![ Main: Magnitude of the GS correlation functions $|\langle
\hat{\mathbf{S}}_0 \hat{\mathbf{S}}_\mathbf{R}\rangle|/S(S+1)$ within a range of separation $|\mathbf{R}| \le 6$ for spin quantum numbers $S=1/2,1$ and $7/2$. Inset: Comparison of $\langle \hat{\mathbf{S}}_0 \hat{\mathbf{S}}_\mathbf{R}\rangle/S(S+1)$ of the minimal and extended version of the RGM for $S=1/2$ (nearest-neighbor correlation not included). []{data-label="Fig6"}](fig6-eps-converted-to.pdf)
![RGM GS correlation lengths corresponding to $q=0$ ($\xi_{\mathbf{Q}_0}$) and $\sqrt{3}\times \sqrt{3}$ ($\xi_{\mathbf{Q}_1}$) ordering. []{data-label="Fig7"}](fig7-eps-converted-to.pdf)
![ Normalized RGM GS structure factor ${\cal S}(\mathbf{q})/S(S+1)$ along the path $\Gamma \to
{\mathbf{Q}_1} \to {\mathbf{Q}_0}
\to \Gamma $ (see inset) for $S=1/2$ and $S=7/2$. []{data-label="Fig9"}](fig9-eps-converted-to.pdf)
Let us turn to the spin-spin correlation functions $\langle
\hat{\mathbf{S}}_0 \hat{\mathbf{S}}_\mathbf{R}\rangle$. In Fig. \[Fig6\], main panel, we show all non-equivalent GS correlators $\langle \hat{\mathbf{S}}_0
\hat{\mathbf{S}}_\mathbf{R}\rangle/S(S+1)$ up to a separation $R = |\mathbf{R}|= 6$ for some selected values of $S$ using a logarithmic scale for $|\langle \hat{\mathbf{S}}_0 \hat{\mathbf{S}}_\mathbf{R}\rangle|/S(S+1)$. In the inset we compare the minimal with the extended version for $S=1/2$ without using a logarithmic scale. Note that the presented data for the minimal version correspond to the results of Bernhard, Canals and Lacroix [@Bernhard2002]. In accordance with Figs. \[Fig2\] and \[Fig3\] for $S=1/2$ the difference between the minimal and the extended version are not tremendous but noticeable. Since for a certain separation $|\mathbf{R}|$ non-equivalent sites exist, more than one data point can appear at one and the same separation $|\mathbf{R}|$. The data suggest that the overall decay of $\ln |\langle \hat{\mathbf{S}}_0 \hat{\mathbf{S}}_\mathbf{R}\rangle/S(S+1)|$ seems to be linear, thus indicating an exponential decay of the correlators. It is also obvious, that the decay is faster the lower the spin quantum number $S$. This observation from Fig. \[Fig6\] is in agreement with results for the correlation lengths $\xi_{\mathbf{Q}_0}$ (corresponding to $q=0$ ordering) and $\xi_{\mathbf{Q}_1}$ (corresponding to $\sqrt{3}\times \sqrt{3}$ ordering), shown in Fig. \[Fig7\] (for the definition of $\xi_{\mathbf{Q}_0}$ and $\xi_{\mathbf{Q}_1}$ see Sec. \[RGM\]). In the extreme quantum spin-half case the correlation lengths are of the order of one lattice spacing as expected in a spin liquid. That is in agreement with known results, e.g., obtained by large-scale density-matrix renormalization-group (DMRG) studies [@Kolley2015]. The RGM data then indicate a power-law increase of both, $\xi_{\mathbf{Q}_0}$ and $\xi_{\mathbf{Q}_1}$, with increasing $S$, see Fig. \[Fig7\]. We find $\xi_{\mathbf{Q}_1} >
\xi_{\mathbf{Q}_0}$ for all $S$, but the difference of both correlation lengths is small.
Now we discuss the GS static magnetic structure factor ${\cal S}(\mathbf{q})$. In Fig. \[Fig8\] we show an intensity plot of ${\cal S}(\mathbf{q})/S(S+1)$ using an extended Brillouin zone, see panel (a) and cf. also [@Kolley2015]. For $S=1/2$ we find the typical pattern [@Laeuchli2009; @Depenbrock2012; @Kolley2015], i.e., the intensity is concentrated along the edge of the extended Brillouin zone, where ${\cal S}(\mathbf{q})$ remains small even at the magnetic $\mathbf{q}$-vectors $\mathbf{Q}_0$ and $\mathbf{Q}_1$ related to the $q=0$ and $\sqrt{3}\times \sqrt{3}$ states. This smooth shape of ${\cal S}(\mathbf{q})$ is related to the fast decay of the spin-spin correlations, see Fig. \[Fig6\]. As increasing $S$ the structure factor develops a more pronounced shape, and pinch points, typical for the classical KHAF [@Zhitomirsky2008], emerge between triangular shaped areas of large intensity, see Fig. \[Fig8\]c. This observation is also obvious from Fig. \[Fig9\], where we show the structure factor along a prominent path in the extended Brillouin zone. As indicated by Figs. \[Fig8\] and \[Fig9\], we find that for all values of $S$ the relation ${\cal S}(\mathbf{Q}_1)>{\cal S}(\mathbf{Q}_0)$ holds. Together with the data for the correlation lengths $\xi_{\mathbf{Q}_0}$ and $\xi_{\mathbf{Q}_1}$ (Fig. \[Fig7\]) we may conclude that $\sqrt{3}\times
\sqrt{3}$ SRO is favored in agreement with previous investigations [@Chubukov:92; @Sachdev_1992; @Henley:1995; @Goetze2011; @Zhito_PRL_XXZ].
From the static GS properties reported above we conclude that, although the magnetic SRO with $\sqrt{3}\times \sqrt{3}$ symmetry becomes more and more pronounced with increasing $S$, within the RGM approach no magnetic LRO for the spin-$S$ KHAF is found. We may compare this finding with known GS results obtained by other methods. Note, however, that for $S>1$ data to compare with are extremely rare. We mention first that within the LSWT the quantum correction of the sublattice magnetization always diverges due to the zero-energy flat band, see, e.g., [@Zhito_PRL_XXZ]. As briefly discussed in the introduction, more sophisticated GS methods such as the CCM and the DMRG yield evidence that for $S=1$ semiclassical magnetic LRO is also lacking [@Goetze2011; @Goetze2015; @Changlani2015; @Liu2015; @Picot2015; @Nishimoto2015]. On the other hand, recent results obtained by CCM, tensor network approaches, and series expansion indicate weak GS $\sqrt{3}\times \sqrt{3}$ LRO for $S=3/2$ [@Goetze2011; @Goetze2015; @Oitmaa2016; @Liu2016]. Previous experience in applying the RGM on frustrated quantum antiferromagnets, see, e.g., [@Barabanov94; @Ihle2001; @Schmalfus2004; @Haertel2013; @Mikheyenkov2013] and references therein, indicate, however, that the implementation of rotational invariance by setting $\langle \hat{S}^z_{i}\rangle=0$ in the equations of motions may overestimate the tendency to melt semiclassical GS magnetic LRO in RGM calculations.
Finite-temperature properties {#finite_T}
-----------------------------
In what follows, as a rule we will present the temperature dependence of physical quantities using a normalized temperature $T/S(S+1)$. This choice ensures a spin-independent behavior of the physical quantities at large temperatures [@Lohmann2011]. Moreover, we mention again that (unless stated otherwise) we present RGM data for the extended version using CCM input (see above).
### Spin-spin correlation functions, specific heat and uniform susceptibility {#subsec_S0SR}
We start with the discussion of the temperature dependence of short-range spin-spin correlation functions $\langle \hat{\mathbf{S}}_0 \hat{\mathbf{S}}_\mathbf{R}\rangle$. We show the absolute values in Fig. \[Fig10\], main panel, for $S=1/2$, $1$, and $7/2$. (Note that the NN correlation is antiferromagnetic, whereas the NNN and NNNN correlation functions are ferromagnetic.) We find that there is a low-temperature region $T/S(S+1) \lesssim 0.1$ where the presented correlation functions are almost temperature independent. This region is largest for the extreme quantum case $S=1/2$. It is also obvious that for $T/S(S+1) < 1$ the magnetic SRO becomes more pronounced as increasing $S$ (cf. also Fig. \[Fig6\]). On the other hand, for $T/S(S+1) > 1$ the curves for various $S$ practically coincide. In the inset of Fig. \[Fig10\] we compare the two versions of the RGM (minimal and extended) as well as the HTE series for $S=1/2$. Obviously, both versions of the RGM agree well with each other. Note, however, that this statement does not hold for larger values of $S$, cf. the discussion in Sec. \[sec:GS\]. The HTE approach for correlation functions is also in good agreement with the RGM data down to $T \sim 0.4$.
Now we turn to the specific heat. For the extreme quantum case $S=1/2$ various methods provide indications for an additional low-temperature peak at about $T=0.1$ [@elstner1994; @Nakamura1995; @tomczak1996thermodynamical; @Lhuillier_thermo_PRL2000; @Bernu2005; @Rigol2007; @Munehisha2014; @Shimokawa2016] due to a set of low-lying singlet states. However, instead of a true maximum a shoulder-like hump may characterize the low-$T$ profile of $C(T)$ [@Bernu2005; @Xi-Chen2017]. It is an open question whether for $S>1/2$ such a feature is still present. Our RGM approach does not show any unconventional feature in the temperature profile of the specific heat at low $T$ for $S=1/2$ and $S=1$, cf. Fig. \[Fig11\]. For $S>1$ a weakly pronounced shoulder-like hump emerges (see the inset of Fig. \[Fig11\]). We argue, that our RGM approach is not able to detect the subtle role of low-lying excitations relevant for the low-temperature physics of the KHAF in the extreme quantum limit of small spin $S$. On the other hand, in the limit of large $S$ the RGM data seem to approach the classical Monte-Carlo data [@Chalker1992; @Huber2001] reasonably well.
![Main panel: Magnitude of the normalized spin-spin correlation functions $|\langle
\hat{\mathbf{S}}_0 \hat{\mathbf{S}}_\mathbf{R}\rangle|/S(S+1)$ as a function of the normalized temperature $T/S(S+1)$ (logarithmic scale) for spin quantum numbers $S=1/2, 1$, and $7/2$ (NN – nearest neighbors; NNN – next-nearest neighbors; NNNN – next-next-nearest neighbors along two $J_1$ bonds). Inset: Magnitude of the spin-spin correlation functions $|\langle
\hat{\mathbf{S}}_0 \hat{\mathbf{S}}_\mathbf{R}\rangle|$ for $S=1/2$ as a function of the temperature $T$ (linear scale): Comparison of the extended (solid) and minimal (dashed) versions of the RGM as well as the 11th-order HTE with subsequent Padé (dashed-dotted). []{data-label="Fig10"}](fig10.pdf)
The temperature dependence of the static uniform susceptibility $\chi_0$ for spin quantum numbers $S=1/2,1,\ldots,7/2$ is shown in Fig. \[Fig12\]. Similar as for the specific heat there is a well-pronounced tendency to shift the typical maximum in $\chi_0(T)$ towards lower values of $T/S(S+1)$ and to enlarge the height of the maximum as increasing $S$. Again, in the limit of large $S$ the RGM data seem to approach the classical Monte-Carlo data [@Reimers1993; @Huber2001] reasonably well. The fact that $\chi_0(T=0)$ is finite, cf. also Fig. \[Fig3\], is in favor of a vanishing gap to magnetic excitations. There is an ongoing controversial discussion of the gap issue for the $S=1/2$ KHAF [@Yan2011; @Depenbrock2012; @Laeuchli2011; @Laeuchli2016; @Iqbal2011; @Iqbal2013; @Xie2017]. However, we do not claim, that our approach is accurate enough at low temperatures in the quantum limit of small $S$ to provide reliable statements on the very existence of an excitation gap.
### Structure factor and correlation lengths {#subsec_strufa}
To get more insight in the magnetic ordering of the KHAF at finite temperatures we investigate the structure factor and the correlation lengths. Some information on magnetic SRO has already been provided in Fig. \[Fig10\]. First we show in Fig. \[Fig13\] an intensity plot of the static structure factor ${\cal S}(\mathbf{q})/S(S+1)$ for $S=1/2$ and $S=3$ for $T/S(S+1)=1.3$ and compare RGM and HTE. The overall impression is that the RGM and HTE approaches yield very similar intensity plots of ${\cal S}(\mathbf{q})/S(S+1)$. The characteristic hexagonal bow-tie pattern (i.e., the intensity is concentrated along the edge of the extended Brillouin zone), which was found at $T=0$, cf. Fig. \[Fig8\], is still present at $T/S(S+1)=1.3$.
Next we show in Fig. \[Fig14\] the static structure factor ${\cal S}(\mathbf{q})/S(S+1)$ along the path $\Gamma \to {\mathbf{Q}_1}
\to {\mathbf{Q}_0}
\to \Gamma $ for $S=1/2$ and $S=7/2$ for $T/S(S+1)=1.5$ (RGM and HTE) and $T=0$ (only RGM, see also Fig. \[Fig9\]). Obviously, the temperature $T/S(S+1)=1.5$ is already large enough, such that all four curves are very close to each other. Although, the weakening of magnetic ordering by thermal fluctuations is evident, the overall shape of the finite-temperature curves is similar to the GS curves, especially the maxima at ${\mathbf{Q}_1}$ ($\sqrt{3}\times\sqrt{3}$ state) and at ${\mathbf{Q}_0}$ ($q=0$ state) are still present, and ${\cal S}(\mathbf{Q}_1)>
{\cal S}(\mathbf{Q}_0)$.
\] for $S=1/2$ and $S=3$ at $T/S(S+1)=1.3$ (left: 9th order HTE, right: RGM). The red (black) circles indicate the expected maxima for a classical $\sqrt{3}\times \sqrt{3}$ ($q=0$) state. []{data-label="Fig13"}](fig13_strufa_contour_plot.pdf)
![ RGM and HTE data for the normalized structure factor ${\cal
S}(\mathbf{q})/S(S+1)$ along the path $\Gamma \to {\mathbf{Q}_1} \to {\mathbf{Q}_0}
\to \Gamma $ for $S=1/2$ and $S=7/2$ at $T/S(S+1)=1.5$. For comparison we also present RGM data for $T=0$ (dashed-dotted lines). []{data-label="Fig14"}](Fig14_sq_t_pathBZ_RGM_HTE_T15-eps-converted-to.pdf)
In experiments, often neutron scattering on powder samples are performed, see, e.g. [@herbertsmithite2009]. Hence, we also present the powder-averaged structure factor ${\cal S}^{\rm
av}(|\mathbf{q}|)/S(S+1)$, i.e., we integrate over all points at equal $q=|\mathbf{q}|$. We show HTE data for ${\cal S}^{\rm av}(|\mathbf{q}|)/S(S+1)$ for $S=1/2$ and $S=7/2$ at various temperatures in Fig. \[Fig15\]. The first broad maximum at $|\mathbf{q}| \sim 4.36$ corresponds to short-ranged antiferromagnetic correlations and its position is in good agreement with experiments on Herbertsmithite [@herbertsmithite2009]. (Note that the separation of NN copper ions in Herbertsmithite is $a=3.4$Å, here we use $a=1$.) While the influence of $T$ on the height of the maxima in $S^{\rm
av}(|\mathbf{q}|)$ is recognizable, the position of the maxima is almost independent of $T$. Thus, from Fig. \[Fig15\] and Fig. \[Fig14\] one can conclude that the type of magnetic SRO found at pretty high temperatures $T/S(S+1)
> 1$ indicate a possible magnetic ordering at low temperatures.
Last but not least we discuss the temperature dependence of the structure factors at the magnetic wave vectors $\mathbf{Q}_0$ ($q=0$ state) and $\mathbf{Q}_1$ ($\sqrt{3}\times \sqrt{3}$ state) and of the corresponding correlation lengths $\xi_{\mathbf{Q}_0}$ and $\xi_{\mathbf{Q}_1}$, see Figs. \[Fig16\] and \[Fig17\]. First we note that the $\sqrt{3}\times \sqrt{3}$ SRO is more pronounced than the $q=0$ SRO for all temperatures $T \ge 0$, i.e., ${\cal S}(\mathbf{Q}_1)|_{T} >{\cal S}(\mathbf{Q}_0)|_{T}$ and $\xi_{\mathbf{Q}_1}(T) >\xi_{\mathbf{Q}_0}(T)$ (cf. also Figs. \[Fig7\] and \[Fig9\] for the GS). As increasing $S$ the SRO becomes more distinct. Only at temperatures $T/S(S+1) \gtrsim 1$ the curves for different $S$ collapse to one universal curve, cf. [@Lohmann2014]. At low temperatures $T < T^*$ we find a plateau-like behavior in the correlation lengths and the structure factors, $\xi_{\mathbf{Q}_i}|_{T<T^*} \approx \xi_{\mathbf{Q}_i}|_{T=0}$ and ${\cal S}(\mathbf{Q}_i)|_{T<T^*} \approx {\cal S}(\mathbf{Q}_i)|_{T=0}$. The region of almost constant correlation lengths and structure factors is largest for $S=1/2$ and it shrinks noticeably as increasing $S$ approaching zero in the classical limit ($\lim_{S \to \infty} T^*/S(S+1) =
0$). To define a reasonable estimate of $T^*$ we chose that value of $T$, where correlation lengths and the structure factors reach $p=99\%$ of its GS values. The corresponding data are shown in Fig. \[Fig18\]. We mention that for the correlation lengths the relation $T^*=a/S(S+1)$ describes the plotted behavior accurately, where $a=0.2$ for $p=99\%$. (Note that the prefactor $a$ increases only slightly to $a=0.28$ as changing $p$ to $p=95\%$.) We may argue that below $T^*$ the quantum fluctuations are more important than thermal fluctuations.
![ Main: RGM data for the correlation length $\xi_{\mathbf{Q}_i}$ (dashed - $\mathbf{Q}_0$; solid - $\mathbf{Q}_1$) for various values of the spin $S$ as a function of the normalized temperature $T/S(S+1)$. Inset: Correlation lengths $\xi_{\mathbf{Q}_i}$ (dashed - $\mathbf{Q}_0$; solid - $\mathbf{Q}_1$) for $S=1/2$ and $S=1$ using an enlarged y-axis.[]{data-label="Fig17"}](Fig17.pdf)
Summary {#sec:sum}
=======
We use two methods to discuss the thermodynamic properties of the kagome Heisenberg antiferromagnet with arbitrary spin $S$, namely the rotational invariant Green’s function method (RGM) and the high-temperature expansion (HTE). Within the RGM we consider GS as well as finite-temperature properties, whereas the HTE is restricted to $T/S(S+1) \gtrsim 1$. Within the RGM approach the model does not exhibit magnetic LRO for all values of $S$. In the extreme quantum case $S=1/2$ the zero-temperature correlation length $\xi(T=0)$ is only of the order of the nearest-neighbor separation. As increasing $S$ the correlation length $\xi(T=0)$ grows according to a power-law in $1/S$. We found that the so-called $\sqrt{3}\times\sqrt{3}$ SRO is favored versus the $q=0$ SRO for all values of $S$. It is worth mentioning that other methods specifically designed for the GS [@Goetze2011; @Goetze2015; @Oitmaa2016; @Liu2016] indicate that GS LRO may appear for $S\ge 3/2$. As known from previous studies the rotational invariant decoupling in the RGM scheme may overestimate the tendency to suppress magnetic order, cf. [@Barabanov94; @Ihle2001; @Schmalfus2004; @Haertel2013; @Mikheyenkov2013] and references therein.
As typical for two-dimensional Heisenberg antiferromagnets, the specific heat and the uniform susceptibility exhibit a maximum related to the size of the exchange coupling $J$. For both quantities, with growing $S$ this maximum moves towards lower values of $T/S(S+1)$ and its height increases. In the limit of large $S$ the RGM data approach the classical curves.
The structure factor ${\cal S}(\mathbf{q})$ shows two maxima at magnetic wave vectors $\mathbf{q}={\mathbf{Q}_i}, i=0,1$, corresponding to the $q=0$ and $\sqrt{3}\times\sqrt{3}$ state, where ${\cal
S}(\mathbf{Q}_1)>{\cal S}(\mathbf{Q}_0)$ holds for all values of $S$ and all temperatures $T \ge 0$. In a finite low-temperature region $T < T^*\approx a/S(S+1), a \approx 0.2$, the magnetic SRO is quite stable against thermal fluctuations, i.e., the correlation lengths and the structure factors ${\cal
S}(\mathbf{Q}_1)$ and ${\cal S}(\mathbf{Q}_0)$ are almost independent of $T$. The powder-averaged structure factor ${\cal S}(|\mathbf{q}|)$ exhibits a broad maximum related to short-ranged antiferromagnetic correlations and its position is in good agreement with experiments on powder samples of Herbertsmithite [@herbertsmithite2009].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank D. Ihle and Paul McClarty for valuable hints.
|
---
abstract: 'We propose a numerical criterion which can be used to obtain accurate and reliable values of the ordering temperatures and critical exponents of spin glasses. Using this method we find a value of the ordering temperature for the $\pm J$ Ising spin glass in three dimensions which is definitely non-zero and in good agreement with previous estimates. We show that the critical exponents of three dimensional Ising spin glasses do not appear to obey the usual universality rules.'
author:
- |
L.W. Bernardi, S. Prakash and I.A. Campbell\
Physique des Solides,\
Université Paris Sud,\
91405 Orsay, France
title: Ordering temperatures and critical exponents in Ising spin glasses
---
The full explanation of the universality rules for critical exponents in second order transitions through the renormalization group theory is one of the most impressive achievements of statistical physics. The universality rules for such transitions state that the critical exponents depend only on the space dimension $d$ and a few basic parameters : the number of order parameter components $n$, the symmetry and the range of the Hamiltonian [@1]. No other parameters are pertinent. In fact it is known that there are exceptions to universality - in certain two dimensional ($2d$) Ising systems with regular frustration the critical exponents vary continuously with the value of a control parameter [@2]. As far as we are aware, no results of this type have been reported in any three dimensional ($3d$) family of Ising systems; it has been tacitly assumed that non-universality is very exceptional.
As compared to standard second order transitions, the situation concerning Ising Spin Glasses (ISGs) is much less clear; indeed the history of ISG simulations has been plagued by technical difficulties associated with long relaxation times. For two decades the very existence of a finite temperature transition in the $3d$ ISG has been hotly contested; as it is obviously essential to have a reliable value of the ordering temperature before obtaining accurate critical exponent estimates, it has been difficult to make stringent numerical tests of universality in $3d$ ISGs.
We will present a numerical criterion which can in favourable cases provide precise and reliable values for the ordering temperature $T_g$ and for the critical exponents of a spin glass, with a moderate level of computational effort. If an independent estimate of the ordering temperature is available the criterion leads to a convenient method for estimating the exponents. We study $3d$ ISGs with various sets of interactions and we conclude from the data that the $3d$ $\pm J$ interaction ISG has a well defined non-zero Tg which can be estimated accurately, and that universality in the usual sense does not hold in $3d$ ISGs.
It would appear probable that glassy transitions in general have a much richer critical behaviour than have conventional second order transitions.
Thus, technically the most difficult problem in numerical ISG studies is the correct identification of the transition temperature $T_g$. For the $3d$ ISG with random $\pm J$ near neighbour interactions on a simple cubic lattice, which has been the subject of a considerable ammount of work, $T_g$ has been estimated in two ways. Ogielski [@3] studied in massive simulations the divergence of the spin glass susceptibility, of the correlation length, and of the relaxation time of the autocorrelation function $$q(t)=\left[<S_i(t)S_i(0)>\right]
\label{eq:1}$$ in order to estimate $T_g$ and the critical exponents. However his analysis has been questioned because of the possibility of ambiguities in the manner of identifying a divergence, if non-conventional temperature dependencies are invoked [@4]. Bhatt and Young [@5] used a finite size scaling technique; they measured the Binder cumulant for the fluctuations of the equilibrium autocorrelation function $$gL=\frac{1}{2}\left[ 3 - \frac{< q^4>}{<q^2>^2}\right]
\label{eq:2}$$ as a function of sample size $L$. The curves $g_L(T)$ for different $L$ should all intersect at $T_g$; in the $3d$ $\pm J$ ISG case the curves indeed intersected but did not appear to fan out below the apparent $T_g$. Only recently have intensive numerical studies shown that a weak fanning out at low temperatures really does occur [@6; @7]. Even with results of high statistical accuracy to hand, Kawashima and Young [@6] give a number of caveats concerning the interpretation of their own data.
We will describe an alternative criterion for determining $T_g$. First, scaling rules tell us [@3] that for a large sample in thermal equilibrium at $T_g$ the relaxation of the autocorrelation function takes the form $$q(t)=\lambda t^{-x}
\label{eq:3}$$
with the exponent $x$ related to the standard static and dynamic exponents $\eta$ and $z$ through $$x=\frac{(d-2+\eta)}{2z}.
\label{eq:4}$$
Secondly, the out of equilibrium relaxation of two randomly chosen replicas $A$ and $B$ of the same sample towards equilibrium at $T_g$ depends on another combination of the same exponents [@8]. The out of equilibrium spin glass susceptibility is defined as $$\chi'_{SG}(t)=\left[<S_i^A(t)S_i^B(t)>^2\right]
\label{eq:5}$$ and it increases with time as $$t^h \ \mathrm{with}\ h = \frac{2-\eta}{z}.
\label{eq:6}$$ Suppose we take $\{T_i\}$, a series of trial values for $T_g$; from measurements of $x$ and $h$ on large samples at each $T_i$ we can deduce from equations \[eq:4\] and \[eq:6\] a set of apparent or effective exponents $$\eta_1(T)=\frac{4x - h(d-2)}{2x+h}
\label{eq:7}$$ $$z(T)=\frac{d}{2x+h}.
\label{eq:8}$$
Finally in another set of simulations on the same system at different \[small\] sample sizes $L$, from standard finite size scaling rules [@5] for the fluctuations in the autocorrelation function in equilibrium at $T_g$ we have $$L^{d-2}<q^2>\ \propto\ L^{-\eta}
\label{eq:9}$$ If we again take a series of trial values of $T_g$ and fit the results using this form at each $T_i$ we will obtain a second series of apparent exponent values $\eta_2(T)$. (This type of fit will only be appropriate close to and below $T_g$; at higher $T$ another factor appears on the right hand side [@5]).
We now plot $\eta_1(T)$ and $\eta_2(T)$ against $T$; for consistency the curves must intersect at the point \[$\eta$, $T_g$\] which represents the true critical exponent $\eta$ and ordering temperature $T_g$ of the system. At this temperature and this temperature only the functional forms of equations \[eq:3\], \[eq:6\] and \[eq:9\] should be exact; at neighbouring temperatures these forms are only approximate but close to $T_g$ they will be adequate to parametrise the numerical data. Once $T_g$ is fixed by the intersection we can obtain $z$ using the $z(T)$ curve given above, and with known $\eta$ and $T_g$ we can go on to fit $<q^2>$ data for temperatures above $T_g$ to obtain the exponent $\nu$. From scaling relations, once we dispose of $\eta$ and $\nu$ all other static exponents are determined.
We show in figure \[fig:1\]
estimates for $\eta_1(T)$ and $\eta_2(T)$ for the $3d$ $\pm J$ ISG calculated using data taken from the literature : $x(T)$ from [@3], $h(T)$ from [@8; @9], and the spin glass susceptibilities for different assumed values of $T_g$; ($T_g=1.0$ from the data given in [@5], $T_g = 1.11$ from [@6], and $T_g=1.175$ from [@3]). There is a well defined crossing point with $T_g=1.165 \pm 0.01$ and $\eta=-0.245
\pm 0.02$. Using the curve for $z(T)$ from equation \[eq:8\] we estimate $z=6.0 \pm 0.2$.
The values obtained in this way are at least as precise as previous estimates and are very close to the central values given by Ogielski [@3] ($T_g=1.175 \pm 0.025$, $\eta=-0.22 \pm 0.05$, $z=6.0\pm0.8$), corroborating his analysis. On the other hand the $T_g$ is marginally outside the error bars quoted by Kawashima and Young ($T_g=1.11 \pm 0.04$) who use extensive Binder cumulant data [@6]. The difficulty in applying this latter method to the $3d$ $\pm J$ ISG case is that the $g_L(T)$ curves lie very close together below $T_g$ so the intersection point is sensitive to small changes in individual $g_L$ curves. Even with extreme statistical accuracy, small corrections to finite size scaling (invoked as a possibility in [@6]) can change the apparent position of the intersection point significantly. The results of ref [@6] could be rendered consistent with the present analysis if the $g_L$ values for the smallest samples studied were affected by corrections to finite size scaling at the 1% level.
The present method is much less sensitive to problems of systematics than are either of the other techniques outlined above. First, both $x$ and $h$ are determined using “large” samples so finite size corrections should be unimportant [@8; @9]. Secondly $h$ is measured out of equilibrium and so is not subject to the problems of long equilibration times. The fact that no preparatory anneal is required also means that the measurements are economical in computer time. The measurements of $x$ need careful equilibration but systematic tests using successsively longer preliminary anneals allow one to obtain reliable values. Numerical data [@3; @10] show that in ISGs $q(t)$ already takes on the asymptotic form, equation \[eq:3\], from quite early times $t \simeq 2$ MCS (Monte Carlo Steps), and that sample to sample variations in the values of $x$ are small so extensive averaging over very large numbers of samples (an essential condition for good $g_L$ data) is unnecessary. Thus the curve $\eta_1(T)$ can be established accurately with moderate numerical effort and minimal systematic error. For the finite size scaling data from which $\eta_2(T)$ is deduced, thorough equilibration is necessary but by studying pairs of replicas [@5] and again testing with increasing anneal times it is easier to obtain accurate values of $<q(t)^2>$ than the combination of moments which constitute the Binder cumulant. Again, the sample to sample variability is much less for $<q(t)^2>$ than for the Binder cumulant. In the $3d$ $\pm J$ ISG the two curves $\eta_1(T)$ and $\eta_2(T)$ intersect cleanly, figure \[fig:1\], so the determination of the crossing point should not be very sensitive to minor deviations from scaling or small statistical uncertainties. Finally, no hypothesis is made concerning the way divergences occur except the essential assumption that standard scaling rules (as opposed to universality rules) hold. The excellent overall agreement between Ogielski’s estimates [@3] and the present ones gives considerable confidence in the general coherence of the standard scaling approach and appear to make any exotic scaling assumption unnecessary.
We therefore consider that both $\eta_i(T)$ curves can be calculated with little in the way of disguised systematic errors; as they stand the $T_g$ and exponent values that we quote should not only be precise but reliable.
We have made further simulations on another $3d$ ISG with $\pm J$ interactions; this is the the fully frustrated system with 20% random bond disorder that we studied in [@10]. We already established an accurate value of $T_g$ ($T_g=0.96$) for this spin glass from Binder cumulant measurements, and we now have measured the exponents $x$ and $h$ at $T_g$ together with an estimate of $\eta$ from the spin glass susceptibility (see Table \[table:1\]). The data are very consistent with each other and lead to an $\eta$ value which is less negative and a $z$ value which is smaller as compared with those of the standard $\pm J$ ISG. This difference already indicates the non-universality of these two exponents in $3d$ ISGs.
We have also carried out extensive simulations on $3d$ ISG systems with different sets of near neighbour interactions. For the $3d$ ISGs with near neighbour Uniform, Gaussian and decreasing Exponential interactions (see [@11] for the definitions of the distributions with the correct normalizations), the data are shown in figure \[fig:1\]. Simulations were done on samples with $L=16$ for $x$, $L=10$ for $h$, and samples from $L=2$ to $L=6$ for $<q(t)^2>$. Careful anneals were carried out where appropriate, checked by the prescription given in [@5]. At each temperature, 10 samples were used for $x$, 500 for $h$ and 2000 to 200 depending on $L$ for $<q(t)^2>$. We estimate that the $\eta_1(T)$ curves are on large enough samples for there to be virtually no finite size correction, so the values can be taken as definitive (apart from statistical errors), but measurements on larger samples could modify the $\eta_2(T)$ curves marginally. It can be seen that the $\eta(T)$ curves again cross cleanly for the Uniform case with a more negative $\eta$ than for the $\pm J$ case. However for the Gaussian and Exponential cases it turns out that the two curves are much more similar to each other making it difficult to identify $T_g$ precisely; for these distributions we have to fall back on an alternative method to estimate $T_g$.
The Migdal-Kadanoff (MK) scaling approach is known to give reasonable values of the ordering temperature for Ising spin glasses [@12; @13; @14]. We have followed the particular method used by Curado and Meunier [@14] but with improved statistical accuracy. It turns out that with a scale factor $b=2$ the MK estimate for the $3d$ $\pm J$ ISG $T_g$ is $1.16 \pm 0.01$, precisely the same as the value we have obtained above from the simulations. This perfect agreement is certainly fortuitous (though in $4d$ where the MK method should be much poorer, the disagreement in $T_g$ between the $b=2$ MK estimate and an accurate simulation value is only 15% [@15]), but we argue that as agreement happens to be excellent for the $\pm J$ case, if we apply the same method with the same scale factor $b$ to other $3d$ ISGs with different sets of interactions, we should obtain $T_g$ estimates which should again be very close to the real values. We obtain MK $T_g$ values which are 1.00, 0.88, and 0.72 for the Uniform, Gaussian and Exponential distributions respectively [@15]. The Uniform distribution value is in good agreement with the simulation value and the other two $T_g$ values are within the range of $T$ where the simulation curves for $\eta_1(T)$ and $\eta_2(T)$ overlap. The Gaussian $T_g$ and $\eta$ are in good agreement with earlier estimates [@5]. Putting uncertainties at $\pm 0.05$ for possible systematic errors in the Gaussian and Exponential MK $T_g$ estimates, we obtain the set of exponent estimates shown in Table \[table:1\].
System $T_g$ $x(T_g)$ $h(T_g)$ $\eta$ $z$
--------- ---------------- ---------- ---------- ----------------- --------------
$\pm J$ $1.165\pm0.01$ 0.064 0.38 $-0.245\pm0.02$ $6.0\pm0.2$
FFd0.2 $0.96\pm0.02$ 0.091 0.437 $-0.12\pm0.02 $ $4.85\pm0.3$
U $1.05\pm0.03$ 0.054 0.41 $-0.375\pm0.03$ $5.8\pm0.5$
G $0.88\pm0.05$ 0.035 0.355 $-0.50\pm0.04$ $7.1\pm0.6$
Exp $0.72\pm0.05$ 0.02 0.275 $-0.62\pm0.12$ $9.5\pm0.7$
: Temperature of transition and critical exponents for several distributions. The distributions are in order ($i$) random $\pm J$ interactions, ($ii$) Fully Frustrated lattice with 20% disorder [@10], ($iii$) random Uniformly distributed interactions, ($iv$) random Gaussian interactions, ($v$) random Decreasing Exponential interactions[]{data-label="table:1"}
According to the usual universality rules, the form of the interaction distribution should not be a pertinent parameter as concerns the critical exponents. Here we find that $3d$ ISG systems which differ only by this distribution function show quite different $\eta$ and $z$ values, Table \[table:1\]. The results indicates a breakdown of conventional universality in $3d$ ISGs.
In order to show that the apparent non-universality is not an artefact, we will turn back to the raw $x$ and $h$ data for the $\pm J$ and Uniform cases. In figure \[fig:2\]
we have plotted the values of these parameters as a function of $T$; the error bars are about $\pm 0.005$ for $h$ and $\pm 0.002$ for $x$. If universality holds $$h(T_g(U)) \equiv h(T_g(J))
\label{eq:10}$$ and $$x(T_g(U)) \equiv x(T_g(J)).
\label{eq:11}$$ By inspection, whatever trial value $T^*$ we choose for $T_g(J)$ within the generous limits $T^*=$ 1.0 to 1.3 provided by the figure, the relation \[eq:10\] leads to us to a $T^*_g(U)$ such that $x(T^*_g(U))$ is considerably smaller than $x(T^*_g(J))$. For instance with $T_g^*(J)=1.16$, $T_g^*(U)=0.88$, $x(T_g^*(J))=0.064$, $x(T_g^*(U))=0.036$. The data cannot satisfy \[eq:10\] and \[eq:11\] simultaneously, demonstrating non-universality.
For the $2d$ regularly frustrated systems which show continuous variation of critical exponents, the breakdown of universality is necessarily associated with the existence of a marginal operator [@16] and it has been pointed out that when breakdown occurs, it does so in Ising systems having more than two ground states [@17] and hence with $n$, the number of components of the order parameter, greater than 1 [@18]. On the Parisi image of finite dimension ISGs [@19], $n$ is essentially infinite; it would be of interest to identify possible marginal operators. We can note that in the regularly frustrated $2d$ systems quoted above, $\nu$ varies continuously but $\eta$ is constant so “weak universality” [@20] still holds. This is not the case for the randomly frustrated systems we have studied.
It would appear that universality breakdown could be much more prevalent than was suspected, and it may well be the rule rather than the exception at spin glass or glass transitions.
We would like to thank D$^r$ N. Kawashima for permission to quote unpublished data. Simulations were carried out thanks to time allocations from IDRIS (Institut du Développement des Ressources en Informatique Scientifique) and TRACS, University of Edinburgh. L.W.B. gratefully acknowledges support from TRACS.
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---
abstract: 'We present the first search for the decay ${D^+_{\mathrm s}}\to \omega e^{+}\nu$ to test the four-quark content of the ${D^+_{\mathrm s}}$ and the $\omega$-$\phi$ mixing model for this decay. We use $586$ $\mathrm{pb}^{-1}$ of $e^{+}e^{-}$ collision data collected at a center-of-mass energy of 4170 MeV. We find no evidence of a signal, and set an upper limit on the branching fraction of $\mathcal{B}({D^+_{\mathrm s}}\to\omega e^+\nu)<$0.20% at the 90% confidence level.'
author:
- 'L. Martin'
- 'A. Powell'
- 'G. Wilkinson'
- 'J. Y. Ge'
- 'D. H. Miller'
- 'I. P. J. Shipsey'
- 'B. Xin'
- 'G. S. Adams'
- 'B. Moziak'
- 'J. Napolitano'
- 'K. M. Ecklund'
- 'J. Insler'
- 'H. Muramatsu'
- 'C. S. Park'
- 'L. J. Pearson'
- 'E. H. Thorndike'
- 'S. Ricciardi'
- 'C. Thomas'
- 'M. Artuso'
- 'S. Blusk'
- 'R. Mountain'
- 'T. Skwarnicki'
- 'S. Stone'
- 'L. M. Zhang'
- 'G. Bonvicini'
- 'D. Cinabro'
- 'A. Lincoln'
- 'M. J. Smith'
- 'P. Zhou'
- 'J. Zhu'
- 'P. Naik'
- 'J. Rademacker'
- 'D. M. Asner'
- 'K. W. Edwards'
- 'K. Randrianarivony'
- 'G. Tatishvili'
- 'R. A. Briere'
- 'H. Vogel'
- 'P. U. E. Onyisi'
- 'J. L. Rosner'
- 'J. P. Alexander'
- 'D. G. Cassel'
- 'S. Das'
- 'R. Ehrlich'
- 'L. Gibbons'
- 'S. W. Gray'
- 'D. L. Hartill'
- 'B. K. Heltsley'
- 'D. L. Kreinick'
- 'V. E. Kuznetsov'
- 'J. R. Patterson'
- 'D. Peterson'
- 'D. Riley'
- 'A. Ryd'
- 'A. J. Sadoff'
- 'X. Shi'
- 'W. M. Sun'
- 'J. Yelton'
- 'P. Rubin'
- 'N. Lowrey'
- 'S. Mehrabyan'
- 'M. Selen'
- 'J. Wiss'
- 'J. Libby'
- 'M. Kornicer'
- 'R. E. Mitchell'
- 'D. Besson'
- 'T. K. Pedlar'
- 'D. Cronin-Hennessy'
- 'J. Hietala'
- 'S. Dobbs'
- 'Z. Metreveli'
- 'K. K. Seth'
- 'A. Tomaradze'
- 'T. Xiao'
title: 'Search for the decay ${D^+_{\mathrm s}}\to\omega e^+\nu$ '
---
Introduction
============
Multiple observations of exotic charmonium states $[1-4]$ have been widely interpreted as four-quark states $[5-13]$. In this analysis we probe the four-quark content of the ${D^+_{\mathrm s}}$ by searching for the decay ${D^+_{\mathrm s}}\to\omega e^+\nu$ (charge conjugate states are implied throughout the article). Assuming that the $\omega$ is a pure two-quark state, its valence quarks are distinct from those of the ${D^+_{\mathrm s}}$, and the decay can proceed through the diagram of Fig. 1, where the $(u\bar{u})$ or $(d\bar{d})$ come from within the ${D^+_{\mathrm s}}$. The initial valence quarks annihilate while a lepton pair is produced. Neither Cabibbo-favored, nor Cabibbo-suppressed decays can contribute to this final state. The study of this specific process was first suggested in Ref. [@bonvi], and Ref. [@gabbiani] estimates the theoretical branching fraction for the analogous decay $B^+\to J/\psi\ \ell^+\nu$.
![ Feynman diagram representing the four-quark semileptonic decay $ {D^+_{\mathrm s}}\to \omega e^{+}\nu$.[]{data-label="fig:psilnu"}](fig1.eps){width="4.0in"}
Recent work by Gronau and Rosner [@rosner] concludes that any value of the branching fraction for ${D^+_{\mathrm s}}\to \omega e^+ \nu$ exceeding $2 \times
10^{-4}$ is unlikely to be explainable by $\omega$-$\phi$ mixing and would provide evidence for non-perturbative effects known as “weak annihilation” (see Ref. [@rosner] for references). An estimate based on comparing hadronic and semileptonic processes gives a branching fraction of $(0.13 \pm
0.05)\%$. This is the first search for this decay mode.
We search for a positron candidate and an $\omega\to \pi^+\pi^-\pi^0$ candidate, which is the dominant decay mode with a branching fraction of 89.2% [@pdb]. Cabibbo-favored decays exist in the same final state, ${D^+_{\mathrm s}}\to\eta e^+\nu$ and ${D^+_{\mathrm s}}\to\phi e^+\nu$, with $\mathcal{B}(\eta\to\pi^+\pi^-\pi^0)=$22.73% and and $\mathcal{B}(\phi\to\pi^+\pi^-\pi^0)=$15.32% [@pdb]. They can play the role of control samples, which are used directly in the analysis in a variety of ways. For example, the effect of certain selection requirements can be readily estimated from any change in the $\eta$ and $\phi$ populations. The two control samples are also well measured using the independent final states $\eta\to \gamma\gamma$ and $\phi\to K^+K^-$. Therefore, this analysis has good statistical sensitivity, and unusually strong control samples using CLEO-c data directly.
The remainder of this article is organized as follows. In Sec. II, the detector, data, and Monte Carlo (MC) samples are described. The data analysis method is described in Sec. III. The fitting procedure is described in Sec. IV. The determination of the branching fraction is discussed in Sec. V.
Detector, data and MC samples
=============================
The data used in the analysis are $e^{+}e^{-}$ collisions at a center-of-mass energy $\sqrt{s}=$4170 MeV. The data are collected by the CLEO-c detector and correspond to an integrated luminosity of $586~\mathrm{pb}^{-1}$, or $0.6\times 10^6$ ${D^+_{\mathrm s}}{D^-_{\mathrm s}}$ inclusive pairs.
The CLEO-c detector is optimized for physics in the charmonium region and is described in detail in Ref. [@cleonim]. The tracking system consists of a central, low mass drift chamber, which wraps directly around the beam pipe, and a main drift chamber inside a solenoidal magnetic field. The particle identification system combines the track information from the gas chambers ($dE/dx$) and the associated ring-imaging Cherenkov detector (RICH) data. The electromagnetic calorimeter consists of CsI crystals, arranged in cylindrical fashion around the drift chamber, to make a barrel at angles given by $|\cos{\theta}|<0.7$, with $\theta$ being the angle measured from the interaction point (IP) with respect to the beam axis. The ends of the cylinder are also instrumented with CsI crystals and are referred to as the end-cap regions.
MC simulations of the known physics processes [@EVTGEN] and of the CLEO-c detector [@GEANT4] are used to estimate backgrounds and calculate signal resolution and efficiencies. The known charm physics processes are included in the $c\bar{c}$ MC simulation. All types of charm backgrounds, dominant in this analysis, are simulated to 20 times the statistics in the data, while the continuum $(u,d,s)$ backgrounds are simulated to 6.6 times the statistics in the data. In the following, where MC results are presented, we multiply the continuum MC sample by 3 to obtain a consistent $\times 20$ normalization. By convention, $c\bar{c}$ MC refers to the charm part of the MC, continuum MC is the non-charm part, and MC is the weighted sum of the two.
The signal MC sample consists of 8$\times 10^5$ ${D^+_{\mathrm s}}\to\omega e^+ \nu$ events generated according to phase space distribution. The same number of events was generated for the ${D^+_{\mathrm s}}\to\eta e^+\nu$ and ${D^+_{\mathrm s}}\to\phi e^+\nu$ control samples.
Further samples for dominant sources of backgrounds were generated. A sample of 2$\times 10^5$ ${D^+_{\mathrm s}}\to\omega\pi^+\pi^0$ events was separately generated, corresponding to about 23.5 times the number of such expected decays in the data. In addition, 5000 events (96 times the data) were generated for the decay chain ${D^+_{\mathrm s}}\to\eta^{\prime} e^+\nu$ followed by $\eta^{\prime}\to \omega\gamma$.
Data selection
==============
About 95% of the $e^+e^- \to {D^+_{\mathrm s}}X$ events are formed through the following exclusive reactions [@cross] $$e^+e^- \to {D^+_{\mathrm s}}D_{\mathrm s}^{\star -}\;{\rm and}\;e^+e^- \to{D^-_{\mathrm s}}D_{\mathrm s}^{\star +}.
\label{eq:reactions}$$ Equal amounts of positive and negative $D_{\mathrm s}^{\star}$ states are produced, and about 95% of them decay to $D_{\mathrm s}\gamma$. The analysis described here selects exclusively both final states in Eq. (\[eq:reactions\]). In the following, we retain both events where the $\gamma$ is associated with the positive side, the signal side due to the convention established in Sec. I, and where the $\gamma$ is associated with the negative side, or tag side. Several kinematic constraints are available, but only those that select both reactions in Eq. (\[eq:reactions\]) are used.
First we search for an exclusively reconstructed hadronic ${D^-_{\mathrm s}}$ candidate, the tag, and a photon candidate. Requiring both of these objects, with three kinematic selections applied, strongly suppresses the backgrounds. The photon candidate, and all tracks and showers which are daughters of the tag candidate, are not used when searching the rest of the event, which, due to the exclusive nature of the analysis, must be the signal candidate. The following are required to form a signal candidate: a positron of opposite charge to the tag, precisely three charged tracks, a $\pi^{0}$ candidate, and net event charge equal to zero. Furthermore, the missing energy and momentum are required to be in a relation consistent with the presence of a nearly massless neutrino. Extra showers in the event are ignored.
The selection is described in detail in the remainder of this section.
Charged and neutral particles selection
---------------------------------------
The criteria for a good track include the requirements that its minimum distance to the IP does not exceed 5 mm in the plane perpendicular to the beam axis and 5 cm along the beam axis. Phase space is limited to $|\cos{\theta}|<0.93$, with $\theta$ being the angle with the beam axis, and momentum $0.05{\ \rm GeV}\ <p<2$ GeV. Good tracks are then selected as pion or kaon using dE/dx and RICH data according to the algorithms described in detail in Ref. [@briere].
Photon candidates are contiguous groups of crystals recording significant energy deposition. They are required to be unmatched to tracks or noisy crystals, and to have a transverse profile consistent with expectations from an electromagnetic shower. The minimum cluster energy is 30 MeV.
Positrons are selected by requiring a good track, and the combined positron probability of the particle ID system and $E/p$ (the ratio between the shower energy associated with the candidate electron and its track momentum) to be greater than 0.8. Positron phase space requirements are $|\cos{\theta}|<0.9$ and $p>0.2$ GeV.
$\pi^0$ and $\eta$ candidates are selected by requiring a photon candidate in the barrel or end-cap, with $|\cos{\theta}|<0.93$. This photon is combined with a second photon, which must not be associated with a noisy crystal, and the invariant mass of the two photons must be within 3 standard deviations of the nominal $\pi^0$ and $\eta$ masses [@pdb].
$K^{0}_{\mathrm S}$ candidates are selected by requiring two oppositely charged tracks. If they are assigned the nominal $\pi^+$ mass [@pdb], their invariant mass must be within 12 MeV of the nominal $K^{0}_{\mathrm S}$ mass [@pdb]. A common vertex is calculated, and it is required to be radially displaced from the IP by at least 3 standard deviations.
\[tab:modes\]
Mode Signal region (GeV) Low sideband (GeV) High sideband (GeV)
---------------------------------------- --------------------- -------------------- ---------------------
$K^0_{\mathrm S} K^- $ \[1.954, 1.983\] \[1.910, 1.939\] \[1.998, 2.026\]
$K^+ K^- \pi^-$ \[1.954, 1.982\] \[1.911, 1.940\] \[1.996, 2.025\]
$K^{\star -}\overline{K}^{\star 0} $ \[1.953, 1.983\] \[1.909, 1.938\] \[1.997, 2.027\]
$\pi^+\pi^-\pi^-$ \[1.955, 1.982\] \[1.913, 1.941\] \[1.996, 2.024\]
$\eta \pi^-$ \[1.940, 2.001\] \[1.892, 1.922\] \[2.019, 2.050\]
$\eta \rho^-$ \[1.940, 1.998\] \[1.885, 1.914\] \[2.021, 2.050\]
$\pi^-\eta^{\prime} (\eta\pi^+\pi^-) $ \[1.944, 1.992\] \[1.885, 1.933\] \[2.004, 2.052\]
$\pi^-\eta^{\prime}(\rho\gamma) $ \[1.944, 1.992\] \[1.886, 1.930\] \[2.002, 2.047\]
: Definitions of $M_{\mathrm{tag}}$ signal and sideband definitions for the tag modes.
Tag-candidate selection
-----------------------
Eight tag decay modes are used and listed in Table \[tab:modes\], using the particle candidates selected according to Sec. III.A. In addition, there are several mode-specific criteria. For ${D^-_{\mathrm s}}\to K^- K^+ \pi^-$ and ${D^-_{\mathrm s}}\to \pi^+ \pi^- \pi^-$, the pion momenta are required to be greater than 0.1 GeV. For ${D^-_{\mathrm s}}\to K^{\star -} \overline{K}^{\star 0}$, only the $(K^{0}_{\mathrm S}\pi^-)(K^+\pi^-)$ channel is considered. The $K^{\star -}$ and $\overline{K}^{\star 0}$ candidate masses are required to be within 100 MeV of the nominal value [@pdb]. For the ${D^-_{\mathrm s}}\to \eta \rho^{-} (\rho^-\to \pi^-\pi^0)$, the $\rho^-$ mass must be within $150$ MeV of the nominal value [@pdb]. For ${D^-_{\mathrm s}}\to \pi^- \eta^{\prime} (\eta^{\prime} \to \rho \gamma), \rho^0\to\pi^+\pi^-$, the $\eta^{\prime}$ mass must be within $20$ MeV of the nominal value [@pdb]. Furthermore, the $(\pi$-$\eta^{\prime})$ helicity angle (defined as the angle $\theta_H$, in the rest frame of the $\rho$, between the momentum of the $\pi^-$ and the momentum of the ${D^-_{\mathrm s}}$) is required to satisfy $|\cos{\theta_H}|<0.8$.
The four-momentum of a tag candidate is defined by $(E_{\mathrm tag},\mathbf{p}_{\mathrm tag})$, with the tag mass defined by $M_{\mathrm{tag}}$. The selection further makes use of the recoil mass ${M_{\mathrm{rec}}}$, defined as $${M_{\mathrm{rec}}}=\sqrt{(E_{\mathrm b}-E_{\mathrm{tag}})^2-(\mathbf{p}_{\mathrm b}-\mathbf{p}_{\mathrm{tag}})^2}.
\label{eq:mrec}$$ Here, $(E_{\mathrm b},\mathbf{p}_{\mathrm b})$ is the four-momentum of the colliding beams. The ${M_{\mathrm{rec}}}$ distribution will peak only for those events where the photon is associated with the signal side, but even when the photon is associated with the tag side, ${M_{\mathrm{rec}}}$ is kinematically constrained so that $|{M_{\mathrm{rec}}}-M^{\star}|<$ 55 MeV, where $M^{\star}$ is the nominal $D^{\star}_{\mathrm{s}}$ mass [@pdb]. Only candidates passing this selection are retained.
The main tag selection is obtained from a 2-D fit described below. Because of the complexity of the fit, the $M_{\mathrm{tag}}$ projection is fitted first, and the fit results used to constrain some of the final 2-D fit nuisance parameters. The $M_{\mathrm{tag}}$ projection is also best suited for side-band background subtraction.
The $M_{\mathrm{tag}}$ distribution is fitted with a double Gaussian function, $G_2$, multiplied by the fitted number of events, $N$, and a first degree polynomial, $A_1$, to describe signal and background $$f(M_{\mathrm{tag}})=NG_2(M_{\mathrm{tag}})+A_1(M_{\mathrm{tag}}).
\label{eq:fitg2}$$ $G_2$ is a probability distribution composed of two Gaussian functions $G(x;\sigma,\mu)$, of unit area, peaking at $\mu$ and with width equal to $\sigma$. Here, the peak is set at ${M_{\mathrm{Ds}}}$, which is the nominal nominal ${D^+_{\mathrm s}}$ mass [@pdb], and the two Gaussians have fractional probabilities $f_1$ and $(1-f_1)$ $$G_2(M_{\mathrm{tag}})= f_1G(M_{\mathrm{tag}};\sigma_1,{M_{\mathrm{Ds}}})+(1-f_1)G(M_{\mathrm{tag}};\sigma_2,{M_{\mathrm{Ds}}}).
\label{eq:defg2}$$
The quantities $\sigma_1$ and $\sigma_2$ are fixed to the value obtained from the fit to the signal MC data. Having obtained $f_1$ from the fit, we construct the variable $\sigma^2_{12}=f_1\sigma_1^2+(1-f_1)\sigma_2^2$. The signal regions are required to be within 2.5$\sigma_{12}$ from the peak position for each mode except for the $(\eta \rho)$ mode where it is selected within 2$\sigma_{12}$.
The $M_{\mathrm{tag}}$ data fit results are listed in Table \[tab:tagfit\]. The rest of the peak fit parameters are listed in Table \[tab:tagfitdg\]. The $M_{\mathrm{tag}}$ distributions for the 8 modes are shown in Fig. \[fig:tagall\]. The sidebands in the $M_{\mathrm{tag}}$ distribution are listed, for each mode, in Table \[tab:modes\].
Having determined the fit parameters for the $M_{\mathrm{tag}}$ distributions, a second kinematic constraint can be imposed using the $MM^{*2}$ variable defined as
$$\label{eq:mmstar2}
MM^{*2}=(E_{\mathrm b}-E_{\mathrm{tag}}-E_{\gamma})^2-(\mathbf{p}_{\mathrm b}-\mathbf{p}_{\mathrm{tag}}-
\mathbf{p}_{\gamma})^2,$$
where $(E_\gamma,\mathbf{p}_\gamma)$ is the photon four-momentum. If the final state is given by Eq. (\[eq:reactions\]), $MM^{*2}$ should peak at ${M_{\mathrm{Ds}}}^2$. The $MM^{*2}$ mass selection criteria are found by a two-dimensional (2-D) binned likelihood fit in the ($MM^{*2},M_{\mathrm{tag}}$) space. Each variable is also kinematically fitted, so that $M_{\mathrm{tag}}$ is the value obtained by constraining $MM^{*2}$ to its nominal value, and vice versa. This procedure improves the signal and also minimizes any correlation between the two variables.
\[tab:tagfit\]
Modes $N_{data}$ Low Sideband High Sideband
---------------------------------------- ---------------- -------------- ---------------
$K^0_{\mathrm S} K^- $ $5828\pm 92$ 1231 958
$K^+ K^- \pi^-$ $25990\pm 285$ 22385 19452
$K^{\star-}\overline{K}^{\star 0} $ $2891\pm 100$ 2783 2647
$\pi^+\pi^-\pi^-$ $8152\pm 369$ 56530 43475
$\eta \pi^-$ $3635\pm 160$ 5727 3379
$\eta \rho^-$ $6877\pm 330$ 26879 14658
$\pi^-\eta^{\prime} (\eta\pi^+\pi^-) $ $2344\pm 70$ 1040 572
$\pi^-\eta^{\prime}(\rho\gamma) $ $4451\pm 337$ 42412 25476
: Number of $D_{s}^{-}$ tag candidates $N_{\mathrm{data}}$ for each mode in the signal region and sidebands for the one-dimensional fit to $M_{\mathrm{tag}}$.
\[tab:tagfitdg\]
Mode $f_1$ $\sigma_1$ (MeV) $\sigma_2$ (MeV)
---------------------------------------- ------- ------------------ ------------------
$K^{0}_{\mathrm S} K^-$ 0.471 4.05 7.00
$K^+K^-\pi^-$ 0.725 3.74 8.92
$K^{\star -} \overline{K}^{\star 0}$ 0.771 3.43 10.65
$\pi^+ \pi^- \pi^-$ 0.899 4.84 9.88
$\eta \pi^-$ 0.650 9.85 15.56
$\eta \rho^-$ 0.574 10.8 18.3
$\pi^-\eta^{\prime} (\eta\pi^+\pi^-) $ 0.590 5.71 13.34
$ \pi^- \eta^\prime (\rho \gamma)$ - 9.60 -
: Signal peak parameters for each tag mode derived from $c\bar{c}$ MC simulation.
![Distribution of $M_{\mathrm{tag}}-{M_{\mathrm{Ds}}}$ of ${D^-_{\mathrm s}}$ candidates, for the different tags: (a) $K^+ K^- \pi^-$; (b) $K^0_{\mathrm S} K^- $; (c) $\eta \pi^-$; (d) $\pi^-\eta^{\prime} (\eta\pi^+\pi^-) $; (e) $\pi^+\pi^-\pi^-$; (f) $K^{\star-}\overline{K}^{\star 0} $; (g) $\eta \rho^-$, and (h) $\pi^-\eta^{\prime}(\rho\gamma) $. The fitted background $A_1$, described in the text, is indicated by the dashed-dotted slope. The signal mass region is indicated by the vertical dotted lines.[]{data-label="fig:tagall"}](fig2.eps){width="6in"}
The 2-D fit is done for each mode separately, and its purpose is to extract the final number of tags for each mode, $N_{i}$, while building on the information obtained in the one-dimensional $M_{\mathrm{tag}}$ fit. The fitting function is $$\label{eq:2dfit}
f(M_{\mathrm{tag}},MM^{*2})= N_{i}G_2(M_{\mathrm{tag}})C(MM^{*2})+G_2(M_{\mathrm{tag}})A_5(MM^{*2})
+A_1(M_{\mathrm{tag}})A_5(MM^{*2}).$$
For each mode, the signal is described by the product of a double Gaussian in the $M_{\mathrm{tag}}$ projection, defined in Eq. (\[eq:defg2\]) and Crystal Ball function [@oreglia], defined as $C$ in the equations, in the $MM^{*2}$ projection, respectively. One of the background components is the combination of a real tag with a random $\gamma$. This type of background ($BG_1$ in Fig. 3 below, and the second term in Eq. (\[eq:2dfit\])) is described by the same double Gaussian $G_2(M_{\mathrm{tag}})$ and a 5th degree polynomial $A_5(MM^{*2})$. The other background ($BG_2$ in Fig. 3 below, and the third term in Eq. (\[eq:2dfit\])) is due to fake tags. The PDF here is the product of a 1st order polynomial ($M_{\mathrm{tag}}$) and a 5th order polynomial ($MM^{*2}$). To simplify the fit, the $M_{\mathrm{tag}}$ projections are fitted using the signal function obtained in the 1-D $M_{\mathrm{tag}}$ fit, but the background parameters are varied.
![ $MM^{*2}$ distributions for the 8 tag modes. Dash-dotted lines are the $BG_2$ background described in the text. Dashed lines are total background $BG_1+BG_2$. (a) $K^+ K^- \pi^-$; (b) $K^0_{\mathrm S} K^- $; (c) $\eta \pi^-$; (d) $\pi^-\eta^{\prime} (\eta\pi^+\pi^-) $; (e) $\pi^+\pi^-\pi^-$; (f) $K^{\star-}\overline{K}^{\star 0} $; (g) $\eta \rho^-$, and (h) $\pi^-\eta^{\prime}(\rho\gamma) $.[]{data-label="fig:mmstar2fig"}](fig3.eps){width="6in"}
\[tab:mmstar2cuts\]
Modes Lower limit ([ GeV]{}$^2$) Upper limit ([ GeV]{}$^2$) $N_i$(data)
---------------------------------------- ---------------------------- ---------------------------- ----------------
$K^0_{\mathrm s} K^- $ 3.7876 3.9539 $3442\pm 138$
$K^+ K^- \pi^-$ 3.7939 3.9510 $15647\pm 271$
$K^{\star-}\overline{K}^{\star 0} $ 3.7505 3.9847 $1707\pm 94$
$\pi^+\pi^-\pi^-$ 3.7701 3.9633 $4595\pm 298$
$\eta \pi^-$ 3.7662 3.9798 $2355\pm 187$
$\eta \rho^-$ 3.7698 3.9632 $3606\pm 640$
$\pi^-\eta^{\prime} (\eta\pi^+\pi^-) $ 3.7409 3.9888 $1716\pm 142$
$\pi^-\eta^{\prime}(\rho\gamma) $ 3.7875 3.9601 $3373\pm 240$
$N_{\mathrm{tag}}$ - - $36441\pm 852$
: $MM^{*2}$ selection range and the number of tags in each mode obtained from the two-dimensional fit. Total number of tags, $N_{\mathrm{tag}}$ is also given. The quoted error is statistical only.
The $MM^{*2}$ distributions are shown in Fig. \[fig:mmstar2fig\]. The $MM^{*2}$ signal regions for each mode are chosen so as to have 95% signal efficiency. The $MM^{*2}$ selection is summarised in Table \[tab:mmstar2cuts\]. Table \[tab:mmstar2cuts\] also lists the final number of tags obtained in each tag mode, $N_i$, as well as the total number of tags, $N_{\mathrm{tag}}$, used to extract the final result.
Signal selection
----------------
The signal is selected by requiring one positron candidate, of charge opposite to the tag charge, two charged pion candidates, of opposite charge, no extra good tracks, and a good $\pi^0$, all selected exclusively of the objects used in the tag. The selection requires a specific number of tracks, and multiple candidates can arise only due to multiple $\pi^0$ candidates. In case of multiple candidates, the $\pi^0$ is selected as follows. Given the photon-photon mass $M_{\gamma\gamma}$, and the calculated mass error $\sigma_{\gamma\gamma}$, the one with the lowest $\chi^2=[(M_{\gamma\gamma}-M_{\pi^0})/\sigma_{\gamma\gamma}]^2$ is chosen. Additional candidate photons are ignored.
The positron, charged pions, and $\pi^0$ are added together to form the four-vector ($E_{\mathrm s},\mathbf{p}_{\mathrm s}$). The measured neutrino candidate mass squared, $MM^2$, is defined as $$MM^{2}=(E_{\mathrm b}-E_{\mathrm{tag}}-E_{\gamma}-E_{\mathrm s})^2-(\mathbf{p}_{\mathrm b}-\mathbf{p}_{\mathrm{tag}}-
\mathbf{p}_{\gamma}-\mathbf{p}_{\mathrm s})^2.$$ The $MM^2$ distributions, with $M_{\mathrm{tag}}$ sideband subtraction, of the two control samples $\eta e^+\nu$ and $\phi e^+\nu$ are shown in Fig. \[fig:mm2eta\]. Based on the shape of $MM^2$, events with $-0.05{\ \rm GeV}^2<MM^2<0.05$ GeV$^2$ are selected for the final analysis.
![Control sample $MM^2$ distributions. (a) solid, ${D^+_{\mathrm s}}\to\eta e^+\nu$ distribution after $M_{\mathrm{tag}}$ sideband subtraction, and $M_{\mathrm{tag}}$ sideband distribution (dotted). (b) solid, ${D^+_{\mathrm s}}\to\eta e^+\nu$ distribution after $M_{\mathrm{tag}}$ sideband subtraction, and $M_{\mathrm{tag}}$ sideband distribution (dotted). []{data-label="fig:mm2eta"}](fig4.eps){width="6in"}
![$M_3$ distribution. Solid: signal selection, after $M_{\mathrm{tag}}$ sideband subtraction. Dotted: $M_{\mathrm{tag}}$ sideband contribution. The arrow shows the location of the $\omega$ nominal mass, Ref. $[17]$.[]{data-label="fig:pipipi0"}](fig5.eps){width="6in"}
The mass of the $\pi^+\pi^-\pi^0$ combination $M_3$ was not used in the candidate selection, and provides the spectrum that is fitted to extract the final result. In Fig. \[fig:pipipi0\], the $M_3$ spectrum is presented, including $M_{\mathrm{tag}}$ sideband contributions. Two peaks are clearly present, at the $\eta$ and $\phi$ masses, with no sign of a signal in the $\omega$ mass region.
The $M_3$ peaks in the signal MC samples are fitted to a Breit-Wigner shape (indicated as $BW$ in the equations), convoluted with a double Gaussian, $$\label{eq:defsig}
s(x)=K \int BW(x_1)G_2(x-x_1)dx_1,$$ where $K$ is a normalization constant to give $s(x)$ a unit area. Table \[tab:dg\] lists the fit results for each of the signal MC samples generated for this analysis.
Decay $f_1$ $\sigma_1$ (MeV) $\sigma_2$ (MeV) R.M.S. (MeV)
------------------ -------- ------------------ ------------------ --------------
$\eta e^+\nu $ 0.8844 3.165 19.85 7.37
$\omega e^+\nu $ 0.8783 5.500 22.52 9.40
$\phi e^+\nu $ 0.8361 5.940 19.83 9.73
: $M_{3}$ signal peak parameters evaluated from signal MC sample. All quantities are defined in the text.[]{data-label="tab:dg"}
The reconstruction efficiency $\epsilon$ for the $\omega e^+\nu$ final state is computed by applying the same requirements to the signal MC events, but correcting for the number of tags found in the data, $$\label{eq:epsil}
\epsilon={1\over N_{\mathrm{tag}}}{\Sigma N_i\epsilon_i},$$ $\epsilon_i$ being the signal MC efficiency for mode $i$. The result is $\epsilon=(5.11\pm 0.15)\%$, with the error due to MC statistics.
Final fit
==========
Figure \[fig:omegamc\] shows only the $M_3$ region used in the fit, which contains ${N_{\mathrm{obs}}}=18$ events. The $\Delta_{M3}=250$ MeV mass window is centered at the nominal $\omega$ mass [@pdb]. In the Fig. 6(a), the data distribution and the fit to the data (described below) are shown. In Fig. 6(b), the MC distribution is shown.
![$M_3$ distribution in the 250 MeV wide region centered at the nominal $\omega$ mass. No $M_{\mathrm{tag}}$ sideband subtraction was used. (a) comparison of data (points) and best fit according to Eq. (9) (line). (b) The MC sample, normalized to 20 times the data statistics. Solid: non-resonant backgrounds. Empty: resonant backgrounds. The arrow shows the location of the $\omega$ nominal mass.[]{data-label="fig:omegamc"}](fig6.eps){width="4.0in"}
Three potential sources of background are considered: non-$D_{\mathrm s}$ backgrounds, $D_{\mathrm s}$ backgrounds where there are non-resonant final states (which have not yet been observed, and are not present in the MC simulation), and backgrounds where there is a true $\omega$. $M_{\mathrm{tag}}$ sideband subtraction only subtracts the first source. A direct fit of a signal and a background component subtracts the first two. The third source of background is subtracted via MC simulation, and is discussed below.
![Statistical only likelihood, normalized to unit area over the positive signal region, for the observed number of signal events in data. The 90% C.L. was computed using only the positive signal region. []{data-label="fig:likeplots"}](fig7.eps){width="4.0in"}
The signal yield is determined by a one parameter unbinned likelihood fit [@pdb]. The free parameter is the total number of signal events $S$. The background level is constrained by the normalization of the probability. $S$ is multiplied by a function of unit area $s(x)$, Eq. (\[eq:defsig\]). The final expression of the unbinned likelihood, $L_u=\Pi_{i}P_{i}$, is obtained from the probabilities $$\label{eq:likeli}
P(M_{3i}|S)=P_i= (S/{N_{\mathrm{obs}}})s(M_{3i})+(1-S/{N_{\mathrm{obs}}})/\Delta_{M3},$$ which correspond to a signal $S$, distributed according to $s(M_3)$, plus a flat background.
Figure \[fig:likeplots\] shows the likelihoods obtained for data, without any peaking background subtraction, in the $S>0$ region. The 90% confidence level (C.L.) is calculated using only the $S>0$ portion of the likelihood. The statistical only upper limit on $S$ at the 90% C.L. is $S_{90}=3.78$ events.
Unbinned likelihood fits, in one dimension, can be tested for goodness of fit using the Cramer-Von Mises test [@cramer], where the goodness of fit parameter is
$$\label{eq:g}
G=\int^{M_{3}^{\mathrm{max}}}_{M_{3}^{\mathrm{min}}}[F(M_3)-F_N(M_3)]^2 dF(M_3).$$
The integral limits are the limits of the fit interval. $F(M_3)$ is the integrated probability function for best-fit parameters, $$\label{eq:f}
F(M_3)=\int_{M_{3}^{\mathrm{min}}}^{M_3} P(M'_3,S_{\mathrm{max}}) dM'_3.$$ Here $S_{\mathrm{max}}=0$, so that $F$ is in fact a straight line. One has $F=(M_{3}-M_{3}^{\min})/\Delta_{M_{3}}$ and $dF(M_3)=dM_{3}/\Delta_{M_{3}}$. $F_N$ is a step function such that $F_N(M_3)=N/{N_{\mathrm{obs}}}$, where $N$ is the rank of the largest event mass which is less than $M_3$. The two functions are shown in Fig. \[fig:cramer\].
A toy MC program was run to generate an ensemble of $10^5$ unbiased experiments. Figure \[fig:cramer\] shows the distribution of $G$ for the ensemble, also shown is the value of $G$ obtained in the fit to data. Only 13.1% of the fits to the generated experiments are better than that to the data. The toy MC program also made it easy to apply the Kolmogorov-Smirnov (KS) test, which simply computes the maximal difference between $F$ and $F_N$. Only 16.0% of the unbiased experiments produced a better KS test than the data. The fit to the data is excellent.
![Cramer-Von Mises test of goodness of fit, for the final fit of this analysis. (a) Comparison of the integrated probability distribution $F$ (dashed) and the step function $F_N$ described in the text (solid). (b) Comparison of the $G$ obtained in this analysis, with a distribution obtained from $10^5$ unbiased toy MC fits.[]{data-label="fig:cramer"}](fig8.eps){width="4.0in"}
Determination of branching fraction and systematic errors
=========================================================
The statistical upper limit on the number of events is translated into a statistical only limit on the branching fraction $\mathcal{B}_{90}$ according to the following equation $$\label{eq:b95}
\mathcal{B}_{90}={S_{90}\over \epsilon N_{\mathrm{tag}}}.$$
There are three quantities on the right hand side of Eq. (\[eq:b95\]), with central values $S_{90}=3.78,\ \epsilon=0.0511$ and $N_{\mathrm{tag}}=36441$, yielding $\mathcal{B}_{90}=0.203$%, which is a purely statistical limit. $N_{\mathrm{tag}}$ has a statistical error of its own, and each of the three quantities in Eq. (\[eq:b95\]) has systematic errors which are discussed below.
Systematic errors
-----------------
Table \[tab:sys\], first row, contains the relevant parameters of the unbinned likelihood (Fig. \[fig:likeplots\]), in the form $\mu\pm\sigma$. $\mu$ is the $S$ value for which $L_u$ is maximal, if one allows also $S<0$ values. It describes the form of the likelihood, but is not used in the determination of the final result.
Systematic errors to $S_{90}$ are also listed in Table \[tab:sys\]. The error associated with the assumed mass and width of $\omega$ is estimated by varying the central values by the uncertainties given in Ref. [@pdb].
The greatest source of $S_{90}$ systematic errors is related to irreducible backgrounds. These also shift the location of the likelihood peak to lower values. Fig. \[fig:omegamc\] shows the background distribution by physical source. All but one of the true $\omega$ are due to the decay chain ${D^+_{\mathrm s}}\to\eta^{\prime} e^+\nu$ with $[\mathcal{B}=(1.12\pm 0.35)\%]$, followed by $\eta^{\prime}\to \omega\gamma$, with $[\mathcal{B}=(3.02\pm 0.33)\%]$ [@pdb]. A dedicated MC simulation for this channel generated 5000 events of which 51 passed all selections. This corresponds to an irreducible background of $(0.53\pm 0.19)$ events. Note that the largest source of error is the semileptonic $\mathcal{B}$ error from Ref. [@pdb]. Therefore, this source of systematics can not be significantly improved with increased simulation statistics.
A second irreducible background comes from $D_{\mathrm s}\to\omega X$ events. Zero events are found in the MC events from the direct decay ${D^+_{\mathrm s}}\to\omega \pi^+$. The $c\bar{c}$ MC significantly underestimates the $(D_{\mathrm s}\to\omega X)$ yield, which is 0.6% in the $c\bar{c}$ MC but 6.1% in data [@roches]. The decay ${D^+_{\mathrm s}}\to\omega\pi^+$ is in the $c\bar{c}$ MC, but no other $\omega n(\pi)$ decays. A dedicated MC for ${D^+_{\mathrm s}}\to\omega\pi^+\pi^0$, which was assumed to saturate the 5.5% difference, was run, and zero events were found. The probability for $n$ background events, given zero MC candidates, is exponential in shape. The systematic error from this source can be represented as $(0.02\pm 0.02)$ in Table \[tab:sys\].
There was one more true $\omega$ event which is in the continuum MC sample, corresponding to one more irreducible background of $0.15\pm 0.15$ events.
\[tab:sys\]
Type Cent. val. (evts.) $\sigma$ (evts.)
--------------------------------------------- -------------------- ------------------
Data fit $-0.25$ 2.21
$\omega$ mass $-$ 0.04
$\omega$ width $-$ 0.006
${D^+_{\mathrm s}}\to \eta^{\prime} e^+\nu$ $-$0.53 0.19
${D^+_{\mathrm s}}\to \omega X$ $-$0.02 0.02
Continuum $-$0.15 0.15
Type Cent. val.(%) $\sigma$ (%)
$N_{\mathrm{tag}}$ stat. $-$ 2.3
$N_{\mathrm{tag}}$ syst. $-$ 2.0
MC statistics $-$ 2.7
$\mathcal{B}(\omega\to 3\pi)$ $-$ 0.8
Tracking $-$ 0.9
$\pi^0$ eff. $-$ 1.0
$\pi^0$ selection variation $-$ 0.5
Positron eff. $-$ 0.6
MC form factor $-$ 0.5
Extra track selection $-$ 0.04
Particle ID $-$ 0.1
: Summary of statistical and systematic errors of $S_{90}$. The first block is the statistical error from the experiment. The second block are errors associated with the quantity $S_{90}$ of Eq. (12) and consists of uncertainties due to the mass and width of the omega, and errors in estimating three irreducible backgrounds as described in the text. The last block are percentage systematic errors associated with $N_{\mathrm{tag}}$ or $\epsilon$.
$N_{\mathrm{tag}}$ was obtained through a fit, with a statistical error of 2.3%. Systematic errors can enter the analysis only through the bias in the choice of fitting function. This can be quantified by varying the fitting function. For each tag mode, the fitting function for the signal was changed, first term of Eq. (\[eq:2dfit\]). The Crystal Ball function was varied in two ways, by keeping the $n$ parameter fixed to its MC fitted values and by changing the $(n,\alpha)$ parameters by one $\sigma$ in a mode specific way. The background was also varied. Instead of a fifth degree polynomial, the data were fitted with a fourth and a sixth degree polynomial. The background was also changed by fixing the amount of $BG_1$ background (described in Sec. III.B) to one standard deviation above or below its central value. Variations of $N_{\mathrm{tag}}$ due to changes in the fitting function were as low as $-1.8\%$ and as high as $+1.5\%$. The assigned systematic $N_{\mathrm{tag}}$ error is 2.0%.
Correlations may affect the fit of the $(M_{\mathrm{tag}},MM^{*2})$ peak, because the fit assumes the two variables are not correlated. To study this, we have computed the $(M_{\mathrm{tag}},MM^{*2})$ correlation coefficient in the signal MC sample, by calculating the correlation coefficient in each tag mode, and then reweighting for the observed number of events in each mode. The result is $\rho=(-1.6\pm 0.9)\%$. The fit error due to remnant correlations is of order $\rho^2$ and is neglected.
Finally, there are the systematic uncertainties on the efficiency to be considered. The $\omega$ branching fraction uncertainty is 0.8% [@pdb]. The tracking efficiency error is a 0.3% Gaussian systematic error per signal track, to be added linearly, totaling 0.9% per event [@naik].
The $\pi^0$ reconstruction efficiency error is 1% [@dobbs], but depends on the exact selection criteria. To estimate the size of the systematics induced by changing selection criteria, the signal MC sample with and without the energy and angular criteria which were used to select photon candidates. There were 41269 reconstructed events with the criteria, and 41868 without the criteria, a difference of 1.5%. There were 101 events instead of 99 in the combined $\eta$ and $\phi$ peaks, a difference of 2%. We assumed a further 0.5% systematic error, listed in Table \[tab:sys\].
The positron reconstruction efficiency is evaluated in a manner similar to Ref. [@dobbs]. Positron efficiencies have been investigated by the Collaboration using a variety of well-known kinematically constrained QED processes. The experimentally measured corrections are convoluted with the positron momentum distribution to obtain the efficiency uncertainty, which is 0.6%. The effects of the extra track cuts and of particle ID cuts can be estimated from the MC sample, by varying or eliminating the cuts. We find errors of 0.04% and 0.1% for the extra track cut and particle ID cuts respectively.
Finally, a different form factor will change the efficiency, mostly because events with low $q^2$ (the positron-neutrino mass) produce lower energy positrons. The signal MC produce phase-space distributed events, and therefore a constant form factor. To evaluate this source of systematics, the form factor was varied by $\pm 20\%$, by reweighting the signal MC events according to the weights
$$w_{\pm i}(q^2_i)=1\pm \frac{0.2(q^2_i-<q^2>)}{<q^2>},$$ with $<q^2>$ being the mean $q^2$ in the signal MC sample. The new efficiencies are 5.08% and 5.14% respectively, to be compared to the given value of 5.11%. A systematic error of 0.5% is assigned to this systematics.
![Final probability distribution for the branching fraction, after convolution of all statistical and systematic errors, as described in the text. []{data-label="fig:finaleff"}](fig9.eps){width="4.0in"}
Determination of the branching fraction
---------------------------------------
To obtain the final result, Gaussian and non-Gaussian errors are convoluted with the non-Gaussian signal $S$ distribution given by the likelihood (Fig. \[fig:likeplots\]), by means of a toy MC program. The procedure is used for several reasons. First, the main source of error, the unbinned likelihood, is non-Gaussian, whereas the smaller sources of error are mostly Gaussian. Second, some sources of error shift the central value of the likelihood, an effect which can be treated exactly by shifting the likelihood on an event-by-event basis. Finally, the exponential and correlated nature of some of the error sources can be reproduced exactly by MC simulation.
A total of $25\times 10^6$ toy experiments are generated, to obtain the probability distribution in Fig. \[fig:finaleff\]. The cumulative effect of the systematic errors is to increase the limit, and the cumulative effect of the irreducible backgrounds is to decrease the limit. Taking into account the systematic uncertainties and the irreducible backgrounds, the upper limit on the branching fraction changes from 0.203% (statistical only) to 0.201%.
Conclusion
==========
We report the first measurement of an upper limit for the branching fraction ${\mathcal B}({D^+_{\mathrm s}}\to\omega e^+\nu)$. We find ${\mathcal B}({D^+_{\mathrm s}}\to\omega e^+\nu)<$0.20% at the 90% C.L., which does not exclude that expected from the model of Ref. [@rosner].
We gratefully acknowledge the effort of the CESR staff in providing us with excellent luminosity and running conditions. D. Cronin-Hennessy thanks the A.P. Sloan Foundation. This work was supported by the National Science Foundation, the U.S. Department of Energy, the Natural Sciences and Engineering Research Council of Canada, and the U.K. Science and Technology Facilities Council.
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|
---
abstract: 'Radiation pressure can be dynamically important in star-forming environments such as ultra-luminous infrared and submillimeter galaxies. Whether and how radiation drives turbulence and bulk outflows in star formation sites is still unclear. The uncertainty in part reflects the limitations of direct numerical schemes that are currently used to simulate radiation transfer and radiation-gas coupling. An idealized setup in which radiation is introduced at the base of a dusty atmosphere in a gravitational field has recently become the standard test for radiation-hydrodynamics methods in the context of star formation. To a series of treatments featuring the flux-limited-diffusion approximation as well as a short-characteristics tracing and M1 closure for the variable Eddington tensor approximation, we here add another treatment that is based on the Implicit Monte Carlo radiation transfer scheme. Consistent with all previous treatments, the atmosphere undergoes Rayleigh-Taylor instability and readjusts to a near-Eddington-limited state. We detect late-time net acceleration in which the turbulent velocity dispersion matches that reported previously with the short-characteristics-based radiation transport closure, the most accurate of the three preceding treatments. Our technical result demonstrates the importance of accurate radiation transfer in simulations of radiative feedback.'
author:
- |
Benny T.-H. Tsang and Miloš Milosavljević\
Department of Astronomy, University of Texas at Austin, Austin, TX 78712, USA
title: Radiation pressure driving of a dusty atmosphere
---
-1cm
[star: formation – ISM: kinematics and dynamics – galaxies: star formation – radiative transfer – hydrodynamics – methods: numerical]{}
Introduction
============
The forcing of gas by stellar and dust-reprocessed radiation has been suggested to reduce star formation efficiency and drive supersonic turbulence and large-scale outflows in galaxies [e.g., @TQM05; @Thompson15; @MQT10; @Murray11; @FaucherGiguere13; @Kuiper15]. Generally the net effect of radiation pressure is to counter the gravitational force and modulate the rate of infall and accretion onto star-forming sites. In its most extreme presentation, radiation pressure accelerates gas against gravity so intensely that the gas becomes unbound. For example, @Geach14 have recently suggested that stellar radiation pressure drives the high-velocity, extended molecular outflow seen in a starburst galaxy at $z = 0.7$. Theoretical and observational evidence thus suggests that radiation may profoundly influence the formation and evolution of star clusters and galaxies. While direct radiation pressure from massive young stars may itself be important in some, especially dust-poor environments [@Wise12], the trapping of the stellar radiation that has been reprocessed by dust grains into the infrared (IR) should be the salient process enabling radiation pressure feedback in systems with the highest star formation rate densities.
Usually, the amplitude of radiative driving of the interstellar medium (ISM) in star-forming galaxies is quantified with the average Eddington ratio defined as the stellar UV (or, alternatively, emerging IR) luminosity divided by the Eddington-limited luminosity computed with respect to the dust opacity. However, in reality, the ISM is turbulent and dust column densities vary widely between different directions in which radiation can escape. The local Eddington ratio along a particular, low-column-density direction can exceed unity even when the average ratio is below unity. @TK14 argue that this is sufficient for radiation pressure to accelerate gas to galactic escape velocities. @AT11 surveyed star forming systems on a large range of luminosity scales, from star clusters to starbursts, and found that their dust Eddington ratios are consistent with the assumption that radiation pressure regulates star formation.
@MQT05 highlighted the importance of radiation momentum deposition on dust grains in starburst galaxies. They showed that the Faber-Jackson relation $L \propto \sigma^{4}$ and the black hole mass-stellar velocity dispersion relation $M_{\rm BH} \propto \sigma^{4}$ could both be manifestations of self-regulation by radiation pressure. @TQM05 argued that radiation pressure on dust grains can provide vertical support against gravity in disks of starburst galaxies if the disks are optically thick to the reprocessed IR radiation. In a study of giant molecular cloud (GMCs) disruption, @MQT10 found that radiation pressure actually dominates in rapidly star-forming galaxies such as ULIRGs and submillimeter galaxies. @Hopkins10 identified a common maximum stellar mass surface density $\Sigma_{\rm max} \sim 10^{11} M_{\odot}$kpc$^{-2}$ in a variety of stellar systems ranging from globular clusters to massive star clusters in starburst galaxies and further to dwarf and giant ellipticals. These systems spanned $\sim7$ orders of magnitudes in stellar mass and $\sim5$ orders of magnitude in effective radius. The universality of maximum stellar mass surface density can be interpreted as circumstantial evidence for the inhibition of gaseous gravitational collapse by radiation pressure.
The preceding studies were based on one-dimensional or otherwise idealized models. To understand the dynamical effects of radiation pressure in a dusty ISM, however, multi-dimensional radiation hydrodynamics (RHD) simulations are required. One specific setup has emerged as the testbed for radiation hydrodynamics numerical methods used in simulating the dusty ISM, specifically in the regime in which the gas (assumed to be thermally coupled to dust) is approximately isothermal and susceptible to compressive, high-mach-number perturbations. @KT12 [hereafter KT12] and @KT13 [hereafter KT13] designed a two-dimensional model setup to investigate the efficiency of momentum transfer from trapped IR radiation to a dusty atmosphere in a vertical gravitational field. Using the flux-limited diffusion (FLD) approximation in the <span style="font-variant:small-caps;">orion</span> code [@Krumholz07], they found that the optically thick gas layer quickly developed thin filaments via the radiative Rayleigh-Taylor instability (RTI). The instability produced clumping that allowed radiation to escape through low-density channels. This significantly reduced net momentum transfer from the escaping radiation to the gas, and the gas collapsed under gravity at the base of the computational box where radiation was being injected.
@Davis14 [hereafter D14] then followed up by simulating the same setup with the <span style="font-variant:small-caps;">athena</span> code [@Davis12] using the more accurate variable Eddington tensor (VET) approximation. They constructed the local Eddington tensor by solving the time-independent radiative transfer equation on a discrete set of short characteristics [@Davis12]. Similar to the simulations of KT12, those of D14 developed filamentary structures that reduced radiation-gas momentum coupling. However, in the long-term evolution of the radiation-pressure-forced atmosphere, D14 detected significant differences, namely, the gas continued to accelerate upward, whereas in KT12, it had settled in a turbulent steady state confined near the base of the box. D14 interpreted this difference of outcome by referring to an inaccurate modeling of the radiation flux in the optically thick-to-thin transition in FLD.
@RT15 [hereafter RT15], simulated the same setup with the new <span style="font-variant:small-caps;">ramses-rt</span> RHD code using the computationally efficient M1 closure for the Eddington tensor. This method separately transports radiation energy density and flux and assumes that the angular distribution of the radiation intensity is a Lorentz-boosted Planck specific intensity. One expects this to provide a significant improvement of accuracy over FLD, however still not approaching the superior accuracy of the short characteristics closure. The M1 results are qualitatively closer to those obtained with the FLD than with the short-characteristics VET. RT15 argue that the differences between FLD and M1 on one hand and the short characteristics VET on the other may be more subtle than simply arising from incorrectly approximating the flux at the optically thick-to-thin transition.
In this paper, we revisit the problem of radiative forcing of a dusty atmosphere and attempt to reproduce the simulations of KT12, D14, and RT15, but now with an entirely different numerical scheme, the implicit Monte Carlo (IMC) method of @Abdikamalov12 originally introduced by @FC71. The paper is organized as follows. In Section \[sec:na\] we review the equations of radiation hydrodynamics and the IMC method. In Section \[sec:assessna\] we then assess the reliability of our approach in a suite of standard radiation hydrodynamics test problems. In Section \[sec:setup\] we describe the simulation setup and details of numerical implementation. We present our results in Section \[sec:results\] and provide concluding reflections in Section \[sec:conclusions\].
Conservation Laws and Numerical Scheme {#sec:na}
======================================
We start from the equations of non-relativistic radiation hydrodynamics. The hydrodynamic conservation laws are $$\begin{aligned}
\label{eqn:mconsrv}
\frac{\partial \rho}{\partial t} +
\nabla \cdot \left( {\rho \mathbf{v}} \right) &= 0,\end{aligned}$$ $$\begin{aligned}
\label{eqn:pconsrv}
\frac{\partial \rho \mathbf{v}}{\partial t} +
\nabla \cdot \left( {\rho \mathbf{v} \mathbf{v}} \right) + \nabla{P} &=
\rho \mathbf{g} + \mathbf{S}, \end{aligned}$$ $$\begin{aligned}
\label{eqn:econsrv}
\frac{\partial \rho E}{\partial t} +
\nabla \cdot [\left( \rho E + P \right) \mathbf{v}] &=
\rho \mathbf{v} \cdot \mathbf{g} + c S_0,\end{aligned}$$ where $\rho$, $\mathbf{v}$, and $P$ are the gas density, velocity, and pressure of the gas and $$\begin{aligned}
E &= e + \frac{1}{2} \mathbf{|v|}^2 \end{aligned}$$ is the specific total gas energy defined as the sum of the specific internal and kinetic energies of the gas. On the right hand side, $\mathbf{g}$ is the gravitational acceleration and $\mathbf{S}$ and $c S_0$ are the gas momentum and energy source terms arising from the coupling with radiation.
The source terms, written here in the lab frame, generally depend on the gas density, thermodynamic state, and velocity. We here investigate a system that steers clear of the dynamic diffusion regime, namely, in our simulations $\tau v/c\ll 1$ is satisfied at all times. Here, $\tau\lesssim 10^2$ is the maximum optical depth across the box and the velocity is non-relativistic $v/c\lesssim 10^{-4}$. Therefore, we can safely drop all $\mathcal{O} (v/c)$ terms contributing to the momentum source density and can approximate the lab-frame momentum source density with a velocity-independent gas-frame expression $$\begin{aligned}
\label{eqn:radpsrc}
\mathbf{S} &= \frac{1}{c}
\int_{0}^{\infty} d\epsilon
\int d\Omega
\left[ k(\epsilon) I(\epsilon,\mathbf{n})
- j(\epsilon, \mathbf{n}) \right]
\mathbf{n} .\end{aligned}$$ Here, $I(\epsilon,\mathbf{n})$ is the specific radiation intensity, $\epsilon$ is the photon energy, $\mathbf{n}$ is the radiation propagation direction, $c$ is the speed of light, $d\Omega$ is the differential solid angle in direction $\mathbf{n}$, and $k(\epsilon)$ and $j(\epsilon, \mathbf{n})$ are the total radiation absorption and emission coefficients. The coefficients are also functions of the gas density $\rho$ and temperature $T$ but we omit these parameters for compactness of notation.
Our energy source term includes the mechanical work per unit volume and time $\mathbf{v} \cdot \mathbf{S}$ that is performed by radiation on gas. Since the gas-frame radiation force density is used to approximate the lab-frame value, the source term is correct only to $\mathcal{O} (v/c)$ $$\begin{aligned}
\label{eqn:radesrc}
c S_0 &= \int_{0}^{\infty} d\epsilon
\int d\Omega
\left[ k(\epsilon) I(\epsilon,\mathbf{n})
- j(\epsilon, \mathbf{n}) \right]
+ \mathbf{v} \cdot \mathbf{S} .\end{aligned}$$ Because in this scheme radiation exerts force on gas but gas does not on radiation, the scheme does not conserve energy and momentum exactly. However, it should be accurate in the non-relativistic, static-diffusion limit; we test this accuracy in Section \[sec:assessna\]. We split the absorption and emission coefficients by the nature of radiative process $$\begin{aligned}
k(\epsilon) = k_{\rm a}(\epsilon)
+ k_{\rm s}(\epsilon),\end{aligned}$$ $$\begin{aligned}
j(\epsilon, \mathbf{n}) = j_{\rm a}(\epsilon)
+ j_{\rm s}(\epsilon, \mathbf{n}).\end{aligned}$$ The subscript ‘a’ refers to thermal absorption and emission, and ‘s’ refers to physical scattering (to be distinguished from the effective scattering that will be introduced in the implicit scheme).
Equations (\[eqn:mconsrv\]–\[eqn:econsrv\]) couple to the radiation subsystem via the radiation source terms defined in Equation (\[eqn:radpsrc\]) and (\[eqn:radesrc\]). Assuming local thermodynamic equilibrium (LTE), the radiation transfer equation can be written as $$\begin{aligned}
\label{eqn:RTE-original}
\frac{1}{c} \frac{\partial I(\epsilon,\mathbf{n})}{\partial t} +
\mathbf{n} \cdot \nabla I(\epsilon,\mathbf{n}) &=&
k_{\rm a}(\epsilon) B(\epsilon) - k(\epsilon) I(\epsilon,\mathbf{n}) \nonumber \\
&+& j_{\rm s}(\epsilon, \mathbf{n}) + j_{\rm ext} (\epsilon,\mathbf{n})\end{aligned}$$ where $B(\epsilon)$ is the Planck function at temperature $T$ and $j_{\rm ext}(\epsilon,\mathbf{n})$ is the emissivity of external radiation sources. Note that the $j_{\rm s}$ term depends implicitly on the specific intensity $I$ which makes Equation (\[eqn:RTE-original\]) an integro-differential equation.
Since the absorption and emission coefficients depend on the gas temperature, and the temperature in turn evolves with the absorption and emission of radiation, the system is nonlinear. We solve the system by operator-splitting (Section \[sec:osscheme\]), by replacing a portion of absorption and emission with effective scattering (thus making the solution implicit; Section \[sec:implicit\]), and by discretizing the radiation field with a Monte-Carlo (MC) scheme (Section \[sec:mcprocedures\]).
Operator-splitting scheme {#sec:osscheme}
-------------------------
Our numerical method is based on the adaptive-mesh refinement (AMR) code <span style="font-variant:small-caps;">flash</span> [@Fryxell00; @Dubey08], version 4.2.2. We use operator-splitting to solve Equations (\[eqn:mconsrv\]–\[eqn:econsrv\]) and (\[eqn:RTE-original\]) in two steps:
1. [*Hydrodynamic update*]{}: Equations (\[eqn:mconsrv\]–\[eqn:econsrv\]) without the radiation source terms $$\begin{aligned}
\label{eqn:mconsrv-hd}
\frac{\partial \rho}{\partial t} +
\nabla \cdot \left( {\rho \mathbf{v}} \right) &= 0,\end{aligned}$$ $$\begin{aligned}
\label{eqn:pconsrv-hd}
\frac{\partial \rho \mathbf{v}}{\partial t} +
\nabla \cdot \left( {\rho \mathbf{v} \mathbf{v}} \right) + \nabla{P} &=
\rho \mathbf{g}, \end{aligned}$$ $$\begin{aligned}
\label{eqn:econsrv-hd}
\frac{\partial \rho E}{\partial t} +
\nabla \cdot [\left( \rho E + P \right) \mathbf{v}] &=
\rho \mathbf{v} \cdot \mathbf{g}\end{aligned}$$ are solved using the <span style="font-variant:small-caps;">hydro</span> module in <span style="font-variant:small-caps;">flash</span>.
2. [*Radiative transport and source deposition update*]{}: Equation (\[eqn:RTE-original\]) coupled to the radiative momentum $$\begin{aligned}
\label{eqn:pconsrv-last}
\rho \frac{\partial \mathbf{v}}{\partial t}
&=
\mathbf{S} \end{aligned}$$ and energy $$\begin{aligned}
\label{eqn:econsrv-rt}
\rho \frac{\partial E}{\partial t} =
c S_0\end{aligned}$$ deposition equations is solved with the implicit method that we proceed to discuss.
Implicit radiation transport {#sec:implicit}
----------------------------
Under LTE conditions, the tight coupling between radiation and gas is stiff and prone to numerical instability. This limits the applicability of traditional, explicit methods unless very small time steps are adopted. The method of @FC71 for nonlinear radiation transport relaxes the limitation on the time step by treating the radiation-gas coupling semi-implicitly. Effectively, this method replaces a portion of absorption and immediate re-emission with elastic scattering, thus reducing the amount of zero-sum (quasi-equilibrium) energy exchange between gas and radiation. Numerous works have been devoted to investigating the semi-implicit scheme’s numerical properties. @Wollaber08 provides a detailed description of the approximations made and presents a blueprint for implementation and stability analysis. @Cheatham10 provides an analysis of the truncation error. Recently, @Roth15 developed variance-reduction estimators for the radiation source terms in IMC simulations.
In this section, we describe a choice of formalism for solving the coupled radiative transport and source term deposition equations. The radiation transport equation is revised to reduce thermal coupling between radiation and gas by replacing it with a pseudo-scattering process. The detailed derivation of the scheme can be found in @FC71 and @Abdikamalov12. Here we reproduce only the main steps. Our presentation follows closely @Abdikamalov12 and the approximations are as in @Wollaber08.
Given an initial specific intensity $I(\epsilon,\mathbf{n},t^{n})$ and gas specific internal energy $e(t^{n})$ at the beginning of a hydrodynamic time step $t^n$, our goal is to solve Equations (\[eqn:RTE-original\]) and (\[eqn:econsrv-rt\]) to compute the time-advanced values $I(\epsilon,\mathbf{n},t^{n+1})$ and $e(t^{n+1})$ at the end of the time step $t^{n+1} = t^{n} + \Delta t$, where $\Delta t$ is the hydrodynamic time step. During this partial update, we assume that the gas density $\rho$ and velocity $\mathbf{v}$ remain constant. For mathematical convenience we introduce auxiliary parametrizations of the thermodynamic variables: the gas internal energy density $$\begin{aligned}
u_{\rm g} = \rho e,\end{aligned}$$ the energy density that radiation would have if it were in thermodynamic equilibrium with gas $$\begin{aligned}
\label{eqn:ur}
u_{\rm r} = \frac{4\pi}{c} \int_{0}^{\infty} B(\epsilon) d\epsilon\end{aligned}$$ (for compactness of notation and at no risk of confusion, we do not explicitly carry dependence on the gas temperature $T$), the normalized Planck function $$\begin{aligned}
b(\epsilon) = \frac{B(\epsilon)}
{4\pi \int_{0}^{\infty} B(\epsilon) d\epsilon},\end{aligned}$$ the Planck mean absorption coefficient $$\begin{aligned}
k_{\rm p} = \frac{\int_{0}^{\infty} k_{\rm a}(\epsilon) B(\epsilon) d\epsilon }
{\int_{0}^{\infty} B(\epsilon) d\epsilon},\end{aligned}$$ and a dimensionless factor, $\beta$, quantifying the nonlinearity of the gas-radiation coupling $$\begin{aligned}
\beta = \frac{\partial u_{\rm r}}{\partial u_{\rm g}} .\end{aligned}$$ At a risk of repetition, we emphasize that the $u_{\rm r}$ defined in Equation (\[eqn:ur\]) is *not* the energy density of the radiation field; it is simply as an alternate parametrization of the gas internal energy density. The absorption coefficient and the gas-temperature Planck function, in particular, can now be treated as functions of $u_{\rm r}$.
Taking the physical scattering to be elastic, Equations (\[eqn:RTE-original\]) and (\[eqn:pconsrv-last\]), and (\[eqn:econsrv-rt\]) can be rewritten as $$\begin{aligned}
\label{eqn:RTE-revised}
\frac{1}{c} \frac{\partial I(\epsilon,\mathbf{n})}{\partial t} +
\mathbf{n} \cdot \nabla I(\epsilon,\mathbf{n}) &=&
k_{\rm a}(\epsilon) b(\epsilon) c u_{\rm r}
- k_{\rm a}(\epsilon) I(\epsilon,\mathbf{n}) \notag \\
&-& k_{\rm s}(\epsilon) I(\epsilon,\mathbf{n})
+ j_{\rm s}(\epsilon, \mathbf{n})\notag\\
&+& j_{\rm ext}(\epsilon,\mathbf{n}) ,
\end{aligned}$$ $$\begin{aligned}
\label{eqn:MEE-revised}
\frac{1}{\beta}
\frac{\partial u_{\rm r}}{\partial t}
+ c k_{\rm p}u_{\rm r}
= \int_{0}^{\infty} d\epsilon \int d\Omega\,
k_{\rm a}(\epsilon) I(\epsilon,\mathbf{n}).\end{aligned}$$ We linearize the equations in $u_{\rm r}$ and $I(\epsilon,\mathbf{n})$ and denote the corresponding constant coefficients with tildes, $$\begin{aligned}
\label{eqn:RTE-revised-tcv}
\frac{1}{c} \frac{\partial I(\epsilon,\mathbf{n})}{\partial t} &=&-
\mathbf{n} \cdot \nabla I(\epsilon,\mathbf{n}) +
\tilde{k}_{\rm a}(\epsilon) \tilde{b}(\epsilon) c u_{\rm r}
- \tilde{k}_{\rm a}(\epsilon) I(\epsilon,\mathbf{n})\notag\\
&-& \tilde{k}_{\rm s}(\epsilon) I(\epsilon,\mathbf{n})
+ j_{\rm s}(\epsilon, \mathbf{n})
+ j_{\rm ext}(\epsilon,\mathbf{n}) ,\end{aligned}$$ $$\begin{aligned}
\label{eqn:MEE-revised-tcv}
\frac{1}{\tilde{\beta}}
\frac{\partial u_{\rm r}}{\partial t} =
-c \tilde{k}_{\rm p} u_{\rm r}+
\int_{0}^{\infty} d\epsilon \int d\Omega\,
\tilde{k}_{\rm a}(\epsilon) I(\epsilon,\mathbf{n}) .\end{aligned}$$ The scattering emission coefficient is $$j_{\rm s} (\epsilon,\mathbf{n}) = \int d\Omega'\, \tilde{k}_{\rm s}(\epsilon) \,\Xi (\epsilon,\mathbf{n},\mathbf{n'}) \,I(\epsilon,\mathbf{n}') ,$$ where $\Xi (\epsilon,\mathbf{n},\mathbf{n'})$ is the elastic scattering kernel. We evaluate the constant coefficients explicitly at $t^n$, the beginning of the time step.
Next, for $t^n\leq t\leq t^{n+1}$, we expand $u_{\rm r}$ to the first order in time $$\label{eqn:urlinear}
u_{\rm r} (t) \simeq u_{\rm r}^n + (t-t^n) {u_{\rm r}^\prime}^n ,$$ where $u_{\rm r}^{n} = u_{\rm r}(t^{n})$ and ${u_{\rm r}^\prime}^n = \partial u_{\rm r}/\partial t\, (t^n)$. It is worth noting that the implicitness in ‘IMC’ refers to that introduced in Equation (\[eqn:urlinear\]). Substituting $u_{\rm r}(t)$ from Equation (\[eqn:urlinear\]) into (\[eqn:MEE-revised-tcv\]), solving for ${u_{\rm r}^\prime}^n$, and substituting the result back into Equation (\[eqn:urlinear\]), we obtain $$\begin{aligned}
\label{eqn:urfinal}
{u}_{\rm r} = f u^{n}_{\rm r} + \frac{1-f}{c \tilde{k}_{\rm p}}
\int_{0}^{\infty} d\epsilon \int d\Omega\,\tilde{k}_{\rm a}(\epsilon) {I}(\epsilon,\mathbf{n}) ,\end{aligned}$$ where $f$ is a time-dependent factor $$\begin{aligned}
f (t)= \frac{1}{1 + (t-t^n) \tilde{\beta} c \tilde{k}_{\rm p}} .\end{aligned}$$ Since it is desirable to work with time-independent coefficients, we approximate $f(t)$ with the so-called Fleck factor that remains constant during the time step $$f\simeq \frac{1}{1 + \alpha \Delta t \tilde{\beta} c \tilde{k}_{\rm p}} ,$$ where $0\leq \alpha\leq 1$ is a coefficient that interpolates between the fully-explicit ($\alpha=0$) and fully-implicit ($\alpha=1$) scheme for updating $u_{\rm r}$. For intermediate values of $\alpha$, the scheme is semi-implicit. The scheme is stable when $0.5 \leq \alpha \leq 1$ [@Wollaber08].
We substitute $u_{\rm r}$ from Equation (\[eqn:urfinal\]) into Equation (\[eqn:RTE-revised\]) to obtain an equation for $I(\epsilon,\mathbf{n})$ in the form known as the implicit radiation transport equation $$\begin{aligned}
\label{eqn:RTE-IMC}
& \frac{1}{c} \frac{\partial I(\epsilon,\mathbf{n})}{\partial t} +
\mathbf{n} \cdot \nabla I(\epsilon,\mathbf{n}) =\notag\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \tilde{k}_{\rm ea}(\epsilon) \tilde{b} (\epsilon) c u_{\rm r}^{n} - \tilde{k}_{\rm ea}(\epsilon) I(\epsilon,\mathbf{n}) - \tilde{k}_{\rm es}(\epsilon) I(\epsilon,\mathbf{n}) \notag \\
&\ \ \ \ \ \ \ \ \ + \frac{\tilde{k}_{\rm a}(\epsilon) \tilde{b}(\epsilon) }{\tilde{k}_{\rm p}}
\int_{0}^{\infty} d\epsilon' \int d\Omega'\,\tilde{k}_{\rm es} (\epsilon') I(\epsilon',\mathbf{n}')
\notag \\
&\ \ \ \ \ \ \ \ \ - \tilde{k}_{\rm s}(\epsilon) I(\epsilon,\mathbf{n}) + {j}_{\rm s}(\epsilon, \mathbf{n}) + j_{\rm ext}(\epsilon, \mathbf{n}) ,\end{aligned}$$ where the effective absorption and scattering coefficients are $$\begin{aligned}
\tilde{k}_{\rm ea}(\epsilon) &= f \,\tilde{k}_{\rm a} (\epsilon), \\
\tilde{k}_{\rm es}(\epsilon) &= (1 - f) \,\tilde{k}_{\rm a} (\epsilon) .\end{aligned}$$
Equation (\[eqn:RTE-IMC\]) admits an instructive physical interpretation. The first two terms on the right-hand side represent the emission and absorption of thermal radiation. Direct comparison with Equation (\[eqn:RTE-revised\]) shows that both terms are now a factor of $f$ smaller. The following two terms containing $\tilde{k}_{\rm es}$ are new; their functional form mimics the absorption and immediate re-emission describing a scattering process. Meanwhile, the physical scattering and external source terms have remained unmodified.
Since the effective absorption $\tilde{k}_{\rm ea}$ and scattering $\tilde{k}_{\rm es}$ coefficients sum to the actual total absorption coefficient $\tilde{k}_{\rm a}$, we can interpret Equation (\[eqn:RTE-IMC\]) as replacing a fraction ($1 - f$) of absorption and the corresponding, energy-conserving fraction of emission by an elastic psedo-scattering process. The mathematical form of the Fleck factor can be rearranged to make this physical interpretation manifest. Assuming an ideal gas equation of state, the radiative cooling time is $$\begin{aligned}
t_{\rm cool} = \frac{4}{c \tilde{\beta} \tilde{k}_{\rm p}}\end{aligned}$$ and the Fleck factor is $$\begin{aligned}
f = \frac{1}{1 + 4 \alpha \Delta t/t_{\rm cool}}.\end{aligned}$$ When $\Delta t /t_{\rm cool} \gg 1$ so that $f \ll 1$, the absorbed radiation is re-radiated within the same time step at zero net change in the gas energy density; the only change is randomization of the radiation propagation direction. The stability of the scheme rests precisely on this reduction of the stiff thermal coupling. However, excessively large time steps can still produce unphysical solutions [@Wollaber08].
After the radiation transport equation has been solved using the IMC method (see Section \[sec:mcprocedures\]), the net momentum and energy exchange collected during the radiative transport solve, which read $$\begin{aligned}
\label{eqn:dep_mom}
\mathbf{S} &=& \frac{1}{c\Delta t} \int_0^\infty d\epsilon\,\tilde{k}(\epsilon) \int d\Omega\int_{t^n}^{t^{n+1}} dt I(\epsilon,\mathbf{n})\,\mathbf{n}\end{aligned}$$ and $$\begin{aligned}
\label{eqn:dep_ener}
c S_0 &=& - 4\pi c u_{\rm r}^{n} \int_0^\infty d\epsilon \,\tilde{k}_{\rm ea}(\epsilon)\, \tilde{b} (\epsilon) \nonumber\\
& & + \frac{1}{\Delta t} \int_0^\infty d\epsilon\, \tilde{k}_{\rm ea}(\epsilon)\int d\Omega\int_{t^n}^{t^{n+1}} dt I(\epsilon,\mathbf{n}) \nonumber\\
& &+\mathbf{v}\cdot\mathbf{S} ,\end{aligned}$$ are deposited in the hydrodynamic variables. Therefore step (ii) in our operator splitting scheme has now been further split into two sub-steps:
(ii$'$) [*Radiative transport and hydrodynamical source term collection*]{}: Solve Equation (\[eqn:RTE-IMC\]) with the IMC method while accumulating the contribution of radiative processes to gas source terms as in Equations (\[eqn:dep\_mom\]) and (\[eqn:dep\_ener\]).
(ii$''$) [*Hydrodynamical source term deposition*]{}: Update gas momentum and energy density using Equations (\[eqn:pconsrv-last\]) and (\[eqn:econsrv-rt\]).
Monte Carlo solution {#sec:mcprocedures}
--------------------
The transition layer between the optically thick and thin regimes strains the adequacy of numerical radiation transfer methods based on low-order closures. In this transition layer, the MC radiative transfer method should perform better than computationally-efficient schemes that discretize low-order angular moments of Equation (\[eqn:RTE-original\]). In the MC approach, one obtains solutions of the radiation transport equation by representing the radiation field with photon packets and modeling absorption and emission with stochastic events localized in space and/or time. This permits accurate and straightforward handling of complicated geometries and, in greater generality than we need here, angle-dependent physical processes such as anisotropic scattering. The specific intensity $I(\epsilon,\mathbf{n})$ is represented with an ensemble of a sufficiently large number of MCPs.[^1]
In the radiation transfer update, starting with the radiation field at an initial time $t^{n}$, we wish to compute the coupled radiation-gas system at the advanced time $t^{n+1}=t^n+\Delta t$. Our MC scheme follows closely that of @Abdikamalov12 and @Wollaber08. The radiation field is discretized using a large number of Monte Carlo particles (MCPs), each representing a collection of photons. We adopt the grey approximation in which we track only the position and the collective momentum of the photons in each MCP. MCPs are created, destroyed, or their properties are modified as needed to model emission, absorption, scattering, and propagation of radiation.
If a finite-volume method is used to solve the gas conservation laws, the physical system is spatially decomposed into a finite number of cells. For the purpose of radiative transport, gas properties are assumed to be constant within each cell. Particles are created using cell-specific emissivities. Each MCP is propagated along a piecewise linear trajectory on which the gas properties (absorption and scattering coefficients) are evaluated locally. Our MC scheme computes an approximation to the solution of Equation (\[eqn:RTE-IMC\]) in two steps: by first creating MCPs based on the emissivities and boundary conditions, and then transporting MCPs through space and time.
### Thermal emission
In Equation (\[eqn:RTE-IMC\]), the term $\tilde{k}_{\rm ea} \tilde{b} c u_{\rm r}^{n}$ on the right-hand side is the frequency-dependent thermal emissivity. Assuming that thermal emission is isotropic ($\tilde{k}_{\rm ea}$ is angle-independent), the total thermal radiation energy emitted by a single cell of gas $\Delta \mathcal{E}$ can be calculated as $$\begin{aligned}
\Delta \mathcal{E}
&= 4\pi \Delta t \Delta V
\int_{0}^{\infty} \tilde{k}_{\rm ea}(\epsilon) B(\epsilon)
d\epsilon ,\end{aligned}$$ where $\Delta t$ is the time step size, $\Delta V$ is the cell volume, and $B(\epsilon)$ is the Planck function at the gas temperature $\tilde{T}(t^n)$. Since we further assume that the opacity is grey (independent of $\epsilon$) then $$\begin{aligned}
\Delta \mathcal{E}
&= c \Delta t \Delta V \tilde{k}_{\rm ea} u_{\rm r}^n .\end{aligned}$$ The net momentum exchange due to thermal emission is zero because the thermal radiation source is isotropic.
We specify that in thermal emission, $\mathcal{N}$ new MCPs are created in each cell in each time step. The energy carried by each new MCP is then $\Delta\mathcal{E}/\mathcal{N}$. The emission time of each such MCP is sampled uniformly within the interval $[t^{n}, t^{n+1}]$. The spatial position of the MCP is sampled uniformly within the cell volume and the propagation direction is sampled uniformly on a unit sphere. Every MCP keeps track of its time $t_i$, current position $\mathbf{r}_i$, momentum $\mathbf{p}_i$, and fraction of the energy remaining since initial emission $\varsigma_i$, where the index $i$ ranges over all the MCPs active in a given hydrodynamic time step. Newly created MCPs are added to the pool of MCPs carried over from previous hydrodynamic time steps.
### Absorption {#sec:absorption}
To minimize noise, we treat absorption deterministically. This ‘continuous absorption’ method is a variance reduction technique common in practical implementations of IMC [@Abdikamalov12; @Hykes09]. Specifically, when an MCP travels a distance $c\delta t_i$ inside a cell with absorption coefficient $\tilde{k}_{\rm ea}$, its momentum is attenuated according to $$\begin{aligned}
\label{eqn:radexp}
\mathbf{p}_i(t_i+\delta t_i) = \mathbf{p}_i(t_i) e^{-\tilde{k}_{\rm ea} c \delta t_i} , \end{aligned}$$ where we denote an arbitrary time interval with $\delta t_i$ to distinguish it from the hydrodynamic time step $\Delta t$.
### Transport {#sec:rt}
In a single hydrodynamic time step, the simulation transports the MCPs through multiple cells. The MCP-specific time remaining until the end of the hydrodynamic time step is $t^{n+1} - t_i$. For each MCP, we calculate or sample the following four distances:
1. The free streaming distance to the end of the hydrodynamic time step $d_{\rm t}=c\,(t^{n+1}-t_i)$.
2. Distance to the next scattering event assuming cell-local scattering coefficients $$\begin{aligned}
d_{\rm s} = -\frac{\ln \xi}{k_{\rm s}+\tilde{k}_{\rm es}} ,\end{aligned}$$ where $\xi$ is a random deviate uniformly distributed on the interval $(0,1]$.
3. Distance to near-complete absorption $d_{\rm a}$ defined as the distance over which only a small fraction $\varsigma_{\rm min}=10^{-5}$ of the initial energy remains.
4. Distance to the current host cell boundary $d_{\rm b}$.
We repeatedly update the four distances, select the shortest one, and carry out the corresponding operation until we reach the end of the hydrodynamic time step $t^{n+1}$. If in such a sub-cycle the shortest distance is $d_{\rm t}$, we translate the MCP by this distance $\mathbf{r}_i\rightarrow \mathbf{r}_i+d_{\rm t}\mathbf{n}_i$, where $\mathbf{n}_i=\mathbf{p}_i/p_i$ is the propagation direction. We also attenuate its momentum according to Equation (\[eqn:radexp\]) and accrue the momentum $-\Delta\mathbf{p}_{i,{\rm a}}$ and energy $|\Delta \mathbf{p}_{i,{\rm a}}|c$ transferred to the gas. If the shortest distance is $d_{\rm s}$, we do the same over this distance, but at the end of translation, we also randomize the MCP’s direction $\mathbf{n}_i\rightarrow\mathbf{n}_i'$ and accrue the corresponding additional momentum $-\Delta \mathbf{p}_{i,{\rm s}}=p_i\,(\mathbf{n}_i'-\mathbf{n}_i)$ and kinetic energy $-\mathbf{v}\cdot\Delta \mathbf{p}_{i,{\rm s}}$ transferred to gas. As a further variance-reduction tactic, given the statistical isotropy of $\mathbf{n}_i'$, we compute the momentum deposited in a scattering event simply as $-\Delta \mathbf{p}_{i,{\rm s}}=-\mathbf{p}_i$. If the shortest distance either is $d_{\rm a}$ or $d_{\rm b}$, we translate the MCP while attenuating its momentum and accruing the deposited energy and momentum. Then we either remove the MCP while instantaneously depositing the remaining momentum and energy to the gas (if the shortest distance is $d_{\rm a}$), or transfer the MCP to its new host cell (or removed the MCP if it has reached a non-periodic boundary of the computational domain).
As the MCPs are transported over a hydrodynamic time step $\Delta t$, the energy and momentum source terms for each cell are accumulated using $$\begin{aligned}
\label{eqn:e_src}
c S_{0} &= \frac{1}{\Delta t\Delta V} \sum ( |\Delta \mathbf{p}_{i,{\rm a}}|c -\mathbf{v}\cdot\Delta \mathbf{p}_{i,{\rm s} }) , \\
\label{eqn:p_src}
\mathbf{S} &= -\frac{1}{\Delta t\Delta V} \sum ( \Delta \mathbf{p}_{i,{\rm a}} + \Delta \mathbf{p}_{i,{\rm s}} ) ,
\end{aligned}$$ where the sums are over all the absorption and scattering events that occurred in a specific computational cell during the hydrodynamic time step. The source terms are then substituted into Equations (\[eqn:pconsrv-last\]) and (\[eqn:econsrv-rt\]) to compute the gas momentum and energy at the end of the hydrodynamic time step.
Assessment of Numerical Algorithm {#sec:assessna}
=================================
To assess the validity of our radiation hydrodynamics implementation, we performed a series of standard tests: a test of radiative diffusion in a scattering medium (Section \[sec:diffusion\_test\]), a test of gas-radiation thermal equilibration (Section \[sec:equilibration\_test\]), a test of thermal wave propagation (Marshak wave; Section \[sec:Marshak\_test\]), and a radiative shock test (Section \[sec:shock\_test\]).
Radiative diffusion {#sec:diffusion_test}
-------------------
![Spherically-averaged radiation energy density profiles in the three-dimensional radiative diffusion test (Section \[sec:diffusion\_test\]). The analytical solutions (solid lines) and the numerical solutions (crosses) are shown at four different times, $(0.2,\,0.6,\,1.2,\,3.2)\times10^{-10}\,\mathrm{s}$. The dashed line and the right axis show the refinement level of the AMR grid. []{data-label="fig:diffusion"}](RadiativeDiffusion.pdf){width="50.00000%"}
Here we test the spatial transport of MCPs across the AMR grid structure in the presence of scattering. In the optically thick limit, radiation transfer proceeds as a diffusion process. The setup is a cubical $L=1\, \mathrm{cm}^{3}$ three-dimensional AMR grid with no absorption and a scattering coefficient of $k_{\rm s} = 600 \rm\, cm^{-1}$. The scattering is assumed to be isotropic and elastic. We disable momentum exchange to preclude gas back-reaction and focus on testing the evolution of the radiation field on a non-uniform grid.
At $t = 0$s, we deposit an initial radiative energy ($\mathcal{E}_{\rm init} = 3.2 \times 10^{6}$erg) at the grid center in the form of 1,177,600 MCPs with isotropically sampled propagation directions. We lay an AMR grid hierarchy such that the refinement level decreases with increasing radius as shown on the right axis of Figure \[fig:diffusion\]. The grid spacing is $\Delta x = 2^{-\ell-2}\,L$, where $\ell$ is the local refinement level. A constant time step of $\Delta t = 2 \times 10^{-12}$s is used and the simulation is run for $4 \times 10^{-10}$s.
The diagnostic is the radiation energy density profile as a function of distance from the grid center and time $\rho e_{\rm rad}(r,t)$. The exact solution in $d$ spatial dimensions is given by $$\begin{aligned}
\rho e_{\rm rad}(r,t) = \frac{\mathcal{E}_{\rm init}}{\left(4 \pi D t\right)^{d/2}}
\exp\left(- \frac{r^{2}}{4 D t}\right),\end{aligned}$$ where $D = c/(d k_{\rm s})$ is the diffusion coefficient.
Figure \[fig:diffusion\] shows the spherically-averaged radiation energy density profile at four times. Excellent agreement of our MC results with the analytical expectation shows that our algorithm accurately captures radiation transport in a scattering medium. We have repeated the test in one and two spatial dimensions and find the same excellent agreement. We have also checked that in multidimensional simulations, the radiation field as represented with MCPs preserves the initial rotational symmetry.
![Evolution of the gas (diamonds) and radiation (squares) energy density in the one-zone radiative equilibration test (Section \[sec:equilibration\_test\]). The exact solution is shown with a solid (gas) and dashed (radiation) line. []{data-label="fig:radeqm"}](RadEqm.pdf){width="46.00000%"}
Radiative equilibrium {#sec:equilibration_test}
---------------------
Here, in a one-zone setup, we test the radiation-gas thermal coupling in LTE. We enabled IMC with an implicitness parameter of $\alpha = 1$. Defining $u_{\rm r}=aT^4$ as in Section \[sec:implicit\], where $a$ is the radiation constant and $T$ is the gas temperature, the stiff system of equations governing the gas and radiation internal energy density evolution is $$\begin{aligned}
\label{eqn:radeqm_ugas}
\frac{d e}{dt} &= k_{\rm a} c \left(e_{\rm rad} - \frac{u_{\rm r}}{\rho}\right), \\
\label{eqn:radeqm_urad}
\frac{d e_{\rm rad}}{dt} &= k_{\rm a} c \left(\frac{u_{\rm r}}{\rho}-e_{\rm rad}\right) ,\end{aligned}$$ where $k_{\rm a}$ is the absorption coefficient. We assume an ideal gas with adiabatic index $\gamma=5/3$.
We perform the one-zone test with parameters similar to those of @TurnerStone01 and @Harries11, namely, the absorption coefficient is $k_{\rm a} = 4.0 \times 10^{-8}$cm$^{-1}$ and the initial energy densities are $\rho e= 10^8$ergcm$^{-3}$ and $\rho e_{\rm rad} = 0$, respectively. The results and the corresponding exact solutions are shown in Figure \[fig:radeqm\]. The MC solution agrees with the exact solutions within $\lesssim 4\%$ throughout the simulation. It shows that the physics of radiation-gas thermal exchange is captured well by our scheme, and that in static media, the scheme conserves energy exactly.
As noted by @Cheatham10, the order of accuracy associated with the IMC method depends on the choice of $\alpha$ and on the specifics of the model system. When the above test problem is repeated with $\alpha = 0.5$, the error is $\lesssim 0.05\%$ because the $\mathcal{O}(\Delta t^{2})$-residuals cancel out and the method is $\mathcal{O}(\Delta t^{3})$-accurate.
Marshak wave {#sec:Marshak_test}
------------
In this test, we simulate the propagation of a non-linear thermal wave, known as the Marshak wave, in a static medium in one spatial dimension [@SuOlson96; @Gonzalez07; @Krumholz07; @Zhang11]. The purpose of this standard test is to validate the code’s ability to treat nonlinear energy coupling between radiation and gas when the gas heat capacity is a function of gas temperature. We employ IMC with $\alpha=1$.
Initially, a static, uniform slab with a temperature of 10K occupying the interval $0 \leq z \leq 15$cm is divided into 256 equal cells. An outflow boundary condition is used on the left and a reflective one on the right. A constant incident flux $F_{\rm inc}=\sigma_{\rm SB} T_{\rm inc}^4$ of $k_{\rm B}T_{\rm inc}= 1\,\mathrm{keV}$ thermal radiation, where $\sigma_{\rm SB}$ and $k_{\rm B}$ are the Stefan-Boltzmann and Boltzmann constants, is injected from the left (at $z = 0$). The gas is endowed with a constant, grey absorption coefficient of $k_{\rm a} = 1\,\mathrm{cm}^{-1}$ and a temperature-dependent volumetric heat capacity of $c_{\rm v} = \alpha T^{3}$. The constant $\alpha$ is related to the Su-Olson retardation parameter $\epsilon$ via $\alpha = 4 a / \epsilon$ and we set $\epsilon = 1$.
![ Radiation energy density profiles in the Marshak wave test problem at times $\theta = (3,\,10,\,20)$ from left to right. Numerical integrations are shown by the data points. The solid lines are the reference solutions of @SuOlson96. []{data-label="fig:marshak-u"}](marshak-u.pdf){width="46.00000%"}
![ The same as Figure \[fig:marshak-u\], but for the gas energy density. []{data-label="fig:marshak-v"}](marshak-v.pdf){width="46.00000%"}
The diagnostics for this test problem are the spatial radiation and gas energy density profiles at different times. @SuOlson96 provided semi-analytical solutions in terms of the dimensionless position $$\begin{aligned}
x = \sqrt{3} k_{\rm a} z\end{aligned}$$ and time $$\begin{aligned}
\theta = \frac{4 a c k_{\rm a} t}{\alpha } .\end{aligned}$$ The radiation and gas internal energy density are expressed in terms of the dimensionless variables $$\begin{aligned}
u(x,\theta) &= \frac{c}{4} \frac{E_{r}(x,\theta)}{F_{\rm inc}}, \\
v(x,\theta) &= \frac{c}{4} \frac{a T(x,\theta)^{4}}{F_{\rm inc}},\end{aligned}$$ where $E_{r}$ and $T$ are the radiation energy density and gas temperature, respectively (note that the relation of $v$ to the specific gas internal energy $e$ is nonlinear).
Figures \[fig:marshak-u\] and \[fig:marshak-v\] show the dimensionless numerical solution profiles at three different times, over-plotting the corresponding semi-analytical solutions of @SuOlson96. At early times, we observe relatively large deviations from the Su-Olson solutions near the thermal wavefront. This is in not surprising given that the solutions were obtained assuming pure radiative diffusion, yet at early times and near the wavefront, where the gas has not yet heated up, the optical depth is only about unity and the transport is not diffusive. @Gonzalez07 observed the same early deviations in their computations based on the M1 closure. At later times when transport is diffusive, both the thermal wave propagation speed and the maximum energy density attained agree well with the Su-Olson solutions.
---------- ---------------------- -------------- ------------------------- --------------- --------- ------------ ---------------- ----------------------- ------------------------------------- ----------------------- ------------------
Run Initial Perturbation $\Sigma$ $g$ [$h_{*}$]{} $t_{*}$ $\tau_{*}$ $f_{\rm E, *}$ $t_{\rm max} / t_{*}$ \[$L_{x}\times L_{y}$\]/[$h_{*}$]{} $\Delta$x/[$h_{*}$]{} $\ell_{\rm max}$
(gcm$^{-2}$) ($10^{-6}$dyneg$^{-1}$) ($10^{-4}$pc) kyr
T10F0.02 sin 4.7 37 0.25 0.045 10 0.02 80 512 $\times$ 256 0.5 7
T03F0.50 sin, $\chi$ 1.4 1.5 6.30 1.1 3 0.5 115 512 $\times$ 2048 1.0 6
---------- ---------------------- -------------- ------------------------- --------------- --------- ------------ ---------------- ----------------------- ------------------------------------- ----------------------- ------------------
\[tab:sim\_par\]
Radiative shock {#sec:shock_test}
---------------
![ Gas (solid curve) and radiation (dashed curve) temperature in the subcritical radiative shock test at $t = 3.8 \times 10^{4}$s. The initial gas velocity is $v = 6$kms$^{-1}$ and the profiles are plotted as a function of $z = x-v t$. []{data-label="fig:subcritical"}](subcritical.pdf){width="48.00000%"}
To finally test the fully-coupled radiation hydrodynamics, we simulate a radiative shock tube. As in the preceding tests, we use IMC with $\alpha=1$, but now, the gas is allowed to dynamically respond to the radiation. We adopt the setup and initial conditions of @Ensman94 and @Commercon11 and simulate both subcritical and supercritical shocks. The setup consists of a one-dimensional $7 \times 10^{10}$cm-long domain containing an ideal gas with a mean molecular weight of $\mu = 1$ and adiabatic index of $\gamma = 7 / 5$. The domain is initialized with a uniform mass density $\rho_{0} = 7.78 \times 10^{-10}$gcm$^{-3}$ and a uniform temperature of $T_{0} = 10$K. The gas has a constant absorption coefficient of $k_{a} = 3.1 \times 10^{-10}$cm$^{-1}$ and a vanishing physical scattering coefficient.
Initially, the gas is moving with a uniform velocity toward the left reflecting boundary. An outflow boundary condition is adopted on the right to allow inflow of gas at fixed density $\rho_{0}$ and fixed temperature $T_{0}$ and also to allow the free escape of radiation MCPs. As gas collides with the reflecting boundary a shock wave starts propagating to the right. The thermal radiation in the compressed hot gas diffuses upstream and produces a warm radiative precursor. The shock becomes critical when the flux of thermal radiation is high enough to pre-heat the pre-shock gas to the post-shock temperature [@Zeldovich67]. We choose the incoming speed to be $v_{0} = 6$kms$^{-1}$ and $20$kms$^{-1}$ in the subcritical and the supercritical shock tests, respectively.
@Mihalas84 provide analytical estimates for the characteristic temperatures of the radiative shocks. For the subcritical case, the post-shock temperature $T_{2}$ is estimated to be $$\begin{aligned}
T_{2} \simeq \frac{2(\gamma - 1) v_{0}^{2}}{R (\gamma + 1)^{2}}.\end{aligned}$$ Using the parameters for the subcritical setup, the analytical estimate gives $T_{2} \simeq 812$K. In our simulation, the post-shock temperature at $t = 3.8 \times 10^{4}$s is $T_{2} \simeq 800$K, which agrees with the analytical solution. The immediate pre-shock temperature $T_{-}$ is estimated to be $$\begin{aligned}
T_{-} \simeq \frac{2(\gamma - 1)}{\sqrt{3}R\rho v}
\sigma_{\rm SB} T_{2}^{4}.\end{aligned}$$ Our simulation gives $T_{-} \sim 300$K while $T_{-}$ is estimated to be $T_{-} = 270$K. Finally, the amplitude of the temperature spike can be estimated to be $$\begin{aligned}
T_{+} \simeq T_{2} + \frac{3 - \gamma}{\gamma + 1} T_{-},\end{aligned}$$ which gives $T_{+} \simeq 990$K. It also close to the value we find, $T_{+} \simeq 1000$K. In both cases, our simulations reproduce the expected radiative precursors. Also, in the supercritical case, the pre-shock and the post-shock temperatures are identical, as expected.
![ The same as Figure \[fig:subcritical\], but for the supercritical radiative shock test at $t = 7.5 \times 10^{3}$s and with initial velocity $v = 20$kms$^{-1}$. []{data-label="fig:supercritical"}](supercritical.pdf){width="48.00000%"}
Setup of radiation-driven atmosphere {#sec:setup}
====================================
We turn to the problem of how radiation drives an interstellar gaseous atmosphere in a vertical gravitational field. The problem was recently investigated by KT12 and KT13, by D14, and by RT15, using the FLD, VET, and M1 closure, respectively. Our aim is to attempt to reproduce these authors’ results, which are all based on low-order closures, using an independent method that does not rely on such a closure. Critical for the hydrodynamic impact of radiation pressure is the extent of the trapping of IR radiation by dusty gas. Therefore we specifically focus on the radiation transfer aspect of the problem and assume perfect thermal and dynamic coupling between gas and dust grains, $T_{\rm g} = T_{\rm d}=T$ and $\mathbf{v}_{\rm g}=\mathbf{v}_{d}=\mathbf{v}$.
We follow the setup of KT12 and D14 as closely as possible. Taking that UV radiation from massive stars has been reprocessed into the IR at the source, we work in the grey approximation in which spectral averaging of the opacity is done only in the IR part of the spectrum. We set the Rosseland $\kappa_{\rm R}$ and Planck $\kappa_{\rm P}$ mean dust opacities to $$\begin{aligned}
\label{eq:Rosseland_Planck}
\kappa_{\rm R,P} = (0.0316,\, 0.1) \left(\frac{T}{10\,K}\right)^{2}\,{\rm cm^{2}\,g^{-1}} .\end{aligned}$$ This model approximates a dusty gas in LTE at $T \le 150$K [@Semenov03]. Diverging slightly from KT12 and D14, who adopted the pure power-law scaling in Equation (\[eq:Rosseland\_Planck\]), to approximate the physical turnover in opacity, we cap both mean opacities to their values at $150$K above this threshold temperature. Overall, our opacity model is reasonable below the dust grain sublimation temperature $\sim$1000K.
The simulation is set up on a two-dimensional Cartesian grid of size $L_x\times L_y$. The grid is adaptively refined using the standard <span style="font-variant:small-caps;">flash</span> second derivative criterion in the gas density. The dusty gas is initialized as a stationary isothermal atmosphere. A time-independent, vertically incident radiation field is introduced at the base of the domain ($y=0$) with a flux vector $F_{*}\hat{\mathbf{y}}$. The gravitational acceleration is $-g\hat{\mathbf{y}}$.
For notational convenience, we define a reference temperature $T_{*} = [F_{*}/(c a)]^{1/4}$, sound speed $ c_{*} = \sqrt{k_{\rm B} T_{*} / (\mu m_{\rm H})}$, scale height $h_{*} = c_{*}^{2}/g$, density $\rho_{*} = \Sigma/h_{*}$ (where $\Sigma$ is the initial average gas surface density at the base of the domain), and sound crossing time $t_{*} = h_{*}/ c_{*}$. In the present setup $F_{*} = 2.54 \times 10^{13}$$L_{\odot}\,\mathrm{kpc}^{-2}$ and the mean molecular weight is $\mu = 2.33$ as expected for molecular hydrogen with a 10% helium molar fraction. The characteristic temperature is $T_{*} = 82$K and the corresponding Rosseland mean opacity is $\kappa_{\rm R, *} = 2.13\,\mathrm{cm}^2\,\mathrm{g}^{-1}$.
Following KT12 and KT13, we adopt two dimensionless parameters to characterize the system: the Eddington ratio $$\begin{aligned}
f_{\rm E, *} = \frac{\kappa_{\rm R, *} F_{*}}{g c} \end{aligned}$$ and the optical depth $$\begin{aligned}
\tau_{*} = \kappa_{\rm R, *} \Sigma .\end{aligned}$$ The atmosphere is initialized at a uniform temperature $T_{*}$. The gas density is horizontally perturbed according to $$\begin{aligned}
\label{eq:initial_density}
\rho(x, y) &= \left[1 + \frac{1 + \chi}{4} \sin\left(\frac{2 \pi x}{\lambda_{x}}\right)\right] \nonumber\\
& \times\begin{cases}
\rho_{*} \,e^{-y/h_{*}}, & \textrm{ if } e^{-y/h_{*}} > 10^{-10} ,\\
\rho_{*} \,10^{-10}, & \textrm{ if } e^{-y/h_{*}} \le 10^{-10} , \\
\end{cases}\end{aligned}$$ where $\lambda_{x} = 0.5 \,L_{x}$. D14 introduced $\chi$, a random variate uniformly distributed on $[-0.25,\,0.25]$, to provide an additional perturbation on top of the sinusoidal profile. If $\chi=0$, the initial density distribution reduces to that of KT12.
KT12 found that at a given $\tau_{*}$, the preceding setup has a hydrostatic equilibrium solution when $f_{\rm E,*}$ is below a certain critical value $f_{\rm E,crit}$. Note that KT12 defined $f_{\rm E,crit}$ assuming the pure power-law opacity scaling in Equation (\[eq:Rosseland\_Planck\]). With our capping of the opacities above 150K, which breaks the dimensionless nature of the KT12 setup, the exact KT12 values for $f_{\rm E,crit}$ cannot be directly transferred to our model. Nevertheless, we use their definition of $f_{\rm E,crit}$ simply to normalize our values of $\tau_{*} $ and $f_{\rm E,*}$. We attempt to reproduce the two runs performed by KT12. The first run T10F0.02 with $\tau_{*} = 10 $ and $f_{\rm E,*} = 0.02 = 0.5\, f_{\rm E,crit}$ lies in the regime in which such a hydrostatic equilibrium solution exists. The second run T03F0.5 with $\tau_{*} = 3$ and $f_{\rm E,*} = 0.5=3.8\, f_{\rm E,crit}$ corresponds to the run performed by both KT12 and D14 that had the smallest ratio $f_{\rm E,*}/f_{\rm E,crit}$ and was still unstable. The latter run probes the lower limit for the occurrence of a dynamically unstable coupling between radiation and gas.
![ Gas density snapshots at four different times in the stable run T10F0.02. The fully simulation domain is larger than shown, 512$\times$256$h_{*}$; here, we only show the bottom quarter. The stable outcome of this run is consistent with the cited literature. []{data-label="fig:stable_dens"}](stable_dens_algae.pdf){width="48.00000%"}
The boundary conditions are periodic in the $x$ direction, reflecting at $y=0$ (apart from the flux injection there), and outflowing (vanishing perpendicular derivative) at $y=L_y$. The reflecting condition does not allow gas flow or escape of radiation. The outflow condition does allow free inflow or outflow of gas and escape of radiation. Unlike the cited treatments, we used non-uniform AMR. The AMR improves computational efficiency early in the simulation when dense gas occupies only a small portion of the simulated domain. In low-density cells, radiation streams almost freely through gas; there, keeping mesh resolution low minimizes the communication overhead associated with MCP handling while still preserving MCP kinematic accuracy. The application of AMR in conjunction with IMC is clearly not essential in two-dimensional, low-dynamic-range setups like the one presented here, but should become critical in three-dimensional simulations of massive star formation; thus, we are keen to begin validating it on simple test problems.
To further economize computational resources, we require a density $\geq 10^{-6}\,\rho_{*}$ for thermal emission, absorption, and scattering calculations; below this density, the gas is assumed to be adiabatic and transparent. We also apply a temperature floor of 10K.
As the simulation proceeds, the total number of MCPs increases. To improve load balance, we limit the maximum number of MCPs allowed in a single computational block ($8\times8$ cells) at the end of the time step to 64, or on average $\sim1$ MCP per cell. (A much larger number of MCPs can traverse the block in the course of a time step.) If the number exceeds this specified maximum at the end of the radiation transport update, we merge some of the MCPs in a momentum- and energy-conserving fashion. We, however, do not properly preserve spatial and higher-angular-moment statistical properties of the groups of MCPs subjected to merging. This deficiency is tolerable in the present simulation where merging takes place only at the lowest level of refinement, where the radiation no longer affects the gas. In future applications, however, we will develop a manifestly more physical MCP merging strategy.
The simulation parameters of the two runs are summarized in Table \[tab:sim\_par\]. The quoted cell width $\Delta x$ is that at the highest level of mesh refinement (the cells are square). Gas with density $\gtrsim 10^{-8}\,\rho_{*}$ always resides at the maximum refinement level $\ell_{\rm max}$ throughout the simulation of duration $t_{\rm max}$.
![ *Top panel:* Time evolution of the mass-weighted mean velocity in the vertical direction in the stable run T10F0.02. *Bottom panel:* The corresponding time evolution of the mass-weighted mean velocity dispersion. The linear dispersions $\sigma_{x}$ and $\sigma_{y}$ are shown with the dotted and dashed lines, respectively. []{data-label="fig:stable_velocity"}](stable_vsigma.pdf){width="48.00000%"}
Results {#sec:results}
=======
{width="100.00000%"}
Stable run T10F0.02
--------------------
Density snapshots at four different times in the simulation are shown in Figure \[fig:stable\_dens\]. The simulation closely reproduces the quantitative results of both FLD (KT12) and VET (D14). Shortly after the beginning of the simulation, the trapping of radiation at the bottom of the domain by the dense dusty gas produces a rise in radiation energy density. As we assume LTE and perfect thermal coupling between gas and dust, the gas temperature increases accordingly. Specifically, after being heated up by the incoming radiation, the effective opacity at the midplane rises by a factor of $\sim10$. The opacity rise enhances radiation trapping and the temperature rises still further to $\sim(3-4)\,T_{*}\sim 300\,\mathrm{K}$. The heating drives the atmosphere to expand upward, but radiation pressure is not high enough to accelerate the slab against gravity. After the initial acceleration, the atmosphere deflates into an oscillatory, quasi-equilibrium state. This outcome is consistent with what has been found with FLD and VET.
To better quantify how the dynamics and the degree of turbulence in the gas compare with the results of the preceding investigations, we compute the mass-weighted mean gas velocity $$\begin{aligned}
\langle \mathbf{v} \rangle = \frac{1}{M} \int_0^{L_y}\int_0^{L_x} \rho(x,y) \mathbf{v} (x,y) dxdy\end{aligned}$$ and linear velocity dispersion $$\begin{aligned}
\sigma_{i} = \frac{1}{M} \int_0^{L_y}\int_0^{L_x}\rho (x,y) (v_{i} (x,y)- \langle v_i \rangle)^{2} dxdy ,\end{aligned}$$ where $M$ is the total mass of the atmosphere and $i$ indexes the coordinate direction. We also define the total velocity dispersion $\sigma = \sqrt{\sigma_{x}^2 + \sigma_{y}^{2}}$.
The time evolution of $\langle v_{y} \rangle$ and the linear dispersions is shown in Figure \[fig:stable\_velocity\]. All the velocity moments are expressed as fractions of the initial isothermal sound speed $c_{*} = 0.54$kms$^{-1}$. Both panels closely resemble those in Figure 2 of D14. Early on at $\sim$10$t_{*}$, radiation pressure accelerates the gas and drives growth in $\langle v_{y} \rangle$, $\sigma_{y}$, and $\sigma_{x}$. After this transient acceleration, $\langle v_{y} \rangle$ executes dampled oscillations about zero velocity (the damping is likely of a numerical origin). The linear velocity dispersions also oscillate, but with smaller amplitudes $\lesssim 0.4\,c_{*}$. The oscillation period in $\sigma_{x}$ is just slightly longer than that reported by D14. The agreement of our and VET results demonstrates the reliability of both radiation transfer methods.
![ *Top panel:* Time evolution of the mass-weighted mean velocity in the vertical direction in the unstable run T03F0.50 (black line). The colored lines are tracks from the cited references (see text and legend). *Bottom panel:* Mass-weighted mean velocity dispersions (see legend). In both panels, the late-time net acceleration and velocity dispersions are in agreement only with the results obtained by D14 with their short characteristics-based VET method. []{data-label="fig:unstable_velocity"}](unstable_vsigma.pdf){width="48.00000%"}
Unstable run T03F0.50
---------------------
To facilitate direct comparison with D14 and RT15, we introduce a random initial perturbation on top of the initial sinusoidal perturbation as in Equation (\[eq:initial\_density\]). The grid spacing $\Delta x$ is twice of that adopted by KT12, D14, and RT15, but as we shall see, this coarser spacing is sufficient to reproduce the salient characteristics of the evolving system. MCP merging is activated at $t = 36\,t_{*}$. Figure \[fig:unstable\_dens\] shows density snapshots at four different times. As in the stable case, the incoming radiation heats the gas at the bottom of the domain and opacity jumps. The flux soon becomes super-Eddington and a slab of gas is lifted upward. At $39\,t_{*}$, fragmentation of the slab by the RTI is apparent; most the gas mass becomes concentrated in dense clumps.
We note that the slab lifting and the subsequent fragmentation are consistently observed in all radiative transfer approaches; differences become apparent only in the long-term evolution. As in the VET simulation (D14), coherent gaseous structures in our simulation continue to be disrupted and accelerated. Qualitatively, radiation drives gas into dense, low-filling-factor filaments embedded in low density $(10^{-3} - 10^{-4})\,\rho_{*}$ gas. At 115$t_{*}$, the bulk of the gas has a net upward velocity and has been raised to altitudes $y\sim 1500\,h_*$.
Figure \[fig:unstable\_velocity\] compares the time evolution of the bulk velocity $\langle v_{y} \rangle$ and velocity dispersions $\sigma_{x,y}$ with the corresponding tracks from the published FLD/VET and M1 simulations (respectively, D14 and RT15). Initially, $\langle v_{y} \rangle$ rises steeply as the gas slab heats up and the incoming flux becomes super-Eddington. All simulations except for the one performed with the M1 closure without radiation trapping (RT15) exhibit a similar initial rise. At $\sim 25\,t_{*}$, the RTI sets in and the resulting filamentation reduces the degree of radiation trapping. This in turn leads to a drop in radiation pressure and $\langle v_{y} \rangle$ damps down under gravity. The transient rise and drop in $\langle v_{y} \rangle$ is observed with all the radiative transfer methods, although the specific times of the acceleration-to-deceleration transition differ slightly. The bulk velocity peaks at $\langle v_{y} \rangle \simeq\,12\,c_{*}$ in IMC and at $\simeq$9$c_{*}$ in VET. The subsequent kinematics differs significantly between the methods. In IMC and VET, the gas filaments rearrange in a way that enables resumption of upward acceleration after $\sim(50-60)\,t_{*}$. At late times, the secondary rise in $\langle v_{y} \rangle$ does not seem to saturate in IMC as it does in VET. Otherwise, the IMC and VET tracks are very similar to each other. In FLD and M1, however, gas is not re-accelerated after the initial transient acceleration. Instead, it reaches a turbulent quasi-steady state in which gas is gravitationally confined at the bottom of the domain and $\langle v_{y} \rangle$ fluctuates around zero.
![ *Top panel:* Time evolution of the volume-weighted Eddington ratio in the unstable run T03F0.50. The colored lines are tracks from the cited references (see text and legend). *Middle panel:* The volume-weighted mean total vertical optical depth. *Bottom panel:* The flux-weighted mean optical depth. []{data-label="fig:unstable_volavg"}](multiplot_volavg.pdf){width="48.00000%"}
The evolution of velocity dispersions in IMC is also in close agreement with VET. Before the RTI onset, the dispersions rise slightly to $\sigma_{y}\gtrsim\,1\,c_{*}$. Once the RTI develops and the slab fragments, $\sigma_{y}$ increases rapidly and $\sigma_{x}$ somewhat more gradually. A drop in $\sigma_{y}$ is observed at $\sim75\,t_{*}$, but after that time, the vertical dispersion rises without hints of saturation. Velocity dispersions at the end of our simulations are consistent with those in VET. In FLD and M1, on the other hand, the asymptotic turbulent quasi-steady states have smaller velocity dispersions $\simeq\,5\,c_{*}$.
To further investigate the coupling of gas and radiation, we follow KT12 and KT13 to define three volume-weighted quantities: the Eddington ratio $$\begin{aligned}
f_{\rm E, V} = \frac{\langle \kappa_{\rm R} \rho F_{y} \rangle_{\rm V}}{c g \rho},\end{aligned}$$ the mean total vertical optical depth $$\begin{aligned}
\tau_{\rm V} = L_{y} \langle \kappa_{\rm R} \rho \rangle_{\rm V},\end{aligned}$$ and the flux-weighted mean optical depth $$\begin{aligned}
\tau_{\rm F} = L_{y} \frac{\langle \kappa_{\rm R} \rho F_{y} \rangle_{\rm V}}
{\langle F_{y} \rangle_{\rm V}},\end{aligned}$$ where $F_{y}$ is the flux in the $y$ direction and $\langle \cdot \rangle_{\rm V} = L_{x}^{-1} L_{y}^{-1} \int_0^{L_y} \int_0^{L_x} \cdot \,dx dy$ denotes volume avearge.
Figure \[fig:unstable\_volavg\] compares the time evolution of the volume-weighted quantities in our IMC run with those in FLD, M1, and VET. The evolution of $f_{\rm E,V}$ in IMC matches both qualitatively and quantitatively that in VET over the entire course of the run. Common to all the simulations except the one carried out with the M1 closure without radiative trapping, the mean Eddington ratio increases from its initial value of $f_{\rm E, V} = 0.5$ to super-Eddington values soon after the simulation beginning. Then it immediately declines toward $f_{\rm E, V} \lesssim 1.5$. After $\sim 20\,t_{*}$, all methods become sub-Eddington, with the M1 with radiative trapping and the FLD exhibiting the most significant decline. Then beyond $\sim 60\,t_{*}$, all simulations attain near-unity Eddington ratios. D14 pointed out that the time evolution of $\langle v_{y} \rangle$ is in general sensitive to the value of $f_{\rm E, V}$, namely, $\langle v_{y} \rangle$ increases when $f_{\rm E,V} > 1$ and decreases otherwise. It is observed that IMC stays slightly super-Eddington at late times, similar to VET. The observed continuous acceleration of the gas with IMC suggests that gas dynamics can be very different between simulations with similar volume-average Eddington rations as long as the simulations are performed with different radiative transfer methods.
The middle panel of Figure \[fig:unstable\_volavg\] shows the evolution of the volume-weighted mean total vertical optical depth $\tau_{\rm V}$. Since this quantity depends only on the gas state but not on the noisier radiation state, the IMC track is smooth. It is a global estimate of the optical thickness of the gas layer and we expect its behavior to be related to that of $f_{\rm E, V}$. The IMC track seems a flattened and downscaled version of the others. This should be an artifact of the precise choice of the opacity law. We cap the opacities $\kappa_{\rm R, P}$ at their values at $T = 150$K, whereas the other authors allow the $\kappa \propto T^{2}$ scaling to extend at $T > 150$K. Therefore, our choice of opacity underestimates the strength of radiation pressure compared to the cited studies, but this discepancy does not appear to affect the hydrodynamic response of the gas.
The bottom panel of Figure \[fig:unstable\_volavg\] shows the ratio $\tau_{\rm F}/\tau_{\rm V}$. Note that $\tau_{\rm F}$ is the true effective optical depth felt by the radiation. Therefore, a small $\tau_{\rm F} / \tau_{\rm V}$ implies a higher degree of flux-density anti-correlation. The evolution of this ratio is similar in all radiation transfer methods.
Conclusions {#sec:conclusions}
===========
We applied the Implicit Monte Carlo radiative transfer method to a standard two-dimensional test problem modeling the radiation hydrodynamics of a dusty atmosphere that is accelerated against gravity by an IR radiation field. The atmosphere is marginally capable of trapping the transiting radiation. We consider this idealized simulation a necessary stepping stone toward characterizing the dynamical impact of the radiation emitted by massive stars and active galactic nuclei. We compare our IMC-derived results with those using low-order closures of the radiative transfer hierarchy that have been published by other groups. Our particle-based approach enables independent validation of the hitherto tested methods.
Sufficiently strong radiation fluxes universally render the atmosphere turbulent, but its bulk kinematics differs between the VET and IMC methods on the one hand and the FLD and M1 methods on the other. We find that the former continue to accelerate the atmosphere against gravity in the same setup in which the latter regulate the atmosphere into a gravitationally-confined, quasi-steady state. This exposes shortcomings of the local closures. Namely, in complex geometries, the FLD seems to allow the radiation to more easily escape through optically thin channels. This can be understood in terms of a de facto artificial re-collimation of the radiation field diffusing into narrow, optically-thin channels from their more optically thick channel walls. In the limit in which the radiation freely streams in the channels, the flux in the channels becomes equal to what it would be for a radiation field in which the photon momenta are aligned with the channel direction. Indeed, D14 argue that in the optically thin regime, the FLD’s construction of the radiation flux is inaccurate in both its magnitude and direction, and has the tendency to reinforce the formation of such radiation-leaking channels.
Whether outflowing or gravitationally-confined, the turbulent atmosphere seems to reach a state approximately saturating the Eddington limit. The nonlinearity arising from the increase of dust opacity with temperature introduces the potential for bi-stability in the global configuration. Subtle differences between numerical closures can be sufficient to force the solution into degenerate, qualitatively different configurations. Robust radiation-hydrodynamic modeling seems to demand redundant treatment with distinct numerical methods including the IMC.
Future work will of course turn to more realistic astrophysical systems. For example, the role of radiation trapping and pressure in massive star forming regions remains a key open problem, both in the context of the nearby [@Krumholz09; @Krumholz12b; @Krumholz14; @Coker13; @Lopez14] and the distant [@Riechers13] universe. Radiative reprocessing by photoionization and dust requires a frequency-resolved treatment of the radiation field as well as a generalization the IMC method to nonthermal processes. The assumption of perfect gas-dust thermal coupling can be invalid and the respective temperatures must be tracked separately. Numerical treatments may be required to resolve dust sublimation fronts [@Kuiper10] and radiation pressure on metal lines [@Tanaka11; @Kuiper13b]. On the small scales of individual massive-star-forming cores, multifrequency radiative transfer may be of essence for robust estimation of the final characteristic stellar mass scale and the astronomically measurable accretion rate [@Yorke02; @Tan14]. Photoionization can set the final stellar masses through fragmentation-induced starvation [@Peters10a]. The star formation phenomenon spans a huge dynamic range that can be effectively treated with telescopic AMR grids constructed to ensure that the local Jeans length is always adequately resolved. It will likely be necessary to invent new acceleration schemes for improving the IMC method’s efficiency in such heterogeneous environments. One promising direction is the introduction of MCP splitting [see, e.g., @Harries15 where MCP splitting is applied in methods developed to simulate radiation transfer in massive star forming systems].
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to the referee M. Krumholz for very helpful comments, to E. Abdikamalov for generously sharing details of his IMC radiative transfer implementation, to C. Ott for inspiring discussions, and to S. Davis and J. Rosdahl for consultation and sharing simulation data with us. B. T.-H. T. is indebted to V. Bromm for encouragements throughout the course of this research. He also acknowledges generous support by The University of Hong Kong’s Hui Pun Hing Endowed Scholarship for Postgraduate Research Overseas. The <span style="font-variant:small-caps;">flash</span> code used in this work was developed in part by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago. We acknowledge the Texas Advanced Computing Center at The University of Texas at Austin for providing HPC resources, in part under XSEDE allocation TG-AST120024. This study was supported by the NSF grants AST-1009928 and AST-1413501.
[^1]: One disadvantage of the MC scheme is low computational efficiency in the optically thick regime where the photon mean free path is short. Efficiency in such regions can be improved by applying the diffusion approximation [@FC84; @Gentile01; @Densmore07]. Recently, @Abdikamalov12 interfaced the IMC scheme at low optical depths with the Discrete Diffusion Monte Carlo (DDMC) method of @Densmore07 at high optical depths. This hybrid algorithm has been extended to Lagrangian meshes [@Wollaeger13]. In the present application, the optical depths are relatively low and shortness of the mean free path is not a limitation.
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abstract: 'We present observational evidence for a galaxy ‘Type’ dependence to the location of a spiral galaxy’s disk parameters in the $\mu_{0}$–$\log h$ plane. With a sample of $\sim$40 Low Surface Brightness galaxies (both bulge- and disk-dominated) and $\sim$80 High Surface Brightness galaxies, the early-type disk galaxies ($\leq$Sc) tend to define a bright envelope in the $\mu_0$–$\log h$ plane, while the late-type ($\geq$Scd) spiral galaxies have, in general, smaller and fainter disks. Below the defining surface brightness threshold for a Low Surface Brightness galaxy (i.e. more than 1 mag fainter than the 21.65 B-mag arcsec$^{-2}$ Freeman value), the early-type spiral galaxies have scale-lengths greater than 8-9 kpc, while the late-type spiral galaxies have smaller scale-lengths. All galaxies have been modelled with a seeing-convolved Sérsic $r^{1/n}$ bulge and exponential disk model. We show that the trend of decreasing bulge shape parameter ($n$) with increasing Hubble type and decreasing bulge-to-disk luminosity ratio, which has been observed amongst the High Surface Brightness galaxies, extends to the Low Surface Brightness galaxies, revealing a continuous range of structural parameters.'
author:
- 'Alister W. Graham'
- 'W.J.G. de Blok'
title: 'A MORPHOLOGICAL-TYPE DEPENDENCE IN THE $\mu_{0}$–$\log h$ PLANE OF SPIRAL GALAXY DISKS'
---
Introduction
============
During the closing decade of the 1900s considerable progress was made in the observation and parameterization of intrinsically faint spiral galaxies (van der Hulst et al. (1993); de Blok et al 1995; Sprayberry et al. 1995; Bothun, Impey, & McGaugh 1997; Beijersbergen, de Blok, & van der Hulst 1999). Galaxies with a central disk surface brightness more than one magnitude fainter than the canonical Freeman (1970) value of 21.65 B-mag arcsec$^{-2}$ were defined to be Low Surface Brightness (LSB) galaxies. Spiral galaxies with a brighter central disk surface brightness have subsequently been labelled High Surface Brightness (HSB) galaxies, and it is these galaxies which typify the Hubble sequence. Although, in reality, this threshold is somewhat arbitrary as observations of many spiral galaxies reveals a smooth continuation in central disk surface brightness values across this boundary. The existence of LSB galaxies – including knowledge of the extreme LSB galaxy ‘Malin 1’ (Bothun et al. 1987, Impey & Bothun 1989) – had of course been known many years earlier (Longmore et al. 1982; Davies et al. 1988), and had been previously predicted, on the grounds of visibility arguments Disney (1976).
In this Letter we combine the data from two LSB galaxy samples with a representative sample of HSB galaxies containing all spiral morphological types. The objective is to explore the tentative suggestion of Graham (2001a) that a morphological type dependence exists in the central disk surface brightness – disk scale-length ($\mu_0$–$\log h$) diagram. Re-analyzing the light-profiles from the diameter-limited sample of 86 spiral galaxies from de Jong & van der Kruit (1994) and de Jong (1996), Graham (2001a) noted that the low surface brightness (high $\mu_0$) small disk scale-length corner of the $\mu_0$–$\log h$ diagram was dominated, in fact, only populated, by late-type spiral galaxy disks. To be somewhat more quantitative, the host galaxies of disks with scale-lengths less than 9 kpc and central surface brightnesses fainter than 19.0 K-mag arcsec$^{-2}$ were all late-type spiral galaxies ($>$Sc). To investigate this further, this same galaxy sample is supplemented here with two of the largest samples of LSB galaxies measured with CCDs. The first is a sample of $\sim$20 ‘disk-dominated’ LSB galaxies from de Blok et al. (1995), and the second is a sample of $\sim$20 ‘bulge-dominated’ LSB spiral galaxies (Beijersbergen et al. 1999).
The selection criteria for the three data samples are reviewed in Section 2, and the derivation of the light-profile model parameters is also discussed there. All of the galaxies have been modelled by us using the same bulge/disk decomposition algorithm. We additionally provide the first quantitative measurement to the prominence, and/or absence, of the bulges in ‘disk-dominated’ LSB galaxies. The resulting distribution of disks in the $\mu_0$–$\log h$ diagram is presented in Section 3.
Data
====
The galaxy sample
-----------------
The data selection procedure, and method of reduction, are described in detail in the above mentioned papers. We will only comment here on the selection criteria – relevant to concerns about possible biases which might result in an under-estimation of early-type spirals in the low surface brightness (high $\mu_0$) small scale-length corner of the $\mu_0$–$\log h$ diagram.
de Jong & van der Kruit (1994) selected ‘undisturbed’ spiral galaxies from the UGC catalog (Nilson 1973) if they had a red minor-axis to major-axis ratio greater than 0.625 (i.e. inclinations less than $\sim$50$^{\rm o}$), a red major-axis $\geq$2.0, and an absolute galactic latitude greater than 25$^{\rm o}$. Telescope time and pointing restrictions resulted in a final sample of 83 spiral galaxies with types ranging from Sa to Sm (plus one S0 and two Irregular galaxies). None of their selection criteria are expected to create a morphological-type bias.
Beijersbergen et al. (1999) selected spiral galaxies from the ESO-LV catalog (Lauberts & Valentijn 1989) to have types ranging from Sa-Sm, inclinations less than 50$^{\rm o}$, and absolute galactic latitudes greater than 15$^{\rm o}$. They additionally required the diameters of the 26 B-mag arcsec$^{-2}$ isophote to be greater than 1and smaller than 3(chip size limitation), and the surface brightness of the disk at the half-light radii to be fainter than 23.8 B-mag arcsec$^{-2}$. From the resulting list of some 600 galaxies, a random sample which had central light concentrations was selected. This prescription certainly should not have biased against the selection of early-type spiral galaxies, but on the contrary selected them if they do indeed exist.
de Blok et al. (1995) selected galaxies from the late-type sample of LSB UGC galaxies from van der Hulst et al. (1993) and the lists by Schombert & Bothun (1988) and Schombert et al (1992). Of the 17 galaxies from the de Blok sample for which we have B-band data, 2 originate from the van der Hulst UGC sample. The others are from the Schombert LSB galaxy lists which were constructed from a visual inspection of sky survey plates, and contain no [*a priori*]{} bias against early-type spiral galaxies. de Blok et al. randomly selected a sub-set of galaxies with inclinations less than 60$^{\rm o}$, central surface brightnesses fainter than 23 B-mag arcsec$^{-2}$, and having single-dish H$_{\rm I}$ observations available. This last criteria is the only possible bias we can identify (given that some early-type spiral galaxies may be gas-poor) ; however, it is expected to be a small effect for the following reason. The morphological make-up of the LSB galaxy catalogs of Schombert et al. (consisting of objects not in the UGC) turned out to be dominated (85%) by late-type galaxies (another 10% are dwarf ellipticals and some 5% unclassifiable). The reason for this composition is that the prominence of the bulge in early-type spiral galaxies had resulted in their prior detection in the first-generation Palomar sky-survey plates used by Nilson. The second-generation sky-survey plates went about a magnitude deeper, and predominantly resulted in the detection of fainter late-type galaxies. Thus, even if the H$_{\rm I}$ requirement excluded [*all*]{} early-type galaxies in the LSB catalogs, this would only be a 5-10% bias, which would equate to the exclusion of 1 or 2 early-type galaxies from our sample.
To summarize, the total galaxy sample is likely to be biased in favor of the inclusion of early-type LSB galaxies due to the selection criteria used by Beijersbergen et al. (1999).
The light-profile model parameters
----------------------------------
The galaxy light-profiles from the three galaxy samples are already published in the respective papers, and we refer the interested reader to these. For the sake of the present analysis, we have applied the same algorithm to every light-profile, simultaneously fitting a seeing-convolved Sérsic (1968) $r^{1/n}$ bulge and a seeing-convolved exponential disk (see Graham 2001b for details). This is an improvement over previous comparative work between the HSB and LSB galaxies because it not only avoids biases between different authors fitting techniques, but provides a more reliable approach to separating the bulge light from that of the disk. Additionally, this is the first time a bulge model has been fitted to the ‘disk-dominated’ sample of LSB galaxies. The structural parameters obtained from this decomposition are: the central disk surface brightness ($\mu_0$), the disk scale-length ($h$), the effective radius of the bulge ($r_e$), and the surface brightness of the bulge at this radius ($\mu_e$), and lastly, the shape parameter $n$ from the $r^{1/n}$ bulge model.
After correcting the heliocentric velocity measurements for Virgo-infall using the 220 model of Kraan-Korteweg (1986), the scale-lengths were converted from arcseconds to kpc using a Hubble constant $H_0$$=$75 km s$^{-1}$ Mpc$^{-1}$. Redshifts for ESO-LV 1590200, ESO-LV 0350110, and ESO-LV 0050050 were obtained from the Parkes telescope HIPASS database (Barnes et al. 2000). Redshifts for the other galaxies came from the respective papers. The surface brightnesses were adjusted for a) $(1+z)^4$ redshift dimming, b) $K$-corrections (using the tables of Poggianti 1997), c) Galactic extinction (Schlegel, Finkbeiner, & Davis 1998; data obtained from NED), and d) inclination corrected. We used the inclination correction 2.5$C$$\log(a/b)$, where $a/b$ is the axis-ratio of the outer disk – assumed to be due to inclination. The value of $C$ was taken to be 0.2 for the $B$-band and 0.5 for the $R$-band (Tully & Verheijen 1997). It is noted that while this correction is necessary for the HSB galaxies, it’s relevance to the LSB galaxies is not as clearly established. Both the HSB and the LSB galaxy samples have inclinations less than 60$^{\rm o}$, and so the maximum correction is 0.15 mag arcsec$^{-2}$ ($B$-band) and $\sim$0.4 mag arcsec$^{-2}$ ($R$-band). The average correction for the LSB galaxies turned out to be only 0.06 ($B$-band) and 0.15 ($R$-band) mag arcsec$^{-2}$, and cannot be a significant bias or explanation for the distribution of points in the $\mu_0$–$\log h$ diagram (Section 3).
We report that only a couple of the late-type LSB galaxies from de Blok et al. (1995) have bulges with central surface brightness values that are brighter than the disk, and even then, only brighter by a few tenths of magnitude. Most of the late-type LSB galaxies have bulges with central surface brightness values that are fainter than that of the disk; their presence signalled by a central bulge in the light-profile that rises less than $0.75(=2.5\log(2))$ mag (and often much less) above the extrapolated disk light-profile. Many of the light-profiles actually show no sign at all of a bulge. In all the late-type LSB galaxies in the sample, the light distribution is overwhelmingly dominated by that of the disk; however, for those LSB galaxies with bulges, we can perform a comparison study with the HSB galaxy sample.
The $\mu_0$–$\log h$ diagram
============================
Shown in Figure \[fig1\] are the central disk surface brightnesses plotted against the associated disk scale-lengths. In general, the disks from early-type spiral galaxies tend to reside, or rather, define, the upper envelope of points in the $\mu_0$–$\log h$ diagram. The disks of late-type spiral galaxies tend to reside, in general, below this envelope. One can also get a feel for the distribution of points by cutting the diagram up further. While we do not wish to suggest anything physical with an arbitrary cut at 22.65 B-mag arcsec$^{-2}$, it is however somewhat revealing to do so. One can see that the early-type spiral galaxies with central disk surface brightnesses fainter than this value have large disk scale-lengths ($>$8-9 kpc) while the LSB late-type spiral galaxies have disk scale-lengths less than this. Yet another way to view the diagram is parallel to lines of constant disk luminosity, which have a slope of 5. One can see that, on average, the brightest early-type spiral galaxies have brighter luminosities than the brightest late-type spirals; a result consistent with the luminosity functions of different spiral galaxy types (Folkes et al. 1999). Furthermore, when one factors in the contribution from the bulge light this trend will be stronger still. The data does however reveal that there is some overlap of galaxy types in the diagram. That is, rather than a strict exclusion of a given population in different parts of the $\mu_0$–$\log h$ diagram. it is more a case of one population dominating over the other. This morphological type separation is actually analogous to the behavior of bulges in the $\mu_e$–$\log r_e$ diagram (Kent 1985; his figure 5b).
Past studies of the $\mu_0$–$\log h$ diagram have either been limited to the upper right of the diagram by the selection-criteria, or have not investigated the morphological-type dependence. Bothun et al. 1997 explored beyond the typical selection-criteria boundaries by including in their Figure 4, in addition to a sample of HSB galaxies, a large sample of LSB galaxies. They revealed a similar distribution of points as seen in Figure \[fig1\], but no morphological-type distinction was made (similarly see, Dalcanton, Spergel, & Summers 1998 figure 4; McGaugh 1998 figure 1). In reviewing the literature, however, we have found that combining figure 3 from Tully & Verheijen (1997) (a $\mu_0$–$\log h$ diagram) with their figure 6 (a plot of $\mu_0$ vs. galaxy type) should reveal the same morphological-type dependence as observed here, lending support to this result.
The upper boundary in the $\mu_0$–$\log h$ plane represents a real decline to the space-density of galaxies with big (large $h$) bright (low $\mu_0$) disks – the selection criteria favors their detection. The presence of this boundary has been used in Graham (2001a, see also Graham 2000a,b) to estimate the degree of opacity in inclined early-type spiral galaxy disks; inclined galaxies with transparent disks are shown to reside above this envelope due to the apparent brightening of their disk with inclination. The lower boundary is however artificial – the result of a cut-off imposed by the galaxy selection criteria . The existence of ultra-low LSB, optically invisible galaxies (Disney 1998), is therefore not excluded. The (solid) lines of constant luminosity shown in Figure \[fig1\] are lines of constant disk luminosity, and therefore do not include possible contributions from the bulge light to the total galaxy light. One may therefore wonder if the upper envelope may in fact represent a boundary of constant ‘galaxy’ luminosity – perhaps the bright end to the luminosity function. To address this thought, we show in Figure \[fig2\] the bulge-to-disk ($B/D$) luminosity ratio, as a function of disk scale-length for every galaxy in the upper envelope delineated by the dotted lines in Figure \[fig1\]. No trend of increasing $B/D$ luminosity ratio with decreasing scale-length is evident, and consequently no support is given to the suggestion that the upper boundary is due to a maximum galaxy luminosity.
In Figure \[fig3\] we have plotted all the $B/D$ luminosity ratios against the bulge shape parameter $n$ (the index from the best-fitting $r^{1/n}$ bulge profile models). These points have been plotted as a function of galaxy type, and also in a way so as to reveal which galaxy sample they have come from. One can see that for both the HSB and LSB galaxies, the shape of the bulge light-profile is correlated with the nature of the host galaxy in the sense that both the $B/D$ luminosity ratio and the galaxy morphological type are correlated with $n$, This result is firmly established for the bulges of HSB spiral galaxies (Andredakis, Peletier, & Balcells 1995; Graham 2001b; Mollenhoff & Heidt 2001), but has not been previously shown for the bulges of LSB spiral galaxies. It would appear that no obvious structural distinction can be made between the LSB and HSB spiral galaxies in this diagram.
We wish to thank Renée Kraan-Korteweg for making her Virgo-centric inflow code available. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The Parkes telescope is part of the Australia Telescope which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
Andredakis, Y.C., Peletier, R.F., & Balcells, M. 1995, , 275, 874 Barnes, D.G., et al. 2000, mnras, in press Beijersbergen, M., de Blok, W.J.G., & van der Hulst, J.M. 1999, A&A, 351, 903 Bothun, G.D., Impey, C.D., Malin, D.F., & Mould, J.R. 1987, , 94, 23 Bothun, G.D., Impey, & McGaugh, S.S. 1997, , 109, 745 Broeils, A.H., 1992, Ph.D. thesis, Univ. Groningen Dalcanton, J.J., Spergel, D.N., & Summers, F.J. 1997, ApJ, 482, 659 Davies, J.I., Phillipps, S., & Disney, M.J. 1988, , 231, 69p de Blok, W.J.G., van der Hulst, J.M., & Bothun G.D. 1995, , 274, 235 de Jong, R.S. 1996, A&AS, 118, 557 de Jong, R.S., & van der Kruit, P.C. 1994, A&AS, 106, 451 Disney, M.J. 1976, Nature, 263, 573 Disney, M.J. 1998, IAU Colloquium 171, ASP Conf. Ser., J.I. Davies, C. Impey, and S. Phillipps, eds., 170, 11 Folkes, S., et al. 1999, , 308, 459 Freeman, K.C. 1970, , 160, 811 Graham, A.W. 2000a, in The Evolution of Galaxies. I - Observational Clues, , eds. Jose M. Vilchez, Grazyna Stasinska, Enrique Perez, in press Graham, A.W. 2000b, in Galaxy Disks and Disk Galaxies, , eds. Enrico M. Corsini, José G. Funes, in press Graham, A.W. 2001a , submitted Graham, A.W. 2001b , 121, 820 Impey, C., & Bothun, G. 1989, , 341, 89 Kent, S.M. 1985, ApJS, 59, 115 Kraan-Korteweg R.C., 1986, A&AS, 66, 255 Lauberts, A., & Valentijn, E.A. 1989, The Surface Photometry Catalogue of the ESO-Uppsala Galaxies (ESO-LV), European Southern Observatory. Longmore, A.J., Hawarden, T.G., Goss, W.M., Mebold, U., & Webster, B.L. 1982, , 200, 325 reference[McG98]{}McGaugh, S. 1998, IAU Colloquium 171, ASP Conf. Ser., J.I. Davies, C. Impey, and S. Phillipps, eds., 170, 19 (astro-ph/9810363) reference[MaH01]{}Mollenhoff, C., & Heidt, J. 2001, A&A, submitted Nilson, P. 1973, Uppsala General Catalog of Galaxies, Ann. Uppsala Astron. Obs., 6 (UGC) Poggianti B.M., 1997, A&AS, 122, 399 Schlegel, D.J., Finkbeiner, D.P., Davis, M., 1998, ApJ, 500, 525 Schombert, J.M., & Bothun, G.D. 1988, AJ, 95, 1389 Schombert, J.M., Bothun, G.D., Schneider, S.E., & McGaugh, S.S. 1992, AJ, 103, 1107 Sèrsic, J.-L. 1968, Atlas de Galaxias Australes (Cordoba: Observatorio Astronomico) Sprayberry, D., Impey, C.D., Bothun, G.D., & Irwin, M.J. 1995, , 109, 558 Tully R.B., Verheijen M.A.W., 1997, ApJ, 484, 145 van der Hulst J.M., Skillman E.D., Smith T.R., Bothun G.D., McGaugh S.S., de Blok W.J.G. 1993, AJ, 106, 548
|
---
abstract: 'We test the predictive power of first-oder reversal curve (FORC) diagrams using simulations of random magnets. In particular, we compute a histogram of the switching fields of the underlying microscopic switching units along the major hysteresis loop, and compare to the corresponding FORC diagram. We find qualitative agreement between the switching-field histogram and the FORC diagram, yet differences are noticeable. We discuss possible sources for these differences and present results for frustrated systems where the discrepancies are more pronounced.'
author:
- 'Helmut G. Katzgraber$^a$, Gary Friedman$^b$, and G. T. Zimányi$^c$'
bibliography:
- 'refs.bib'
title: Fingerprinting Hysteresis
---
Introduction
============
The conventional methods [@he:92; @proksh:94; @hedja:94] to characterize magnetic interactions in hysteretic systems, such as the $\delta M$ method [@che:92; @el-hilo:92], utilize isothermal remanent magnetization (IRM) and dc demagnetization remanence (DCD) curves based on the Wohlfarth relation [@wohlfarth:58]. Recently, FORC diagrams [@pike:99; @katzgraber:02b] have been introduced to study hysteretic systems. Their extreme sensitivity has helped to “fingerprint” several experimental systems as well as theoretical models ranging from geological samples and recording media to paradigmatic models of random magnets and spin glasses [@katzgraber:02b].
In this work we perform numerical simulations of random magnets (and spin glasses) in order to test the predictive power of FORC diagrams by comparing to a histogram of up- and down-switching fields of the underlying switching units along the major hysteresis loop.
The aforementioned re-parametrization of the major hysteresis loop (switching-field histogram) displays the information carried by the major loop in a more comprehensive way and provides a good comparison to the FORC diagram. We find, that the major-loop behavior predicts the minor-loop behavior captured by FORC diagrams well. We present a comparison of both distributions and discuss some differences between them. We argue that switching-field histograms are useful to study hysteretic systems in more detail than with conventional methods due to their simplicity and ease to compute.
Model & Algorithm {#model}
=================
The Hamiltonian of the random-field Ising model (RFIM) is given by [@ji:92] $${\mathcal H}= \sum_{\langle i,j \rangle} J_{ij}S_iS_j -
\sum_i h_i S_i - H \sum_i S_i \,.
\label{eq:hamilton}$$ Here $S_i = \pm 1$ are Ising spins on a square lattice of size $N = L^3$ in three dimensions with periodic boundary conditions. The interactions between the spins are uniform ($J_{ij} = 1$) and nearest-neighbor, and $H$ represents the externally applied field. Disorder is introduced into the model by coupling the spins to site-dependent random fields $h_i$ drawn from a Gaussian distribution with zero mean and standard deviation $\sigma_{\rm R}$.
We simulate the zero-temperature dynamics of the RFIM by changing the external field $H$ in small steps starting from positive saturation. After each field step we compute the local field $f_i$ of each spin: $$f_i=\sum_{j} J_{ij}S_j - H -h_i\; .
\label{eq:local_field}$$ A spin is unstable if it points opposite to its local field, i.e., $f_i \cdot S_i < 0$. We then flip a randomly chosen unstable spin and update the local fields at neighboring sites. This procedure is repeated until all spins are stable.
For the rest of this work we set $L = 50$ ($N =
125000$ spins) and $\sigma_{\rm R} = 5.0$, unless otherwise specified. The different figures are calculated by averaging over 5000 disorder realizations in order to reduce finite-size effects.
FORC Diagrams {#forc}
=============
In oder to calculate an FORC diagram, a family of First Order Reversal Curves (FORCs) with different reversal fields $H_{\rm R}$ is measured, with $M(H, H_{\rm R})$ denoting the resulting magnetization as a function of the applied and reversal fields. Computing the mixed second order derivative [@dellatorre:99; @pike:99] $$\rho(H, H_{\rm R})= -\frac{1}{2}
[{\partial}^2 M/{\partial} H {\partial} H_{\rm R}]
\label{eq:rho}$$ and changing variables to $H_{\rm c}=(H-H_{\rm R})/2$ and $H_{\rm b}=(H+H_{\rm R})/2$, the local coercivity and bias respectively, yields the “FORC distribution” $\rho(H_{\rm b}, H_{\rm c})$. FORC diagrams resemble the commonly known Preisach diagrams [@preisach:35; @mayergoyz:86], yet they are model-independent and therefore more general.
![ FORC diagram of the RFIM for disorder $\sigma_R =
5.0$, well above the critical disorder [@perkovic:99] in three dimensions. Note the pronounced vertical feature at $H_{\rm c} = 1$ with a wake extending to $H_{\rm c} = 0$ which corresponds to multi-domain nucleation in the sample. The dots along the $H_{\rm b}$-axis are numerical noise (no data smoothing). \[fig1\] ](forc-d5.0.eps)
Figure \[fig1\] shows an FORC diagram of the RFIM at high disorder strength ($\sigma_{\rm R} = 5.0 >
\sigma_{\rm crit} \approx 2.16$) [@perkovic:99]. Note the vertical ridge at $H_{\rm c} \approx 1$ reminiscent of domain-wall nucleation [@drossel:98]. A vertical cross-section of the ridge ($H_{\rm c} = 1$) mirrors the distribution of the applied random fields, because these can be viewed as a distribution of random biases acting on the spins when $\sigma_{\rm R} \gg
\sigma_{\rm crit}$. We have tested this in detail by selecting the random fields from a box distribution. The resulting FORC diagram is qualitatively similar to the Gaussian case, yet a vertical cross-section of the ridge is box-shaped. This is not evident by studying the major hysteresis loop for different disorder-distribution shapes and illustrates the advantages of the FORC method over conventional approaches for studying hysteretic systems.
Switching-Field Histograms {#sfh}
==========================
In order to test the predictive power of FORC diagrams, we simulate the RFIM with the zero-temperature dynamics described in Sec. \[model\] and store the up- and down-switching fields of the spins along the major hysteresis loop. We then create a histogram of the number of flipped spins for a given pair of up- ($H_{\uparrow}$) and down-switching fields ($H_\downarrow$). By changing the variables to the coercivity \[$H_{\rm c} =
(H_{\uparrow} - H_\downarrow)/2$\] and bias \[$H_{\rm b}
= (H_{\uparrow} + H_\downarrow)/2$\] of each spin, we obtain a distribution of the coercivities and biases of the spins in the system along the major hysteresis loop.
![ Switching-field histogram of the three-dimensional RFIM with $\sigma_{\rm R} = 5.0$. Note the close resemblance to the FORC diagram presented in Fig. \[fig1\]. Because no derivatives of the data are required, the contours are much smoother than in the case of an FORC diagram. \[fig2\] ](sfh-d5.0.eps)
Figure \[fig2\] shows the switching-field histogram (SFH) for the RFIM. One can see a close resemblance with the corresponding FORC diagram presented in Fig. \[fig1\]. In order to better compare FORC diagram and SFH, in Fig. \[fig3\] we present the absolute difference between both diagrams.
![ Absolute difference between the FORC diagram presented in Fig. \[fig1\] and the corresponding SFH in Fig. \[fig2\] for the three-dimensional RFIM with $\sigma_R = 5.0$. The details are discussed in the text. \[fig3\] ](diff-d5.0.eps)
Even though the SFH and the FORC diagram of the RFIM differ slightly (see Fig. \[fig3\]), the main characteristics representing the underlying physical properties of the model are the same (vertical ridge representing multi-domain nucleation). It is interesting that a zeroth-order reversal curve (the major hysteresis loop) contains possibly all the necessary information to reconstruct the first order reversal curves of the system (the FORC diagram).
The differences found between the FORC diagram and the SFH could be due to numerical error in the derivatives of the FORCs because noise is amplified in numerical derivatives considerably. In addition, the differences could be attributed to either hysteron correlations or the failure of the (simple) Preisach picture of hysterons. The latter would imply that a generalization of “classical” hysterons is required.
Figure \[fig2\] also illustrates how the re-parametrization of the major hysteresis loop in terms of an SFH shows more details about the microscopic structure of the system. The gained information is similar to the information provided by an FORC diagram, yet the computation of an SFH is [*considerably*]{} faster than calculating an FORC diagrams (generally $\sim 10^2$ times faster) and involves no numerical derivatives of the data, thus reducing numerical error.
Frustrated Systems {#frustr}
==================
As the random-field Ising model is a random magnet with no frustration, we also calculate the FORC diagram and SFH for the 3D Edwards-Anderson Ising spin glass [@binder:86] (EASG). Due to frustration, a spin can flip more than twice along the full hysteresis loop. With the current definition of the SFH this is not taken into account and differences to an FORC diagram are expected.
The Hamiltonian of the Edwards-Anderson Ising spin glass is given by Eq. (\[eq:hamilton\]) where the $J_{ij}$ are nearest-neighbor interactions chosen according to a Gaussian distribution with zero mean and standard deviation unity, and $h_i = 0$ $\forall
i$. $H$ represents the externally applied magnetic field and periodic boundary conditions are applied. For the simulations we use the zero-temperature algorithm presented in Sec. \[model\]. Frustration is introduced by the random signs of the interactions $J_{ij}$.
Figure \[fig4\] shows the FORC diagram of the EASG. One can see a pronounced ridge along the $H_{\rm
c}$-axis together with an asymmetric feature at small coercivities. The underlying details of the EASG FORC diagram have been discussed elsewhere [@katzgraber:02b].
![ FORC Diagram of the EASG. Note the ridge along the $H_c$-axis. Data for 5000 disorder realizations and $N
= 50^3$ spins. \[fig4\] ](forc-ea.eps)
In Fig. \[fig5\] the SFH of the EASG is shown. Note that the asymmetry present in the FORC diagram in Fig. \[fig4\] is lost. The weight of the asymmetric part of the FORC diagram shifts to the ridge at $H_b =
0$.
![ SFH of the EASG. While the ridge along the $H_c$-axis is qualitatively conserved, the SFH shows drastic differences to the FORC diagram presented in Fig. \[fig4\]. In particular, the asymmetry with respect to the horizontal axis is lost. \[fig5\] ](sfh-ea.eps)
Although some of the features in the FORC diagram (Fig. \[fig4\]) of the EASG are missing in the corresponding SFH (Fig. \[fig5\]), the horizontal ridge reminiscent of the underlying reversal-symmetry of the Hamiltonian [@katzgraber:02b] is conserved. In particular, by comparing the FORC diagram and SFH one can study the effects of frustration on the hysteretic behavior of a spin glass.
Conclusions
===========
By re-parameterizing the major hysteresis loop with a switching-field histogram we show that for systems with no frustration (random-field Ising model) the SFH closely resembles the FORC diagram. Small differences can be attributed to numerical error in the calculation of an FORC diagram, hysteron correlations, or the breakdown of the hysteron picture.
SFHs show more details about the system than the major hysteresis loop and are considerably faster to computer than FORC diagrams. Therefore they are an efficient alternative in order to study the microscopic distributions of coercivity and bias of the switching units.
Because the switching fields of the underlying microscopic switching units have to be recorded for the computation of an SFH, the method is in general limited to numerical studies of hysteretic systems. Experimental applicability might be possible with synthetic particulate samples [@liu:02] where the individual switching units can be traced during the magnetic field sweep.
We also present results on the (frustrated) Edwards-Anderson Ising spin glass. We show that there are clear differences between the FORC diagram and the SFH because SFHs do not take into account multiple switching events of the spins, a hallmark of spin glasses. We suggest these differences can be used to quantify the effects of frustration in FORC diagrams.
The FORC and SFH methods promise to be powerful tools to “fingerprint” hysteretic systems. Still, the breadth of information they provides remains to be understood fully.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank F. Pázmándi and R. T. Scalettar for discussions.
|
---
abstract: 'The final state interactions (FSI) in $\Delta S=-1$, $\Delta C=\pm 1$ decays of $B$-meson are discussed. The rescattering corrections are found to be of order of $15-20\%$. The strong interaction phase shifts are estimated and their effects on $CP$- asymmetry are discussed.'
address: |
National Centre for Physics and Physics Department, Quaid-i-Azam University,\
Islamabad, Pakistan.
author:
- Fayyazuddin
title: 'Final State Interactions and $\Delta S=-1,$ $\Delta C=\pm 1$ $B$–decays'
---
Introduction
============
Direct $CP$-violation in the decays $B\rightarrow f$ and $\bar{B}\rightarrow
\bar{f}$ depends on the strong final state interactions. In fact the $CP$-asymmetry parameter vanishes in the limit of no strong phase shifts. The purpose of this paper is to study the $\Delta C=\pm 1,$ $\Delta S=-1$ $B$-decays taking into account the final state interactions [@f1; @f2; @f3; @f4; @f5]. Such decays are described by the effective Lagrangians $$\begin{aligned}
{\cal L}_{eff} &=&\frac{G_F}{\sqrt{2}}V_{cb}V_{us}^{*}[\bar{s}\gamma ^\mu
(1-\gamma _5)u][\bar{c}\gamma _\mu (1-\gamma _5)b] \nonumber \\
{\cal L}_{eff} &=&\frac{G_F}{\sqrt{2}}V_{ub}V_{cs}^{*}[\bar{s}\gamma ^\mu
(1-\gamma _5)c][\bar{u}\gamma _\mu (1-\gamma _5)b] \label{01}\end{aligned}$$ Both these Lagrangians have $\Delta I=\frac 12$. The weak phase in the Wolfenstein parameterization [@f6] of CKM matrix [@f7] is given by $$\frac{V_{ub}V_{cs}^{*}}{V_{cb}V_{us}^{*}}=\sqrt{\rho ^2+\eta ^2}e^{i\gamma
},\qquad \sqrt{\rho ^2+\eta ^2}=0.36\pm 0.09 \label{02}$$ It is quite difficult to reliably estimate the final state interactions in weak decays. The problem is somewhat simplified by using isospin and $SU(3)$ symmetry in discussing the strong interaction effects. We will fully make use of these symmetries. Moreover we note in Regge phenomenology, the strong interactions scattering amplitudes can be written in terms of Pomeron exchange and exchange of $\rho -A_2$ and $\omega -f$ trajectories in $t$-channel. The problem is further simplified if there is an exchange degeneracy. In fact this is the case here. In $s(u)$ channel only the states with quark structure $s\bar{c}$ $(c\bar{s})$ can be exchanged ($s,$ $u,$ $t,$ are Mandelstam variables). An important consequence of this is that since $%
\bar{K}D$ has quantum numbers $C=1,$ $S=-1$, only a state with structure $c%
\bar{s}$ ($C=1,$ $S=1,$ $Q=+1)$ can be exchanged in $u$-channel where as no exchange is allowed in the $s$-channel (exotic). On the other hand $\bar{K}%
\bar{D}$ ($C=-1,$ $S=-1$) state is non-exotic and we can have an exchange of a state with structure $s\bar{c}$ ($C=-1,$ $S=-1,$ $Q=-1$) in $s$-channel and a state with structure $d\bar{c}$ or $u\bar{c}$ ($C=-1,$ $S=0$) in $u$-channel for a quasi-elastic channel such as $\bar{K}^0\bar{D}^0\rightarrow
\pi ^{+}D_s^{-}$ or $K^{-}\bar{D}^0$($\bar{K}^0D^{-}$)$\rightarrow \pi
^0D_s^{-}$. In Regge Phenomenology, exotic $u$-channel implies exchange degeneracy i.e. in $t$-channel $\rho -A_2$ and $\omega -f$ trajectories are exchange degenerate. Taking into account $\rho -\omega $ degeneracy, all the elastic or quasielastic scattering amplitudes can be expressed in terms of two amplitudes which we denote by $F_P$, $F_\rho $ and $\bar{F}_\rho
=e^{-i\pi \alpha \left( t\right) }F_\rho $, where $F_P$ is given by pomeron exchange, $F_\rho $ by particle exchange trajectory for which $\alpha _\rho
\left( t\right) =\alpha _{A_2}\left( t\right) =\alpha _\omega \left(
t\right) =\alpha _f\left( t\right) =\alpha \left( t\right) $. We have argued above, that elastic and quasi-elastic scattering amplitudes can be calculated fairly accurately. This combined with the following physical picture [@f8; @f9] gives us a fairly reliable method to estimate the effect of rescattering on weak decays. In the weak decays of $B$-measons, the $b$ quark is converted into $b\rightarrow c+q+\bar{q}$, $b\rightarrow
u+q+\bar{q}$; since for the [@f8; @f9] tree graph the configuration is such that $q$ and $\bar{q}$ essentially go together into the color singlet state with the third quark recoiling, there is a significant probability that the system will hadronize as a two body final state. This physical picture has been put on a strong theoretical basis in [@f10; @r1]. In this picture the strong phase shifts are expected to be small at least for tree amplitude.
Now the discontinuity or imaginary part of decay amplitude is given by [@r2; @f11] $$%TCIMACRO{\func{Im} }
%BeginExpansion
\mathop{\rm Im}%
%EndExpansion
A_f=\sum_nM_{fn}^{*}A_n \label{03}$$ where $M_{fn}$ is the scattering amplitude for $f\rightarrow n$. According to above picture the important contribution to the decay amplitude $%
A_f\equiv A\left( B\rightarrow f\right) $ in Eq. (\[03\]) is from those two body decays of $B$ which proceed through tree graphs. Thus it is reasonable to assume that the decays proceeding through the following chains $$\begin{aligned}
\bar{B}^0 &\rightarrow &K^{-}D^{+}\rightarrow \bar{K}^0D^0 \nonumber \\
\bar{B}_s^0 &\rightarrow &K^{-}D_s^{+}\rightarrow \pi ^{-}D^{+} \label{04}\end{aligned}$$ and $$\begin{aligned}
\bar{B}^0 &\rightarrow &\pi ^{+}D_s^{-}\rightarrow \bar{K}^0\bar{D}^0
\nonumber \\
B^{-} &\rightarrow &\pi ^0D_s^{-}\rightarrow \bar{K}^0D^{-} \nonumber \\
&\rightarrow &\eta D_s^{-}\rightarrow K^{-}\bar{D}^0 \nonumber \\
\bar{B}_s^0 &\rightarrow &K^{+}D_s^{-}\rightarrow \pi ^{+}D^{-} \label{05}\end{aligned}$$ may make a significant contribution to the decay amplitudes. Note that the intermediate channels are those channels for which the decay amplitude is given by tree amplitude for which the factorization anstaz is on a strong footing [@r1; @r2].
In view of above arguments, the dominant contribution in Eq. (\[03\]) is from a state $n=\acute{f}$ where the decay amplitude $A_{\acute{f}}$ is given by the tree graph. For the quasi elastic channels listed in Eqs. (\[04\]) and (\[05\]), the scattering amplitude $M_{f\acute{f}}$ is given by the $\rho $-exchange amplitude $F_\rho $ or $\bar{F}_\rho $. The purpose of this paper is to calculate the rescattering correction to $A_f$ by the procedure outlined above. Final state interactions considered from this point of view will be labelled as FSI (A). In an alternative point of view, labelled (B) one lumps all channels other the elastic channel in one category. In the random phase approximation of reference [@f10], one may take the parameter $\rho =\left[ \overline{\left| A_n^2\right| }\right]
^{1/2}/\left. A_f\right. ,$ $n\neq f$ nearly $1$ for the color suppressed decays but $\rho $ small for the two body decays dominated by tree graphs. In this paper the observational effects of final state interactions will be analyzed from the point of view (A).
Decay Amplitudes Decomposition
==============================
The amplitudes for various $\bar{B}$ decays are listed in Table1. They are characterized according to decay topologies: (1) a color-favored tree amplitude $T$, (2) a color-suppressed tree amplitude $C$, (3) an exchange amplitude $E$, and (4) an annihilation amplitude $A$. The isospin decomposition of relevant decays along with strong interaction phases are also given in Table1. Isospins and $SU(3)$ symmetry gives the following relationships for various decay amplitudes $$\begin{aligned}
A_{-+}+A_{00} &=&A_{-0} \nonumber \\
B_{-s^{+}}-B_{-+} &=&A_{-+} \label{61} \\
\bar{A}_{-0}-\bar{A}_{0-} &=&\bar{A}_{00} \nonumber \\
\sqrt{2}\bar{A}_{0s}-\sqrt{6}\bar{A}_{8s^{-}} &=&2\bar{A}_{0-} \nonumber \\
\bar{B}_{+s^{-}}-\bar{B}_{+-} &=&\bar{A}_{+s^{-}} \nonumber \\
\bar{B}_{+-}-\sqrt{6}\bar{B}_{80} &=&2\bar{A}_{00} \label{07}\end{aligned}$$ First we note that in the naive factorization ansatz, we have the following relations between the amplitudes of various topologies[@f12] $$\begin{aligned}
\frac CT &=&\left( \frac{a_2}{a_1}\right) \frac{%
f_DF_0^{B-K}(m_D^2)(m_B^2-m_K^2)}{f_KF_0^{B-D}(m_K^2)(m_B^2-m_D^2)}
\nonumber \\
&\approx &\left( \frac{a_2}{a_1}\right) (0\cdot 72) \label{08} \\
\frac ET &=&\left( \frac{a_2}{a_1}\right) \frac{%
f_{B_s}F_0^{D_s-K}(m_B^2)(m_{D_s}^2-m_K^2)}{%
f_KF_0^{B_s-D_s}(m_K^2)(m_{B_s}^2-m_{D_s}^2)} \nonumber \\
&\approx &\left( \frac{a_2}{a_1}\right) (0\cdot 08) \label{09} \\
\frac{\bar{T}}T &\approx &0\cdot 72\sqrt{\rho ^2+\eta ^2} \label{10}\end{aligned}$$
\[101\] $$\begin{aligned}
\frac{\bar{C}}T &=&\left( \frac{a_2}{a_1}\right) \frac{%
f_DF_0^{B-K}(m_D^2)(m_B^2-m_K^2)}{f_{D_s}F_0^{B-\pi }(m_{D_s}^2)(m_B^2-m_\pi
^2)} \nonumber \\
&\approx &\left( \frac{a_2}{a_1}\right) \label{11a} \\
\frac{\bar{A}}{\bar{T}} &=&\frac{f_BF_0^{D-K}(m_B^2)(m_D^2-m_K^2)}{%
f_{D_s}F_0^{B-\pi }(m_{D_s}^2)(m_B^2-m_\pi ^2)} \nonumber \\
&\approx &0\cdot 08 \label{11b}\end{aligned}$$ where $a_2/a_1\approx 0.2-0.3$. The numerical values have been obtained using $f_D\approx 200$ MeV, $f_{D_s}\approx 240$ MeV, $f_K\approx 158$ MeV, $%
f_B\approx 180$ MeV.
$$\begin{aligned}
\frac{F_0^{B-K}\left( m_D^2\right) }{F_0^{B-D}\left( m_K^2\right) } &\approx
&0.05\approx \frac{F_0^{D_s-K}\left( m_B^2\right) }{F_0^{B_s-D_s}\left(
m_K^2\right) } \nonumber \\
F_0^{B-K}\left( m_D^2\right) &\approx &F_0^{B-\pi }\left( m_{D_s}^2\right)
\label{12}\end{aligned}$$
Since the amplitudes $C$ ($\bar{C}$), $E$, $\bar{A}$ are suppressed relative to tree amplitude, they are subject to important corrections due to rescattering.
Rescattering
============
In order to calculate rescattering corrections and to obtain $s$-wave strong phases, we consider the scattering processes $$\begin{aligned}
P_a+\bar{D} &\rightarrow &P_b+\bar{D} \\
P_a+D &\rightarrow &P_b+D\end{aligned}$$ where $P_a$ is a pseudoscalar octet. Using $SU(3)$, the scattering amplitude can be written as $$M=\chi ^{\dagger }\left[ \bar{F}_1\frac 12[\lambda _b,\lambda _a]+\bar{F}_2%
\frac 12\{\lambda _b,\lambda _a\}+\bar{F}_3\delta _{ba}\right] \chi
\label{13}$$ where $\chi $ is an $SU(3)$ triplet $$\chi =\left(
\begin{array}{l}
\bar{D}^0 \\
D^{-} \\
D_s^{-}
\end{array}
\right) \label{14}$$ For the process $P_a+D\rightarrow P_b+D$, we replace $\bar{F}_i$ ($i=1,2,3$) by $F_i$ and $\chi $ by the triplet $$\chi =\left(
\begin{array}{l}
D^0 \\
D^{+} \\
D_s^{+}
\end{array}
\right) \label{15}$$ In order to express the scattering amplitudes in terms of Regge trajectories, it is convenient to define two amplitudes $$\begin{aligned}
M^{+} &=&P+f+A_2 \nonumber \\
&=&-C_P\frac{e^{-i\pi \alpha _P(t)/2}}{\sin \pi \alpha _P(t)/2}\left( \frac s%
{s_0}\right) ^{\alpha _P(t)}+\left[ -C_f\frac{1+e^{-i\pi \alpha _f(t)}}{\sin
\pi \alpha _f(t)}\left( \frac s{s_0}\right) ^{\alpha _f(t)}-C_{A_2}\frac{%
1+e^{-i\pi \alpha _{A_2}(t)}}{\sin \pi \alpha _{A_2}(t)}\left( \frac s{s_0}%
\right) ^{\alpha _{A_2}(t)}\right] \label{16} \\
M^{-} &=&\rho +\omega =\left[ C_\omega \frac{1-e^{-i\pi \alpha _\omega (t)}}{%
\sin \pi \alpha _\omega (t)}\left( \frac s{s_0}\right) ^{\alpha _\omega
(t)}+C_\rho \frac{1+e^{-i\pi \alpha _\rho (t)}}{\sin \pi \alpha _\rho (t)}%
\left( \frac s{s_0}\right) ^{\alpha _\rho (t)}\right] \label{17}\end{aligned}$$ Due to exchange degeneracy, for linear Regge trajectories $$\begin{aligned}
\alpha _\rho \left( t\right) &=&\alpha _{A_2}\left( t\right) =\alpha _\omega
\left( t\right) =\alpha _f\left( t\right) =\alpha _0\left( t\right) +\acute{%
\alpha}t \label{18} \\
C_f &=&C_\omega ;\qquad C_{A_2}=C_\rho \nonumber \\
C_\omega &=&C_\rho \label{19}\end{aligned}$$ We take[@f11; @f13] $\alpha _0=0.44\approx 1/2$ and $\acute{\alpha}=0.94$GeV$^{-2}\approx 1$GeV and for the pomeron $\alpha _P\left( t\right) =\alpha
_P\left( 0\right) +\acute{\alpha}_Pt$, $\alpha _P\left( 0\right)
=1.08\approx 1$ and $\acute{\alpha}_P\approx 0.25$GeV$^{-2}$. In particular for the processes $K^{-}D^0\rightarrow K^{-}D^0$ and $K^{-}\bar{D}%
^0\rightarrow K^{-}\bar{D}^0$,we get $$\begin{aligned}
M(K^{-}D^0 &\rightarrow &K^{-}D^0)=P+(f-\omega )+(A_2-\rho ) \nonumber \\
&=&iC_P\left( \frac s{s_0}\right) e^{bt}-2(C_\omega +C_\rho )\frac 1{\sin
\pi \alpha (t)}\left( \frac s{s_0}\right) ^{\alpha (t)} \nonumber \\
&=&iC_P\left( \frac s{s_0}\right) e^{bt}-4C_\rho \frac 1{\sin \pi \alpha (t)}%
\left( \frac s{s_0}\right) ^{\alpha (t)} \nonumber \\
&=&F_P+F_\rho \label{20}\end{aligned}$$ $$\begin{aligned}
M(K^{-}\bar{D}^0 &\rightarrow &K^{-}\bar{D}^0)=P+(f+\omega )+(A_2+\rho )
\nonumber \\
&=&F_P+e^{-i\pi \alpha (t)}F_\rho =F_P+\bar{F}_\rho \label{21}\end{aligned}$$ where $b=\acute{\alpha}_P\ln (s/s_0)$. We take [@r2] $C_P=5$. In order to estimate $C_\rho $, we note that $SU(3)$ gives $$\begin{aligned}
\gamma _{\rho K^{+}K^{-}} &=&-\gamma _{\rho K^0\bar{K}^0}=\frac 12\gamma
_{\rho \pi ^{+}\pi ^{-}}=\frac 12\gamma _0 \nonumber \\
\gamma _{\omega K^{+}K^{-}} &=&\gamma _{\omega K^0\bar{K}^0}=\frac 12\gamma
_0 \nonumber \\
\gamma _{\rho D^{+}D^{-}} &=&-\gamma _{\rho D^0\bar{D}^0}=-\gamma _{\omega
D^{+}D^{-}}=-\gamma _{\omega D^0\bar{D}^0} \label{22}\end{aligned}$$ For our purpose, we will take $\gamma _{\rho D^{+}D^{-}}\approx \gamma
_{\rho K^{+}K^{-}}=\frac 12\gamma _0$, so that $$C_\rho =\gamma _{\rho K^{+}K^{-}}\gamma _{\rho D^0\bar{D}^0}=-\frac 14\gamma
_0^2 \label{23}$$ For $\gamma _0^2$, we use that value $\gamma _0^2=72$ as given in reference [@f13].
Now using Eq. (\[13\]) and Eqs. (\[20\]) and (\[21\]), we can express all the elastic or quasi-elastic scattering amplitudes in terms of the Regge amplitudes $F_P$ and $F_\rho $. These amplitudes are given in TableII.
We are now in a position to discuss the rescattering corrections to the decay amplitudes. From Eq. (\[03\]), the two particle unitarity gives [@f11; @f13].
$$\begin{aligned}
DiscA(\bar{B}^0 &\rightarrow &\bar{K}^0D^0)=\frac 1{32\pi }\frac{\left| \vec{%
p}\right| }s\int d\Omega M^{*}(\bar{K}^0D^0\rightarrow K^{-}D^{+})A(\bar{B}%
^0\rightarrow K^{-}D^{+}) \nonumber \\
&\approx &\frac 1{16\pi s}\int_{-2\left| \vec{p}\right| ^2}^0dtM^{*}(\bar{K}%
^0D^0\rightarrow K^{-}D^{+})A(\bar{B}^0\rightarrow K^{-}D^{+})\end{aligned}$$
where we have put $\left| \vec{p}\right| \simeq \frac 12\sqrt{s}$. Now using TableII and Eqs. (\[20\]) and (\[23\]), we get $$\begin{aligned}
DiscA(\bar{B}^0 &\rightarrow &\bar{K}^0D^0)=\gamma _0^2\frac 1{16\pi s}\frac{%
A(\bar{B}^0\rightarrow K^{-}D^{+})}{\sin \pi \alpha _0}\times \left(
s/s_0\right) ^{\alpha _0}\int_{-2\left| \vec{p}\right| ^2}^0dte^{t\acute{%
\alpha}\ln s/s_0} \nonumber \\
&=&\frac{\gamma _0^2}{16\pi }\left( s/s_0\right) ^{\alpha _0-1}\frac 1{\ln
(s/s_0)}A(\bar{B}^0\rightarrow K^{-}D^{+}) \label{24}\end{aligned}$$ where in evaluating the integral in Eq. (\[24\]), we have put $\sin \pi
\alpha (t)=\sin \pi \alpha _0=\sin \frac \pi 2=1$ and $\acute{\alpha}s_0=1$ i.e. $s_0=1$GeV$^2.$
We now use dispersion relation [@f11; @f13; @f14] to obtain $$A(\bar{B}^0\rightarrow \bar{K}^0D^0)_{FSI}=\frac{\gamma _0^2}{16\pi }\frac{A(%
\bar{B}^0\rightarrow K^{-}D^{+})}{\ln (m_B^2/s_0)}\frac{\sqrt{s_0}}{m_B}%
\frac 1\pi \int_{\left( m_D+m_K\right) ^2}^\infty \left( s/s_0\right)
^{\alpha _0-1}\frac{ds}{s-m_B^2} \label{25}$$ where in $\ln (s/s_0)$, we have put $s=m_B^2$. Noting that $\alpha _0\approx
1/2$, we get $$\begin{aligned}
A(\bar{B}^0 &\rightarrow &\bar{K}^0D^0)_{FSI}=\frac{\gamma _0^2}{16\pi }%
\frac 1{\ln (m_B^2/s_0)}\frac{\sqrt{s_0}}{m_B}\frac 1\pi \left[ i\pi +\ln
\frac{1+x}{1-x}\right] A(\bar{B}^0\rightarrow K^{-}D^{+}) \nonumber \\
&\equiv &\epsilon e^{i\theta }A(\bar{B}^0\rightarrow K^{-}D^{+}) \label{26}\end{aligned}$$ where $$\begin{aligned}
x &=&\frac{m_D+m_K}{m_B}\simeq 0\cdot 447,m_B=5\cdot 279 \nonumber \\
\epsilon &=&\frac{\gamma _0^2}{16\pi }\frac 1{\ln (m_B^2/s_0)}\frac{\sqrt{s_0%
}}{m_B}\left[ 1+\frac 1{\pi ^2}\left( \ln \frac{1+x}{1-x}\right) ^2\right]
^{1/2} \nonumber \\
&=&\gamma _0^2\left[ 1\cdot 18\times 10^{-3}\right] \simeq 0\cdot 08
\nonumber \\
\theta &=&\tan ^{-1}\left[ \frac \pi {\ln \frac{1+x}{1-x}}\right] \simeq 73^0
\label{27}\end{aligned}$$ >From Eqs. (\[61\], \[07\]) and TableII, following the same procedure, we can eaisly calculate the rescattering corrections for other decays. Hence after taking into account rescattering corrections to the decay amplitudes, we get
\[28\] $$\begin{aligned}
A_{00} &=&a_{00}e^{i\delta _{00}}+\epsilon e^{i\theta }a_{-+}e^{i\delta
_{-+}} \label{28a} \\
A_{-+} &=&a_{-+}e^{i\delta _{-+}} \label{28b} \\
A_{-0} &=&a_{-0}e^{i\delta _{-0}}+\epsilon e^{i\theta }a_{-+}e^{i\delta
_{-+}} \label{28c} \\
B_{-s^{+}} &=&b_se^{i\delta _s} \label{28d} \\
B_{-+} &=&b_{1/2}e^{i\delta _{1/2}}+\frac 12(1+i)\epsilon e^{i\theta
}b_se^{i\delta _s} \label{28e}\end{aligned}$$
\[29\] $$\begin{aligned}
\bar{A}_{00} &=&(\bar{a}_{00}e^{i\bar{\delta}_{00}}+\epsilon e^{i\theta }%
\bar{a}e^{i\bar{\delta}})e^{i\gamma } \label{29a} \\
\bar{A}_{-0} &=&(\bar{a}_{-0}e^{i\bar{\delta}_{-0}}+\frac 12\epsilon (1-%
\frac i3)\bar{a}e^{i\bar{\delta}})e^{i\gamma } \label{29b} \\
\bar{A}_{0-} &=&(\bar{a}_{0-}e^{i\bar{\delta}_{0-}}-\frac 12\epsilon (1+%
\frac i3)\bar{a}e^{i\bar{\delta}})e^{i\gamma } \label{29c} \\
B_{+s^{-}} &=&\bar{b}_se^{i\bar{\delta}_s} \label{29d} \\
\bar{B}_{-+} &=&\bar{b}_{1/2}e^{i\bar{\delta}_{1/2}}+\frac 12(1+i)\epsilon
e^{i\theta }\bar{b}_se^{i\bar{\delta}_s} \label{29e}\end{aligned}$$ The phase factor $i$ arises due to the phase factor $e^{i\pi \alpha (t)}$ ($%
\bar{F}_\rho =\ e^{i\pi \alpha (t)}F)$ $[$we can also write $(1+i)\frac 12=%
\frac 1{\sqrt{2}}e^{i\frac \pi 4},$ $(1\pm \frac i3)=\frac{\sqrt{10}}3e^{\pm
i\phi },$ $\phi =\tan ^{-1}1/3=18^0]$
Strong interaction phase shifts
===============================
For the $s$-wave scattering, the $l=0$ partial wave scattering amplitude $f$ is given by
$$f=\frac 1{16\pi s}\int_{-4\vec{p}^2}^0M(s,t)dt \label{30}$$
Hence from Eqs. (\[20\]) and (\[21\]), we get for $\sqrt{s}=5\cdot 279$GeV, $$\begin{aligned}
f_P &=&\frac 1{16\pi s}\frac{iC_P}b\left( \frac s{s_0}\right) \simeq 0.12i
\label{31} \\
f_\rho &=&\frac{\gamma _0^2}{16\pi }\frac 1{\ln \left( \frac s{s_0}\right) }%
\left( \frac s{s_0}\right) ^{1/2}\simeq 0.08 \label{32} \\
\bar{f}_\rho &=&\frac{\gamma _0^2}{16\pi }(-i)\frac 1{\ln \left( \frac s{s_0}%
\right) -i\pi }\left( \frac s{s_0}\right) ^{1/2}\simeq 0.04+0.04i \label{33}\end{aligned}$$ Using Table II and Eqs. (\[31\]-\[33\]), we can determine the $s$-wave scattering amplitude $f$ and $S$-matrix, $S=1+2if\equiv \eta e^{2i\Delta }$ for each individual scattering process. They are given in Table III.
Now, using Eq. (\[05\]) we get $$%TCIMACRO{\func{Re} }
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
A_f\left( 1-\eta e^{-2i\Delta }\right) -i%
%TCIMACRO{\func{Im} }
%BeginExpansion
\mathop{\rm Im}%
%EndExpansion
A_f\left( 1+\eta e^{2i\Delta }\right) =\frac 1i\sum_{n\neq f}A_nS_{nf}^{*}
\label{34}$$ Taking the absolute square of Eq. (\[34\]), we obtain, writing $A_f=\left|
A_f\right| e^{i\delta _f}$: $$\left| A_f\right| ^2\left[ \left( 1+\eta ^2\right) -2\eta \cos 2\left(
\delta _f-\theta \right) \right] =\sum_{n,\acute{n}\neq f}A_nS_{nf}^{*}A_{%
\acute{n}}^{*}S_{\acute{n}f} \label{35}$$ Using the random phase approximation of reference[@r2]: $$\sum_{n,\acute{n}\neq f}A_nS_{nf}^{*}A_{\acute{n}}^{*}S_{\acute{n}%
f}=\sum_{n\neq f}\left| A_n\right| ^2\left| S_{nf}\right| ^2=\overline{%
\left| A_n\right| }^2\left( 1-\eta ^2\right) \label{36}$$ we obtain from Eqs. (\[35\]) and (\[36\]):
\[37\] $$\begin{aligned}
\tan ^2\left( \delta _f-\Delta \right) &=&\left( \frac{1-\eta }{1+\eta }%
\right) \frac{\left[ \rho ^2-\left( \frac{1-\eta }{1+\eta }\right) \right] }{%
\left[ 1-\rho ^2\left( \frac{1-\eta }{1+\eta }\right) \right] } \label{37a}
\\
\rho ^2 &=&\frac{\overline{\left| A_n\right| }^2}{\left| A_f\right| ^2},%
\frac{1-\eta }{1+\eta }\leq \rho ^2\leq \frac{1+\eta }{1-\eta } \label{37b}\end{aligned}$$ Except for the parameter $\rho $, everything is known from the TableIII. For the moment let us evaluate the final state phase shifts $\delta _f$ from Eq. (\[37\]) for three values of $\rho ^2$ viz $\rho ^2=\frac{\left( 1-\eta
\right) }{\left( 1+\eta \right) },$ $0\cdot 25$ and$1$. These phase shifts are tabulated in Table IV for various decays.
The following remarks are in order. Irrespective of parameter $\rho $, it follows from the above analysis that
$$\begin{aligned}
\bar{\delta}_s &=&\delta _s \nonumber \\
\bar{\delta}_{1/2} &=&\delta _{1/2} \label{39}\end{aligned}$$
It is reasonable to assume that $\rho $ is of the same order for all the favored decays; it may be different for the suppressed decays but within a set it does not differ much. Under this assumption it follows from Table IV $$\begin{aligned}
\bar{\delta} &=&\delta _{-+} \nonumber \\
\bar{\delta}_{-0} &=&\bar{\delta}_{0-} \nonumber \\
&&\bar{\delta}_{00=}\delta _{00} \label{40}\end{aligned}$$ Further if we take the point of view consistent with that of reference [@f8] i.e. for the decay amplitudes dominated by tree graphs the final state interactions are negligible, ($\rho $: small) then we may conclude $$\begin{aligned}
\delta _{-+} &=&\bar{\delta}\leq 7^0 \nonumber \\
\delta _{-0} &\leq &13^0 \nonumber \\
\delta _s &=&\bar{\delta}_s\leq 10^0 \label{41}\end{aligned}$$ To proceed further, we first consider the case (A): For the color suppressed decays, since we have already taken into account the rescattering corrections c.f. Eqs. (\[28\]) and (\[29\]), it is reasonable to take the same value of $\rho $ for all the decays ($\rho \ll 1$). Then from Table IV, we have $$\begin{aligned}
\bar{\delta}_{00=}\delta _{00} &=&\delta _{-+}\leq 7^0 \nonumber \\
\bar{\delta}_{-0} &=&\bar{\delta}_{0-}\leq 10^0 \nonumber \\
\bar{\delta}_{1/2} &=&\delta _{1/2}\leq 10^0 \label{42}\end{aligned}$$ For the case (B): The phase shifts for color favored decays will be small as given in Eq. (\[41\]); but for the color suppressed decays $\rho $ would be of the order $1$ and the phase shifts for these decays will be in the range of $20^0$ as given in Table IV. For this case, in Eqs. (\[28\]) and (\[29\]), $\epsilon $ will be taken to zero as the effect of rescattering is supposed to be reflected in the parameter $\rho =1$.
Observational Consequences of final state interactions (FSI)
============================================================
We note that the decay amplitudes after taking into account FSI are given in Eqs. (\[28\]) and (\[29\]). From these equations, we obtain (neglecting terms of order $\epsilon ^2$)
\[43\] $$\begin{aligned}
a_{-0}^2 &=&a_{-+}^2+a_{00}^2+2a_{-+}a_{00}\cos \left( \delta _{-+}-\delta
_{00}\right) \label{43a} \\
\Gamma \left( B^{-}\rightarrow K^{-}D^0\right) &=&a_{-0}^2+2\epsilon
a_{-0}a_{-+}\cos \left( \theta +\delta _{-+}-\delta _{-0}\right) \label{43b}
\\
\Gamma \left( \bar{B}^0\rightarrow \bar{K}^0D^0\right) &=&a_{00}^2+2\epsilon
a_{00}a_{-+}\cos \left( \theta +\delta _{-+}-\delta _{00}\right) \label{43c}
\\
\Gamma \left( \bar{B}^0\rightarrow K^{-}D^{+}\right) &=&a_{-+}^2 \label{43d}\end{aligned}$$
\[44\] $$\begin{aligned}
\bar{a}_{-0}^2 &=&\bar{a}_{00}^2+\bar{a}_{0-}^2+2\bar{a}_{00}\bar{a}%
_{0-}\cos \left( \bar{\delta}_{00}-\bar{\delta}_{0-}\right) \label{44a} \\
\Gamma \left( B^{-}\rightarrow K^{-}\bar{D}^0\right) &\simeq &\bar{a}_{-0}^2+%
\frac{\sqrt{10}}3\epsilon \bar{a}_{-0}\bar{a}\cos \left( \theta -\phi +\bar{%
\delta}-\bar{\delta}_{-0}\right) \label{44b} \\
\Gamma \left( B^{-}\rightarrow \bar{K}^0D^{-}\right) &\simeq &\bar{a}_{0-}^2-%
\frac{\sqrt{10}}3\epsilon \bar{a}_{0-}\bar{a}\cos \left( \theta +\phi +\bar{%
\delta}-\bar{\delta}_{0-}\right) \label{44c} \\
\Gamma \left( \bar{B}^0\rightarrow \bar{K}^0\bar{D}^0\right) &\simeq &\bar{a}%
_{00}^2+2\epsilon \bar{a}_{00}\bar{a}\cos \left( \theta +\bar{\delta}-\bar{%
\delta}_{00}\right) \label{44d} \\
\Gamma \left( B^{-}\rightarrow \pi ^0D_s^{-}\right) &=&\bar{a}^2 \label{44e}\end{aligned}$$ First we note from Eqs. (\[28\]), (\[29\]), (\[43a\]) (\[44a\]) that in the absence of rescattering the amplitudes $\left| A_{-0}\right| $, $%
\left| A_{-+}\right| $, $\left| A_{00}\right| $, and $\left| \bar{A}%
_{-0}\right| $, $\left| \bar{A}_{00}\right| $, $\left| \bar{A}_{0-}\right| $ will form two closed triangles. Any deviation from triangular relations would indicate rescattering.
Now using the factorization(see TableI) and Eqs. (\[08\]–\[101\]), we obtain $a_{00}/a_{-+}=C/T\approx 0.72\,(a_2/a_1)$, $\bar{a}_{00}/\bar{a}=%
\bar{C}/\bar{T}\approx (a_2/a_1)$, $\bar{a}_{0-}/\bar{a}=\bar{A}/\bar{T}%
\approx 0.08$ and noting that $\epsilon =0.08$, $a_2/a_1\sim \lambda =0\cdot
22$, $\theta =73^0$ and $\phi =18^0$, we note that FSI corrections are in the range of $15-20\%$, except for $B^{-}\rightarrow \bar{K}^0D^{-}$, (c.f. Eq. (\[43c\]) where it is almost zero, since $\theta +\phi \simeq 90^0$. However we note that the effect of FSI is of considerable importance for the decays $\bar{B}_s^0\rightarrow \pi ^{-}D^{+}$ and $\bar{B}%
_s^0\rightarrow \pi ^{+}D^{-}$ as both these decays in the absence of FSI are extremely suppressed as these decays occur through $W$-exchange diagram (c.f. TableI). Taking into account FSI, we obtain from Eqs. (\[28\]), (\[29\]) and (\[09\])
$$\begin{aligned}
\frac{\Gamma \left( \bar{B}_s^0\rightarrow \pi ^{-}D^{+}\right) }{\Gamma
\left( \bar{B}_s^0\rightarrow K^{-}D_s^{+}\right) } &=&\frac{b_{1/2}^2}{b_s^2%
}\left[ 1+\sqrt{2}\epsilon \frac{b_s}{b_{1/2}}\cos \left( \theta +\frac \pi 4%
+\delta _s-\delta _{1/2}\right) +\frac 12\epsilon ^2\frac{b_s^2}{b_{1/2}^2}%
\right] \nonumber \\
&\simeq &\sqrt{2}\epsilon \left( E/T\right) \cos \left( \theta +\frac \pi 4%
+\delta _s-\delta _{1/2}\right) +\frac 12\epsilon ^2 \nonumber \\
&\simeq &2.3\times 10^{-3}=\frac{\Gamma \left( \bar{B}_s^0\rightarrow \pi
^{+}D^{-}\right) }{\Gamma \left( \bar{B}_s^0\rightarrow K^{+}D_s^{-}\right) }
\label{45}\end{aligned}$$
In the absence of FSI, this ratio has the value $3\cdot 1\times 10^{-4}$.
We now discuss the effect of FSI on CP-asymmetry. We define $$\begin{aligned}
{\cal A}_{\mp } &=&\frac{\Gamma \left( B^{-}\rightarrow K^{-}D_{\mp
}^0\right) -\Gamma \left( B^{+}\rightarrow K^{+}D_{\mp }^0\right) }{\Gamma
\left( \bar{B}^0\rightarrow K^{-}D^{+}\right) } \nonumber \\
&=&\pm 2\sin \gamma \left[
%TCIMACRO{\func{Re} }
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
A_{-0}%
%TCIMACRO{\func{Im} }
%BeginExpansion
\mathop{\rm Im}%
%EndExpansion
\bar{A}_{-0}-%
%TCIMACRO{\func{Im} }
%BeginExpansion
\mathop{\rm Im}%
%EndExpansion
A_{-0}%
%TCIMACRO{\func{Re} }
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
\bar{A}_{-0}\right] /\left| A_{-+}\right| ^2 \label{46} \\
{\cal R}_{\mp } &=&\frac{\Gamma \left( B^{-}\rightarrow K^{-}D_{\mp
}^0\right) +\Gamma \left( B^{+}\rightarrow K^{+}D_{\mp }^0\right) }{\Gamma
\left( \bar{B}^0\rightarrow K^{-}D^{+}\right) } \nonumber \\
&=&\left\{ \left| A_{-0}\right| ^2+\left| \bar{A}_{-0}\right| ^2\mp 2\cos
\gamma \left[
%TCIMACRO{\func{Re} }
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
A_{-0}%
%TCIMACRO{\limfunc{Re} }
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
\bar{A}_{-0}+%
%TCIMACRO{\func{Im} }
%BeginExpansion
\mathop{\rm Im}%
%EndExpansion
A_{-0}%
%TCIMACRO{\func{Im} }
%BeginExpansion
\mathop{\rm Im}%
%EndExpansion
\bar{A}_{-0}\right] \right\} /\left| A_{-+}\right| ^2 \label{47}\end{aligned}$$ where $D_{\mp }^0=\frac{\left( D^0\mp \bar{D}^0\right) }{\sqrt{2}}$ and weak phase $e^{i\gamma }$ has been taken out. For $\bar{B}^0$ and $B^0$ decays, $%
{\cal A}_{\mp }^0$, ${\cal R}_{\mp }^0$ can be obtained by changing $%
A_{-0}\rightarrow A_{00}$ and $\bar{A}_{-0}\rightarrow \bar{A}_{00}$ in Eqs. (\[46\], \[47\]). Then using Eqs. (\[28\]) and (\[29\]), we get
\[48\] $$\begin{aligned}
{\cal A}_{\mp } &=&\pm 2\sin \gamma \left[ -f\bar{r}\sin (\delta _{-0}-\bar{%
\delta}_{-0})-\epsilon \bar{r}\sin (\theta +\delta _{-+}-\bar{\delta}_{-0})+%
\frac{\sqrt{10}}6\epsilon f\bar{f}\sin (\theta -\phi +\bar{\delta}-\delta
_{-0})\right] \label{48a} \\
{\cal R}_{-}-{\cal R}_{+} &=&\mp 4\cos \gamma \left[ f\bar{r}\cos (\delta
_{-0}-\bar{\delta}_{-0})+\epsilon \bar{r}\cos (\theta +\delta _{-+}-\bar{%
\delta}_{-0})+\frac{\sqrt{10}}6\epsilon f\bar{f}\cos (\theta -\phi +\bar{%
\delta}-\delta _{-0})\right] \label{48b}\end{aligned}$$
\[49\] $$\begin{aligned}
{\cal A}_{\mp }^0 &=&\pm 2\sin \gamma \left[ -r_0\bar{r}_0\sin (\delta _{00}-%
\bar{\delta}_{00})+\epsilon \bar{f}r_0\sin (\theta +\bar{\delta}-\delta
_{00})-\epsilon \bar{r}_0\sin (\theta +\delta _{-+}-\bar{\delta}%
_{00})+\epsilon ^2\bar{f}\sin (\bar{\delta}-\delta _{-+})\right] \label{49a}
\\
{\cal R}_{-}^0-{\cal R}_{+}^0 &=&\mp 4\cos \gamma \left[ r_0\bar{r}_0\cos
(\delta _{00}-\bar{\delta}_{00})+\epsilon \bar{f}r_0\cos (\theta +\bar{\delta%
}-\delta _{00})+\epsilon \bar{r}_0\cos (\theta +\delta _{-+}-\bar{\delta}%
_{00})\right] \simeq \mp 4\cos \gamma \left[ r_0\bar{r}_0\cos (\delta _{00}-%
\bar{\delta}_{00})\right] \label{49b}\end{aligned}$$ where
$$\begin{aligned}
f &=&\frac{a_{-0}}{a_{-+}}=\left( 1+C/T\right) \simeq 1.22 \nonumber \\
\bar{r} &=&\frac{\bar{a}_{-0}}{a_{-+}}=\left( \frac{\bar{C}+\bar{A}}{\bar{T}}%
\right) \frac{\bar{T}}T=\sqrt{\rho ^2+\eta ^2}\left( 0.72\right) \times 0.30
\nonumber \\
r_0 &=&\frac{a_{00}}{a_{-+}}=\frac CT=\left( a_2/a_1\right) \nonumber \\
\bar{r}_0 &=&\frac{\bar{a}_{00}}{a_{-+}}=\frac{\bar{C}}{\bar{T}}\left( \frac{%
\bar{T}}T\right) =\sqrt{\rho ^2+\eta ^2}\left( a_2/a_1\right) \left(
0.72\right) \nonumber \\
\bar{f} &=&\frac{\bar{a}}{a_{-+}}=\frac{\bar{T}}T=\sqrt{\rho ^2+\eta ^2}%
\left( 0.72\right) \label{50}\end{aligned}$$
As is clear from Eqs. (\[48a\]) and (\[49a\]), the FSI corrections tend to cancel each other; in ${\cal A}_{\mp }^0$ the cancellation is almost complete and one gets ${\cal A}_{\mp }^0\simeq 0$, where as for ${\cal A}%
_{\mp }$ one gets the value $\left( 1\times 10^{-3}\right) \sin \gamma $. >From Eqs. (\[48b\]) and (\[49b\]), using Eq. (\[50\]) and phase shifts from TableIV (cf first column), we get $$\begin{aligned}
{\cal R}_{-}-{\cal R}_{+} &\simeq &\left( 0.42\right) \cos \gamma \nonumber
\\
{\cal R}_{-}^0-{\cal R}_{+}^0 &\simeq &\left( 0.23\right) \cos \gamma
\label{51}\end{aligned}$$ Let us now discuss the direct CP-violation for $B_s$-decay. Defining $B_{\mp
}^0=\frac 1{\sqrt{2}}\left( B_s^0\mp \bar{B}_s^0\right) $, we get using Eqs. (\[28d\]) and (\[29e\])
\[11\] $$\begin{aligned}
\frac{2\left| A\left( B_{\mp }\rightarrow K^{+}D_s^{-}\right) \right|
^2-b_s^2-\bar{b}_s^2}{2b_s\bar{b}_s} &=&\mp \left[ \cos \left( \gamma
-\delta _s+\bar{\delta}_s\right) \right] \label{52a} \\
\frac{2\left| A\left( B_{\mp }\rightarrow K^{-}D_s^{+}\right) \right|
^2-b_s^2-\bar{b}_s^2}{2b_s\bar{b}_s} &=&\mp \left[ \cos \left( \gamma
+\delta _s-\bar{\delta}_s\right) \right] \label{52b}\end{aligned}$$ Since $\delta _s=\bar{\delta}_s$, it implies
$$\Gamma \left( B_{\mp }\rightarrow K^{+}D_{s}^{-}\right) =\Gamma \left(
B_{\mp }\rightarrow K^{-}D_{s}^{+}\right) \label{53}$$
Our result $\delta _{s}=\bar{\delta}_{s}$, has important implication for determining the phase $\gamma $ discussed in reference [@f15].
Finally we discuss the time dependent analysis of $B$ decays to get information about weak phase $\gamma $. Following the well known procedure [@f16], [@f17] the time dependent decay rate for $B^0(t)$ and $\bar{B%
}^0(t)$ are given by $$\begin{aligned}
{\cal A}(t) &\equiv &\frac{[\Gamma _f(t)+\Gamma _{\bar{f}}(t)]-[\bar{\Gamma}%
_f(t)+\bar{\Gamma}_{\bar{f}}(t)]}{\Gamma _f(t)+\bar{\Gamma}_f(t)} \nonumber
\label{57} \\
&\simeq &-2\frac{\sin (\Delta m_Bt)\sin (2\beta +\gamma )\times
[\left\langle f\left| H\right| B^0\right\rangle ^{*}\left\langle f\left|
H\right| \bar{B}^0\right\rangle +\left\langle f\left| H\right| \bar{B}%
^0\right\rangle ^{*}\left\langle f\left| H\right| B^0\right\rangle ]}{\left|
\left\langle f\left| H\right| B^0\right\rangle \right| ^2+\left|
\left\langle f\left| H\right| \bar{B}^0\right\rangle \right| ^2} \label{54}\end{aligned}$$ and $$\begin{aligned}
{\cal F}(t) &=&\frac{[\Gamma _f(t)+\Gamma _{\bar{f}}(t)]-[\bar{\Gamma}%
_f(t)+\Gamma _{\bar{f}}(t)]}{\Gamma _f(t)+\bar{\Gamma}_f(t)} \nonumber \\
&=&\frac{-2}{\left| \left\langle f\left| H\right| B^0\right\rangle \right|
^2+\left| \left\langle f\left| H\right| \bar{B}^0\right\rangle \right| ^2}%
\left[ \cos (\Delta m_Bt)[\left| \left\langle f\left| H\right| \bar{B}%
^0\right\rangle \right| ^2-\left| \left\langle f\left| H\right|
B^0\right\rangle \right| ^2]\right. \nonumber \label{55} \\
&&\left. -i\sin (\Delta m_Bt)\cos (2\beta +\gamma )[\left\langle f\left|
H\right| B^0\right\rangle ^{*}\left\langle f\left| H\right| \bar{B}%
^0\right\rangle -\left\langle f\left| H\right| \bar{B}^0\right\rangle
^{*}\left\langle f\left| H\right| B^0\right\rangle ]\right] \label{55}\end{aligned}$$ where $$\Gamma _f,_{\bar{f}}(t)\equiv \Gamma \left( B^0(t)\rightarrow f,\bar{f}%
\right) ,\bar{\Gamma}_f,_{\bar{f}}(t)\equiv \Gamma \left( \bar{B}%
^0(t)\rightarrow f,\bar{f}\right) \label{56}$$ Taking $f\equiv K_sD^0$ and $\bar{f}\equiv K_s\bar{D}^0$ and using Eqs. (\[29\]) and (\[30\]) we get $${\cal A}(t)=-4\frac{\Gamma \left( \bar{B}^0\rightarrow K^{-}D^{+}\right)
(a_2/a_1)(C/T)\sqrt{\rho ^2+\eta ^2}}{\Gamma \left( \bar{B}^0\rightarrow
\bar{K}^0D^0\right) +\Gamma \left( \bar{B}^0\rightarrow \bar{K}^0\bar{D}%
^0\right) }\left[ \sin (\Delta m_Bt)\sin (2\beta +\gamma )\times Y\right]
\label{57}$$ $$\begin{aligned}
&&{\cal F}(t)+2\cos (\Delta m_Bt)\frac{\Gamma \left( \bar{B}^0\rightarrow
\bar{K}^0D^0\right) -\Gamma \left( \bar{B}^0\rightarrow \bar{K}^0\bar{D}%
^0\right) }{\Gamma \left( \bar{B}^0\rightarrow \bar{K}^0D^0\right) +\Gamma
\left( \bar{B}^0\rightarrow \bar{K}^0\bar{D}^0\right) } \nonumber
\label{58} \\
&=&-4\left[ \frac{\Gamma \left( \bar{B}^0\rightarrow K^{-}D^{+}\right)
(a_2/a_1)(C/T)\sqrt{\rho ^2+\eta ^2}}{\Gamma \left( \bar{B}^0\rightarrow
\bar{K}^0D^0\right) +\Gamma \left( \bar{B}^0\rightarrow \bar{K}^0\bar{D}%
^0\right) }\sin (\Delta m_Bt)\cos (2\beta +\gamma )\times Z\right]
\label{58}\end{aligned}$$ where $$Y=\left[ \cos (\delta _{00}-\bar{\delta}_{00})+\epsilon (a_1/a_2)\cos
(\theta +\bar{\delta}-\delta _{00})+\epsilon (a_1/a_2)\epsilon \cos (\theta
+\delta _{-+}-\bar{\delta}_{00})+\epsilon ^2(a_1/a_2)^2\cos (\bar{\delta}%
-\delta _{-+})\right] \label{59}$$ $$Z=\sin (\delta _{00}-\bar{\delta}_{00})+\epsilon \left( \frac{a_1}{a_2}%
\right) \sin (\theta +\bar{\delta}-\delta _{00})+\epsilon \left( \frac{a_1}{%
a_2}\right) \sin (\theta +\delta _{-+}-\bar{\delta}_{00})+\epsilon ^2\left(
\frac{a_1}{a_2}\right) ^2\sin (\bar{\delta}-\delta _{-+}) \label{60}$$ >From Eqs. (\[57\]) and (\[58\]) it follows that the experimental measurements of ${\cal A}(t)$ and ${\cal F}(t)$ would give $\sin (2\beta
+\gamma )Y$ and $\cos (2\beta +\gamma )Z$. Note that Eqs. (\[57\]) and (\[58\]) directly gives $\tan (2\beta +\gamma )Y/Z$. However, the relations $\bar{\delta}=\delta _{-+},$ $\delta _{00}=\bar{\delta}_{00}$ do not depend on the detail of the model. In this case $$Z=2\epsilon \left( \frac{a_1}{a_2}\right) \sin \theta \approx 0.69
\label{61}$$ $$Y=1+2\epsilon \left( \frac{a_1}{a_2}\right) \cos \theta +\epsilon
^2(a_1/a_2)^2\approx 1.34 \label{62}$$ where, we have used $\epsilon =0.08$, $\theta =73^0$ and $a_2/a_1\approx
0.22 $ to give an order of magnitude for $Y$ and $Z$. It is clear that the dominant contribution to $Z$ comes from the rescattering, where for $Y$, the rescattering corrections are in the range of $33\%$.
For time dependent decays of $B_s^0,$ one can get $${\cal A}_s(t)\equiv \frac{\Gamma _{f_s}(t)-\bar{\Gamma}_{\bar{f}_s}(t)}{%
\Gamma _{f_s}(t)+\bar{\Gamma}_{f_s}(t)}=\frac{b_s\bar{b}_s}{b_s^2+\bar{b}_s^2%
}\sin (\Delta m_{B_s}t)[S+\bar{S}] \label{63}$$ $$\begin{aligned}
{\cal F}_s(t) &\equiv &\frac{[\Gamma _{f_s}(t)+\bar{\Gamma}_{\bar{f}%
_s}(t)]-[\Gamma _{\bar{f}_s}(t)+\bar{\Gamma}_{f_s}(t)]}{\Gamma _{f_s}(t)+%
\bar{\Gamma}_{f_s}(t)} \nonumber \\
&=&2\left[ \frac{(b_s^2-\bar{b}_s^2)\cos (\Delta m_{B_s}t)+b_s\bar{b}_s(S-%
\bar{S})\sin (\Delta m_{B_s}t)}{b_s^2+\bar{b}_s^2}\right] \label{64}\end{aligned}$$ where $f_s=K^{+}D_s^{-}$, $\bar{f}_s=K^{-}D_s^{+}$ and $$\begin{aligned}
S &=&\sin (2\phi _{Ms}+\gamma +\delta _s-\bar{\delta}_s) \nonumber \\
\bar{S} &=&\sin (2\phi _{Ms}+\gamma -\delta _s+\bar{\delta}_s) \label{65}\end{aligned}$$ Since $b_s^2$ and $\bar{b}_s^2$ are given by the decay widths $\Gamma \left(
\bar{B}_s^0\rightarrow K^{-}D_s^{+}\right) $ and $\Gamma \left( \bar{B}%
_s^0\rightarrow K^{+}D_s^{-}\right) $ respectively, it is clear from Eqs. (\[63\]) and (\[64\]) that it is possible to determine $S$ and $\bar{S}$ from the experimental value of ${\cal A}_s(t)$ and ${\cal F}_s(t)$. However for $B_s^0$, $\phi _{Ms}\simeq 0$. But $\delta _s=\bar{\delta}_s$ follows from general arguments (cf. Eq. (\[39\])), it is therefore reasonable to use $\delta _s=\bar{\delta}_s$. In this case $S=\bar{S}$ ,it is then possible to determine $\sin \gamma $, using Eq. (\[63\]).
Finally we give an estimate of the CP asymmetry parameter ${\cal A}(t)$ and $%
{\cal A}_s(t)$, using $\sqrt{\rho ^2+\eta ^2}=0.36$. Then we find from Eqs. (\[59\]), (\[61\]) and (\[63\]) $$\begin{aligned}
{\cal A}(t) &\approx &-4\frac{\sqrt{\rho ^2+\eta ^2}}{1+\rho ^2+\eta ^2}%
\left[ \sin (\Delta m_Bt)\sin (2\beta +\gamma )\times 1.34\right] \\
&\approx &-1.94\sin (\Delta m_Bt)\sin (2\beta +\gamma )\end{aligned}$$ $$\begin{aligned}
{\cal A}_s(t) &\approx &2\frac{\sqrt{\rho ^2+\eta ^2}\bar{T}/T}{1+\left(
\rho ^2+\eta ^2\right) \bar{T}/T}\sin (\Delta m_{B_s}t)\sin \gamma \\
&\approx &0.49\sin (\Delta m_{B_s}t)\sin \gamma\end{aligned}$$
To conclude: The rescattering corrections are of the order of $15-20\%$; except for $\bar{B}_s^0\rightarrow \pi ^{-}D^{+}$ and $\bar{B}%
_s^0\rightarrow \pi ^{+}D^{-}$ where they are greater than the decay amplitude given by $W$-exchange graph. The direct $CP$-asymmetry parameter is of the order of $10^{-3}\sin \gamma $ for $B^{-}\rightarrow K^{-}D_{\mp
}^0$ and $B^{+}\rightarrow K^{+}D_{\mp }^0$ decays as the rescattering correction tend to cancel each other for these decays. But for the time dependent $CP$-asymmetry we get the value $-1.94\sin (2\beta +\gamma )$ for $%
B^0\rightarrow K_sD^0$, $K_s\bar{D}^0$ decays. For $B_s^0\rightarrow K^{\pm
}D_s^{\mp }$ decays our analysis gives the strong phase shifts $\delta _s=%
\bar{\delta}_s$ and we get time dependent $CP$-asymmetry of the order $%
\left( 0\cdot 49\right) \sin \gamma $ which may be used to extract the weak phase $\gamma $ in future experiments. Finally the formalism developed for final state interactions in this paper is also applicable for the $\Delta
S=0,$ $\Delta C=\pm 1$ $B$–decays
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$$\begin{array}{c}
\text{{\bf Table I}: Amplitudes for }\Delta C=\pm 1\text{ and }\Delta S=-1%
\text{ decay modes of }\bar{B}. \\
\Delta C=+1,\Delta S=-1\text{ decays} \\
\begin{tabular}{|c|c|c|}
\hline
Mode & Amplitude & A$_{\text{topoloy}}$ \\ \hline
$B^{-}\rightarrow K^{-}D^{0}$ & $A_{-0}=a_{-0}e^{i\delta _{-0}}=2A_{1}$ & $%
Te^{i\delta _{T}}+Ce^{i\delta _{C}}$ \\ \hline
$\bar{B}^{0}\rightarrow K^{-}D^{+}$ & $A_{-+}=a_{-+}e^{i\delta
_{-+}}=A_{1}+A_{0}$ & $Te^{i\delta _{T}}$ \\ \hline
$\bar{B}^{0}\rightarrow \bar{K}^{0}D^{0}$ & $A_{00}=a_{00}e^{i\delta
_{00}}=A_{1}-A_{0}$ & $Ce^{i\delta _{C}}$ \\ \hline
$\bar{B}_{s}^{0}\rightarrow K^{-}D_{s}^{+}$ & $B_{-s^{+}}=b_{s}e^{i\delta
_{s}}$ & $Te^{i\delta _{T}}+Ee^{i\delta _{E}}$ \\ \hline
$\bar{B}_{s}^{0}\rightarrow \pi ^{-}D^{+}$ & $B_{-+}=b_{1/2}e^{i\delta
_{1/2}}=B_{1/2}$ & $Ee^{i\delta _{E}}$ \\ \hline
$\bar{B}_{s}^{0}\rightarrow \pi ^{0}D^{0}$ & $B_{00}=\frac{1}{\sqrt{2}}%
b_{1/2}e^{i\delta _{1/2}}=\frac{1}{\sqrt{2}}B_{1/2}$ & $\frac{1}{\sqrt{2}}%
Ee^{i\delta _{E}}$ \\ \hline
\end{tabular}
\\
\Delta C=-1,\Delta S=-1\text{ decays} \\
\begin{tabular}{|c|c|c|}
\hline
Mode & Amplitude & A$_{\text{topoloy}}$ \\ \hline
$\bar{B}^{0}\rightarrow \bar{K}^{0}D^{0}$ & $\bar{A}_{00}=\bar{a}%
_{00}e^{i\left( \bar{\delta}_{00}+\gamma \right) }=2\bar{A}_{1}$ & $\bar{C}%
e^{i\left( \bar{\delta}_{C}+\gamma \right) }$ \\ \hline
$B^{-}\rightarrow K^{-}\bar{D}^{0}$ & $\bar{A}_{-0}=\bar{a}_{-0}e^{i\left(
\bar{\delta}_{-0}+\gamma \right) }=\bar{A}_{1}+\bar{A}_{0}$ & $\left( \bar{C}%
e^{i\bar{\delta}_{C}}+\bar{A}e^{i\bar{\delta}_{A}}\right) e^{i\gamma }$ \\
\hline
$\bar{B}^{0}\rightarrow \bar{K}^{0}D^{-}$ & $\bar{A}_{0-}=\bar{a}%
_{0-}e^{i\left( \bar{\delta}_{0-}+\gamma \right) }=-\bar{A}_{1}+\bar{A}_{0}$
& $\bar{A}e^{i\left( \bar{\delta}_{A}+\gamma \right) }$ \\ \hline
$\bar{B}^{0}\rightarrow \pi ^{+}D_{s}^{-}$ & $\bar{A}_{+s^{-}}=\bar{a}%
e^{i\left( \bar{\delta}+\gamma \right) }$ & $\bar{T}e^{i\left( \bar{\delta}%
_{T}+\gamma \right) }$ \\ \hline
$B^{-}\rightarrow \pi ^{0}D_{s}^{-}$ & $\bar{A}_{0s^{-}}=\frac{1}{\sqrt{2}}%
\bar{A}_{+s^{-}}$ & $\frac{1}{\sqrt{2}}\bar{T}e^{i\left( \bar{\delta}%
_{T}+\gamma \right) }$ \\ \hline
$B^{-}\rightarrow \eta _{8}D_{s}^{-}$ & $\bar{A}_{8s^{-}}=\bar{a}%
_{8}e^{i\left( \bar{\delta}_{8}+\gamma \right) }$ & $\left( \frac{1}{\sqrt{6}%
}Te^{i\bar{\delta}_{T}}-\sqrt{\frac{2}{3}}\bar{A}e^{i\bar{\delta}%
_{A}}\right) e^{i\gamma }$ \\ \hline
$\bar{B}_{s}^{0}\rightarrow K^{+}D_{s}^{-}$ & $\bar{B}_{+s^{-}}=\bar{b}%
_{s}e^{i\left( \bar{\delta}_{s}+\gamma \right) }$ & $\left( \bar{T}e^{i\bar{%
\delta}_{T}}+\bar{E}e^{i\bar{\delta}_{E}}\right) e^{i\gamma }$ \\ \hline
$\bar{B}_{s}^{0}\rightarrow \pi ^{+}D^{-}$ & $\bar{B}_{+-}=\bar{b}%
_{1/2}e^{i\left( \bar{\delta}_{1/2}+\gamma \right) }=\bar{B}_{1/2}$ & $\bar{E%
}e^{i\left( \bar{\delta}_{E}+\gamma \right) }$ \\ \hline
$\bar{B}_{s}^{0}\rightarrow \pi ^{0}\bar{D}^{0}$ & $\bar{B}_{00}=\frac{1}{%
\sqrt{2}}\bar{b}_{1/2}e^{i\left( \bar{\delta}_{1/2}+\gamma \right) }=\frac{1%
}{\sqrt{2}}\bar{B}_{1/2}$ & $\frac{1}{\sqrt{2}}\bar{E}e^{i\left( \bar{\delta}%
_{E}+\gamma \right) }$ \\ \hline
\end{tabular}
\end{array}$$
$$\begin{aligned}
\text{{\bf Table II}} &:&\text{{\bf \ }Scattering amplitudes for various
scattering processes as given by SU(3) [Eq. (\ref{13})]. } \\
&&\text{The last column gives the amplitudes in terms of Regge exchanges} \\
&&\left. F_{P}=iC_{P}(s/s_{0})e^{bt},\text{ \quad }F_{\rho }=\gamma _{0}^{2}%
\frac{1}{\sin \pi \alpha (t)}(s/s_{0})^{\alpha (t)},\quad C_{P}\approx 5,%
\text{ \quad }\gamma _{0}^{2}\approx 72\right. \\
&&
\begin{tabular}{|c|c|c|}
\hline
Scattering Prpocesses & Scattering amplitude & $
\begin{array}{c}
\text{Scattering amplitude in terms of Regge amplitudes} \\
F_{P}\text{ and }F_{\rho }\text{ Eqs. (\ref{20}) and (\ref{21}).}
\end{array}
$ \\ \hline
$K^{-}D^{+}\rightarrow K^{-}D^{+}$ & $-\frac{2}{3}F_{2}+F_{3}$ & $F_{P}:\bar{%
K}^{0}D^{0}\rightarrow \bar{K}^{0}D^{0}$ \\ \hline
$K^{-}D^{0}\rightarrow K^{-}D^{0}$ & $F_{1}+\frac{1}{3}F_{2}+F_{3}$ & $%
F_{P}+F_{\rho }$ \\ \hline
$K^{-}D^{+}\rightleftharpoons \bar{K}^{0}D^{0}$ & $F_{1}+F_{2}$ & $F_{\rho }$
\\ \hline
$K^{-}D_{s}^{+}\rightarrow K^{-}D_{s}^{+}$ & $-F_{1}+\frac{1}{3}F_{2}+F_{3}$
& $F_{P}+e^{-i\pi \alpha (t)}F_{\rho }:\pi ^{-}D^{+}\rightarrow \pi
^{-}D^{+} $ \\ \hline
$K^{-}D_{s}^{+}\rightarrow \pi ^{-}D^{+}$ & $-F_{1}+F_{2}$ & $e^{-i\pi
\alpha (t)}F_{\rho }:\sqrt{2}\left( K^{-}D_{s}^{+}\rightarrow \pi
^{0}D^{0}\right) $ \\ \hline
$K^{-}\bar{D}^{0}\rightarrow K^{-}\bar{D}^{0}$ & $\bar{F}_{1}+\frac{1}{3}%
\bar{F}_{2}+\bar{F}_{3}$ & $F_{P}+e^{-i\pi \alpha (t)}F_{\rho }:\bar{K}%
^{0}D^{-}\rightarrow \bar{K}^{0}D^{-}$ \\ \hline
$\bar{K}^{0}D^{-}\rightleftharpoons \bar{K}^{0}D^{-}$ & $\bar{F}_{1}+\bar{F}%
_{2}$ & $e^{-i\pi \alpha (t)}F_{\rho }$ \\ \hline
$\bar{K}^{0}\bar{D}^{0}\rightarrow \bar{K}^{0}\bar{D}^{0}$ & $-\frac{2}{3}%
\bar{F}_{2}+\bar{F}_{3}$ & $F_{P}$ \\ \hline
$\pi ^{+}D_{s}^{-}\rightarrow \pi ^{+}D_{s}^{-}$ & $-\frac{2}{3}\bar{F}_{2}+%
\bar{F}_{3}$ & $F_{P}:\pi ^{0}D_{s}^{-}\rightarrow \pi ^{0}D_{s}^{-}$ \\
\hline
$\eta _{8}D_{s}^{-}\rightarrow \eta _{8}D_{s}^{-}$ & $\frac{2}{3}\bar{F}_{2}+%
\bar{F}_{3}$ & $F_{P}+\left( 1+e^{-i\pi \alpha (t)}\right) F_{\rho }$ \\
\hline
$\pi ^{+}D_{s}^{-}\rightarrow \bar{K}^{0}\bar{D}^{0}$ & $-\bar{F}_{1}+\bar{F}%
_{2}$ & $F_{\rho }$ \\ \hline
$\pi ^{0}D_{s}^{-}\rightarrow K^{-}\bar{D}^{0}$ & $\frac{1}{\sqrt{2}}\left( -%
\bar{F}_{1}+\bar{F}_{2}\right) $ & $\frac{1}{\sqrt{2}}F_{\rho }:-\left( \pi
^{0}D_{s}^{-}\rightarrow \bar{K}^{0}D^{-}\right) $ \\ \hline
$\eta _{8}D_{s}^{-}\rightarrow \bar{K}^{0}D^{-}$ & $-\frac{1}{\sqrt{6}}%
\left( 3\bar{F}_{1}+\bar{F}_{2}\right) $ & $\frac{1}{\sqrt{6}}(1-2e^{-i\pi
\alpha (t)})F_{\rho }:\eta _{8}D_{s}^{-}\rightarrow K^{-}\bar{D}^{0}$ \\
\hline
$K^{+}D_{s}^{-}\rightarrow K^{+}D_{s}^{-}$ & $\bar{F}_{1}+\frac{1}{3}\bar{F}%
_{2}+\bar{F}_{3}$ & $F_{P}+e^{-i\pi \alpha (t)}F_{\rho }:\pi
^{+}D^{-}\rightarrow \pi ^{+}D^{-}$ \\ \hline
$K^{+}D_{s}^{-}\rightarrow \pi ^{+}D^{-}$ & $\bar{F}_{1}+\bar{F}_{2}$ & $%
e^{-i\pi \alpha (t)}F_{\rho }:\sqrt{2}\left( K^{+}D_{s}^{-}\rightarrow \pi
^{0}\bar{D}^{0}\right) $ \\ \hline
\end{tabular}\end{aligned}$$
$$\begin{aligned}
\text{{\bf Table III}} &:&\text{ Partial wave }l=0\text{ scattering
amplitude }f\text{ for elastic scattering; }S=1+2if=\eta e^{2i\Delta } \\
&&
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Scattering process & $f$ & $S$ & $\eta $ & $\Delta $ & $\frac{1-\eta }{%
1+\eta }$ \\ \hline
$K^{-}D^0\rightarrow K^{-}D^0$ & $0.08+0.12i$ & $0.76+0.16i$ & $0.78$ & $6^0$
& $0.12$ \\ \hline
$K^{-}D^{+}\rightarrow K^{-}D^{+}$ & $0.12i$ & $0.76$ & $0.76$ & $0$ & $0.14$
\\ \hline
$\bar{K}^0D^0\rightarrow \bar{K}^0D^0$ & $0.12i$ & $0.76$ & $0.76$ & $0$ & $%
0.14$ \\ \hline
$K^{-}D_s^{+}\rightarrow K^{-}D_s^{+}$ & $0.04+0.16i$ & $0.68+0.08i$ & $0.68$
& $3.5^0$ & $0.19$ \\ \hline
$\pi ^{-}D^{+}\rightarrow \pi ^{-}D^{+}$ & $0.04+0.16i$ & $0.68+0.08i$ & $%
0.68$ & $3.5^0$ & $0.19$ \\ \hline
$\bar{K}^0\bar{D}^0\rightarrow \bar{K}^0\bar{D}^0$ & $0.12i$ & $0.76$ & $%
0.76 $ & $0$ & $0.14$ \\ \hline
$\pi ^{+}D_s^{-}\rightarrow \pi ^{+}D_s^{-}$ & $0.12i$ & $0.76$ & $0.76$ & $%
0 $ & $0.14$ \\ \hline
$\pi ^0D_s^{-}\rightarrow \pi ^0D_s^{-}$ & $0.12i$ & $0.76$ & $0.76$ & $0$ &
$0.14$ \\ \hline
$K^{-}\bar{D}^0\rightarrow K^{-}\bar{D}^0$ & $0.04+0.16i$ & $0.68+0.08i$ & $%
0.68$ & $3.5^0$ & $0.19$ \\ \hline
$\bar{K}^0D^{-}\rightarrow \bar{K}^0D^{-}$ & $0.04+0.16i$ & $0.68+0.08i$ & $%
0.68$ & $3.5^0$ & $0.19$ \\ \hline
$
\begin{array}{c}
K^{+}D_s^{-}\rightarrow K^{+}D_s^{-}, \\
\pi ^{+}D^{-}\rightarrow \pi ^{+}D^{-}
\end{array}
$ & $0.04+0.16i$ & $0.68+0.08i$ & $0.68$ & $3.5^0$ & $0.19$ \\ \hline
\end{tabular}\end{aligned}$$
$$\begin{aligned}
&&\left. \text{{\bf Table IV}: Final state strong interaction phase shift }%
\delta _f\text{ for }\rho =\sqrt{\frac{1-\eta }{1+\eta }},\text{ 0.5 and 1}%
\right. \\
&&
\begin{tabular}{cccc}
phase shift in degree $\rho $ & $\sqrt{\frac{1-\eta }{1+\eta }}$ & $0.5$ & $%
1 $ \\
$\delta _{-0}$ & $6$ & $13$ & $25$ \\
$\delta _{-+}$ & $0$ & $7$ & $20$ \\
$\delta _{00}$ & $0$ & $7$ & $20$ \\
$\delta _s$ & $3$ & $10$ & $23$ \\
$\delta _{1/2}$ & $3$ & $10$ & $23$ \\
$\bar{\delta}_{00}$ & $0$ & $7$ & $20$ \\
$\bar{\delta}$ & $0$ & $7$ & $20$ \\
$\bar{\delta}_{-0}$ & $3$ & $10$ & $23$ \\
$\bar{\delta}_{0-}$ & $3$ & $10$ & $23$ \\
$\bar{\delta}_s$ & $3$ & $10$ & $23$ \\
$\bar{\delta}_{1/2}$ & $3$ & $10$ & $23$%
\end{tabular}\end{aligned}$$
|
---
abstract: 'We study the asymptotic behavior of the Schrödinger equation in the presence of a nonlinearity of Hartree type in the semi-classical regime. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution without altering the rapid oscillations. We show the validity of the WKB-analysis when the potential in the nonlinearity is singular around the origin. No new resonant wave is created in our model, this phenomenon is inhibited due to the nonlinearity. The nonlocal nature of this latter leads us to build our result on a high-frequency averaging effects. In the proof we make use of the Wiener algebra and the space of square-integrable functions.'
address: |
Univ. Montpellier 2\
Mathématiques, UMR 5149\
CC 051\
34095 Montpellier\
France
author:
- Lounes MOUZAOUI
title: '**High-frequency averaging in the semi-classical singular Hartree equation**'
---
[^1]
Introduction
============
In this paper we are interested in studying of the Schrödinger equation in the presence of a *Hartree type nonlinearity*
$$\label{l'equa2}
i{\varepsilon}\partial_{t} u^{{\varepsilon}} +
\frac{{\varepsilon}^{2}}{2} \Delta u^{{\varepsilon}} =
{\varepsilon}\lambda (K*{|u^{{\varepsilon}}|}^{2})u^{{\varepsilon}},$$
with ${\varepsilon}>0$, $u^{{\varepsilon}}\in \mathcal{S}'(I \times \mathbb{R}^{d},
\mathbb{C}), d \in \{1,2,3\}, I $ interval of $\mathbb{R}$, $\lambda \in {\mathbb{R}}, \mbox{and } K(x) = \frac{1}{|x|^{\gamma}}$ with $0<\gamma<d$. This model is of physical application. It appears in many physical phenomena like in describing superfluids (see [@Berloff99]). The presence of ${\varepsilon}$ in this equation is to show the microscopic/macroscopic scale ratio. For small ${\varepsilon}>0$, the scaling of $\eqref{l'equa2}$ corresponds to the *semi-classical* regime, i.e. the regime of *high-frequency* solutions $u^{{\varepsilon}}(t,x)$. This system contains the Schrödinger-Poisson system ($d=3, \gamma=1$), a model which is used in studying the quantum transport in semi-conductor devices (see [@MR1387456; @semi-cond-ref; @Sconducteur2]). Many authors have interested to study the asymptotic behavior of in this case like in where it is made use of Wigner measure techniques. Several papers like [@MR2290279; @art7] are devoted to study the strong asymptotic limit of the solutions of with Hartree nonlinearity in the case of a *single-phase* WKB initial data in the form $$u^{{\varepsilon}}_{0}(x)=\alpha^{{\varepsilon}}(x)e^{i\varphi(x)/{\varepsilon}},$$ with $\varphi(x)\in {\mathbb{C}}$ ${\varepsilon}$-independent and $\alpha^{{\varepsilon}}(x) \in {\mathbb{R}}.$
Due to the small parameter ${\varepsilon}$ in front of the nonlinearity, we consider a *weakly nonlinear regime.* This means that the nonlinearity does not affect the geometry of the propagation (see Section 3), and technically means that it does not show up in the eikonal equation, but only in the transport equations determining the amplitudes. The object of this paper is to construct an approximate solution $u^{\varepsilon}_{app}$ for the exact solution $u^{\varepsilon}$ of subject to an initial data of WKB type, given by a superposition of ${\varepsilon}$-oscillatory plane waves, i.e.
$$\label{cond_init}
\left.u^{{\varepsilon}}_{app}\right|_{t=0}(x)=u^{{\varepsilon}}_{0}(x)=\sum_{j\in {\mathbb{N}}} \alpha_{j}(x)e^{i\kappa_{j}\cdot x/\varepsilon}.$$
We begin the paper by showing the existence of an exact solution to in $L^{2}({\mathbb{R}}^{d})\cap W({\mathbb{R}}^{d})$ where $W({\mathbb{R}}^{d})$ denotes the *Wiener algebra* $$W({\mathbb{R}}^{d}) = \mathcal{F}(L^{1}({\mathbb{R}}^{d}))=\lbrace f \in \mathcal{S}'({\mathbb{R}}^{d},{\mathbb{C}}), \|f\|_{W}:=\|{\widehat}{f}\|_{L^{1}({\mathbb{R}}^{d})} < +\infty\rbrace,$$ and where $$(\mathcal{F}f)(\xi)={\widehat}{f}(\xi)=\frac{1}{(2\pi)^{d/2}} \int_{{\mathbb{R}}^{d}} f(x)e^{-ix\cdot \xi} \, \mathrm{d}x.$$ In the next step, we construct an approximate solution in the form $$u^{{\varepsilon}}_{app}(t,x) = \sum_{j\in {\mathbb{N}}} a_{j}(t,x)e^{i\phi_{j}(t,x)/{\varepsilon}},$$ in $L^{2}({\mathbb{R}}^{d})\cap W({\mathbb{R}}^{d})$ by determining the amplitudes $a_{j}$ and the phases $\phi_{j}$, then we proceed to study its stability to justify our construction.
Fourier transform of the potential $K$ is found to be $${\widehat}{K}(\xi)=\frac{C_{d,\gamma}}{|\xi|^{d-\gamma}},$$ (see \[3, Proposition 1.29\]) so under the general assumptions of [@MR2731651] where the considered kernel $K$ is such that $(1+|\xi|){\widehat}{K}(\xi) \in L^{\infty}({\mathbb{R}}^{d})$ we can not deduce the well posedness of . It is a critical case for our model because we do not know how to construct an exact local solution in $W({\mathbb{R}}^{d})$ due to the singularity of ${\widehat}{K}$ around the origin $({\widehat}{K} \notin L^{\infty})$.
To define the framework of the amplitudes $(a_{j})_{j\in {\mathbb{N}}}$ we introduce the following definitions.
\[def\_espace\] For $d \in \lbrace 1,2,3 \rbrace$ and $0< \gamma < d $ we define $n\in{\mathbb{N}}$ as follows $$n = \left\{
\begin{array}{ll}
2 & \mbox{if }\, d=1 \mbox{ or } 2 \\
2 & \mbox{if }\, d=3 \mbox{ and } \gamma \in [1,3[\\
3 & \mbox{if }\, d=3 \mbox{ and } \gamma \in ]0,1[.\\
\end{array}
\right.$$ We define the space $$Y({\mathbb{R}}^{d}):=\lbrace f \in L^{2}\cap W, \partial^{\eta}f \in L^{2}\cap W,
\forall \eta \in {\mathbb{N}}^{d}, |\eta| {\leqslant}n\rbrace,$$ equipped with the norm $$\|f\|_{Y({\mathbb{R}}^{d})} = \sum_{0{\leqslant}|\eta| {\leqslant}n} \|\partial^{\eta}f\|_{L^{2}\cap W}.$$ We set $$E({\mathbb{R}}^{d})=\lbrace a=(a_{j})_{j\in {\mathbb{N}}}\; | \;(a_{j})_{j\in {\mathbb{N}}} \in \ell^{1}({\mathbb{N}}, Y)\rbrace,$$ which is a Banach space when it is equipped with the norm $$\|a\|_{E({\mathbb{R}}^{d})}=\sum_{j\in {\mathbb{N}}} \|a_{j}\|_{Y}.$$
Now, we can state the main theorem of this work.
\[mainTh\] For $d \in \lbrace 1,2,3\rbrace$ and $0<\gamma <d$, consider the Cauchy problem , subject to initial data $u^{{\varepsilon}}_{0}$ of the form , with initial amplitudes $(\alpha_{j})_{j\in {\mathbb{N}}} \in E({\mathbb{R}}^{d})$. We assume that there exists $\delta >0$ such that
$$\inf\lbrace|\kappa_{k}-\kappa_{m}|, k,m
\in {\mathbb{N}}, k \neq m\rbrace \geqslant \delta > 0.$$
Then, for all $T >0$ there exists $C, {\varepsilon}_{0}(T) >0$, such that for any $ {\varepsilon}\in ]0, {\varepsilon}_{0}]$, the exact solution to satisfies $u^{{\varepsilon}} \in L^{\infty}([0,T];L^{2}\cap W)$ and can be approximated by
$$\|u^{{\varepsilon}}- u^{{\varepsilon}}_{app}\|_{L^{\infty}([0,T];L^{2}\cap W)} {\leqslant}C{\varepsilon}^{\beta},$$ where $ \beta=\min\lbrace 1,d-\gamma \rbrace $ and where $u^{{\varepsilon}}_{app}$ is defined by $$u^{{\varepsilon}}_{app}(t,x)=\sum_{j\in {\mathbb{N}}} \alpha_{j}(x-t\kappa_{j})e^{iS_{j}(t,x)}e^{i\kappa_{j}\cdot x /{\varepsilon}- i|\kappa_{j}|^{2}/ 2{\varepsilon}},$$ with $S_{j} \in {\mathbb{R}}$ defined in .
In particular, for $d=3$ and $\gamma=1$ we obtain an approximation result for Schrödinger-Poisson equation with $\beta=1$. This confirms in a general way what was guessed in for the Schrödinger-Poisson system. An important feature of our justification consists in the absence of phenomena of phase resonances like we can see later.
Existence and uniqueness
========================
We consider the following Cauchy problem
$$\label{eq:(SP')}
i\partial_{t} u +
\frac{1}{2}\Delta u =
\lambda (K*{|u|}^{2})u,$$
subject to an initial data $u_{0} \in L^{2}({\mathbb{R}}^{d}) \cap W({\mathbb{R}}^{d})$, with $K(x)= \frac{1}{|x|^{\gamma}}, 0<\gamma <d$. For $0 <\gamma<d$, the Fourier transform of $K$ for $\xi \in {\mathbb{R}}^{d}$ is $${\widehat}{K}(\xi)= \frac{C_{d,\gamma}}{|\xi|^{d-\gamma}}.$$ In the following we denote $\mathcal{K}_{1}= {\widehat}{K}(\xi) \mathds{1}_{[|\xi| {\leqslant}1]} $ and $\mathcal{K}_{2}= {\widehat}{K}(\xi) \mathds{1}_{[|\xi| > 1]}$, so, ${\widehat}{K}=\mathcal{K}_{1}+\mathcal{K}_{2}$ with $\mathcal{K}_{1} \in L^{p}({\mathbb{R}}^{d})$ for all $p\in [1,\frac{d}{d-\gamma}[$ and $\mathcal{K}_{2} \in L^{q}({\mathbb{R}}^{d})$ for all $q \in ]\frac{d}{d-\gamma}, +\infty]$. Let $g(u)=(K*|u|^{2})u$. For ${\varepsilon}>0$ we set $$U^{{\varepsilon}}(t)=e^{i{\varepsilon}\frac{t}{2} \Delta},$$ with $U(t):=U^{1}(t)$. We recall some important properties of $W({\mathbb{R}}^{d}).$
\[W\_algebra\] Wiener algebra space $W({\mathbb{R}}^{d})$ enjoys the following properties (see [@MR2607351; @MR2515782]):
i. $W({\mathbb{R}}^{d})$ is a Banach space, continuously embedded into $L^{\infty}({\mathbb{R}}^{d})$.
ii. $W({\mathbb{R}}^{d})$ is an algebra, in the sense that the mapping $ (f,g)\mapsto fg$ is continuous from $W({\mathbb{R}}^{d})^{2}$ to $ W({\mathbb{R}}^{d})$, and moreover $$\forall f,g \in W({\mathbb{R}}^{d}), \; \|fg\|_{W} {\leqslant}\|f\|_{W}\|g\|_{W}.$$
iii. For all $t \in {\mathbb{R}}$, $U^{{\varepsilon}}(t)$ is unitary on $W({\mathbb{R}}^{d})$.
iv. For all $s > \frac{d}{2}$ there exists a positive constant $C(s,d)$ such that for all $f\in H^{s}({\mathbb{R}}^{d}) $ $$\|f\|_{W} {\leqslant}C(s,d) \|f\|_{H^{s}}.$$
The following theorem ensures the existence and uniqueness of the solution to :
\[exist\_uniq\] We consider the above initial value problem with $u_{0} \in L^{2}({\mathbb{R}}^{d}) \cap W({\mathbb{R}}^{d})$. Then there exists $T >0$ depending on $\|u_{0}\|_{L^{2}\cap W}$ and a unique solution $u \in C([0,T]; L^{2}({\mathbb{R}}^{d})\cap W({\mathbb{R}}^{d}))$ to .
The following lemma will be useful to prove the above theorem :
\[lemm\_W\] Let $h \in L^{1}\cap W, f_{1}, f_{2} \in L^{2}\cap W $ and $K$ like defined as above. Then
$$\label{ineg4}
\|K*h\|_{W} {\leqslant}\|\mathcal{K}_{1}\|_{L^{1}} \|h\|_{L^{1}} + \|\mathcal{K}_{2}\|_{L^{\infty}} \|h\|_{W},$$
in particular we have $$\label{inge4}
\|g(f_{1})-g(f_{2})\|_{L^{2}\cap W} {\leqslant}C (\|f_{1}\|^{2}_{L^{2}\cap W}+
\|f_{2}\|^{2}_{L^{2}\cap W})\|f_{1}-f_{2}\|_{L^{2}\cap W},$$ for some positive constant $C$.
By definition of the $W$ norm : $$\begin{aligned}
\|K*h\|_{W} &=C\|(\mathcal{K}_{1}+\mathcal{K}_{2}) {\widehat}{h}\|_{L^{1}}\\
& {\leqslant}C\|\mathcal{K}_{1}\|_{L^{1}} \|\widehat{h}\|_{L^{\infty}} + C\|\mathcal{K}_{2}\|_{L^{\infty}} \|\widehat{h}\|_{L^{1}}\\
&{\leqslant}C\|\mathcal{K}_{1}\|_{L^{1}} \|h\|_{L^{1}} + C\|\mathcal{K}_{2}\|_{L^{\infty}} \|h\|_{W}.
\end{aligned}$$ Moreover, for $f_{1}, f_{2} \in L^{2}\cap W$ $$\|g(f_{1})-g(f_{2})\|_{L^{2}\cap W}= \phantom{\|(K*(|f_{1}|^{2}-|f_{2}|^{2}))f_{1}\|_{L^{2}\cap W}
+\|(K*|f_{2}|^{2})(f_{1}-f_{2})\|_{L^{2}\cap W}.}$$ $$\begin{aligned}
&=\|(K*(|f_{1}|^{2}-|f_{2}|^{2}))f_{1}+(K*|f_{2}|^{2})(f_{1}-f_{2})\|_{L^{2}\cap W}\\
&{\leqslant}\|(K*(|f_{1}|^{2}-|f_{2}|^{2}))f_{1}\|_{L^{2}\cap W}+\|(K*|f_{2}|^{2})(f_{1}-f_{2})\|_{L^{2}\cap W}.
\end{aligned}$$ We control the last two norms. We have $$\|(K*|f_{2}|^{2})(f_{1}-f_{2})\|_{L^{2}\cap W}=\phantom{\|(K*|f_{2}|^{2})(f_{1}-f_{2})\|_{L^{2}}+ \|(K*|f_{2}|^{2})(f_{1}-f_{2})\|_{W}}$$ $$\begin{aligned}
&=\|(K*|f_{2}|^{2})(f_{1}-f_{2})\|_{L^{2}}+\|(K*|f_{2}|^{2})(f_{1}-f_{2})\|_{W}\\
&{\leqslant}\|K*|f_{2}|^{2}\|_{L^{\infty}}\|f_{1}-f_{2}\|_{L^{2}}\|K*|f_{2}|^{2}\|_{W}\|f_{1}-f_{2}\|_{W} \\
&{\leqslant}C \|K*|f_{2}|^{2}\|_{W}\|f_{1}-f_{2}\|_{L^{2}\cap W}\\
&{\leqslant}C \|\mathcal{K}_{1}\|_{L^{1}} \|f_{2}\|^{2}_{L^{2}}\|f_{1}-f_{2}\|_{L^{2}\cap W} +C\|\mathcal{K}_{2}\|_{L^{\infty}} \|f_{2}\|^{2}_{W}\|f_{1}-f_{2}\|_{L^{2}\cap W},\\
\end{aligned}$$ where we have used and the embedding of $W({\mathbb{R}}^{d})$ into $L^{\infty}({\mathbb{R}}^{d})$. Remark that
$$|f_{1}|^{2}-|f_{2}|^{2}= \frac{1}{2} (f_{1}-f_{2})(\overline{f_{1}}+\overline{f_{2}})+
\frac{1}{2} (\overline{f_{1}}-\overline{f_{2}})(f_{1}+f_{2}).$$ From and Lemma \[W\_algebra\] iii, we have
$$\|(K*(|f_{1}|^{2}-|f_{2}|^{2}))f_{1}\|_{L^{2}\cap W} {\leqslant}\phantom{\|K*(|f_{1}|^{2}-|f_{2}|^{2})\|_{L^{\infty}}\|f_{1}\|_{L^{2}\cap W}
+\|K*(|f_{1}|^{2}-|f_{2}|^{2})\|_{W}\|f_{1}\|_{L^{2}\cap W}}$$ $$\begin{aligned}
&{\leqslant}\|K*(|f_{1}|^{2}-|f_{2}|^{2})\|_{L^{\infty}}\|f_{1}\|_{L^{2}\cap W}
+\|K*(|f_{1}|^{2}-|f_{2}|^{2})\|_{W}\|f_{1}\|_{L^{2}\cap W}\\
&{\leqslant}C\|K*(|f_{1}|^{2}-|f_{2}|^{2})\|_{W}\|f_{1}\|_{L^{2}\cap W} \\
&{\leqslant}C\|K*((f_{1}-f_{2})(\overline{f_{1}}+
\overline{f_{2}}))\|_{W}\|f_{1}\|_{L^{2}\cap W}\\
&\phantom{pp}+C\|K*((\overline{f_{1}}-\overline{f_{2}})(f_{1}+f_{2}))\|_{W}\|f_{1}\|_{L^{2}\cap W}\\
&{\leqslant}\|\mathcal{K}_{1}\|_{L^{1}} \|(f_{1}-f_{2})(\overline{f_{1}}+
\overline{f_{2}})\|_{L^{1}} \|f_{1}\|_{L^{2}\cap W}\\
&\phantom{pp}+C\|\mathcal{K}_{2}\|_{L^{\infty}} \|(f_{1}-f_{2})(\overline{f_{1}}+
\overline{f_{2}})\|_{W} \|f_{1}\|_{L^{2}\cap W}\\
&\phantom{pp}+ C\|\mathcal{K}_{1}\|_{L^{1}} \|(\overline{f_{1}}-\overline{f_{2}})(f_{1}+f_{2})\|_{L^{1}}\|f_{1}\|_{L^{2}\cap W}\\
&\phantom{pp}+ C\|\mathcal{K}_{2}\|_{L^{\infty}} \|(\overline{f_{1}}-\overline{f_{2}})(f_{1}+f_{2})\|_{W}\|f_{1}\|_{L^{2}\cap W}\\
&{\leqslant}C\|f_{1}\|^{2}_{L^{2}\cap W}\|f_{1}-f_{2}\|_{L^{2}\cap W}
+C\|f_{1}\|_{L^{2}\cap W}\|f_{2}\|_{L^{2}\cap W}\|f_{1}-f_{2}\|_{L^{2}\cap W}\\
&\phantom{pp}+C\|f_{2}\|^{2}_{L^{2}\cap W}\|f_{1}-f_{2}\|_{L^{2}\cap W}\\
&{\leqslant}C (\|f_{1}\|^{2}_{L^{2}\cap W}+
\|f_{2}\|^{2}_{L^{2}\cap W})\|f_{1}-f_{2}\|_{L^{2}\cap W},
\end{aligned}$$ where we have used the Cauchy-Schwartz inequality. The desired control follows easily.
Let $T>0$ to be specified later. We set
$$X= \lbrace u \in C([0,T]; L^{2} \cap W),
\|u\|_{L^{\infty}([0,T]; L^{2}\cap W)} {\leqslant}2\|u_{0}\|_{L^{2}\cap W} \rbrace.$$
Duhamel’s formulation of reads
$$u(t)= U(t)u_{0} -i\lambda \int_{0}^{t} U(t-\tau)(K*|u|^{2})u(\tau)\; \mathrm{d}\tau.$$ We denote by $\Phi(u)(t)$ the right hand side in the above formula and $G(u)=\Phi(u)(t)- U(t)u_{0}$. For $q\geq 1$ and a space $S$ we define $\|u\|_{L^{q}_{T}S}:= \|u\|_{L^{q}([0,T]; S)}$ for $u\in L^{q}([0,T]; S)$. Let $ u\in X $. We have $$\begin{aligned}
\|\Phi(u)\|_{L^{\infty}_{T}L^{2}\cap L^{\infty}} & {\leqslant}\|\Phi(u)(t)\|_{L^{\infty}_{T}L^{2}} + \|\Phi(u)(t)\|_{L^{\infty}_{T}W},
\end{aligned}$$ and $$\begin{aligned}
\|\Phi(u)\|_{L^{\infty}_{T}L^{2}} &{\leqslant}\|u_{0}\|_{L^{2}} +
\|G(u)\|_{L^{\infty}_{T}L^{2}}\\
& {\leqslant}\|u_{0}\|_{L^{2}}+ C\|g\|_{L^{1}_{T}L^{2}}\\
&{\leqslant}\|u_{0}\|_{L^{2}}+ C\|K*|u|^{2}\|_{L^{\infty}_{T}L^{\infty}} \|u\|_{L^{\infty}_{T}L^{2}} T.
\end{aligned}$$ We apply Lemma \[lemm\_W\] after replacing $h$ by $|u|^{2}$. We obtain by the embedding of $W$ into $L^{\infty}$ $$\begin{aligned}
\|K*|u|^{2}\|_{L^{\infty}} &{\leqslant}C \|K*|u|^{2}\|_{W}\\
&{\leqslant}C\|\mathcal{K}_{1}\|_{L^{1}}\||u|^{2}\|_{W} + C\|\mathcal{K}_{2}\|_{L^{\infty}} \||u|^{2}\|_{L^{1}}\\
&{\leqslant}C\|u\|^{2}_{L^{2}\cap W}.
\end{aligned}$$ So we have $$\begin{aligned}
\|\Phi(u)\|_{L^{\infty}_{T}L^{2}}
&{\leqslant}\|u_{0}\|_{L^{2}}+ C\|u\|^{2}_{L^{\infty}_{T}L^{2}\cap W} \|u\|_{L^{\infty}_{T}L^{2}} T\\
&{\leqslant}\|u_{0}\|_{L^{2}}+ C\|u\|^{3}_{L^{\infty}_{T}L^{2}\cap W} T.
\end{aligned}$$ Moreover, always by applying Lemma \[lemm\_W\] with $h=|u|^{2}$ we obtain $$\begin{aligned}
\|\Phi(u)(t)\|_{W}& {\leqslant}\|u_{0}\|_{W}+ \int_0^t \|K*|u|^{2}(\tau)\|_{W}\|u(\tau)\|_{W} \, \mathrm{d}\tau\\
&{\leqslant}\|u_{0}\|_{W}+ C\int_0^t \|u(\tau)\|^{3}_{L^{2}\cap W}\, \mathrm{d} \tau,\\
\end{aligned}$$ and thus $$\begin{aligned}
\|\Phi(u)\|_{L^{\infty}_{T}W} &{\leqslant}\|u_{0}\|_{W}+ C \|u\|^{3}_{L^{\infty}_{T}L^{2}\cap W}T.\\
\end{aligned}$$ Finally we have $$\|\Phi(u)\|_{L^{\infty}_{T}L^{2}\cap W} {\leqslant}\|u_{0}\|_{L^{2}\cap W}
+ C\|u_{0}\|_{L^{2}\cap W}^{3}T.$$ By reducing sufficiently $T$ (depending on $\|u_{0}\|_{L^{2}\cap W}$) we get for all $u\in X$, $ \|\Phi(u)(t)\|_{L^{\infty}_{T}L^{2}} {\leqslant}2 \|u_{0}\|_{L^{2}\cap W}$. Moreover, for $u, v \in X$ we have $$\|\Phi(u)-\Phi(v)\|_{L^{\infty}_{T}L^{2}\cap W} {\leqslant}\|G(u)-G(v)\|_{L^{\infty}_{T}L^{2}}
+ \|G(u)-G(v)\|_{L^{\infty}_{T}W}.$$ For some positive constant $C$ we have $$\begin{aligned}
\|G(u)-G(v)\|_{L^{\infty}_{T}L^{2}} &{\leqslant}C\|g(u)-g(v)\|_{L^{1}_{T}L^{2}}\\
&{\leqslant}C\|g(u)-g(v)\|_{L^{\infty}_{T}L^{2}}T\\
&{\leqslant}C\|g(u)-g(v)\|_{L^{\infty}_{T}L^{2}\cap W}T.\\
\end{aligned}$$ Moreover $$\begin{aligned}
\|G(u)-G(v)\|_{W} &{\leqslant}\int_0^t \|g(u)(\tau)-g(v)(\tau)\|_{W} \, \mathrm{d}\tau,\\
\end{aligned}$$ and $$\begin{aligned}
\|G(u)-G(v)\|_{L^{\infty}_{T} W} &{\leqslant}\|g(u)-g(v)\|_{L^{\infty}_{T}W}T\\
& {\leqslant}\|g(u)-g(v)\|_{L^{\infty}_{T}L^{2}\cap W}T.
\end{aligned}$$ By replacing $f_{1}, f_{2}$ by $u, v$ respectively in Lemma \[lemm\_W\] we obtain $$\begin{aligned}
\|\Phi(u)-\Phi(v)\|_{L^{\infty}_{T}L^{2}\cap W} &{\leqslant}C\|g(u)-g(v)\|_{L^{\infty}_{T}L^{2}\cap W}T\\
&{\leqslant}C \|u_{0}\|_{L^{2}\cap W}^{2} \|u-v\|_{L^{\infty}_{T}L^{2}\cap W}T.
\end{aligned}$$ Choosing $T$ possibly smaller (still depending on $\|u_{0}\|_{L^{2}\cap W}$) we deduce that $\Phi$ is a contraction from $X$ to $X$. Thus $\Phi$ has a unique fixed point $u\in X$ and Theorem $\ref{exist_uniq}$ follows.
Derivation of the approximate solution
======================================
We consider the rescaled version of : $$\label{eq:SPultim}
\ i\varepsilon \partial_{t} u^{\varepsilon} +
\frac{{\varepsilon}^{2}}{2} \Delta u^{\varepsilon}=
\varepsilon \lambda (K*{|u^{\varepsilon}|}^{2})u^{\varepsilon}.$$ We seek an approximation of solutions to in the form $$u^{{\varepsilon}}_{app}(t,x)=\sum_{j\in {\mathbb{N}}} a_{j}(t,x) e^{i\phi_{j}(t,x)/{\varepsilon}}.$$ We begin by proceeding formally. We plug the ansatz above into . This yields $$\label{eq_sol_app}
i{\varepsilon}\partial_{t}u^{{\varepsilon}}_{app}+\frac{{\varepsilon}^{2}}{2} \Delta u^{{\varepsilon}}_{app}-
{\varepsilon}(K*|u^{{\varepsilon}}_{app}|^{2})u^{{\varepsilon}}_{app}= \sum_{k=0}^{2} {\varepsilon}^{k}Z^{{\varepsilon}}_{k}
+{\varepsilon}(W^{{\varepsilon}}+r^{{\varepsilon}}),$$ with $$Z^{{\varepsilon}}_{0}= -\sum_{j\in {\mathbb{N}}}(\partial_{t}\phi_{j}+\frac{1}{2}|\nabla\phi_{j}|^{2})a_{j}e^{i\phi_{j}/{\varepsilon}},$$ $$Z^{{\varepsilon}}_{1}= i\sum_{j\in {\mathbb{N}}} (\partial_{t}a_{j}+\nabla\phi_{j}\nabla a_{j}+
\frac{1}{2}a_{j}\Delta\phi_{j})e^{i\phi_{j}/{\varepsilon}},$$ and $$\label{term_r2}
Z^{{\varepsilon}}_{2}=\frac{1}{2}\sum_{j\in {\mathbb{N}}} \Delta a_{j}e^{i\phi_{j}/{\varepsilon}}.$$ Other terms appear due to the presence of the nonlinearity which are $$\label{pas_rest}
W^{{\varepsilon}}= -\sum_{j\in {\mathbb{N}}}(K*\sum_{\ell\in {\mathbb{N}}}|a_{\ell}|^{2})a_{j}e^{i\phi_{j}/{\varepsilon}},$$ and $$\label{rest}
r^{{\varepsilon}}= -\sum_{j\in {\mathbb{N}}}(K*\sum_{\tiny \begin{matrix}k,\ell \in {\mathbb{N}}\\
k\neq \ell\end{matrix}} (a_{k}\overline{a_{\ell}}e^{i(\phi_{k}-\phi_{\ell})/{\varepsilon}}))a_{j}e^{i\phi_{j}/{\varepsilon}}.$$ We aim to eliminate all equal powers of ${\varepsilon}$. Hence, by setting $Z^{{\varepsilon}}_{0}=0$ we obtain the eikonal equation, whose solution is explicitly given by $$\label{eikonale}
\phi_{j}(t,x)= \kappa_{j}\cdot x-\frac{t}{2} |\kappa_{j}|^{2}.$$ Next, we set $Z^{{\varepsilon}}_{1}+W^{{\varepsilon}}=0$ without including $r^{{\varepsilon}}$. This latter will constitute the first term error, the second will be $Z^{{\varepsilon}}_{2}$. We obtain for all $j\in{\mathbb{N}}$
$$\label{eq_amplitude}
\partial_{t}a_{j}+ \kappa_{j}\cdot \nabla a_{j} = -i(K*\sum_{\ell\in {\mathbb{N}}}|a_{\ell}|^{2})a_{j}, \quad
a_{j}(0)=\alpha_{j},$$
where we have used the fact that $\Delta\phi_{j}=0.$
\[lemm\_ampl\] The transport equation with initial amplitudes $(\alpha_{j})_{j\in {\mathbb{N}}} \in E({\mathbb{R}}^{d})$ admits a unique global-in-time solutions $a=(a_{j})_{j\in {\mathbb{N}}}\in C([0,\infty[;E({\mathbb{R}}^{d}))$, which can be written in the form $$\label{form_amplitude}
a_{j}(t,x)=\alpha_{j}(x-t\kappa_{j})e^{iS_{j}(t,x)},$$ where $$\label{ampli_puissance}
S_{j}(t,x)= -\int_{0}^{t}(K*\sum_{\ell \in {\mathbb{N}}} |\alpha_{\ell}(x +(\tau-t)\kappa_{j}-\tau\kappa_{\ell})|^{2})\;\mathrm{d}\tau.$$
We multiply with $\overline{a_{j}}$. We obtain $$\label{dd}
\overline{a_{j}}\partial_{t}a_{j}+ \kappa_{j}\cdot (\overline{a_{j}}\nabla a_{j}) = -i(K*\sum_{\ell\in {\mathbb{N}}}|a_{\ell}|^{2})|a_{j}|^{2}
\in i{\mathbb{R}}.$$ But $$2\mathrm{Re}(\overline{a_{j}}\partial_{t}a_{j}+ \kappa_{j}\cdot (\overline{a_{j}}\nabla a_{j}))
=(\partial_{t}+\kappa_{j}\cdot \nabla)|a_{j}|^{2}.$$ So, we deduce from that $(\partial_{t}+\kappa_{j}\cdot \nabla)|a_{j}|^{2}=0$, which yields $$|a_{j}(t,x)|^{2}=|\alpha_{j}(x-t\kappa_{j})|^{2}.$$ and follows for some real function $S_{j}$. To determine $S_{j}$ we inject into . We get $$i(\partial_{t}+\kappa_{j}\cdot\nabla)S_{j}(t,x)\alpha_{j}(x-t\kappa_{j})=-i(K*\sum_{\ell\in {\mathbb{N}}}|\alpha_{j}(x-t\kappa_{\ell})|^{2})
\alpha_{j}(x-t\kappa_{j}).$$ It suffices to impose $$\partial_{t}(S_{j}(t,x+t\kappa_{j}))=-K*\sum_{\ell\in {\mathbb{N}}}|\alpha_{j}(x-t(\kappa_{\ell}-\kappa_{j}))|^{2},$$ which yields $$S_{j}(t,x+t\kappa_{j})=-\int_{0}^{t} K*\sum_{\ell\in {\mathbb{N}}}|\alpha_{j}(x-\tau(\kappa_{\ell}-\kappa_{j}))|^{2}\;\mathrm{d}\tau,$$ and finally we have $$S_{j}(t,x)=-\int_{0}^{t} K*\sum_{\ell\in {\mathbb{N}}}|\alpha_{j}(x+(\tau-t)\kappa_{j})-\tau\kappa_{\ell}|^{2}\;\mathrm{d}\tau.$$
Global in time existence of $a_{j}$’s is not trivial from their explicit formula like we can see in the following. We rewrite in its integral form
$$\label{form_integ_aj}
a_{j}(t,x)= \alpha_{j}(x-t\kappa_{j}) + \int_{0}^{t} \mathcal{N}(a)_{j}\, (\tau, x+(\tau-t)\kappa_{j})\mathrm{d}\tau,$$
where the nonlinearity $\mathcal{N}$ is given by $$\mathcal{N}(a)_{j}=-i\bigg(K*\sum_{\ell\in {\mathbb{N}}} |a_{\ell}|^{2}\bigg)a_{j}.$$ For $a\in E({\mathbb{R}}^{d})$ we have by definition $$\begin{aligned}
\|\mathcal{N}(a)\|_{E} &= \sum_{j\in {\mathbb{N}}} \|\mathcal{N}(a)_{j}\|_{Y}
=\sum_{j\in {\mathbb{N}}} \sum_{|\eta|{\leqslant}n}\|\partial^{\eta}\mathcal{N}(a)_{j}\|_{L^{2}\cap W}.
\end{aligned}$$ Let $|\eta|{\leqslant}n$. Leibnitz formula yields $$\begin{aligned}
\|\partial^{\eta}\mathcal{N}(a)_{j}\|_{L^{2}\cap W}
=&\|\partial^{\eta}\bigg((K*\sum_{\ell\in {\mathbb{N}}} |a_{\ell}|^{2})a_{j}\bigg)\|_{L^{2}\cap W}\\
{\leqslant}&\sum_{\theta{\leqslant}\eta} C^{\theta}_{\eta} \|(K*\sum_{\ell \in {\mathbb{N}}} \partial^{\theta}|a_{\ell}|^{2})\;\partial^{\eta-\theta}a_{j}\|_{L^{2}\cap W}\\
{\leqslant}&C \sum_{|\theta|{\leqslant}n} \|(K*\sum_{\ell \in {\mathbb{N}}} \partial^{\theta}|a_{\ell}|^{2})\|_{W}\|a_{j}\|_{Y}\\
{\leqslant}&C \|a_{j}\|_{Y} \bigg(\sum_{\ell\in {\mathbb{N}}}\|\mathcal{K}_{1}\|_{L^{1}}
\sum_{|\theta|{\leqslant}n}\|\partial^{\theta}|a_{\ell}|^{2}\|_{L^{1}}\\
&+\|\mathcal{K}_{2}\|_{L^{\infty}}
\sum_{|\theta|{\leqslant}n}\|\partial^{\theta}|a_{\ell}|^{2}\|_{W}\bigg)\\
{\leqslant}&C \|a_{j}\|_{Y}\sum_{\ell\in {\mathbb{N}}}(\|\mathcal{K}_{1}\|_{L^{1}}+\|\mathcal{K}_{2}\|_{L^{\infty}}) \|a_{\ell}\|^{2}_{Y}\\
{\leqslant}&C\|a_{j}\|_{Y}\|a\|^{2}_{E}.
\end{aligned}$$ Finally we have $$\|\mathcal{N}(a)\|_{E}{\leqslant}C\|a\|^{3}_{E}.$$ This shows that $\mathcal{N}(a)$ defines a continuous mapping from $E^{3}$ to $E$ and by the standard Cauchy-Lipschitz theorem for the ordinary differential equations, a local-in-time existence results immediately follows. Now, we have to show that the solution $a(t)=(a_{j}(t))_{j\in {\mathbb{N}}} $ is global in time. Let $[0,T_{\max}[$ be the maximal time interval where $(a_{j})_{j\in {\mathbb{N}}}$ is defined. From we have for $|\eta|{\leqslant}n$ and $t\in [0,T_{\max}[$ $$\begin{aligned}
\partial^{\eta}a_{j}(t)=&\;\partial^{\eta}\alpha_{j}(\cdot-t\kappa_{j}) +
\int_{0}^{t} \partial^\eta\mathcal{N}(a)_{j}(\tau,\cdot+(\tau-t)\kappa_{j})\;\mathrm{d}\tau\\
=&\;\partial^{\eta}\alpha_{j}(\cdot-t\kappa_{j})\\
&+\int_{0}^{t}\sum_{\theta{\leqslant}\eta}C^{\theta}_{\eta} (K*\sum_{\ell \in {\mathbb{N}}} \partial^{\theta}|\alpha_{\ell}(\cdot-t\kappa_{j})|^{2})\;\partial^{\eta-\theta}a_{j}(\tau,\cdot+(\tau-t)\kappa_{j})\;\mathrm{d}\tau.\\
\end{aligned}$$ Taking $L^{2}\cap W$ norm yields $$\|\partial^{\eta}a_{j}(t)\|_{L^{2}\cap W} {\leqslant}\phantom{\|\partial^{\eta}\alpha_{j}\|_{L^{2}\cap W}+C\sum_{\theta{\leqslant}\eta} \int_{0}^{t}
\|K*\sum_{\ell \in {\mathbb{N}}} \partial^{\theta}|\alpha_{\ell}(\cdot-t\kappa_{j})|^{2}\|_{W}
\|\partial^{\eta-\theta}a_{j}(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau}$$ $$\begin{aligned}
&{\leqslant}\|\partial^{\eta}\alpha_{j}\|_{L^{2}\cap W}+ C\sum_{\theta{\leqslant}\eta} \int_{0}^{t}
\|K*\sum_{\ell \in {\mathbb{N}}} \partial^{\theta}|\alpha_{\ell}(\cdot-t\kappa_{j})|^{2}\|_{W}
\|\partial^{\eta-\theta}a_{j}(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau\\
&{\leqslant}\|\partial^{\eta}\alpha_{j}\|_{L^{2}\cap W}
+C\sum_{\theta{\leqslant}\eta} \int_{0}^{t} \|K*\sum_{\ell \in {\mathbb{N}}} \partial^{\theta}|\alpha_{\ell}|^{2}\|_{W}
\|\partial^{\eta-\theta}a_{j}(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau\\
&{\leqslant}\|\partial^{\eta}\alpha_{j}\|_{L^{2}\cap W}
+C (\|\mathcal{K}_{1}\|_{L^{1}}+\|\mathcal{K}_{2}\|_{L^{\infty}})\sum_{\ell\in {\mathbb{N}}}\|\alpha_{\ell}\|^{2}_{Y} \int_{0}^{t}\|a_{j}(\tau)\|_{Y}\;\mathrm{d}\tau\\
&{\leqslant}\|\partial^{\eta}\alpha_{j}\|_{L^{2}\cap W}+C\int_{0}^{t}\|a_{j}(\tau)\|_{Y}\;\mathrm{d}\tau,\\
\end{aligned}$$ and so $$\begin{aligned}
\|a_{j}(t)\|_{Y}=\sum_{|\eta|{\leqslant}n} \|\partial^{\eta} a_{j}(t)\|_{L^{2}\cap W}
{\leqslant}\|\alpha_{j}\|_{Y} +C\int_{0}^{t}\|a_{j}(\tau)\|_{Y}\;\mathrm{d}\tau.
\end{aligned}$$ By Gronwall lemma $$\|a_{j}(t)\|_{Y}{\leqslant}\|\alpha_{j}\|_{Y}e^{Ct}.$$ for some positive constant $C$ independent of $j$ and $t$. After summing with respect to $j$ we obtain $$\|a(t)\|_{E}{\leqslant}\|\alpha\|_{E}e^{Ct}.$$ The growth of $\|a(t)\|_{E}$ is at most exponential, and thus can not explode at finite time. We deduce that $T_{\max}=\infty$ and $a=(a_{j})_{j\in {\mathbb{N}}}\in C([0,\infty[;E({\mathbb{R}}^{d}))$.
Estimations on the remainder
============================
We will estimate in $ L^{2}\cap W$ the term $r^{{\varepsilon}}$ and $Z^{{\varepsilon}}_{2}$. To this end, we assume $(\alpha_{j})_{j\in {\mathbb{N}}} \in \ell^{1}({\mathbb{N}}, Y({\mathbb{R}}^{d}))$.
\[estim\_error\] Let $r^{{\varepsilon}}$ be defined by with the plane-wave phases $\phi_{j}$ given by . We have the following bound:
$$\label{control_R}
\|r^{{\varepsilon}}\|_{L^{2}\cap W}{\leqslant}C\delta^{\gamma-d}\|a\|_{E}^{3}{\varepsilon}^{d-\gamma},$$
$$\label{control_X2}
\|Z^{{\varepsilon}}_{2}\|_{L^{2}\cap W} {\leqslant}\|a\|_{E},$$
where $\delta$ is like defined in Theorem \[mainTh\].
Inequality is obvious. We recall the definition of $r^{{\varepsilon}}$:
$$r^{{\varepsilon}}= -\sum_{j\in {\mathbb{N}}}(K*\sum_{\tiny \begin{matrix}k,\ell \in {\mathbb{N}}\\
k\neq \ell\end{matrix}} a_{k}\overline{a_{\ell}}e^{i(\phi_{k}-\phi_{\ell})/{\varepsilon}})a_{j}e^{i\phi_{j}/{\varepsilon}}.$$
We estimate $W$ norm of $r^{{\varepsilon}}$, we obtain
$$\begin{aligned}
\|r^{{\varepsilon}}\|_{W} &= \|\sum_{j\in {\mathbb{N}}}(K*\sum_{\tiny \begin{matrix}k,\ell \in {\mathbb{N}}\\
k\neq \ell\end{matrix}} a_{k}\overline{a_{\ell}}e^{i(\phi_{k}-\phi_{\ell})/{\varepsilon}})
a_{j}e^{i\phi_{j}/{\varepsilon}}\|_{W}\\
&{\leqslant}\sum_{j\in {\mathbb{N}}}\|K*\sum_{\tiny \begin{matrix}k,\ell \in {\mathbb{N}}\\
k\neq \ell\end{matrix}} a_{k}\overline{a_{\ell}}e^{i(\phi_{k}-\phi_{\ell})/{\varepsilon}}\|_{W}\|a_{j}\|_{W} \\
&{\leqslant}\sum_{\tiny \begin{matrix}k,\ell \in {\mathbb{N}}\\
k\neq \ell\end{matrix}} \|K*a_{k}\overline{a_{\ell}}e^{i(\phi_{k}-\phi_{\ell})/{\varepsilon}}\|_{W}
\sum_{j\in {\mathbb{N}}} \|a_{j}\|_{W}.
\end{aligned}$$
Let us look more closely to the $W$ norm of the convolution above. For $b$ sufficiently regular and $\omega\in {\mathbb{R}}^{d}$, we denote $I^{{\varepsilon}}(x)= K*(be^{i\omega\cdot x/{\varepsilon}})$. Remark that
$$\begin{aligned}
{\widehat}{I^{{\varepsilon}}}& = (2\pi)^{d/2} {\widehat}{K}\;{\widehat}{be^{i\omega\cdot x/{\varepsilon}}}\\
&= (2\pi)^{d/2}{\widehat}{K}\;{\widehat}{b}(\cdot-\frac{\omega}{{\varepsilon}}).
\end{aligned}$$
So, if we take the $W$ norm of $I^{{\varepsilon}}$ we obtain $$\begin{aligned}
\|I^{{\varepsilon}}\|_{W}=&\; \|{\widehat}{I^{{\varepsilon}}}\|_{L^{1}}\\
=&\;C \|{\widehat}{K}\, {\left\langle}\xi - \frac{\omega}{{\varepsilon}}{\right\rangle}^{m}{\left\langle}\xi - \frac{\omega}{{\varepsilon}}{\right\rangle}^{-m} {\widehat}{b}(\cdot -\frac{\omega}{{\varepsilon}})\|_{L^{1}}\\
{\leqslant}&\;C\|\mathcal{K}_{1}\,{\left\langle}\xi - \frac{\omega}{{\varepsilon}}{\right\rangle}^{m}{\left\langle}\xi - \frac{\omega}{{\varepsilon}}{\right\rangle}^{-m}{\widehat}{b}(\cdot -\frac{\omega}{{\varepsilon}})\|_{L^{1}}\\
&+\;C\|\mathcal{K}_{2}\,{\left\langle}\xi - \frac{\omega}{{\varepsilon}}{\right\rangle}^{m}{\left\langle}\xi - \frac{\omega}{{\varepsilon}}{\right\rangle}^{-m}{\widehat}{b}(\cdot -\frac{\omega}{{\varepsilon}})\|_{L^{1}} \\
{\leqslant}&\;C\|\mathcal{K}_{1}\|_{L^{1}}\|{\left\langle}\xi - \frac{\omega}{{\varepsilon}}{\right\rangle}^{m}{\widehat}{b}(\cdot -\frac{\omega}{{\varepsilon}})\|_{L^{\infty}}
\sup_{|\xi|{\leqslant}1}{\left\langle}\xi-\frac{\omega}{{\varepsilon}}{\right\rangle}^{-m}\\
&+\;C\|{\left\langle}\xi - \frac{\omega}{{\varepsilon}}{\right\rangle}^{m}{\widehat}{b}(\cdot -\frac{\omega}{{\varepsilon}})\|_{L^{1}}
\sup_{|\xi|> 1}\bigg(|\xi|^{\gamma-d}{\left\langle}\xi-\frac{\omega}{{\varepsilon}}{\right\rangle}^{-m}\bigg).\\
\end{aligned}$$
By Peetre inequality we have for all $m{\leqslant}d-\gamma$ and $|\xi| > 1$
$$\begin{aligned}
|\xi|^{d-\gamma}{\left\langle}\xi - \frac{\omega}{{\varepsilon}}{\right\rangle}^{m} \geqslant |\xi|^{d-\gamma}\frac{{\left\langle}\frac{\omega}{{\varepsilon}}{\right\rangle}^{m}}{{\left\langle}\xi{\right\rangle}^{m}}
\geqslant C{\left\langle}\frac{\omega}{{\varepsilon}}{\right\rangle}^{m}\geqslant C{\varepsilon}^{-m},
\end{aligned}$$ and thus $$\sup_{|\xi|> 1}(|\xi|^{\gamma-d}{\left\langle}\xi-\frac{\omega}{{\varepsilon}}{\right\rangle}^{-m}) {\leqslant}C{\varepsilon}^{m}{\leqslant}C{\varepsilon}^{d-\gamma},$$ Moreover $$\begin{aligned}
{\left\langle}\xi -\frac{\omega}{{\varepsilon}}{\right\rangle}^{m} \geqslant \frac{{\left\langle}\frac{\omega}{{\varepsilon}}{\right\rangle}^{m}}{{\left\langle}\xi{\right\rangle}^{m}}
\geqslant {\left\langle}\frac{\omega}{{\varepsilon}}{\right\rangle}^{m}
\geqslant C {\varepsilon}^{-m},
\end{aligned}$$ and $$\sup_{|\xi|{\leqslant}1}{\left\langle}\xi-\frac{\omega}{{\varepsilon}}{\right\rangle}^{-m}{\leqslant}C{\varepsilon}^{d-\gamma}.$$ So, for $m=d-\gamma$ $$\begin{aligned}
\label{inegI}
\|I^{{\varepsilon}}\|_{W} {\leqslant}C\bigg(\|\mathcal{K}_{1}\|_{L^{1}} \|{\left\langle}\xi{\right\rangle}^{d-\gamma}{\widehat}{b}\|_{L^{\infty}} + C\|\mathcal{K}_{2}\|_{L^{\infty}}\|{\left\langle}\xi{\right\rangle}{\widehat}{b}\|_{L^{1}}\bigg) |\omega|^{d-\gamma}{\varepsilon}^{d-\gamma}.
\end{aligned}$$
Hence, for all $k,\ell \in {\mathbb{N}}, k\neq \ell$
$$\begin{aligned}
\|K*a_{k}\overline{a_{\ell}}e^{\frac{\phi_{k}-\phi_{\ell}}{{\varepsilon}}}\|_{W}
{\leqslant}&\;C\|\mathcal{K}_{1}\|_{L^{1}} \|{\left\langle}\xi{\right\rangle}^{d-\gamma}{\widehat}{a_{k}\overline{a_{\ell}}}\|_{L^{\infty}}|\kappa_{k}-\kappa_{\ell}|^{d-\gamma}{\varepsilon}^{d-\gamma}\\
&+ C\|\mathcal{K}_{2}\|_{L^{\infty}}\|{\left\langle}\xi{\right\rangle}^{d-\gamma}{\widehat}{a_{k}\overline{a_{\ell}}}\|_{L^{1}} |\kappa_{k}-\kappa_{\ell}|^{d-\gamma}{\varepsilon}^{d-\gamma}\\
{\leqslant}&\;C\delta^{\gamma-d}\|\mathcal{K}_{1}\|_{L^{1}} \|{\left\langle}\xi{\right\rangle}^{d-\gamma}{\widehat}{a_{k}\overline{a_{\ell}}}\|_{L^{\infty}}{\varepsilon}^{d-\gamma}\\
&+\; C\delta^{\gamma-d}\|\mathcal{K}_{2}\|_{L^{\infty}}\|{\left\langle}\xi{\right\rangle}^{d-\gamma}{\widehat}{a_{k}\overline{a_{\ell}}}\|_{L^{1}} {\varepsilon}^{d-\gamma}.\\
\end{aligned}$$ Remark that $$\begin{aligned}
\|{\left\langle}\xi{\right\rangle}^{d-\gamma}{\widehat}{a_{k}\overline{a_{\ell}}}\|_{L^{\infty}}
&{\leqslant}\|{\left\langle}\xi{\right\rangle}^{n}{\widehat}{a_{k}\overline{a_{\ell}}}\|_{L^{\infty}}\\
&{\leqslant}C\sum_{|\eta|{\leqslant}n} \|{\widehat}{\partial^{\eta} (a_{k}\overline{a_{\ell}})}\|_{L^{\infty}}\\
&{\leqslant}C\sum_{|\eta|{\leqslant}n} \|\partial^{\eta} (a_{k}\overline{a_{\ell}})\|_{L^{1}}\\
&{\leqslant}C\sum_{|\eta|{\leqslant}n} \|\partial^{\eta} a_{k}\|_{L^{2}}\sum_{|\eta|{\leqslant}n} \|\partial^{\eta} a_{\ell}\|_{L^{2}}\\
&{\leqslant}C \|a_{k}\|_{Y} \|a_{\ell}\|_{Y}.
\end{aligned}$$ Moreover $$\begin{aligned}
\|{\left\langle}\xi{\right\rangle}^{d-\gamma}{\widehat}{a_{k}\overline{a_{\ell}}}\|_{L^{1}}
&{\leqslant}\|{\left\langle}\xi{\right\rangle}^{n}{\widehat}{a_{k}\overline{a_{\ell}}}\|_{L^{1}}\\
&{\leqslant}C\sum_{|\eta|{\leqslant}n} \|{\widehat}{\partial^{\eta} (a_{k}\overline{a_{\ell}})}\|_{L^{1}}\\
&{\leqslant}C\sum_{|\eta|{\leqslant}n} \|\partial^{\eta} (a_{k}\overline{a_{\ell}})\|_{W}\\
&{\leqslant}C\sum_{|\eta|{\leqslant}n} \|\partial^{\eta} a_{k}\|_{W}\sum_{|\eta|{\leqslant}n} \|\partial^{\eta} a_{\ell}\|_{W}\\
&{\leqslant}C \|a_{k}\|_{Y} \|a_{\ell}\|_{Y}.
\end{aligned}$$
Thus we obtain
$$\begin{aligned}
\|K*a_{k}\overline{a_{\ell}}e^{\frac{\phi_{k}-\phi_{\ell}}{{\varepsilon}}}\|_{W}
& {\leqslant}C\delta^{\gamma-d}(\|\mathcal{K}_{1}\|_{L^{1}}
+ C\|\mathcal{K}_{2}\|_{L^{\infty}})\|a_{k}\|_{Y} \|a_{\ell}\|_{Y}{\varepsilon}^{d-\gamma}.\\
\end{aligned}$$
In view of the above inequality, we deduce $$\begin{aligned}
\|r^{{\varepsilon}}\|_{W}
&{\leqslant}\sum_{\tiny \begin{matrix}k,\ell \in {\mathbb{N}}\\
k\neq \ell\end{matrix}} C\delta^{\gamma-d}(\|\mathcal{K}_{1}\|_{L^{1}}
+ C\|\mathcal{K}_{2}\|_{L^{\infty}})\|a_{k}\|_{Y} \|a_{\ell}\|_{Y}{\varepsilon}^{d-\gamma}
\sum_{j\in {\mathbb{N}}} \|a_{j}\|_{Y}\\
&{\leqslant}C\delta^{\gamma-d}(\|\mathcal{K}_{1}\|_{L^{1}}
+ C\|\mathcal{K}_{2}\|_{L^{\infty}})\sum_{k\in {\mathbb{N}}} \|a_{k}\|_{Y}\sum_{\ell\in {\mathbb{N}}} \|a_{\ell}\|_{Y}
\sum_{j\in {\mathbb{N}}} \|a_{j}\|_{Y}{\varepsilon}^{d-\gamma}\\
&{\leqslant}C\delta^{\gamma-d}(\|\mathcal{K}_{1}\|_{L^{1}}
+ C\|\mathcal{K}_{2}\|_{L^{\infty}})\|a\|_{E}^{3}{\varepsilon}^{d-\gamma}.
\end{aligned}$$
To control $L^{2}$ norm of $r^{{\varepsilon}}$ it suffices to remark that
$$\begin{aligned}
\|r^{{\varepsilon}}\|_{L^{2}} &= \|\sum_{j\in {\mathbb{N}}}(K*\sum_{\tiny \begin{matrix}k,\ell \in {\mathbb{N}}\\
k\neq \ell\end{matrix}} a_{k}\overline{a_{\ell}}e^{i(\phi_{k}-\phi_{\ell})/{\varepsilon}})
a_{j}e^{i\frac{\phi_{j}}{{\varepsilon}}}\|_{L^{2}}\\
&{\leqslant}\sum_{j\in {\mathbb{N}}}\|K*\sum_{\tiny \begin{matrix}k,\ell \in {\mathbb{N}}\\
k\neq \ell\end{matrix}} a_{k}\overline{a_{\ell}}e^{i(\phi_{k}-\phi_{\ell})/{\varepsilon}}\|_{L^{\infty}}\|a_{j}\|_{L^{2}} \\
&{\leqslant}C\sum_{\tiny \begin{matrix}k,\ell \in {\mathbb{N}}\\
k\neq \ell\end{matrix}} \|K*a_{k}\overline{a_{\ell}}e^{i(\phi_{k}-\phi_{\ell})/{\varepsilon}}\|_{W}
\sum_{j\in {\mathbb{N}}} \|a_{j}\|_{Y}.
\end{aligned}$$ and in the same way as previously we get
$$\begin{aligned}
\|r^{{\varepsilon}}\|_{L^{2}} & {\leqslant}C\delta^{\gamma-d}(\|\mathcal{K}_{1}\|_{L^{1}}
+ \|\mathcal{K}_{2}\|_{L^{\infty}})\|a\|_{E}^{3}{\varepsilon}^{d-\gamma}.
\end{aligned}$$ Finally $$\begin{aligned}
\|r^{{\varepsilon}}\|_{L^{2}\cap W} & {\leqslant}C\delta^{\gamma-d}(\|\mathcal{K}_{1}\|_{L^{1}}
+ \|\mathcal{K}_{2}\|_{L^{\infty}})\|a\|_{E}^{3}{\varepsilon}^{d-\gamma}.
\end{aligned}$$
Justification of the approach
=============================
In this section we show that the solution $u^{{\varepsilon}}_{app}$ is a good approximation of the exact solution in the sense of Theorem \[mainTh\]. We assume that assumptions of Theorem \[mainTh\] are verified. At this stage we have constructed an approximate solution $$u^{{\varepsilon}}_{app} = \sum_{j\in {\mathbb{N}}} a_{j}(t,x)e^{i\phi(t,x)/{\varepsilon}},$$ in $C([0,\infty[;E({\mathbb{R}}^{d}))$ where the corresponding $\phi_{j}$’s are like constructed in and the profiles are given by Lemma \[lemm\_ampl\]. Let $T^{{\varepsilon}}>0$ the time of local existence of the exact solution $u^{{\varepsilon}}$ to given by the fixed point argument. We consider a fixed $T>0$ and we introduce the error between the exact solution $u^{{\varepsilon}}$ and the approximate solution $u^{{\varepsilon}}_{app}$ : $$w^{{\varepsilon}}=u^{{\varepsilon}}-u^{{\varepsilon}}_{app}.$$ We have in particular $$u^{{\varepsilon}}_{app} \in C([0,T], L^{2}\cap W),$$ so there exists $R>0$ independent of ${\varepsilon}$ such that $$\|u^{{\varepsilon}}_{app}(t)\|_{L^{2}\cap W}{\leqslant}R, \; \forall t \in [0,T].$$ Since $u^{{\varepsilon}} \in C([0,T^{{\varepsilon}}],L^{2}\cap W)$ and $w^{{\varepsilon}}(0)=0$, there exists $t^{{\varepsilon}}$ such that $$\label{so_long_as_w}
\|w^{{\varepsilon}}(t)\|_{L^{2}\cap W}{\leqslant}R,$$ for $t\in [0,t^{{\varepsilon}}]$. So long as holds, we infer $$\begin{aligned}
\|w^{{\varepsilon}}(t)\|_{L^{2}\cap W}{\leqslant}&\;|\lambda|\int_{0}^{t} \|\left(g(u^{{\varepsilon}}+w^{{\varepsilon}})-g(u^{{\varepsilon}}_{app})\right)(\tau)\|_{L^{2}\cap W}
\; \mathrm{d}\tau \\
&+ |\lambda|{\varepsilon}\int_{0}^{t} \|Z^{{\varepsilon}}_{2}(\tau)\|_{L^{2}\cap W} \;\mathrm{d}\tau
+\;|\lambda|\int_{0}^{t} \|R(\tau)\|_{L^{2}\cap W} \; \mathrm{d}\tau\\
{\leqslant}&\;C\int_{0}^{t} \|w^{{\varepsilon}}(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau+C{\varepsilon}\int_{0}^{t} \mathrm{d}\tau
+ C{\varepsilon}^{d-\gamma}\int_{0}^{t} \mathrm{d}\tau\\
{\leqslant}&\;C\int_{0}^{t} \|w^{{\varepsilon}}(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau + C{\varepsilon}^{\beta}\int_{0}^{t} \mathrm{d}\tau,
\end{aligned}$$ where we have used Lemma \[lemm\_W\] and Proposition \[estim\_error\], with $\beta=\min \lbrace 1,d-\gamma\rbrace$. Gronwall lemma implies that so long as holds, $$\begin{aligned}
\|w^{{\varepsilon}}(t)\|_{L^{2}\cap W} &{\leqslant}C(e^{Ct}-1) {\varepsilon}^{\beta},
\end{aligned}$$ for all $t\in [0,t^{{\varepsilon}}]$. Choosing ${\varepsilon}\in ]0,{\varepsilon}_{0}]$ with ${\varepsilon}_{0}$ sufficiently small, we see that remains true for $t\in [0,T]$ and Theorem \[mainTh\] follows.
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[^1]: This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01).
|
---
author:
- |
Matti Pitkänen\
*, Torkkelinkatu 21 B 39, 00530 Helsinki, Finland\
*
date: '30. June 1995'
title: 'p-Adic TGD: Mathematical ideas.'
---
Abstract
The mathematical basis of p-adic Higgs mechanism discussed in papers hep-th@xxx.lanl.gov 9410058-62 is considered in this paper. The basic properties of p-adic numbers, of their algebraic extensions and the so called canonical identification between positive real numbers and p-adic numbers are described. Canonical identification induces p-adic topology and differentiable structure on real axis and allows definition of definite integral with physically desired properties. p-Adic numbers together with canonical identification provide analytic tool to produce fractals. Canonical identification makes it possible to generalize probability concept, Hilbert space concept, Riemannian metric and Lie groups to p-adic context. Conformal invariance generalizes to arbitrary dimensions since p-adic numbers allow algebraic extensions of arbitrary dimension. The central theme of all developments is the existence of square root, which forces unique quadratic extension having dimension $D=4$ and $D=8$ for $p>2$ and $p=2$ respectively. This in turn implies that the dimensions of p-adic Riemann spaces are multiples of $4$ in $p>2$ case and of $8$ in $p=2$ case.
Note:
The .eps files representing p-adic fractals discussed in the text as well as MATLAB programs needed to generate the fractals are supplied by request. The commands attaching .eps files to text are in the text but preceided by comment signs: please remove these signs.
Introduction
============
There are a lot of speculations about the role of p-adic numbers in Physics [@Padstring; @Narstrings; @Padvira]. In [@padrev] one can find a review of the work done. In general the work is related to quantum theory and based on assumption that quantum mechanical state space is ordinary complex Hilbert space. This is not absolutely necessary since p-adic unitarity and probability concepts make sense [@padprob]. What is however essential is some kind of correspondence between p-adic and real numbers since the predictions of, say, p-adic quantum mechanics should be expressed in terms of real numbers. The formulation of physical theory using p-adic state space and p-adic dynamical variables requires also the construction of p-adic differential and integral calculus. Also the p-adic counterpart of Riemann geometry as well as group theory are needed. In this chapter the aim is to carry out these generalizations.
The key observation behind all developments to be represented in the sequel is very simple: there is canonical correspondence between p-adic numbers and nonnegative real numbers given by “pinary” expansion of real number: positive real number $x= \sum x_np^n$ ($x=0,1,..,p-1$, $p$ prime) is mapped to p-adic number $\sum x_np^{-n}$. This canonical correspondence allows to induce p-adic topology and differentiable structure to the real axis. p-Adically differentiable functions define typically fractal like real functions via canonical identification so that p-adic numbers provide analytic tool for producing fractals. p-adic numbers allow algebraic extensions of arbitrary dimension and the concept of complex analyticity generalizes to p-adic analyticity. The fact that real continuity implies p-adic continuity implies that real physics can emerge above some length scale $L_p$ as an excellent approximation of underlying p-adic physics.
The canonical correspondence makes possible to generalize the concepts of inner product, integration, Hilbert space, Riemannian metric, Lie group theory and Quantum mechanics to p-adic context in a relatively straightforward manner. Essentially the fractal counterparts of all these structures are obtained in this manner. A possible reason for the practical absence of p-adic physics is probably that the existence and importance of the canonical correspondence has not been realized. The successfull p-adic description of Higgs mechanism relies heavily on canonical correspondence. In later chapters it will be found that the concepts of p-adic probability and unitarity make sense and one can associate with p-adic probabilities unique real probabilities using canonical correspondence and this predicts novel physical effects.
The topics of the chapter are following:\
i) p-Adic numbers, their algebraic extensions and canonical identification are described. The existence of square root of p-adically real number is necessary in many applications of p-adic numbers (p-adic group theory, p-adic unitarity, Riemannian geometry) and its existence implies unique algebraic extension, which is 4-dimensional in $p>2$ case and 8-dimensional in $p=2$ case.\
ii) p-Adic valued inner product necessary for various generalizations is introduced.\
iii) The concepts of p-adic differentiability and analyticity are introduced and the fractal properties of p-adically differentiable functions as well as nondeterminism of p-adic differential equations are demonstrated. It is also shown that period doubling property is characteristic feature of 2-adically differentiable functions.\
iii) The concept of p-adic valued integration is defined: this concept is necessary in order to formulate p-adic variation principles.\
iv) formulate p-adic Riemannian needed in TGD: eish applications: the existence of p-adic inner product and p-adic valued integration is essential for these developments. The dimensions of p-adic Riemann spaces are multiples of $4$ ($p>2$) or $8$ ($p=2$). It is hardly an accident that these dimensions are spacetime and imbedding space dimensions in TGD.\
v) consider some characteristic details related to p-adic counterparts of Lie-groups
p-Adic numbers
==============
p-Adic numbers ($p$ is prime: 2,3,5,... ) can be regarded as a completion of rational numbers using norm which, is different from the ordinary norm of real numbers [@2adic]. p-Adic numbers are representable as power expansion of the prime number $p$ of form:\
$$\begin{aligned}
x&=& \sum_{k \ge k_0} x(k) p^k , \ x(k) =0,....,p-1\end{aligned}$$
The norm of a p-adic number given by
$$\begin{aligned}
\vert x \vert &= &p^{-k_0(x)}\end{aligned}$$
Here $k_0(x)$ is the lowest power in the expansion of p-adic number. The norm differs drastically from the norm of ordinary real numbers since it depends on the lowest pinary digit of the p-adic number only. Arbitrarily high powers in the expansion are possible since the norm of p-adic number is finite also for numbers, which are infinite with respect to the ordinary norm. A convenient representation for p-adic numbers is in the form
$$\begin{aligned}
x&=& p^{k_0} \varepsilon (x)\end{aligned}$$
where $\varepsilon (x)=k+....$ with $0<k<p$, is p-adic number with unit norm and analogous to the phase factor $exp(i\phi)$ of complex number.
The distance function $d(x,y)= \vert x-y\vert_p$ defined by p-adic norm possesses a very general property called ultrametricity:
$$\begin{aligned}
d(x,z)&\le&max\{ d(x,y),d(y,z)\}\end{aligned}$$
The properties of the distance function make it possible to decompose the $R_p$ into a union of disjoint sets using the criterion that $x$ and $y$ belong to same class if the distance between $x$ and $y$ satisfies the condition
$$\begin{aligned}
d(x,y)&\le& D\end{aligned}$$
This division of the metric space into classes has following properties:\
a) Distances between the members of two different classes $X$ and $Y$ do not depend on the choice of points $x$ and $y$ inside classes. One can therefore speak about distance function between classes.\
b) Distances of points $x$ and $y$ inside single class are smaller than distances between different classes.\
c) Classes form a hierarchical tree.
Notice that the concept of ultrametricity emerged to Physics in the models for spin glassess and is believed to have also applications in biology [@Parisi]. The emergence of p-adic topology as effective topology of spacetime would make ultrametricy property basic feature of Physics at long length scales.
Canonical correspondence between p-adic and real numbers
========================================================
There exists a natural continuous map $Id: R_p \rightarrow R_+$ from p-adic numbers to non-negative real numbers given by the “pinary” expansion of the real number for $x\in R$ and $y \in R_p$ this correspondence reads
$$\begin{aligned}
y&=& \sum_{k>N} y_k p^k\rightarrow x=\sum_{k<N} y_k p^{-k}
\nonumber\\
y_k &\in& \{0,1,..,p-1\}\end{aligned}$$
This map is continuous as one easily finds out. There is however a little difficulty associated with the definition of the inverse map since the pinary expansion like also desimal expansion is not unique ($1= 0.999...$) for real numbers $x$, which allow pinary expansion with finite number of pinary digits
$$\begin{aligned}
x&=&\sum_{k=N_0}^{N} x_k p^{-k}\nonumber\\
x&=& \sum_{k=N_0} ^{N-1} x_k p^{-k}+
(x_N-1) p^{-N} +(p-1)p^{-N-1}\sum_{k=0,..} p^{-k} \nonumber\\
\\end{aligned}$$
The p-adic images associated with these expansions are different
$$\begin{aligned}
y_1&=&\sum_{k=N_0}^{N} x_k p^{k}\nonumber\\
y_2&=& \sum_{k=N_0}^{N-1} x_k p^k+
(x_N-1) p^{N} +(p-1)p^{N+1}\sum_{k=0,..} p^{k} \nonumber\\
&=& y_1 + ( x_N-1)p^N -p^{N+1}\end{aligned}$$
so that the inverse map is either two-valued for p-adic numbers having expansion with finite pinary digits or single valued and discontinuous and nonsurjective if one makes pinary expansion unique by choosing the one with finite pinary digits. The finite pinary digit expansion is natural choice since in applications one always must use pinary cutoff in real axis. Furthermore. p-adicity is good approximation only below (rather than above, as thought originally) some length scale, which means pinary cutoff on real axis.
What about the p-adic counterpart of negative real numbers? In TGD:eish applications this correspondence is not needed since canonical identification is used only in the direction $R_p \rightarrow R$. Furthermore, it is always possible to choose the real coordinates of finite spacetime region so that coordinate variables are nonnegative so that the problem disappears.
The canonical identification map will be crucial for the proposed applications of p-adic numbers. The topology induced by this map in the set of positive real numbers differs from ordinary topology. The difference is easily understood by interpreting the p-adic norm as a norm in the set of real numbers. The norm is constant in each interval $[p^k,p^{k+1})$ (see Fig. \[Norm\]) and is equal to the usual real norm at the points $x=
p^k$: the usual linear norm is replaced with a piecewise constant norm. This means that p-adic topology is coarser than the usual real topology and the higher the value of $p$ is, the coarser the resulting topology is above given length scale. This hierarchical ordering of p-adic topologies will be central feature as far as the proposed applications of the p-adic numbers are considered.
Ordinary continuity implies p-adic continuity since the norm induced from p-adic topology is rougher than ordinary norm. This means that physical system can be genuinely p-adic below certain length scale $L_p$ and become in good approximation real, when length scale resolution $L_p$ is used in its description. TGD:eish applications rely on this assumption. p-Adic continuity implies ordinary continuity from right as is clear already from the properties of p-adic norm (the graph of the norm is indeed continuous from right). This feature is one clear signature of p-adic topology.
The linear structure of p-adic numbers induces corresponding structure in the set of positive real numbers and p-adic linearity in general differs from the ordinary concept of linearity. For example, p-adic sum is equal to real sum only provided the summands have no common pinary digits. Furthermore, the condition $x+_p y <max\{x,y\}$ holds in general for the p-adic sum of real numbers. p-Adic multiplication is equivalent with ordinary multiplication only provided that either of the members of the product is power of $p$. Moreover one has $x
\times_p y <x \times y$ in general. The p-Adic negative $-1_p$ associated with p-adic unit 1 is given by $(-1)_p= \sum_k (p-1) p^k$ and defines p-adic negative for each real number $x$. An interesting possibility is that p-adic linearity might replace ordinary linearity in strongly nonlinear systems so that nonlinear systems would look simple in p-adic topology.
Canonical correspondence is quite essential in TGD:eish applications. Canonical identification makes it possible to define p-adic valued definite integral and this is cornerstone of TGD:eish applications. Canonical identification is in key role in the successfull predictions of the elementary particle masses. Canonical identification make also possible to undestand the connection between p-adic and real probabilities. These and many other succesfull applications suggests that canonical identification is involved with some deeper mathematical structure. The following inequalities hold true:
$$\begin{aligned}
(x+y)_R&\leq& x_R+y_R\nonumber\\
\vert x\vert_p \leq (xy)_R&\leq &x_Ry_R\end{aligned}$$
where $\vert x \vert_p$ denotes p-adic norm. These inequalities can be generalized to case of $(R_p)^n $ ( linear space over p-adic numbers).
$$\begin{aligned}
(x+y)_R&\leq& x_R+y_R\nonumber\\
\vert \lambda \vert_p\vert y\vert_R \leq (\lambda y)_R&\leq &\lambda_Ry_R\end{aligned}$$
where the norm of the vector $x\in T_p^n$ is defined in some manner. The case of Euclidian space suggests the definition
$$\begin{aligned}
(x_R)^2 &=& (\sum_n x_n^2 )_R\end{aligned}$$
These inequilities resemble those satisfied by vector norm. The only difference is failure of linearity in the sense that the norm of scaled vector is not obtained by scaling the norm of original vector. Ordinary situation prevails only if scaling corresponds to power of $p$. Amusingly, the p-adic counterpart of Minkowskian norm
$$\begin{aligned}
(x_R)^2 &=& (\sum_k x_k^2- \sum_l x_l^2 )_R\end{aligned}$$
produces nonnegative norm. Clearly the p-adic space with this norm is analogous to future light cone.
These observations suggests that the concept of normed space or Banach space might have generalization and physically the generalization might apply to the description of nonlinear system. The nonlinearity would be concentrated in the nonlinear behaviour of the norm under scaling.
Algebraic extensions of p-adic numbers
======================================
Real numbers allow only complex numbers as an algebraic extension. For p-adic numbers algebraic extensions of arbitrary dimension are possible [@2adic]. The simplest manner to construct (n+1)-dimensional extensions is to consider irreducible polynomials $P_n(t)$ in $R_p$ assumed to have rational coefficients: irreduciblity means that polynomial does not possess roots in $R_p$ so that one cannot decompose it into a product of lower order $R_p$ valued polynomials. Denoting one of the roots of $P_n(t)$ by $\theta $ and defining $\theta^0= 1$ the general form of the extension is given by
$$\begin{aligned}
Z&=& \sum_{k=0,..,n-1}x_k \theta^k\end{aligned}$$
Since $\theta$ is root of the polynomial in $R_p$ it follows that $\theta^n$ is expressible as sum of lower powers of $\theta$ so that these numbers indeed form n-dimensional linear space with respect to the p-adic topology.
Especially simple odd dimensional extensions are cyclic extensions obtained by considering the roots of the polynomial
$$\begin{aligned}
P_n(t) &=&t^n+ \epsilon d\nonumber\\
\epsilon&=& \pm 1\end{aligned}$$
For $n=2m+1$ and $(n=2m, \epsilon=+1)$ the irreducibility of $P_n(t)$ is guaranteed if $d$ does not possess $n$:th root in $R_p$. For $(n=2m,\epsilon=-1)$ one must assume that $d^{1/2}$ does not exist p-adically. In this case $\theta$ is one of the roots of the equation
$$\begin{aligned}
t^n &=& \pm d\end{aligned}$$
where $d$ is p-adic integer with finite number of pinary digits. It is possible although not necessary to identify roots as complex numbers. There exists $n$ complex roots of $d$ and $\theta$ can be chosen to be one of the real or complex roots satisfying the condition $\theta^n =\pm d$. The roots can be written in the general form
$$\begin{aligned}
\theta& = &d^{1/n} exp(i\phi (m)), \ m=0,1,....,n-1\nonumber\\
\phi (m)&= &\frac{m2\pi}{n} \ or \ \frac{m\pi}{n}\end{aligned}$$
Here $d^{1/n}$ denotes the real root of the equation $\theta^n=d$. Each of the phase factors $\phi (m)$ gives rise to algebraically equivalent extension: only the representation is different. Physically these extensions need not be equivalent since the identification of p-adic numbers with complex numbers plays fundamental role in the applications. The cases $\theta^n = \pm d$ are physically and mathematically quite different.
The norm of an algebraically extended p-adic number $x$ can be defined as some power of the ordinary p-adic norm of the determinant of the linear map $ x: ^eR_p^n \rightarrow ^eR_p^n$ defined by the multiplication with $x$: $y \rightarrow xy$
$$\begin{aligned}
N(x) &=& \vert det(x)\vert^{\alpha}, \ \alpha >0 \nonumber\\
\\end{aligned}$$
The requirement that norm is homogenous function of degree one in the components of the algebraically extended 2-adic number (like also the standard norm of $R^n$ ) implies the condition $\alpha=1/n$, where $n$ is the dimension of the algebraic extension.
The canonical correspondence between the points of $R_+$ and $R_p$ generalizes in obvious manner: the point $\sum_k x_k\theta ^k$ of algebraic extension is identified as the point $ (x^0_R, x^1_R,...,x^k_R, ..,)$ of $R^n $ using the pinary expansions of the components of p-adic number. The p-adic linear structure of the algebraic extension induces linear structure in $R_+^n$ and p-adic multiplication induces multiplication for the vectors of $R_+^n$. An exciting possibility is that p-adic linearity might replace ordinary linearity in strongly nonlinear systems.
Algebraic extension allowing square root on p-adic real axis
============================================================
The existence of square root of «real« p-adic number is a common theme in various applications of p-adic numbers.\
a) The p-adic generalization of the representation theory of ordinary groups and Super Kac Moody and Super Virasoro algebras exists provided an extension of p-adic numbers allowing square roots of «real« p-adic numbes is used. The reason is that matrix elements of the raising and lowering operators in Lie-algebras as well as oscillator operators typically involve square roots.\
b) The existence of square root of «real« p-adic number is also necessary ingredient in the definition of p-adic unitarity and quantum probability concepts since the solution of the requirement that $p_{mn}=
S_{mn}\bar {S}_{mn} $ is p-adically real leads to expressions involving square roots.\
c) p-Adic Riemannian geometry necessitates the existence of square root of «real« p-adic numbers since the definition of the infinitesimal length $ds=
\sqrt{g_{ij}dx^idx^j} $ involves square root.\
What is important is that only the square root of p-adically real numbers is needed: the square root need not exist outside the real axis. It is indeed impossible to find finite dimensional extension allowing square root for all numbers of the extension. For $p>2$ the minimal dimension for algebraic extension allowing square roots near real axis is $D=4$. For $p=2 $ the dimension of the extension is $D=8$.
For $p>2$ the form of the extension can be derived by the following arguments.\
a) For $p>2 $ p-adic number $y$ in the range $(0,p-1)$ allows square root only provided there exists p-adic number $x\in \{0,p-1\}$ satisfying the condition $y= x^2 \ mod \ p$. Let $x_0$ be the smallest integer, which does not possess p-adic square root and add the square root $\theta$ of $x_0$ to the number field. The numbers in the extension are of the form $x+
\theta y$. The extension allows square root for every $x\in \{0,p-1\}$ as is easy to see. p-adic numbers $mod \ p$ form a finite field $G(p,1)$ [@2adic] so that any p-adic number $y$, which does not possess square root can be written in the form $ y= x_0 u$, where $u$ possesses square root. Since $\theta$ is by definition the square root of $x_0$ then also $y$ possesses square root. The extension does not depend on the choice of $x_0$.
The square root of $-1$ does not exist for $ p \ mod \ 4 = 3$ [@Number] and $p=2 $ but the addition of $\theta$ gurantees its existence automatically. The existence of $\sqrt{-1}$ follows from the existence of $\sqrt{p-1}$ implied by the extension by $\theta$. $\sqrt{(-1+p) -p}$ can be developed in power in powers of $p$ and series converges since the p-adic norm of coefficients in Taylor series is not larger than $1$. If $p-1$ doesn not possess square one can take $\theta$ to be equal to $\sqrt{-1}$.
b\) The next step is to add square root of $p$ so that extension becomes 4-dimensional and arbitrary number in the extension can be written as
$$\begin{aligned}
Z&=& (x+\theta y) +\sqrt {p}( u +\theta v)\end{aligned}$$
This extension is natural for p-adication of spacetime surface so that spacetime can be regarded as a number field locally. An important point to notice that the extension guarantees the existence of square for «real« p-adic numbers only.
c\) In $p=2$ case 8-dimensional extension is needed to define square roots. The addition of $\sqrt{2}$ implies that one can restrict the consideration to the square roots of odd 2-adic numbers. One must be careful in defining square roots by the Taylor expansion of square root $\sqrt{x_0+x_1} $ since $n$:th Taylor coefficient is proportional to $2^{-n}$ and possesses 2-adic norm $2^n$. If $x_0$ possesses norm $1$ then $x_1$ must possess norm smaller than $1/8$ for series to converge. By adding square roots $\theta_1=\sqrt{-1},\theta_2= \sqrt{2}$ and $\theta_3=\sqrt{3}$ and their products one obtains 8-dimensional extension. In TGD imbedding space $H=M^4_+ \times CP_2$ can be regarded locally as 8-dimensional extension of p-adic numbers. It is probably not an accident that the dimensions of minimal extensions allowing square roots are the space time and imbedding space dimensions of TGD.
By construction any p-adically real number in the extension allows square root. The square root for an arbitrary number sufficiently near real axis can be defined through Taylor series expansion of the square root function $\sqrt{Z}$ in point of p-adic real axis. The subsequent considerations show that the p-adic square root function does not allow analytic continuation to $R^4$ and the points of extension allowing square root form a set consisting of disjoint converge cubes of square root function forming structure resembling lightcone.
p-Adic square root function for $p>2$
-------------------------------------
The study of the properties of the series representation of square root function shows that the definition of square root function is possible in certain region around real p-adic axis. What is nice that this region can be regarded as the p-adic counterpart of the future light cone defined by the condition
$$\begin{aligned}
N_p(Im(Z))&<& N_p(t=Re(Z))=p^k\end{aligned}$$
where the real p-adic coordinate $t=Re(Z)$ is identified as time coordinate and the imaginary part of the p-adic coordinate is identified as spatial coordinate. p-Adic norm for four-dimensional extension is analogous to ordinary Euclidian distance. p-Adic light cone consists of «cylinders« parallel to time axis having radius $N_p(t)= p^k$ and length $p^{k-1}(p-1)$: at points $t= p^k$. As a real space (recall the canonical correspondence) the cross section of the cylinder corresponds to parallelpiped rather than ball.
The result can be understood heuristically as follows.\
a) For four-dimensional extension allowing square root ($p>2$) one can construct square root at each p-adically real point $x(k,s)= sp^k$, $s=1,...,p-1$, $k\in Z$. The task is to show that by using Taylor expansion one can define square root also in some neighbourhood of each of these points and find the form of this neighbourhood.\
b) Using the general series expansion of the square root function one finds that the convergence region is p-adic ball defined by the condition
$$\begin{aligned}
N_p(Z-sp^k) &\leq &R(k)\end{aligned}$$
and having radius $R(k) = p^d, d \in Z$ around the expansion point.\
c) A purely p-adic feature is that the convergence spheres associated with two points are either disjoint or identical! In particular, the convergence sphere $B(y)$ associated with any point inside convergence sphere $B(x)$ is identical with $B(x)$: $B(y)= B(x)$. The result follows directly from the ultrametricity of the p-adic norm. The result means that stepwise analytic continuation is not possible and one can construct square root function only in the union of p-adic convergence spheres associated with the p-adically real points $x(k,s)=sp^k$.\
d) By the scaling properties of the square root function the convergence radius $R(x(k,s))\equiv R(k)$ is related to $R(x(0,s))\equiv R(0)$ by the scaling factor $p^{-k}$:
$$\begin{aligned}
R(k) &=& p^{-k}R(0)\end{aligned}$$
so that convergence sphere expands as a function of p-adic time coordinate. The study of convergence reduces to the study of the series at points $x=s=1,...,k-1$ with unit p-adic norm.\
e) Two neighbouring points $x=s$ and $x=s+1$ cannot belong to same convergence sphere: this would lead to contradiction with basic results of about square root function at integer points. Therefore the convergence radius satisfies the condition
$$\begin{aligned}
R(0)&<&1\end{aligned}$$
The requirement that convergence is achieved at all points of the real axis implies
$$\begin{aligned}
R(0)&=&\frac{1}{p}\nonumber\\
R(p^ks)&=& \frac{1}{p^{k+1}}\end{aligned}$$
If the convergence radius is indeed this then the region, where square root is defined corresponds to a connected light cone like region defined by the condition $ N_p(Im(Z))= N_p(Re(Z))$ and $p>2$-adic space time is p-adic counterpart of $M^4$ light cone. If convergence radius is smaller the convergence region reduces to a union of disjoint p-adic spheres with increasing radii.
How the p-adic light cone differs from the ordinary light cone can be seen by studying the explicit form of the p-adic norm for $p>2$ square root allowing extension $Z=x+iy+\sqrt{p}(u+iv)$
$$\begin{aligned}
N_p(Z) &=& (N_p(det(Z)))^{\frac{1}{4}}\nonumber\\
&=& (N_p((x^2+y^2)^2+2p^2((xv-yu)^2+
(xu-yv)^2)+p^4(u^2+v^2)^2))^{\frac{1}{4}}\nonumber\\
\\end{aligned}$$
where $det(Z)$ is the determinant of the linear map defined by multiplication with $Z$. The definition of convergence sphere for $x=s$ reduces to
$$\begin{aligned}
N_p(det(Z_3))&=&N_p(y^4+2p^2y^2(u^2+v^2)+p^4(u^2+v^2)^2))<1\end{aligned}$$
For physically interesting case $p \ mod \ 4=3$ the points $(y,u, v)$ satisfying the conditions
$$\begin{aligned}
N_{p}(y)&\leq&\frac{1}{p}\nonumber\\
N_p(u)&\leq& 1 \nonumber\\
N_p(v)&\leq &1\end{aligned}$$
belong to the sphere of convergence: it is essential that for all $u$ and $v$ satisfying the conditions one has also $N_p(u^2+v^2)\leq
1$. By the canonical correspondence between p-adic and real numbers the real counterpart of the sphere $r=t$ is now parallelpiped $0\leq y<1,0\leq
u<p,0\leq v<p$, which expands with average velocity of light in discrete steps at times $t=p^k$.
The emergence of p-adic light cone as a natural p-adic coordinate space is in nice accordance with the basic assumptions about the imbedding space of TGD and shows that big bang cosmology might basically related to the existence of p-adic square root! The result gives also support for the idea that p-adicity is responsible for the generation of lattice structures (convergence region for any function is expected to be more or less parallelpiped like region).
A peculiar feature of the p-adic light cone is the instantaneous expansion of 3-space at moments $t_p= p^k$. A possible physical interpretation is that p-adic light cone or rather the individual convergence cube of the light cone represents the time development of single maximal quantum coherent region at p-adic level of topological condensate (probably there are many of them). The instantaneous scaling of the size of region by factor $\sqrt{p}$ at moment $t_R= p^{k/2}$ corresponds to a phase transition and thus to quantum jump. Experience with p-adic QFT indeed shows that $L_p=\sqrt{p}L_0$ ,$L_0\simeq 10^4\sqrt{G}$ appears as infrared cutoff length for the p-adic version of standard model so that p-adic continuity is replaced with real continuity (implying p-adic continuity) above the length scale $L_p$. The idea that larger p-adic length scales $p^{k/2}L_p$, $k>0$, would form quantum coherent regions for physically most interesting values of $p$ is probably unrealistic and $L_p$ probably gives a typical size of 3-surface at p:th condensate level. Of course, already this hypothesis is far from trivial since $L_p$ can have arbitrarily large values so that arbitrarily large quantum coherent systems would be possible.
$p=M_{127}$, the largest physically interesting Mersenne prime, provides an interesting example:\
i) $p=M_{127}$-Adic light cone does not make sense for time and length scales smaller than length scale defined by electron Compton length and QFT below this length scale makes sense.\
ii) The first phase transition would happen at time of order $10^{-1}$ seconds, which corresponds to length scales of order $10^7 $ meters.\
iii) The next phase transition takes place at $t_R\simeq 10^{11}$ light years and corresponds to the age of the Universe.
Recent work with the p-adic field theory limit of TGD has shown that the convergence cube of p-adic square root function having size $L_p= \frac{L_0}{\sqrt{p}}$, $L_0= 1.824\cdot 10^4\sqrt{G}$, serves as a natural quantization volume for p-adic counterpart of standard model. An open question is whether also larger convergence cubes serve as quantization volumes or whether $L_p$ gives natural upper bound for the size of p-adic 3-surfaces. The original idea that p-adic manifolds, constructed by gluing together pieces of p-adic light cone together along their sides, could be used to build Feynmann graphs with lines thickened to 4-manifolds has turned out to be not useful for physically most interesting (large) values of $p$.
Convergence radius for square root function
-------------------------------------------
In the following it will be shown that the convergence radius of $\sqrt{t+Z}$ is indeed nonvanishing for $p>2$. The expression for the Taylor series of $\sqrt{t+Z}$ reads as
$$\begin{aligned}
\sqrt{t+Z}&=& = \sqrt{x}\sum_n a_n\nonumber\\
a_n&=& (-1)^n\frac{(2n-3)!!}{2^nn!} x^n\nonumber\\
x&=&\frac{Z}{t}
\end{aligned}$$
The necessary criterion for the convergence is that the terms of the power series approach to zero at the limit $n\rightarrow \infty$. The p-adic norm of $n$:th term is for $p>2$ given by
$$\begin{aligned}
N_p(a_n)&=& N_p(\frac{(2n-3)!!}{n!}) N_p(x^n)<N_p(x^n)N_p(\frac{1}{n!})\end{aligned}$$
The dangerous term is clearly the $n!$ in the denominator. In the following it will be shown that the condition
$$\begin{aligned}
U&\equiv &\frac{N_p(x^n)}{N_p(n!)} <1 \ for \ \ N_p(x)<1\end{aligned}$$
holds true. The strategy is as follows:\
a) The norm of $x^n$ can be calculated trivially: $ N_p(x^n) =p^{-Kn}, K\ge 1$.\
b) $N_p(n!) $ is calculated and an upper bound for $U$ is derived at the limit of large $n$.
### p-Adic norm of $n!$ for $p>2$
Lemma 1: Let $n= \sum_{i=0}^{k}n(i)p^i$, $ 0\le n(i)<p$ be the p-adic expansion of $n$. Then $N_p(n!)$ can be expressed in the form
$$\begin{aligned}
N_p(n!)&=& \prod_{i=1}^{k} N(i)^{n(i)}\nonumber\\
N(1)&=&\frac{1}{p}\nonumber\\
N(i+1)&=& N(i)^{p-1}p^{-i}\end{aligned}$$
An explicit expression for $N(i)$ reads as
$$\begin{aligned}
N(i)&=& p^{-\sum_{m=0}^{i} m(p-1)^{i-m} }\end{aligned}$$
Proof: $n!$ can be written as a product
$$\begin{aligned}
N_p(n!) &=& \prod_{i=1}^{k}X(i,n(i) )\nonumber\\
X(k,n(k))&=& N_p((n(k)p^k)!)\nonumber\\
X(k-1,n(k-1))&=&N_p(\prod_{i=1}^{n(k-1)p^{k-1}}(n(k)p^k+i))=
N_p( (n(k-1)p^{k-1})!)\nonumber\\
X(k-2,n(k-2))&=&N_p(\prod_{i=1}^{n(k-2)p^{k-2}}(n(k)p^k+n(k-1)p^{k-1}+i
))\nonumber\\
&=& N_p((n(k-2)p^{k-2})!)\nonumber\\
X(k-i,n(k-i))&=& N_p((n(k-i)p^{k-i})!)\end{aligned}$$
The factors $X(k,n(k))$ reduce in turn to the form
$$\begin{aligned}
X(k,n(k))&=& \prod_{i=1}^{n(k)}Y(i,k )\nonumber\\
Y(i,k)&=& \prod_{m=1}^{p^k} N_p(ip^k+m)\end{aligned}$$
The factors $Y(i,k)$ in turn are indentical and one has
$$\begin{aligned}
X(k,n(k))&=& X(k)^{n(k)}\nonumber\\
X(k)&=& N_p(p^k!)\end{aligned}$$
The recursion formula for the factors $X(k)$ can be derived by writing explicitely the expression of $N_p(p^k!)$ for a few lowest values of $k$:\
1) $X(1)= N_p(p!) = p^{-1}$\
2) $ X(2) = N_p(p^2!)= X(1)^{p-1}p^{-2} $ ( $p^2!$ decomposes to $p-1$ products having same norm as $p!$ plus the last term equal to $p^2$.\
i) $ X(i)= X(i-1)^{p-1}p^{-i}$
Using the recursion formula repeatedly the explicit form of $X(i)$ can be derived easily. Combining the results one obtains for $N_p(n!)$ the expression
$$\begin{aligned}
N_p(n!)&=& p^{-\sum_{i=0}^{k}n(i) A(i)}\nonumber\\
A(i)&=& \sum_{m=1}^{i} m(p-1)^{i-m}\end{aligned}$$
The sum $A(i)$ appearing in the exponent as the coefficient of $n(i)$ can be calculated by using geometric series
$$\begin{aligned}
A(i)&=& (\frac{p-1}{p-2})^2
(p-1)^{i-1}(1+ \frac{i}{(p-1)^{i+1}}- \frac{(i+1)}{(p-1)^i})\nonumber\\
&\leq& (\frac{p-1}{p-2})^2(p-1)^{i-1}\end{aligned}$$
### Upper bound for $N_p(\frac{x^n}{n!})$ for $p>2$
By using the expressions $n= \sum_i n(i)p^i$, $N_p(x^n)=p^{-Kn}$ and the expression of $N_p{n!}$ as well as the upper bound
$$\begin{aligned}
A(i)
&\leq& (\frac{p-1}{p-2})^2(p-1)^{i-1}\end{aligned}$$
for $A(i)$ one obtains the upper bound
$$\begin{aligned}
N_p(\frac{x^n}{n!}) \leq
p^{-\sum_{i=0}^{k}n(i) p^i(K-(\frac{(p-1)}{(p-2)} )^2
(\frac{(p-1)}{p})^{i-1})} \nonumber\\
\\end{aligned}$$
It is clear that for $N_p(x)<1$ that is $K\ge1$ the upper bound goes to zero. For $p>3$ exponents are negative for all values of $i$: for $p=3$ some lowest exponents have wrong sign but this does not spoil the convergence. The convergence of the series is also obvious since the real valued series $\frac{1}{1-\sqrt{N_p(x)}}$ serves as majorant.
$p=2$ case
----------
In $p=2$ case the norm of a general term in the series of the square root function can be calculated easily using the previous result for the norm of $n!$:
$$\begin{aligned}
N_p(a_n)&=& N_p(\frac{(2n-3)!!}{2^nn!}) N_p(x^n)= 2^{-(K-1)n+\sum_{i=1}^{k}
n(i)\frac{i(i+1)}{2^{i+1} }}\end{aligned}$$
At the limit $n\rightarrow \infty$ the sum term appearing in the exponent approaches zero and convergence condition gives $ K>1$ so that one has
$$\begin{aligned}
N_p(Z)&\equiv& (N_p(det(Z)))^{\frac{1}{8}}\leq\frac{1}{4}\end{aligned}$$
The result does not imply disconnected set of convergence for square root function since the square root for half odd integers exists:
$$\begin{aligned}
\sqrt{s+\frac{1}{2}}= \frac{\sqrt{2s+1}}{\sqrt{2}}\end{aligned}$$
so that one can develop square as series in all half odd integer points of p-adic real axis. As a consequence the structure for the set of convergence is just the 8-dimensional counterpart of the p-adic light cone. Spacetime has natural binary structure in the sense that each $N_p(t)=
2^k$ cylinder consists of two identical p-adic 8-balls (parallelpipeds as real spaces). Since $\sqrt{Z}$ appears in the definition of the fermionic Ramond fields one might wonder whether once could interpret this binary structure as a geometric representation of half odd integer spin. The coordinate space associated with spacetime representable as a four-dimensional subset of this light cone inherits the light cone structure.
p-Adic inner product and Hilbert spaces
----------------------------------------
Concerning the physical applications of complex p-adic numbers the problem is that p-adic norm is not bilinear in its arguments and therefore it does not define inner product and angle. One can however consider a generalization of the ordinary complex inner product $\bar{z} z$ to p-adic valued inner product. It turns out that p-adic quantum mechanics in the sense as it is used in p-adic TGD can be based on this inner product.
Restrict the consideration to minimal extension allowing square roots near real axis ($p>2$) and denote the complex conjugate of $Z$ with $Z_c$ and by $\hat{Z}$ the conjugate of $Z$ under the conjugation $\sqrt{p}\rightarrow -\sqrt{p}$: $Z\rightarrow \hat{Z} = x+\theta x-
\sqrt{p}(u+\theta v)$. The inner product in the 4-dimensional extension of p-adic numbers reads as
$$\begin{aligned}
\langle Z,Z\rangle &=& Z_cZ+ \hat{Z}_c\hat{Z}
=2(x^2+y^2+ p(u^2+v^2))\end{aligned}$$
This inner product is bilinear and symmetric, defines p-adically real norm and vanishes only if $Z$ vanishes. This inner product leads to p-adic generalization of unitarity and probability concept. The solution of the unitarity condition $\sum_k S_{mk}\bar{S}_{nk}= \delta (m,n)$ involves square root operations and therefore the minimal extension for the Hilbert space is $4$-dimensional in $p>2$ case and $8$-dimensional in $p=2$ case. The physically most interesting consequences of this result are encountered in p-adic quantum mechanics.
The inner product associated with minimal extension allowing square root near real axis provides a natural generalization of the real and complex Hilbert spaces respectively. Instead of real or complex numbers square root allowing algebraic extension extension appears as the multiplier field of the Hilbert space and one can understand the points of Hilbert space as infinite sequences $(Z_1,Z_2,...,Z_n,....) $, where $Z_i$ belongs to the extension. The inner product $\sum_k \langle Z^1_k,Z_2^k\rangle $ is completely analogous to the ordinary Hilbert space inner product.
A particular example of p-adic Hilbert space is obtained as a generalization of the space of complex valued functions $f: R^n\rightarrow C$. The inner product for functions $f_1$ and $f_2$ is just the previously defined inner product $\langle f_1(x) ,f_2(x)\rangle$ combined with integration over $R_p^n$: the definition of p-adic integration will be considered later in detail.
The inner product allows to define the concepts of length and angle for two vectors in p-adic extension possessing either p-adic or ordinary complex values. This implies that the concepts and of p-adic Riemannian metric, Kähler metric and conformal invariance become possible.
p-Adic Numbers and Finite Fields
--------------------------------
Finite fields (Galois fields) consists of finite number of elements and allow sum, multiplication and division. A convenient representation for the elements of a finite field is as the roots of the polynomial equation $t^{p^m}-t=0 \ mod \ p$ , where $p$ is prime, $m$ an arbitrary integer and $t$ is element of a field of characteristic $p$ ($pt=0$ for each $t$). The number of elements in finite field is $p^m $, that is power of prime number and the multiplicative group of a finite field is group of order $p^m-1$. $G(p,1)$ is just cyclic group $Z_p$ with respect to addition and $G(p,m)$ is in rough sense $m$:th Cartesian power of $G(p,1)$ .
The elements of the finite field $G(p,1)$ can be identified as the p-adic numbers $0,...,p-1$ with p-adic arithmetics replaced with modulo p arithmetics. Finite fields $G(p,m)$ can be obtained from m-dimensional algebraic extensions of p-adic numbers by replacing p-adic arithmetics with modulo p arithmetics. In TGD context only the finite fields $G(p>2,2)$ , $p \ mod \ 4=3$ and $G(p=2,4)$ appear naturally. For $p>2$, $p \ mod \ 4 =3$ one has: $x+iy+\sqrt{p}(u+iv) \rightarrow x_0+iy_0 \in G(p,2) $.
As far as applications are considered the basic observation is that the unitary representations of p-adic scalings $x\rightarrow p^kx$ $k \in Z$ lead naturally to finite field structures. These representations reduce to representations of finite cyclic group $Z_m$ if $x\rightarrow p^m x$ acts trivially on representation functions for some value of $m$, $m=1,2,..$. Representation functions, or equivalently the scaling momenta $k=0,1,...,m-1$ labeling them, have a structure of cyclic group. If $m \neq p $ is prime the scaling momenta form finite field $G(m,1)=Z_m$ with respect to summation and multiplication modulo $m$. The construction p-adic field theory shows that also the p-adic counterparts of ordinary planewaves carrying p-adic momenta $k=0,1...,p-1$ can be given the structure of Finite Field $G(p,1)$: one can also define complexified planewaves as square roots of the real p-adic planewaves to obtain Finite Field $G(p,2)$.
p-Adic differential calculus
============================
It would be nice to have a generalization of the ordinary differential and integral calculus to p-adic case. Instead of trying to guess directly the formal definition of p-adic differentiability it is better to guess what kind of functions $f:R_p \rightarrow R$ might be natural candidates for p-adically differentiable functions and then try to find whether the concept of p-adic differentiability makes sense. There are several candidates for p-adically differentiable functions.\
a) p-Analytic maps $R_p \rightarrow R_p$ representable as power series of p-adic argument induce via the canonical identification maps $R_+ \rightarrow R_+$. These maps are well defined for algebraic extensions of p-adic numbers, too and induce p-analytic maps $R_+^n \rightarrow R_+^n$ via the canonical correspondence. These functions correspond to ordered fractals.\
b) A second candidate is obtained as a generalization of canonical identification map $R_p \rightarrow R$: $Y_D(x)= \sum x_k p^{-kD}$, where $D $ is so call anomalous dimension. The corresponding map $R\rightarrow R$ is given by $\sum_k x_k p^{-k}
\rightarrow \sum x_k p^{-kD}$: $D=1$ gives identity map. These functions are not differentiable in the strict sense of the word and give rise to chaotic fractals, which resemble Brownian functions.
p-Analytic maps
---------------
p-analytic maps $g: R_p \rightarrow R_p$ satisfy the usual criterion of differentiability and are representable as power series
$$\begin{aligned}
g(x)&=& \sum_k g_k x^k\end{aligned}$$
Also negative powers are in principle allowed. The rules of p-adic differential calculus are formally identical to those of the ordinary differential calculus and generalize in trivial manner for algebraic extensions.
The class of p-adically constant functions (in the sense that p-adic derivative vanishes) is larger than in real case: any function depending on finite number of positive pinary digits of p-adic number and of arbitrary number of negative pinary digits is p-adically constant. This becomes obvious, when one considers the definition of p-adic derivative: when the increment of p-adic coordinate becomes sufficiently small p-adic constant doesn’t detect the variation of $x$ since it depends on finite number of positive p-adic pinary digits only. p-adic constants correspond to real functions, which are constant below some length scale $\Delta x=
2^{-n}$. As a consequence p-adic differential equations are nondeterministic: integration constants are arbitrary functions depending on finite number of positive p-adic pinary digits. This feature is central as far applications are considered.
p-Adically analytic functions induce maps $R_+ \rightarrow R_+$ via the canonical identification map. The simplest manner to get some grasp on their properties is to plot graphs of some simple functions (see Fig. \[square\] for the graph of p-adic $x^2$ and for Fig. \[oneoverx\]) for the graph of p-adic $1/x$). These functions have quite characteristic features resulting from the special properties of p-adic topology:\
a) p-Analytic functions are continuous and differentiable from right: this peculiar asymmetry is a completely general signature of p-adicity. As far as time dependence is considered the interpretation of this property as mathematical counterpart of irreversibelity looks natural. This suggests that the transition from reversible microscopic dynamics to irreversible macroscopic dynamics corresponds to the transition from the ordinary topology to effective p-adic topology.\
b) There are large discontinuities associated with the points $x=p^n$. This implies characteristic theshold phenomena. Consider a system whose output $f(n)$ is function of input, which is integer $n$. For $n<p$ nothing peculiar happens but for $n=p$ the real counterpart of the output becomes very small for large values of $p$. In biosystems threshold phenomena are typical and p-adicity might be the key in their understanding. The discontinuities associated with powers of $p=2$ are indeed encountered in many physical situations. Auditory experience has the property that given frequency $\omega_0$ and its multiples $2^k \omega_0$, octaves, are experienced as same frequency suggesting the auditory response function for a given frequency $\omega_0$ is 2-adicallly analytic function. Titius-Bode law states that the mutual distances of planets come in powers of $2$, when suitable unit of distance is used. In turbulent systems period doubling spectrum has peaks at frequencies $\omega = 2^k \omega_0$.\
c) A second signature of p-adicity is “p-plicity” appearing in the graph of simple p-analytic functions. As an example, consider the graph of p-adic $x^2$ demonstrating clearly the decomposition into $p$ steps at each interval $[p^k,p^{k+1})$.\
d) The graphs of p-analytic functions are in general ordered fractals as the examples demonstrate. For example, power functions $x^n$ are selfsimilar (the values of the function at some any interval $(p^k,p^{k+1})$ determines the function completely) and in general p-adic $x^n$ with nonnegative (negative) $n$ is smaller (larger) than real $x^n$ expect at points $x= p^n$ as the graphs of p-adic $x^2$ and $1/x$ show (see Fig. \[oneoverx\]) These properties are easily understood from the properties of p-adic multiplication. Therefore the first guess for the behaviour of p-adically analytic function is obtained by replacing $x$ and the coefficients $g_k$ with their p-adic norms: at points $x= p^n$ this approximation is exact if the coefficients of the power series are powers of $p$. This step function approximation is rather reasonable for simple functions such as $x^n$ as the figures demonstrate. Since p-adically analytic function can be approximated with $f(x)\sim f(x_0)+b(x-x_0)^n$ or as $a(x-x_0)^n$ (allowing nonanalyticity at $x_0$) around any point the fractal associated with p-adically analytic function has universal geometrical form in sufficiently small length scales.
p-Adic analyticity is well defined for the algebraic extensions of $R_p$, too. The figures \[real\] and \[imag\] visualize the behaviour of real and imaginary parts of two adic $z^2$ function as function of real $x$ and $y$ coordinates in the parallelpiped $I^2$,$I= [1+2^{-7},2-2^{-7}]$. An interesting possibility is that the order parameters describing various phases of physical system are p-adically differentiable functions. The p-analyticity would therefore provide a means for coding the information about ordered fractal structures.
The order parameter could be one coordinate component of a p-adically analytic map $R^n\rightarrow R^n$, $n=3,4$. This is analogous to the possibility to regard the solution of Laplace equation in 2 dimensions as a real or imaginary part of an analytic function. A given region $V$ of the order parameter space corresponds to a given phase and the volume of ordinary space occupied by this phase corresponds to the inverse image $g^{-1}(V)$ of $V$. Very beautiful images are obtained if the order parameter is the the real or imaginary part of p-adically analytic function $f(z)$. A good example is p-adic $z^2$ function in the parallelpiped $[a,b]\times [a,b]$, $a=1+2^{-9}$, $ b=2-2^{9}$ of $C$-plane. The value range of the order parameter can divided into, say, $16$ intervals of same length so that each interval corresponds to a unique color. The resulting fractals possess features, which probably generalize to higher dimensional extensions.\
a) The inverse image is ordered fractal and possessess lattice/ cell like structure, with the sixes of cells appearing in powers of $p$. Cells are however not identical in analogy with the differentiation of biological cells.\
b) p-Analyticity implies the existence of local vector valued order parameter given by the p-analytic derivative of $g(z)$: the geometric structure of the phase portrait indeed exhibits the local orientation clearly.\
c) In a given resolution there appear 0,1, and 2-dimensional structures and also defects inside structures. In 3-dimensional situation rather rich structures are to be expected.
Even more beautiful structures are obtained by adding some disorder: for instance, the composite map $z(x,y)= Y_D(x^2-y^2) $ for $D=1/2$ for $p=2$, where the function $Y_D(x)$ is defined in the next section gives rise to extremely beatiful fractal using the previous description. Noncolored pictures cannot reproduce the beauty of these fractals not suggested by the expectations based on the appearence of the graphs of $Y_D(x)$ (see Fig. \[yydee\]) (the MATLAB programs needed to generate p-adic fractals are supplied by request for interested reader).
These observations suggests that p-analyticity might provide a means to code the information about ordered fractal structures in the spatial behaviour of order parameters (such as entzyme concentrations in biosystems). An elegant manner to achieve this is to use purely real algebraic extension for 3-space coordinates and for the order parameter: the image of the order parameter $ \Phi= \phi_1+ \phi_2\theta +\phi_3\theta^2$ under the canonical identification is real and positive number automatically and might be regarded as concentration type quantity.
Functions $Y_D$
---------------
p-Analytic functions give rise to ordered fractals. One can find also functions describing disordered fractals. The simplest generalization of the identification of real and p-adic numbers to a chaotic fractal is the following one
$$\begin{aligned}
Y_D(x_p)&=& \sum_n x(n) p^{-nD}\end{aligned}$$
where $D$ is constant. $D=1$ gives identication map. $Y_D$ defines in obvious manner a map $R_+\rightarrow R_+$ via the canonical identification map. To each pinary digit there corresponds the power $p^{-kD}$ so that a change of single pit induces change of form $p^{-kD}$ nonlinear in the increment $\vert dx_p \vert $.
One can generalize the definition of $Y_D$ . The anomalous dimension $D$ can be p-adic constant and therefore depend on finite number of positive p- adic pinary digits of $x_p$ and the most general definition of $Y_D$ reads
$$\begin{aligned}
Y_D(x_p)&=& \sum_n x(n) p^{-nD(x_{<n+1})}\nonumber\\
D&=& D(x_p)\nonumber\\
x_{<n} &=& \sum_{k<n} x_kp^k\end{aligned}$$
Here it is essential to assume that the anomalous dimension associated with n:th pinary digit is the value of anomalous dimension associated with n:th pinary digit cutoff of $x$: otherwise p-adic continuity is lost. This generalization allows also fractal functions, which become ordinary smooth functions in sufficiently small length scales: the only assumption needed is that $D(x)$ approaches $D=1$, when the number of pinary digits of $x$ becomes large. The definition of the functions $Y_D$ generalizes in trivial manner to higher dimensional case, the anomalous dimensions being now p-adically constant functions of all p-adic coordinates.
Although the functions $Y_D$ are not differentiable in the strict sense of the word they have the property that if $x_p$ has finite number of nonvanishing pinary digits then for sufficiently small increment $dx_p$ so that $x_p$ and $dx_p$ have no common pinary digits one has just
$$\begin{aligned}
Y_D(x_p+dx_p)- Y_D(x_p) &=& Y_D (dx_p)\end{aligned}$$
$Y_D(dx_p)$ might be called $D$-differential with anomalous dimension $D$. This differential maps the tangent space of p-adic numbers to the tangent space of real numbers in fractal like manner in the sense that if p-adic and real tangent spaces are identified in canonical manner then $D$-differential induces nonlinear fractal like map of real tangent space to itself. The functions $Y_D(dx_p)$ are therefore good candidate for a fractal like generalization of linear differential $dx_p$.
The local anomalous dimension $D$ corresponds to the so called Lifschitz-Hölder exponent $\alpha$ encountered in the theory of multifractals [@Frac]. Multifractals are decomposed into union of fractals with various fractal dimensions by decomposing the range $S$ of fractal function to a union $ \cup_{\alpha} S_{\alpha}$ of disjoint sets $S_{\alpha}$ : $S_{\alpha}$ consists of points of $S$, for which the anomalous dimension $D$ has fixed value $\alpha$. One can associate to each set $S_{\alpha}$ its own fractal dimension and this decomposition plays important role in fractal analysis of the empirical data. The values of $D>1$ and $D<1$ one correspond to “antifractal” (ordinary derivative vanishes) and fractal (ordinary derivative is divergent) behavior respectively.
The simplest manner to see the fractality properties is to plot the graph of $Y_D$. The general features of the graph (see. Fig. \[yydee\] ) are following:\
a) $Y_D$ is continuous from right and there are sharp discontinities associated with the points $x= p^m$. The graph of $Y_D$ is selfsimilar if $D$ is constant. The value of $p$ reflects itself as a characteristic “p-peakedness” for $D<1$ and “p-stepness” for $D<1$.\
b) For $D<1$ $Y_D$ is surjective but not injective. This is seen as typical fluctuating behaviour resembling that associated with Brownian motion. If $D<1$ is constant there is infinite number of preimages associated with a given point $y$.\
c) For $D>1$ $Y_D$ is constant almost everywhere, nonsurjective, and increases monotonically.\
d) $x=0$ and $x=1$ are fixed points common to all $Y_D$. These points are attractors for $D<1$ and repellors for $D>1$. If $D<1$ $Y_D$ has also additional fixed points $x>1$ in the neighbourhood $x=1$.
The graph of $Y_D$, $D<1$ resembles that of Brownian motion. The following arguments suggest that there is more than a mere analogy involved and that functions $Y_D$ with p-adically constant $D$ combined with ordinary differentiable functions might provide a description for random processess.\
a) $Y_D$ equals to $p^{kD}$ at points $x=p^D$. For $D=1/2$ this means that $Y_D$ is analogous to the root mean square distance $d(t)=\sqrt{\langle r^2\rangle } (t)$ from origin in Brownian motion, which behaves as $ d \propto \sqrt{t}$.\
b) In Brownian motion $d(t)$ is not differentiable function at origin: $d(t) \propto t^{1/2}$. The same holds true for $Y_D$,$D=1/2$ at each point $x$ so that $Y_D$ in certain sense provides a simulation of Brownian motion.\
c) $Y_D$ is only the simplest example of Brownian looking motion and as such too simple to describe realistic situations. It is however possible to form composites of $Y_D$ and p-adically differentiable functions as well as ordinary differentiable functions, which both are right differentiable with anomalous dimension $D=1$. These functions contain also p-adic constants, which depend on finite number of pinary digits of $t$ in arbitrary manner so that nondeterminism results. These features suggest that the functions $Y_D$ provide basic element for the description of Brownian processess.\
d) There is no obvious reason to exclude values of $D$ different from $D=1/2$ and this means that the concept of Brownian motion generalizes.\
e) Since Brownian motion can be regarded as Gaussian process (the value of the increment of $x$ obeys Gaussian distribution) it seems that also higher dimensional Gaussian processes possessing as their graphs Brownian surfaces could be described by using the higher dimensional algebraic extensions of p-adic numbers and corresponding higher dimensional extensions of $Y_D$. The deviation of $D$ from $D=1/2$ might correspond to anomalous dimensions deriving from the non-Gaussian behaviour implied by interactions.
One can form also functional composites of $Y_D$ and anomalous dimensions are multiplicative in this process: $y_{D_1} \circ y_{D_2}$ possessess anomalous dimension $D= D_1 \times D_2$. For $D_i <1$ the functional composition in general implies more chaotic behaviour. It must be emphasized, that functions $Y_D$ (not very nice objects!) do not have appear in any applications of this book.
p-Adic integration
===================
The concept of p-adic definite integral can be defined for functions $R_p\rightarrow C$ [@padrev] using translationally invariant Haar measure for $R_p$. In present context one is however interested in definining p-adic valued definite integral for functions $f: R_p\rightarrow R_p$: target and source spaces could of course be also some some algebraic extensions of p-adic numbers. What makes the definition nontrivial is that the ordinary definition as the limit of Riemann sum doesn«t work: Riemann sum approaches to zero in p-adic topology and one must somehow circumvent this difficulty. Second difficulty is related to the absence of well ordering for p-adic numbers. The problems are avoided by defining integration essentially as the inverse of differentation and using canonical correspondence to define ordering for p-adic numbers.
The definition of p-adic integral functions defining integration as inverse of differentation operation is straightforward and one obtains just the generalization of standard calculus. For instance, one has $\int z^n = \frac{z^{n+1}}{(n+1)}+ C$ and integral of Taylor series is obtained by generalizing this. One must however notice that the concept of integration constant generalizes: any function $R_p\rightarrow R_p$ depending on finite number of pinary digits only, has vanishing derivative.
Consider next definite integral. The absence of well ordering implies that the concept of integration range $(a,b)$ is not well defined as purely p-adic concept. A possible resolution of the problem is based on canonical identification. Consider p-adic numbers $a$ and $b$. It is natural to define $a$ to be smaller than $b$ if the canonical images of $a$ and $b$ satisfy $a_R<b_R$. One must notice that $a_R=b_R$ does not imply $a=b$ since the inverse of the canonical identification map is two-valued for real numbers having finite number of pinary digits. For two p-adic numbers $a,b$ with $a<b$ one can define the integration range $(a,b)$ as the set of p-adic numbers $x$ satisfying $a\leq x\leq b$ or equivalently $a_R\leq x_R\leq b_R$. For a given value of $x_R$ with finite number of pinary digits one has two values of $x$ and $x$ can be made unique by requiring it to have finite number of pinary digits.
One can define definite integral $\int_a^b f(x)dx$ formally as
$$\begin{aligned}
\int_a^b f(x)dx&=& F(b)-F(a)\nonumber\\\end{aligned}$$
where $F(x)$ is integral function obtained by allowing only ordinary integration constants and $b_R>a_R$ holds true. One encounters however problem, when $a_R=b_R$ and $a$ and $b$ are different. Problem is avoided if integration limits are assumed to correspond p-adic numbers with finite number of pinary digits.
One could perhaps relate the possibility of p-adic integration constants depending on finite number of pinary digits to the possibility to decompose integration range $[a_R,b_R]$ as $a=x_0<x_1<....x_n=b$ and to select in each subrange $[x_k,x_{k+1}]$ the inverse images of $x_k\leq x\leq x_{k+1}$, with $x$ having finite number of pinary digits in two different manners. These different choices correspond to different integration paths and the value of the integral for different paths could correspond to the different choices of p-adic integration constant in integral function. The difference between a given integration path and ’standard’ path is simply the sum of differences $F(x_k)-F(y_k)$, $(x_k)_R= (y_k)_R$.
This definition has several nice features:\
a) Definition generalizes in obvious manner to higher dimensional case.\
g) Standard connection between integral function and definite integral holds true and in higher dimensional case the integral of total divergence reduces to integral over boundaries of integration volume. This property guarantees that p-adic action principle leads to same field equations as its real counterpart. It this in fact this property, which drops other alternatives from consideration.\
c) Integral is linear operation and additive as a set function.\
d) The basic results of real integral calculus generalize as such to p-adic case.
There is however a problem related to the generalization of the integral to the case of non-analytic functions. For instance, the so called number theoretic plane waves defined as functions $a^{kx}$ with $a\in \{1,p-1\}$ is so called primitive root satisfying $a^{p-1}=1$ and $k \in
Z$, are p-adic counterparts of ordinary plane waves and nonanalytic functions of $x$. The construction of field theory limit of TGD is based on Fourier analysis using p-adic planewaves. It is difficult to avoid the use of these functions in construction of p-adic version of perturbative QFT. What one needs is definition of integral guaranteing orthogonality of the p-adic plane waves in suitable integration range. A formal integration using the integration formula gives factor
$$\begin{aligned}
\int_0^{p-1} a^{kx}dx&=& \frac{1}{ln(a)}(a^{k(p-1)}-1)=0, \ k=1,...,p-1,
\nonumber\\ \int_0^{p-1}a^{kx}dx_{\vert k=0} &=& \int_0^{p-1} dx = p-1\end{aligned}$$
Although the factor $ln(a)$ is ill defined p-adically this does not matter since integral vanishes for $k\neq 0$: for $k=O$ the integral is in well defined. p-Adic planewaves are not differentiable in ordinary sense but the differentation can be defined purely algebraically as multiplication with p-adic momentum.
p-Adic manifold geometry
=========================
In the following the concepts of p-Adic Riemannian and conformal geometries are considered.
p-Adic Riemannian geometry
--------------------------
It is possible to generalize the concept of (sub)manifold geometry to p-adic (sub)manifold geometry. The formal definition of p-adic Riemannian geometry is based on p-adic line element $ds^2= g_{kl} dx^kdx^l$. Lengths and angles are defined in the usual manner and their definition involves square root $ds$ of the line element. The existence of square roots forces quadratic extension of p-adic numbers allowing square roots. As found the extension is $4$-dimensional for $p>2$ and $8$-dimensional in $p=2$ case. This extension in question must appear as coefficient ring of p-adic tangent space so that p-adic Riemann spaces must be locally cartesian powers of $4-$ ($p>2$) or 8-dimensional ($p=2$) extension. Therefore spacetime and imbedding space dimensions of TGD emerge very naturally in p-adic context.
The definition of pseudo-Riemannian metric poses problem: it seems that one should be able to make distinction between negative and positive p-adic numbers. A possible manner to make this distinction is to p-adic numbers with unit norm to be positive or negative according to whether they are squares or not. This definition makes sense if $-1$ does not possess square root: this is true for $p \ mod \ 4 = 3$. This condition will be encountered in most applications of p-adic numbers. At analytic level the definition generalizes in obvious manner: what is required that the components of the metric are p-adically real numbers. The p-adic counter part of the Minkowski metric can be defined as
$$\begin{aligned}
ds^2_p &=& (dm^0)^2 - ((dm^1 )^2+(dm^2 )^2+(dm^3 )^2)\end{aligned}$$
The real image of this line element under canonical identification is nonnegative but metric allows to define the p-adic counterpart of $ M^4$ lightcone as the surface $(m^0)^2 - ((m^1 )^2+(m^2 )^2+(m^3 )^2)=0$ and this surface can be regarded as a fractal counterpart of the ordinary light cone. Furthermore, this metric allows the p-adic counterpart of Lorentz group as its group of symmetries.
An interesting possibility is that one could define the length of a fractal curve («coast line of Britain«) using p-adic Riemannian geometry. A possible model of this curve is obtained by identifying ordinary real plane with its p-adic counterpart via canonical identification and modelling the fractal curve with p-adically continuous or even analytic curve $x=x(t)$. The real counterpart of this curve is certainly fractal and need not have well defined length. The p-adic length of this curve can be defined as p-adic integral of $s_p= \int ds$ and its real counterpart $s_R$ obtained by canonical identification can be defined to be the real length of the curve.
The concept of p-adic Riemann manifold as such is not quite enough for the mathematization of the topological condensate concept. Rather, topological condensate can be regarded as a surface obtained by glueing together p-adic spacetime regions with different values of $p$ together along their boundaries. Each region is regarded as submanifold p-adic counterpart of $H= M^4_+\times CP_2$. A natural manner to perform the gluing operation is to use canonical identification to map the boundaries of two regions $p_1$ and $p_2$ to real imbedding space $H$ and to require that $p_1$ and $p_2$ boundary points correspond to same point in $H$.
p-Adic conformal geometry
--------------------------
It would be nice to have a generalization of ordinary conformal geometry to p-adic context. The following considerations and results of p-adic TGD suggest that the induced Kähler form defining Maxwell field on spacetime surface could be the basic entity of 4-dimensional conformal geometry rather than metric. If the existence of square root is required the dimension of this geometry is $D=4$ of $D=8$ depending on the value of $p$. In the following it is assumed that the extension used is the minimal extension allowing square root and $p \ mod \ 4=3$ condition holds so that imaginary unit belongs to the generators of the extension.
In 2-dimensional case line element transforms by a conformal scale factor in p-analytic map $Z\rightarrow f(Z)$. In four-dimensional case this requirement leads to degenerate line element
$$\begin{aligned}
ds^2&=& g(Z,Z_c,...) dZdZ_c \nonumber\\
&=& g(Z,Z_c,..) (dx^2+dy^2+p(du^2+dv^2)+ 2\sqrt{p}(dxdu+dydv))\end{aligned}$$
where the conformal factor $g(Z,Z_c,..)$ is invariant under complex conjugation. The metric tensor associated with the line element does not possess inverse. This is obvious from the fact that line element depends on two coordinates $Z,Z_c$ only so that p-adic conformal metric is effectively 2-dimensional rather than 4-dimensional. It therefore seems that one must give up conformal covariance requirement for line element.
In two-dimensional conformal geometry angles are simplest conformal invariants and are expressible in terms of the inner product. In 4-dimensional case one can define invariants, which are analogous to angles. Let $A$ and $B$ be two vectors in 4-dimensional quadratic extension allowing square root. Denote A (B) and its various conjugates by $A_i$ ($B_i$), $i=1,2,3,4$. Define phase like quantities $X_{ij}=$ «$exp(i2\Phi_{ij})$« between $A$ and $B$ by the following formulas
$$\begin{aligned}
X_{ij}&\equiv & \frac{A_i A_j B_kB_l }{\sqrt{A_1A_2A_3A_4}
\sqrt{B_1B_2B_3B_4}}\end{aligned}$$
where $i,j,k,l$ is permutation of $1,2,3,4$. Each quantity $X_{ij}$ is invariant under one of the conjugations ${}_c$, $\hat{}$ or $\hat{}_c$ and $X_{ij}$ has values in 2-dimensional subspace of the 4-dimensional extension. As in ordinary case the angles are invariant under conjugation and this means that only $3$ angle like quantities exists: this is in accordance with the fact that 3-angles are needed to specify the orientation of the vector $A$ with respect to the vector $B$.
One can define also more general invariants using four vectors $A,B,C,D$ and permutations $i,j,k,l$ and $r,s,t,u$ of $1,2,3,4$
$$\begin{aligned}
U_{ijkl}&=& \frac{X_{ijkl}}{X_{rstu}}\nonumber\\
X_{ijkl}&\equiv&A_iB_jC_kD_l\end{aligned}$$
The number of the functionally independent invariants is reduced if various conjugates of invariants are not counted as different invariants. If 2 or 3 vectors are identical one obtains as special case invariants associated with 3 and 2 vectors. If there are only two vectors the number of the functionally independent invariants is $6$.
There exists quadratic conformal covariants associated with tensors of weight two. The general form of the covariant is given by
$$\begin{aligned}
X&=&g^{ij:kl} A_{ij} B_{kl}\end{aligned}$$
The tensor $g^{ij:kl}$ has the property that in complex coordinates $Z,\bar{Z},\hat{Z},\bar{\hat{Z}}$ the only nonvanishing components of the tensor have $i\neq j\neq k\neq l$. This guarantees multiplicative transformation property in conformal transformations $Z\rightarrow W(Z)$:
$$\begin{aligned}
X (W) &= & \frac{dW}{dZ}
\frac{d\bar{W}}{d\bar{Z}}
\frac{d\hat{W}}{d\hat{Z}}
\frac{d\bar{\hat{W}}}{d\bar{\hat{Z}}} X(Z)\end{aligned}$$
The simplest example of tensor $g^{ij:kl}$ is permutation symbol and the instanton density of any gauge field defines p-adic conformal covariant (the quantity is actually $Diff^4$ invariant).
The Kähler form of $CP_2$ is self dual but this property in general does not hold true for the induced Kähler form defining Maxwell field on spacetime surface. Kähler action density (Maxwell action) formed formed from the induced Kähler form on spacetime surface is in general not p-adic conformal invariant as such whereas the ’instanton density’ is conformal invariant. It turns out that if $CP_2$ complex coordinate (4-dimensional extension) is p-adically analytic function of $M^4$ complex coordinate then the induced Kähler form is self dual in the approximation that the induced metric is flat and one can express Kähler action density as
$$\begin{aligned}
J^{\alpha\beta}J_{\alpha\beta}&=& \epsilon^{\alpha\beta\gamma\delta}
J_{\alpha\beta}J_{\gamma\delta}\end{aligned}$$
This quantity satisfies the conditions guaranteing multiplicative transformation property under p-adic conformal transformations. What is nice that p-dically analytic maps define approximate extremals of Kähler action: action density however vanishes identically in flat metric approximation. An interesting open problem is whether one could find more general extremals of Kähler action satisfying the condition
$$\begin{aligned}
J^{\alpha\beta}&=& g^{\alpha\beta\gamma\delta}J_{\gamma\delta}
\end{aligned}$$
such that the tensor $g^{...}$ satisfies the required conditions but does not reduce to the permutation symbol.
Whether 4-dimensional p-adic conformal invariance plays role in p-adic TGD is not clear. It turns out the entire $Diff^(M^4)$ rather than only $Conf(M^4)$ acts as approximate symmetries of Kähler action (broken only by gravitational effects) and that it is this larger invariance, which seems to be relevant for the dynamics of the interior of spacetime surface. It is p-adic counterpart of the ordinary 2-dimensional conformal invariance on boundary components of 3-surface, which plays key role in the calculation of particle masses.
p-Adic symmetries
=================
The most basic level questions physicist can ask about p-adic numbers are related to symmetries. It seems obvious that the concept of Lie-group generalizes: nothing prevents from replacing the real or complex representation spaces associated with the definitions of classical Lie-groups with linear space associated with some algebraic extension of p-adic numbers: the defining algebraic conditions, such as unitarity or orthogonality properties, make sense for algebraically extended p-adic numbers, too. In case of orthogonal groups one must replace the ordinary real inner product with p-adically real inner product $\sum_k X_k^2$ in a Cartesian power of a purely real extension of p-adic numbers: it should be emphasized that this inner product must be p-adic valued. In the unitary case one must consider complexification of a Cartesian power of purely real extension with inner produc $\sum \bar{Z_k} Z_k$. It should be emphasized however that the p-adic inner product differs from the ordinary one so that the action of, say, p-adic counterpart of rotation group in $R_p^3$ induces in $R^3$ an action, which need not have much to do with ordinary rotations so that the generalization is physically highly nontrivial. For very large values of $p$ there are however good reasons to expect that locally the action of these groups resembles the action of their real counter parts.
A simple example is provided by the generalization of rotation group $SO(2)$. The rows of a rotation matrix are in general $n$ orthonormalized vectors with the property that the components of these vectors have p-adic norm not larger than one. In case of $SO(2)$ this means the the matrix elements $a_{11}=a_{22}=a,a_{12}=-a_{21}=b$ satisfy the conditions
$$\begin{aligned}
a^2+b^2&=&1 \nonumber\\
\vert a\vert_p &\leq &1 \nonumber\\
\vert b\vert_p &\leq &1\end{aligned}$$
One can formally solve $a$ as $a=\sqrt{1-b^2}$ but the solution doesn’t exists always. There are various possibilities to define the orthogonal group.\
a) One possibility is to allow only those values of $a$ for which square root exists as p-adic number. In case of orthogonal group this requires that both $b= sin(\Phi)$ and $a=cos(\Phi)$ exist as p-adic numbers. If one requires rurther that $a$ and $b$ make sense also as ordinary rational numbers, they define Pythagorean triangle with integer sides and the group becomes discrete and cannot be regarded as Lie-group. Pythagorean triangles emerge for any rational counterpart of any Lie-group.\
b) Other possibility is to allow an extension of p-adic numbers allowing square root. The minimal extensions has dimension $4$ (8) for $p>2$ ($p=2$). Therefore spacetime dimension and imbedding space dimension emerge naturally as minimal dimensions for spaces, where p-adic $SO(2)$ acts ’stably’. The requirement that $a$ and $b$ are real is necessary unless one wants complexification of the $so(2)$ and gives constraints on the values of group parameters and again Lie-group property is expected to be lost.\
c) Lie-group property is guaranteed if the allowed group elements are expressible as exponents of Lie-algebra generator $Q$. $g(t)= exp(iQt)$. This exponents exists only provided the p-adic norm of $t$ is smaller than one. If one uses square root allowing extension one can require that $t$ satisfies $\vert t\vert \leq p^{-n/2}$, $n>0$ and one obtains a decreasing hierarchy of groups $G_1,G_2,..$. For physically interesting values of $p$ (typically of order $p=2^{127}-1$ ) the real counterparts of the transformations of these groups are extremely near to unit element of group. These conclusions hold true for any group. An especially interesting example physically is the group of ’small’ Lorentz transformations with $t=O(\sqrt{p})$. If the rest energy of the particle is of order $O(\sqrt{p})$: $E_0= m= m_0\sqrt{p}$ (as it turns out) then the Lorentz boost with velocity $\beta= \beta_0\sqrt{p}$ gives particle with energy $E=
m/\sqrt{1-\beta_0^2p}= m(1+\frac{\beta_0^2p}{2}+..)$ so that $O(p^{1/2})$ term in energy is Lorentz invariant. This suggests that nonrelativistic regime corresponds to small Lorentz transformations whereas in genuinely relativistic regime one must include also the discrete group of ’large’ Lorentz transformation with rational transformations matrices.
[99]{} R.B.J.T. Allenby and E.J. Redfern (1989), [*Introduction to Number Theory with Computing*]{}, Edward Arnold P. Bak, C. Tang and K. Wiesenfeld (1988) Phys. Rev. A, Vol. 38, No 1, p. 364 M. Barnsley (1988),[*Fractals Everywhere*]{}, Academic Press Z.I. Borevich and I.R. Shafarevich (1966) ,[*Number Theory*]{}, Academic Press. L. Brekke and P.G. O. Freund (1993), [*p-Adic Numbers in Physics*]{}, Phys. Rep. vol. 233, No 1 G. Cherbit (Ed.) (1987), [*Fractals*]{}, p. 145, J. Wiley and Sons. U. Dudley (1969), [*Elementary Number Theory*]{}, W.H. Freeman and Company J. Feder (1988), [*Fractals*]{}, Plenum Press, New York P.G..O. Freund, M. Olson (1987), Phys. Lett. 199B, 186-190 J-L. Gervais (1988), Phys. Lett. B ,vol 201, No 3, p. 306 C.Itzykson,Hubert Saleur,J-B.Zuber (Editors)(1988):[*Conformal Invariance and Applications to Statistical Mechanics*]{}, Word Scientific M. Kaku (1991),[*Strings, Conformal Fields and Topology*]{}, Springer Verlag A. Yu. Khrennikov, [*p-Adic Probability and Statistics*]{}, Dokl. Akad Nauk, 1992 vol 433 , No 6 S. Lundqvist, N.H. March, M. P. Tosi (1988), [*Order and Chaos in Nonlinear Physical Systems*]{},p. 295, Plenum Press, New York, B. Mandelbrot (1977), [*The Fractal Geometry of Nature*]{}, Freeman, New York G. Parisi (1992) [*Field Theory, Disorder and Simulations*]{}, World Scientific M. Pitkänen (1990) [*Topological Geometrodynamics*]{} Internal Report HU-TFT-IR-90-4 (Helsinki University). Summary of the recent state of TGD (1991) of Topological Geometrodynamics. I.V. Volovich (1987), Class. Quantum. Grav. 4 P.Wilzcek (1981), Phys. Rev. Lett. 48, p. 1144
|
---
author:
- 'Run-Ran Liu'
- Ming Li
- 'Chun-Xiao Jia'
- 'Bing-Hong Wang'
title: 'Cascading failures in coupled networks with both inner-dependency and inter-dependency links'
---
Introduction {#introduction .unnumbered}
============
In the past decade, the robustness of isolated networks has been extensively studied [@nw3; @nw8; @nw9]. Recently, based on the motivation that many real-world complex systems, such as physical, social, biological, and infrastructure systems, are becoming significantly more dependent on each other, the robustness of coupled networks has been studied by means of percolation in interdependent networks [@idn1]. In these works, the inter-dependency links have been proposed to represent the dependencies of nodes between different networks. Consequently, the failure of a node will result in the failure of the node connected to it by a dependency link. It has been recognized that the inter-dependency makes the coupled system more fragility than a single network [@idn1; @vespignani2010], especially for the system with multiple networks coupled together [@Gao2011a; @Gao2011b; @non1; @non2], and demonstrates a discontinuous percolation transition.
Along this pioneering work, interdependent networks with different topological properties, coupling method and attack strategies have been studied extensively in the past few years, such as partially dependency [@idn2], inter-similarity [@idn3; @idn4], multiple support-dependency relations [@idn5], targeted attack [@idn6] and localized attack[@loc1; @loc2], assortativity [@idn8; @Valdez; @assor], clustering [@cluster1; @cluster2], degree distribution [@de1; @de2], and spatially embedded networks[@idn9; @Bashan2013; @Shekhtman; @Danziger]. All these works further demonstrate that the fragility of the networks when they are dependent on each other.
On the other hand, to reflect the strongly dependency of units inside a system, percolation in networks with inner-dependency links has also attracted a great attention [@sdl1; @lim1]. Similar with the interdependent networks, the iterative process of cascading failures caused by connectivity and dependency links will also lead to a discontinuous percolation transition, rather than the well-known continuous phase transition in isolated networks, which has a devastating effect on the network stability. Furthermore, with a view to that more than two nodes depend on each other, dependency group is often used to replace the dependency link in the study of percolation in isolated networks with dependency [@sdl2; @sdl3; @lim2; @lim3].
However, the previous studies of the percolation in networks with dependency are all based on the assumption that the networks contain either inner-dependency links or inter-dependency links [@rvn]. For a real network system, some nodes may depend on nodes outside the networks, and some inside. That is to say that the inner- and inter-dependency links could exists in a coupled networks system simultaneously. For example, in a trading network, some companies may depend on each other due to supply and demand balance. On the other hand, some companies could depend on some units in a financial network, which forms by banks, investors, and so on. Although the effects of the two types of dependencies on the network stability have been explored separately, there is still lack of unified understanding of various robustness properties of the coupled networks due to the coaction of the two types of links. In this paper, we will develop a model to study the robustness of such networks, i.e., networks with both inner- and inter-dependency links.
This paper is organized as follows. In the next section, we will give the model and general formalism using generating function techniques. After that, we will give study our model on coupled random networks system and coupled scale-free networks as examples. At the same time, the simulation results will be presented to test the analysis results. In the last section, we will summary our findings in this paper.
Results {#results .unnumbered}
=======
Model and general formalism {#model-and-general-formalism .unnumbered}
---------------------------
We consider two coupled networks $A$ and $B$ with degree distributions $p_{k}^{A}$ and $p_{k}^{B}$, respectively, and each node has exactly one dependency link (inner- or inter- dependency link), where the dependency link means that the two nodes connected by it depend on each other, one of which fails, the other will fail too. Assuming that the two networks have the same size $N$, there are $N$ dependency links in the network system. Specifically, a fraction $\beta$ of the dependency links are set as the inter-dependency links, others are the inner-dependency links. For inter-dependency links, the two stubs (nodes) are chosen randomly in the two networks, respectively, and in the same networks for inner-dependency links. When $\beta \rightarrow 0$, there is no dependency between the two networks and the model will reduce to the model of the single network with dependency link density $q=1$ in ref.[@sdl1]. When $\beta \rightarrow 1$, our model will reduce to the original model of interdependent network proposed in ref.[@idn1].
We want to study the robustness of such coupled system after an initial attack of a fraction, $1-p$, of nodes in network $A$. The failure of a node in network $A$ will lead to the failure of its dependency partner no matter it is in network $A$ or network $B$, even though it still connects to the network by connectivity links. The failures of nodes in network $B$ have the similar consequence. On the other hand, the failures of nodes or their connectivity links may also cause the other nodes to disconnect from the networks, which is also considered as failure. Therefore, after the initial attack in network $A$, the two cascading processes (dependency and connectivity) will occur alternately in networks $A$ and $B$ until no further splitting and node removal can occur.
Here, we focus on the size of the giant component of the two networks, $S^A$ and $S^B$, which are the probability that a randomly chosen node belongs to the giant component of the final network $A$ or $B$, respectively. Note that $S^A$ is generally different from $S^B$ due to the initial node removal. To solve this model as the method used in refs.[@fs1; @fs2], we need two auxiliary parameters $R^A$ and $R^B$, which give the probability that the node, arriving at by following a randomly chosen link in network $A$ or $B$, belongs to the giant component of the final network $A$ or $B$. Then, in the steady state, $S^A$ satisfies $$S^{A} = p^{2}(1-\beta) (f^{A})^{2}+ p \beta f^{A}f^{B}. \label{Sa}$$ Here, $f^{A}=1-G_{0}^{A}(1-R^{A})$ and $f^{B}=1-G_{0}^{B}(1-R^{B})$ with $G_{0}^{A}(x)=\sum_kp_{k}^{A}x^k$ and $G_{0}^{B}(x)=\sum_kp_{k}^{B}x^k$ denoting the corresponding generating functions of the degree distributions of networks $A$ and $B$, respectively. Obviously, $f^{A}$ ($f^{B}$) means the probability that a randomly chosen node in network $A$ ($B$) belongs to the giant component of network $A$ ($B$)[@nm]. Since the two stubs of a dependency link are chosen randomly, $(f^{A})^2$ and $f^{A}f^{B}$ express that a node in network $A$ and its dependency partner in network $A$ or $B$ (with a fraction $\beta$ or $1-\beta$) belongs to the giant component, simultaneously. In addition, $p^2$ expresses that the node and its dependency partner in network $A$ are preserved after the initial removal.
Similarly, $S^{B}$ can be written as $$S^{B} = (1-\beta) (f^{B})^{2}+ p \beta f^{A}f^{B}. \label{Sb}$$ Since the initial attack only takes place in network $A$, the first term of the right side of eq.(\[Sb\]) is different with that of eq.(\[Sa\]).
To solve eqs.(\[Sa\]) and (\[Sb\]), we need the equations for $R^A$ and $R^B$, which can be obtained by considering the branch process in the two networks [@nm], $$\begin{aligned}
R^A &=& p^{2} (1-\beta)[1-G_{1}^A(1-R^A)][1-G_{0}^A(1-R^A)] + p \beta [1-G_{1}^A(1-R^A)][1-G_{0}^B(1-R^B)], \\ \label{Ra}
R^B &=& (1-\beta)[1-G_{1}^B(1-R^B)][1-G_{0}^B(1-R^B)]+ p \beta [1-G_{1}^B(1-R^B)][1-G_{0}^A(1-R^A)], \label{Rb}\end{aligned}$$ where $G_{1}^A(x)=\sum_{k}p_{k}^{A}kx^{k-1}/\langle k\rangle^A =G^{A\prime}_{0}(x)/G^{A\prime}_{0}(1)$ is the corresponding generating function of the underlying branching processes of network $A$, and the brackets $\langle \cdots \rangle$ denote an average over the degree distribution $p_{k}^{A}$. Similarly, $G_{1}^B(x)=\sum_{k}p_{k}^{B}kx^{k-1}/\langle k\rangle^B =G^{B\prime}_{0}(x)/G^{B\prime}_{0}(1)$. Given arbitrary degree distributions $p_{k}^A$, $p_{k}^B$ and the fraction of initial removal $1-p$, we can solve eqs. (\[Sa\])-(\[Rb\]) to obtain the order parameters $S^A$ and $S^B$.
Random networks {#random-networks .unnumbered}
---------------
Next, we will study two coupled random networks with the same Poisson degree distribution $p_k=\frac{e^{-\langle k \rangle}\langle k \rangle ^k}{k!}$ in details [@tmgt], where $\langle k \rangle$ is the average degree. In this case, the generating functions of the two networks take a simple form $G_{0}^A(x)=G_{1}^A(x)=G_{0}^B(x)=G_{1}^B(x)=e^{-\langle k \rangle (1-x)}$. Therefore, we have $R^{A}=S^{A}$ and $R^{B}=S^{B}$. This yields $$\begin{aligned}
S^A &= &p^{2} (1-\beta)(1-e^{-\langle k \rangle S^A})^{2} + p \beta (1-e^{-\langle k \rangle S^A})(1-e^{-\langle k \rangle S^B}), \label{Saer} \\
S^B &= &(1-\beta)(1-e^{-\langle k \rangle S^B})^{2} + p\beta (1-e^{-\langle k \rangle S^A})(1-e^{-\langle k \rangle S^B}). \label{Sber}\end{aligned}$$ For $\beta=0$, one obtains $S^A= p^2 (1-e^{-\langle k\rangle S^A})^2$ and $S^B= (1-e^{-\langle k\rangle S^B})^2$. This covers the equations found in refs. [@sdl2; @sdl3]. In this case, the percolation transition of network $A$ is discontinuous, and network $B$ has nothing to do with the fraction of initial preserved nodes $p$. For another case $\beta=1$, one can also find that $S^A=S^B=p (1-e^{-\langle k\rangle S^A})^2$, which coincides with the result of the interdependent networks[@idn1].
Next, we discuss the solution of eqs.(\[Saer\]) and (\[Sber\]) to obtain the percolation properties of this system. In general, eqs. (\[Saer\]) and (\[Sber\]) have a trivial solution at $(S^{A}=0,S^{B}=0)$, which means that the two networks $A$ and $B$ are completely fragmented. In addition, there is another trivial solution $(S^{A}=0,S^{B}>0)$ for eqs.(\[Saer\]) and (\[Sber\]) as the initial node removal is only for network $A$. Let $S^{A}=0$ in eq.(\[Sber\]), we can get the trivial solution of $S^{B}$, $$S^{B}_{0}=(1-\beta)(1-e^{-\langle k \rangle S^{B}_{0}})^{2}. \label{sb0}$$ Here, we use $S^{B}_{0}$ instead of $S^{B}$ to avoid confusion. As the numerical solution of eq.(\[sb0\]) shown in Fig.\[f1\], above a critical point $\beta_{c}'\approx1-2.4554/\langle k\rangle$, the minimum values $S^{B}_{0}=0$, which is equivalent to the trivial solution $(S^{A}=0,S^{B}=0)$, and means network $B$ is completely fragmented with the fragmented of network $A$. And below the critical point $\beta_{c}'$, $S^{B}_{0}>0$, which means that network $B$ is still functioning, although network $A$ is completely fragmented.
\[0.6\][![ (Colour online) The minimum values of $S^{B}$, labeled as $S^{B}_{0}$, as a function of the parameter $\beta$ for different average degrees. The value of $S^{B}_{0}$ jumps from $S^{B}_{0}\approx1.2564/\langle k\rangle$ to zero abruptly at the critical point $\beta_{c}'\approx1-2.4554/\langle k\rangle$. The lines denote the numerical solutions and the symbols denote the simulation results from $20$ time realizations on networks with $10^5$ nodes.[]{data-label="f1"}](fig1.eps "fig:")]{}
In order to discuss the nontrivial solutions, we construct two functions based on eqs. (\[Saer\]) and (\[Sber\]), $$\begin{aligned}
W_{1}(S^A,S^B) &= &S^A - p^{2} (1-\beta)(1-e^{-\langle k \rangle S^A})^{2} - p \beta (1-e^{-\langle k \rangle S^A})(1-e^{-\langle k \rangle S^B}),\\
W_{2}(S^A,S^B) &= &S^B - (1-\beta)(1-e^{-\langle k \rangle S^B})^{2} - p \beta (1-e^{-\langle k \rangle S^A})(1-e^{-\langle k \rangle S^B}).\end{aligned}$$ The nontrivial solution of $S^A$ and $S^B$ can be presented by the crossing points of the cures $W_{1}(S^A,S^B)=0$ and $W_{2}(S^A,S^B)=0$ in the $S^A-S^B$ plane for any given values of $p$, $\langle k\rangle$ and $\beta$ as shown in Fig.\[f2\].
When $\beta >\beta_c'$, we find that cures $W_{1}=0$ and $W_{2}=0$ have a tangent point with $S^A_c>0$ and $S^B_c>S^B_0=0$ (see panels $(a)-(c)$ of Fig.\[f2\]). This indicates that the system undergoes a discontinuous percolation transition when $\beta >\beta_c'$. For $\beta<\beta_c'$, $S^B_0>0$, there exists two cases shown in panels $(d)-(f)$ and $(g)-(i)$ of Fig.\[f2\], respectively. For panels $(g)-(i)$, the tangent point of cures $W_{1}=0$ and $W_{2}=0$ appears with $S^A_c>0$ and $S^B_c>S^B_0>0$, which indicates the system also undergoes a discontinuous percolation transition for $\beta=0.2$. However, for $\beta=0.4$ ($(d)-(f)$ of Fig.\[f2\]), the nontrivial cross point of cures $W_{1}=0$ and $W_{2}=0$ appears at $S^A_c=0$ and $S^B_c=S^B_0>0$. This means that the system undergoes a continuous percolation transition, when $\beta$ is larger than a certain value $\beta_c(<\beta_c')$.
\[0.6\][![ (Colour online) Graphical solutions for eqs.(\[Saer\]) and (\[Sber\]) with $\langle k\rangle=8$. (a)-(c), $\beta=0.8>\beta_c'$, $p_c\approx 0.2513$ with nonzero $S^A_c$ and $S^B_c$. (d)-(f), $\beta=0.4<\beta_c'$, $p_c\approx 0.3136$ with $S^A_c=0$ and nonzero $S^B_c$. (g)-(i), $\beta=0.2<\beta_c'$. $p_c\approx 0.4539$ with nonzero $S^A_c$ and $S^B_c$. []{data-label="f2"}](fig2.eps "fig:")]{}
In the following, we try to obtain the two tricritical points of the system as indicated in Fig.\[f2\]. In general, we can keep $S^B$ constant in function $W_1$, and check the behaviours of the order parameter $S^A$. In this way, it is easy to know that the critical point $p_c$ must satisfy the derivative of equation $W_{1}(S^A,S^B)=0$ with respect to $S^A$, that is $$2(1-\beta) \langle k\rangle p_c^2e^{-\langle k\rangle S^A_c}(1-e^{-\langle k\rangle S^A_c})+\beta \langle k\rangle p_c e^{-\langle k\rangle S^A_c} (1-e^{-\langle k\rangle S^B_c}) = 1. \label{w1d}$$ It is obvious that this equation will hold for the value $(S^A_c,S^B_c)$. For the discontinuous percolation transition, we don’t know the simple form of $(S^A_c,S_c^B)$, which can be obtained numerically as shown in Fig.\[f2\]. So, we put our attention to the continuous percolation transition, for which $S_c^A=0$ and $S_c^B=S^B_0$. A simple calculation will tell us that $S^B_0=0$ does not make eq.(\[w1d\]) true. Conclusion can be drawn that the continuous percolation transition can only be found when $\beta<\beta_c'$, i.e., $\beta_c'$ is one of the tricritical points.
As discussed earlier, when $\beta<\beta_{c}'$, the system does not always take a continuous percolation transition. This phenomenon is similar with the findings in refs.[@idn2; @lim1; @lim2]. As shown in these papers, this type of tricritical point also satisfies $d^2W_{1}(S^A,S^B)/d(S^{A})^2=0$. Note that at the tricritical point, the conditions of continuous and discontinuous percolation transitions are satisfied simultaneously. Hence, we have $$\langle k\rangle (1-e^{-\langle k\rangle S^B_0})^{2}\beta_c^{2}+2\beta_c-2=0.$$ That is $$\beta_{c}=\frac{\sqrt{1+2\langle k\rangle(1-e^{-\langle k\rangle S^B_0})^{2}}-1}{\langle k\rangle(1-e^{-\langle k\rangle S^B_0})^{2}}, \label{betac}$$ where $S^B_0$ can be obtained by eq.(\[sb0\]). Above all, the system demonstrates a continuous percolation transition for $\beta_c<\beta<\beta_{c}'$, and discontinuous percolation transition for $\beta<\beta_c$ or $\beta>\beta_{c}'$.
In addition, we can also get the continuous percolation transition point from eq.(\[w1d\]) by letting $S^A_c=0$ and $S^B_c=S^B_0$, $$p_{c}^{II}=\frac{1}{\beta \langle k\rangle (1-e^{-\langle k\rangle S^B_0})}. \label{pc2}$$ For discontinuous percolation transition, the critical point $p_{c}^{I}$ can be obtained numerically as shown in Fig.\[f2\].
Since $S^B_0$ decreases with the increase of $\beta$ as shown in Fig.\[f1\], there is a typical $\beta^*$ that minimizes the critical point $p_{c}^{II}$ (see eq.(\[pc2\])), which corresponds to the optimal robustness of the system. The optimal solution $\beta^*$ can also be obtained numerically by eqs.(\[sb0\]) and (\[pc2\]), some simulation results will be shown later.
Furthermore, we can find that with the decreasing of average degree, $\beta_c$ increases and $\beta_c'$ decreases. As a result, the two tricritical points can merge together when the average degree is less than a typical value $\langle \tilde{k}\rangle$, i.e., the continuous percolation transition disappears when $\langle k\rangle$ less than $\langle\tilde{k}\rangle$. This typical value $\langle \tilde{k}\rangle$ can be easily found by letting $\beta_{c}=\beta_{c}'$. Substituting $\beta_{c}\approx1-2.4554/\langle k\rangle$ and $S^B_0\approx1.2564/\langle k\rangle$ into eq.(\[pc2\]), we can get the typical average degree $\langle\tilde{k}\rangle\approx 5.5533$.
Scale-free networks {#scale-free-networks .unnumbered}
-------------------
For scale-free networks, the degree distribution is $P(k)\sim k^{-\lambda}(k_{min} \leq k \leq k_{max})$, where $k_{min}$ and $k_{max}$ are the lower and upper bounds of the degree, respectively, and $\lambda$ is the power law exponent. The sizes of the giant components $S^{A}$ and $S^{B}$ can be solved numerically by using the theoretical framework developed in eqs. (\[Sa\]) and (\[Sb\]). Since the sizes of giant components $S^{A}$ and $S^{B}$ depend on the auxiliary parameters $R^A$ and $R^B$ directly, we can discuss the phase transition of the system by using the parameters $R^A$ and $R^B$. In order to locate the tricritical points $\beta_c$ and $\beta_{c}'$ for two coupled scale-free networks, we use the similar methods as the coupled random networks. We keep $R^B$ constant in eq. (\[Ra\]), and check the behaviours of the order parameter $R^A$. At the critical point $p_c$, we have $$p_c^2 (1-\beta) \{G^{A\prime}_{1}(1-R^{A}_{c})[1-G^{A}_{0}(1-R^{A}_{c})]+[1-G^{A}_{1}(1-R^{A}_{c})]G^{A\prime}_{0}(1-R^{A}_{c})\}+p_c \beta G^{A\prime}_{1}(1-R^{A}_{c})[1-G^{B}_{0}(1-R^{B}_{c})]=1. \label{sfw1d}$$ For the continuous percolation transition, $R_c^A=0$ and $R_c^B=R^B_0$ with $R^B_0\neq0$. When $R^B_0=0$, eq. (\[sfw1d\]) cannot hold any more, and we can conclude that $\beta_{c}'$, at which $R^B_{0}$ jumps to zero, is also one of the tricritical points. At this time, we can get the continuous percolation transition point from eq.(\[sfw1d\]) $$p^{II}=\frac{1}{\beta \frac{\langle k(k-1)\rangle}{\langle k\rangle} [1-G^{B}_{0}(1-R^{B}_{0})]}. \label{sfpc2}$$ Similar to the coupled random networks, $\beta_{c}'$ and $R_0^B$ can be solved numerically by letting $R_c^A=0$ in eq. (\[Rb\]), therefore, we have $$R_{0}^{B}=(1-\beta)[1-G^{B}_{1}(1-R^{B}_{0})][1-G^{B}_{0}(1-R^{B}_{0})]. \label{sfsb0}$$
At the other tricritical point $\beta_{c}$, the conditions of continuous and discontinuous percolation transitions are satisfied simultaneously, i.e., $\beta_{c}$ makes the first and the second order derivative of eq. (\[Ra\]) with respective $S^{A}$ hold at the percolation transition point $p_{c}$. Hence we have $$[1-G^{B}_{0}(1-R^{B}_{c})]^{2}\frac{\langle k(k-1)(k-2) \rangle}{\langle k\rangle^{2}}\beta_{c}^{2}+2\beta_{c}-2=0.$$ The critical point $\beta_{c}$ is $$\beta_{c}=\frac{\sqrt{1+2[1-G^{B}_{0}(1-R^{B}_{c})]^{2}\frac{\langle k(k-1)(k-2) \rangle}{\langle k\rangle^{2}}}-1}{[1-G^{B}_{0}(1-R^{B}_{c})]^{2}\frac{\langle k(k-1)(k-2) \rangle}{\langle k\rangle^{2}}}. \label{sfbetac}$$
By plugging the degree distribution for scale-free networks into the generating functions, we can get the theoretical values for the tricritical point $\beta_{c}$, the second order percolation points $p^{II}$, as well as the numerical solution for $\beta_{c}'$. Similar to random networks, we cannot get the analytical expressions for the first order percolation transition points, but they can be solved numerically by eq. (\[Ra\]).
Simulation results and discussion {#simulation-results-and-discussion .unnumbered}
---------------------------------
We firstly show how the giant component sizes $S^A$ and $S^B$ vary in dependence on the fraction of initial preserved nodes $p$ for both coupled random networks and coupled scale-free networks by simulation and theory in Fig.\[f3\]. One can find that the analytical results are in agreement with the simulation results well. For the results of coupled random networks, one can find that the giant component size $S^A$ of network $A$ emergences abruptly when $p$ exceeds a threshold $p_{c}^{I}$ for $\beta=0.2$, $\beta=0.8$ and $\beta=1$. However, for $\beta=0.4$ and $\beta=0.6$, the giant component size $S^A$ of network $A$ increases continuously as $p$ exceeds a threshold $p_{c}^{II}$. The phenomena of network $B$ are similar, but a nonzero $S^B$ below the critical point for $\beta<\beta'_{c}$. For two coupled scale-free networks, the results are similar to the random networks, but different crtical points and tricritical points. As the scale-free networks we used in Fig. \[f3\], $\langle k(k-1)\rangle$ is divergence for a network with infinite size. Hence, according to eq.\[sfpc2\], the second order critical point $p_c^{II}\rightarrow0$.
\[0.6\][![ (Colour online) The sizes of the giant components $S^A$ and $S^B$ *vs.* $p$. Panels (a) and (b) show the results for network $A$ and network $B$ in coupled random networks with $\langle k \rangle =8$, respectively. Panels (c) and (d) show the results for network $A$ and network $B$ in coupled scale-free networks with $k_{min}=4$, $k_{max}=316$ and $\lambda=2.7$, respectively. The solid lines show the theoretical predictions, and the symbols represent simulation results from $20$ time realizations on networks with $10^5$ nodes.[]{data-label="f3"}](fig3.eps "fig:")]{}
From Fig.\[f3\], we can also find that the threshold $p_{c}$ first decreases and then increases along with the increasing of $\beta$ for both coupled random networks and coupled scale-free networks, which can be further validated in Fig.\[f4\]. Since the impact of initial removal is different for networks $A$ and $B$, the significance of the phenomenon is also slight different. For network $A$ suffered attack, its robustness can be optimized by arranging the ratio of inter-dependency links and inner-dependency links properly. For network $B$, the impacts of the initial node removal can be reduced by decreasing the fraction of inter-dependency links, however, more inner-dependency links will also reduce the stability of network $B$ itself. Note that all the second critical points of SF networks shown in Fig.\[f4\] will be zero, when the network size tends to infinite.
\[0.6\][![ (Colour online) The critical point $p_c$ for different values of $\beta$. Panel (a) shows the results for coupled random networks with different average degree. For $\langle k\rangle=8$, the first tricritical point $\beta_{c}=0.3929$ and the second tricritical point $\beta_{c}'=0.6931$. For $\langle k\rangle=6$, $\beta_{c}=0.4511$ and $\beta_{c}'=0.5908$. For $\langle k\rangle=4$, the two tricritical points are merged together and the coupled networks always demonstrate discontinuous percolation transition. The theoretical prediction for the continuous percolation transition points $p_{c}^{II}$ are the results of eq.(\[pc2\]) and the discontinuous percolation transition points $p_c^I$ are obtained as the way shown in Fig.\[f2\]. Panel (b) shows the results for coupled scale-free networks with different lower bounds $k_{min}$ and the same upper bound $k_{min}=316$. For $k_{min}=2$, $\beta_{c}=0.1068$ and $\beta_{c}'=0.2633$. For $k_{min}=3$, $\beta_{c}=0.1266$ and $\beta_{c}'=0.5427$. For $k_{min}=4$, $\beta_{c}=0.1355$ and $\beta_{c}'=0.6685$. In both panels, the symbols represent simulation results from $20$ time realizations on networks with $10^5$ nodes, and the solid lines represent the theoretical predictions. []{data-label="f4"}](fig4.eps "fig:")]{}
The phase diagrams of the systems, including coupled random networks and coupled scale-free networks, are shown in Fig.\[f4\] by both simulation and analysis. We use the simulation method developed by Parshani *et al.* to estimate the discontinuous percolation transition points [@sdl1]. That is the number of iterative failures (NOI) sharply increases with approaching the critical point $p^{I}_{c}$. For the continuous transition, we calculate the point of maximum fluctuation for the size of the giant component to estimate the critical transition point [@idn8]. From Fig.\[f4\], one can find that the simulation and theoretical results are consistent well, and there is an optimal $\beta^*$ to maximize the system robustness for both coupled random networks and scale-free networks. This shows that a suitable arrangement of the dependency links will suppress the prorogation of failure within and among networks, simultaneously. Furthermore, this finding also indicates that the high density of dependency links could not always lead to a discontinuous percolation transition as the previous studies [@idn2; @sdl1]. In addition, for coupled random networks, one can also find that the crossover of the two types of percolation transitions disappears as our theory prediction, when the average degree is below $\langle\tilde{k}\rangle\approx 5.5533$. For coupled scale-free networks, the crossover of the two types of percolation transitions can also disappear, the condition for which depends on the degree distributions of the coupled networks. Furthermore, critical exponents of a percolation system depend on its dimension [@percolation]. For random graphs and scale-free networks, they can be regarded as infinite dimensional systems, and their critical exponents are mean field and belong to the same universality class.
Conclusions {#conclusions .unnumbered}
===========
In this paper we have studied the cascading failures in coupled networks with each node has a inner-dependency or inter-dependency link. Through simulation and theoretical study, we found that there exists an optimal value of $\beta^*$ leading to the most robust coupled networks for both random networks and scale-free networks, where $\beta$ is the fraction of the nodes have inter-dependency links.
More interestingly, we found that the high density of dependency links does not always lead to a discontinuous percolation transition as the previous studies. For random coupled networks, as long as the average degree of the network exceeds a typical $\langle\tilde{k}\rangle\approx5.5533$, the system will demonstrate a continuous percolation transition for $\beta_{c}<\beta<\beta_{c}'$, where the two tricritical points $\beta_{c}$ and $\beta_{c}'$ can be obtained exactly by our theoretical method. These results reveal that the number of dependency links is not the only factor that affects the robustness of the coupled networks, and a suitable arrangement of the dependency links will suppress the prorogation of failure within and among networks, simultaneously. We think that this nontrivial combined effect of the two types dependency links shown in this work will facilitate the design of resilient infrastructures.
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Acknowledgements {#acknowledgements .unnumbered}
================
This work is funded by: The National Natural Science Foundation of China (Grant Nos.: 11305042, 61503355, 61403114, 11275186). RRL and CXJ acknowledge the support of the research start-up fund of Hangzhou Normal University (Grant Nos.: 2011QDL29, 2011QDL31). ML is also supported by the Fundamental Research Fund for the Central Universities. CXJ is also supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ14F030009.
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
RRL, ML, CXJ, and BHW conceived and designed the research. RRL and CXJ carried out the numerical simulations. RRL and ML developed the theory and wrote the manuscript.
Additional Information {#additional-information .unnumbered}
======================
The authors declare no competing financial interests.
|
---
abstract: |
Approximate solutions representing the gravitational-electrostatic balance of two arbitrary point sources in general relativity have led to contradictory arguments in the literature with respect to the condition of balance. Up to the present time, the only known exact solutions which can be interpreted as the non-linear superposition of two spherically symmetric (Reissner-Nordström) bodies without an intervening strut has been for critically charged masses, $M^2_i =
Q^2_i$. In the present paper, an exact electrostatic solution of the Einstein-Maxwell equations representing the exterior field of two arbitrary charged Reissner-Nordström bodies in equilibrium is studied. The invariant physical charge for each source is found by direct integration of Maxwell’s equations. The physical mass for each source is invariantly defined in a manner similar to which the charge was found. It is shown through numerical methods that balance without tension or strut can occur for non-critically charged bodies. It is demonstrated that other authors have not identified the correct physical parameters for the mass and charge of the sources. Further properties of the solution, including the multipole structure and comparison with other parameterizations, are examined.
address: |
Department of Physics and Astronomy, University of Victoria\
P.O. Box 3055, Victoria, B.C. V8W 3P6 (Canada)
author:
- 'G. P. Perry and F. I. Cooperstock'
title: |
Electrostatic Equilibrium of Two Spherical\
Charged Masses in General Relativity
---
Introduction
============
In a recent paper by Bonnor [@bonnor1], the equilibrium conditions for a charged test particle in the field of a spherically symmetric charged mass (Reissner-Nordström solution) were investigated. He found that the classical condition for equilibrium $$\label{classical}
M_1 M_2 = Q_1 Q_2$$ for which the separation between the particles is arbitrary, was neither necessary nor sufficient for electrostatic balance of two spherical masses. This is in conflict with the earlier results of Barker and O’Connell [@barker] and Ohta and Kimura [@ohta] who used different approximation methods. Barker and O’Connell claimed that in the post-Newtonian approximation, the equation $$(M_1 Q_2 - M_2 Q_1)(Q_1-Q_2) = 0$$ had to be satisfied in addition to (\[classical\]). Ohta and Kimura claimed that in the post-post-Newtonian approximation, the necessary and sufficient condition for balance is that each mass be “critically” charged, $$M_i = |Q_i| \ \ \ \ \ i=1,2 \label{critical}$$ and balance can occur for arbitrary separation of the sources. Up to the present time, the problem of gravitational-electrostatic balance of two spherical bodies in general relativity without an intervening Weyl line singularity (strut or tension) has been solved [*exactly*]{} only for critically charged masses [@coopcruz; @papapetrou; @tomimatsu]. A balance solution was originally thought to have been found [@coopcarminati] within the Herlt class for both sources having $M_i > \left|Q_i\right|$, but it was subsequently shown that the intervening line singularity could not be removed [@perry]. Kramer [@kramer] presented an exact solution for the electrostatic counterpart of the double Kerr-NUT solution with zero spin parameter. He found that condition (\[classical\]) holds for electrostatic balance. However he stated that his solution cannot be interpreted as the non-linear superposition of two Reissner-Nordström solutions and thus the masses are not spherically symmetric.
In the present paper, an exact electrostatic solution of the Einstein-Maxwell equations representing the exterior field of two arbitrary charged non-linearly superposed Reissner-Nordström sources in equilibrium is given. It is obtained with the aid of Sibgatullin’s [@sib] method for constructing the complex Ernst potentials [@ernst1]. It is mathematically equivalent to the solutions of Manko [*et al*]{} [@manko] and Chamorro [*et al*]{} [@chamorro], henceforth referred to as papers I and papers II respectively, (with their spin parameters set to zero) and they are all special cases of the general mathematical solution given by Ernst [@ernst2]. It is of primary importance that the parameters in the solution be related to a [*physical*]{} set of parameters in order for any subsequent analysis of the solution to have any significant physical meaning. For a physical set of parameters, one would prefer to use the individual masses and charges of each source and the distance between the sources. The invariant charge enclosed by a spacelike hypersurface can be found by the direct integration of Maxwell’s equations. For space-times with a timelike Killing vector, a conserved quantity which can be interpreted as the contribution to the total mass from each body can be invariantly defined in analogy with the charge (See, for example, Refs. and ). This paper follows Kramer [@kramer] for the definition of the individual mass of each body. In Section \[massandcharge\], the integrals of charge and mass are given and they are applied to the Weyl-class solution for two Reissner-Nordström bodies in Section \[weylsection\]. Section \[non-weyl\] presents the solution for a parameterization of the non-Weyl-class double Reissner-Nordström solution based on the Weyl-class parameterization. These are then compared to the parameterizations proposed in papers I and II. It is shown that the parameterizations employed in papers I and II do not represent the physical masses or charges of the individual sources even in the Weyl-class limit (except for the special case of identical bodies in paper I). Due to the complexities of the parameterization, a rendering of the solution in terms of the individual masses and charges as given in Section \[massandcharge\] has not yet been accomplished. However, numerical analysis of the physical masses and charges is possible for a given set of parameters. In Section \[equilibsec\], balance without a strut or tension for numerical values of the physical mass and charge is examined. It is found that there are balance conditions for which neither body is critically charged and the Newtonian balance condition does not hold. This is in accordance with Bonnor’s [@bonnor1] test particle analysis. The dependence of the balance condition on the separation of the bodies is not yet known. A discussion of the results and conclusions are given in Sections \[discussion\] and \[conclusion\].
Mass and Charge {#massandcharge}
===============
For a static axially symmetric space-time, the mass $M_i$ and charge $Q_i$ of a source inside a closed 2–surface $\sigma_i$ are given by the integrals [@landau; @kramer] $$\label{mass}
M_i \equiv -\frac{1}{8\pi} \oint_{\sigma_i} \!\! K^{ab} \sqrt{-g} \,
df^*_{ab}\,$$ $$\label{charge}
Q_i = -\frac{1}{8\pi} \oint_{\sigma_i} \!\! F^{ab} \sqrt{-g} \,
df^*_{ab}$$ where $$K^{ab} \equiv \xi^{a;b}+\Phi F^{ab}\, .$$ The timelike Killing vector is $\xi^a$, $F_{ab}$ is the electromagnetic field tensor, $\Phi$ is the electrostatic potential, $g$ is the determinant of the metric and $df^*_{ab}$ is the dual to the surface element 2-form $df^{ab}$, $$df^{*}_{ab} = \frac{1}{2} e_{abcd}df^{cd}$$ (here $e_{abcd}$ is the flat space Levi–Civita permutation symbol). The above integral conservation laws follow from the local conservation laws $$F^{ab}_{\ \ ;b;a} = 0 \ \ \ \ \ \ K^{ab}_{\ \ ;b;a} = 0,$$ the first, following from the conservation of charge and the second from the existence of the timelike Killing vector $\xi^a$ and the restriction to a static axially symmetric space-time metric. Since the Einstein–Maxwell equations also imply $$F^{ab}_{\ \ ;b} = 0 \ \ \ \ \ \ K^{ab}_{\ \ ;b} = 0,$$ in a source free region, any deformation of the surface $\sigma_i$ in the electrovacuum region outside the sources does not change the values of the integrals $M_i$ and $Q_i$.
The Weyl-Class Two Body Solution {#weylsection}
================================
To investigate the structure of space-times with two sources, the Weyl-class double Reissner-Nordström solution provides a suitable yet mathematically uncumbersome framework from which to proceed. The solution is easily found through the method presented in Ref. . The metric for a static axially symmetric space-time can be written in the canonical form $$\label{metric}
ds^2 = e^w dt^2 - e^{v-w}\left(d\rho^2+dz^2\right) - \rho^2
e^{-w}d\phi^2,$$ where $w$ and $v$ are functions of the cylindrical coordinates $\rho$ and $z$. The Weyl-class solutions are characterized by the metric function $w$ being a function of the electrostatic potential, i.e. $w
= w(\Phi)$ so that the gravitational and electrostatic equipotential surfaces overlap. For asymptotically flat boundary conditions, the unique functional relationship between $e^w$ and $\Phi$ is [@weyl] $$\label{weylclass}
e^w = 1 - 2\frac{m_{\mbox{\tiny T}}}{q_{\mbox{\tiny T}}} \Phi + \Phi^2,$$ where $\Phi$ is the electrostatic potential and $m_{\mbox{\tiny T}}$ and $q_{\mbox{\tiny T}}$ are the total mass and charge respectively. The solution representing two “undercharged” ($M_i >
\left| Q_i \right|$) Reissner-Nordström bodies (or “black holes”) is given by $$\Phi = a\frac{f-1}{a^2 f -1},$$ where $$f = \left(\frac{R_1+R_2 - 2 l_1}{R_1+R_2 + 2 l_1}\right)
\left(\frac{R_3+R_4 - 2 l_2}{R_3+R_4 + 2 l_2}\right)\, ,$$ $$\begin{aligned}
R^2_1 & \equiv & (z - d - 2 l_1)^2 + \rho^2 \, , \\
R^2_2 & \equiv & (z - d )^2 + \rho^2 \, , \\
R^2_3 & \equiv & (z + d )^2 + \rho^2 \, , \\
R^2_4 & \equiv & (z + d + 2 l_2)^2 + \rho^2 \, . \end{aligned}$$ The constant parameters $2d$ and $2l_1,2l_2$ are the coordinate distance between the horizons and the “lengths” of the horizons (Weyl “rods”) respectively (see Fig. \[fig1\]). The parameter $a$ is defined through the equation $$\frac{1+a^2}{a} = \frac{2 m_{\mbox{\tiny T}}}{q_{\mbox{\tiny T}}} .$$ The metric function $e^w$ is found through Eq. (\[weylclass\]). The metric function $e^v$ is $$e^v = \frac{\left(R_1+R_2\right)^2-4l_1^2}{4 R_1 R_2} \cdot
\frac{\left(R_3+R_4\right)^2-4l_2^2}{4 R_3 R_4} \cdot
\left[ \frac{\left(\left(l_1+l_2+d\right) R_1 + \left(l_2+d\right) R_2
- l_1 R_4\right)d}{\left(\left(l_1+d\right) R_1 + R_2 d -l_1 R_3
\right) \left(l_2 + d\right)}\right]^2 \; .$$ Choosing the surface $\sigma_1$ to encompass body 1 and the surface $\sigma_2$ to encompass body 2 of Fig. \[fig1\], the mass and charge integrals of Eqs. (\[mass\]) and (\[charge\]) yield $$\begin{array}{rclrcl}
M_1 & = &{\displaystyle \frac{1+a^2}{1-a^2} }\, l_1 & \mbox{\ \ \ }
Q_1 & = &{\displaystyle \frac{2a}{1-a^2} }\, l_1 \\ \mbox{ } \\
M_2 & = &{\displaystyle \frac{1+a^2}{1-a^2} }\, l_2 & \mbox{\ \ \ }
Q_2 & = &{\displaystyle \frac{2a}{1-a^2} }\, l_2
\end{array}$$ The above form of the individual mass and charge for each Reissner-Nordström body is similar to the form proposed in Ref. for the mass and charge decomposition of two charged Curzon particles. It was stated in Ref. that the conjectured charge decomposition for both the double Reissner-Nordström and Curzon cases were verified by direct calculation through Eq. (\[charge\]). It is straightforward to verify that Eq. (\[mass\]) yields the conjectured mass decomposition for the double Curzon solution. Because of the functional relationship between the gravitational potential and the electrostatic potential, not all of the parameters $M_1,M_2,Q_1,Q_2$ are independent. Thus the Weyl-class is also characterized by the constraint $$\label{weylconstraint}
M_1 Q_2 = M_2 Q_1.$$ Removal of the line singularity between the bodies yields Eq. (\[classical\]) as an additional condition on the parameters. As a result, the parameters also satisfy Eq. (\[critical\]). Thus equilibrium without a strut or tension occurs for “critically” charged sources and this balance is found to be independent of the separation distance [@coopcruz].
Non-Weyl parameterizations {#non-weyl}
==========================
Generalizing the Weyl-class double Reissner-Nordström solution to the case in which the gravitational and electrostatic equipotential surfaces no longer overlap has usually been attempted through the means of generating techniques (see, for example, Refs. and ). In these techniques, new solutions are generated from old ones rather than by solving the equations directly. Recently, considerable interest has focused upon a method [@sib] which constructs the Ernst potentials [@ernst1] from initial data on the symmetry axis. The complex Ernst potentials ${\cal E}(\rho,z) $ and $\Psi(\rho,z)$ of all stationary axisymmetric electrovacuum space-times with axis data of the form $${\cal E}(z,\rho=0) = \frac{U-W}{U+W}, \ \ \ \Psi (z,\rho=0) =
\frac{V}{U+W} ,$$ where $$U = z^2 + U_1 z + U_2$$ $$V = V_1 z + V_2$$ $$W = W_1 z + W_2$$ and $U_1,U_2,V_1,V_2,W_1,W_2$ are complex constants, have been found [@ernst2]. However, a mathematical solution to the Einstein-Maxwell field equations does not imply a well understood physical interpretation of the solution. Sibgatullin’s method of constructing the Ernst potentials aids in obtaining the physically meaningful parameterization which is sought for the two-body case in question.
In Sibgatullin’s method, it is required that the Ernst potentials along the $z$-axis be specified. Our choice was [@perrycoop1; @perrycoop2] $${\cal E}(\rho=0,z) \equiv e(z) =
1-\frac{2(m_1(z+z_2) + m_2(z+z_1))}{(z+z_1+m_1)(z+z_2+m_2) -q_1 q_2},$$ $$\label{ourernst}
\Psi (\rho=0,z) \equiv F(z) =
\frac{q_1(z+z_2) + q_2(z+z_1)}{(z+z_1+m_1)(z+z_2+m_2) -q_1 q_2}.$$ It has the form of the Weyl-class double Reissner-Nordström axis data. If the additional Weyl-class constraint $$\label{weylcondition}
m_1 q_2 - m_2 q_1 = 0$$ is placed on the functions $e(z)$ and $F(z)$, then Sibgatullin’s method yields the Weyl-class double Reissner-Nordström solution (in an alternate form to Ref. ) and the parameters $m_1,m_2,q_1,q_2$ are the physical masses and charges as defined by Eqs. (\[mass\]–\[charge\]) (i.e., $M_1=m_1,Q_1=q_1,M_2=m_2,Q_2=q_2$). For the solution of two Weyl-class Reissner-Nordström black holes (given in Section \[weylsection\]), Fig. \[fig1\] shows the coordinate positions of the centers of the “rods” as $d+l_1$ for body 1 and $-d-l_2$ for body 2. The parameters $z_1,z_2$ identify the negative of the coordinate positions of the centers of the “rods”, i.e., $$\begin{aligned}
z_1 & = & -d-l_1, \\
z_2 & = & d + l_2.\end{aligned}$$ If condition (\[weylcondition\]) is not imposed, $w\neq w(\Phi)$, i.e., the gravitational and electrostatic equipotential surfaces no longer overlap. In Section \[equilibsec\] it will be shown that the parameters $m_1,m_2,q_1,q_2$ then no longer carry the suggested physical meaning and the parameters $z_1,z_2$ no longer coincide with the centers of the “rods” when the Weyl-class constraint (\[weylcondition\]) is not imposed.
The full Ernst potentials ${\cal E}(\rho,z)$ and $\Psi(\rho ,z)$ for the axis data of Eq. (\[ourernst\]), expressed in terms of the cylindrical coordinates $(\rho,\,z)$, are found to be (the details of the method can be found in Refs. and in the review article ) $$\label{ernstpot}
{\cal E} = \frac{A-B}{A+B}, \ \ \ \Psi = \frac{C}{A+B},$$ where $$A \equiv \sum_{i < j}^4 a_{ij} r_i r_j, \ \
B \equiv \sum_{i =1}^4 b_i r_i , \ \
C \equiv \sum_{i =1}^4 c_i r_i ,
\ \ r_n \equiv \sqrt{\rho^2 + (z-\alpha_n)^2} ,\ \ \ (n = 1
\rightarrow 4).$$ The constants $\alpha_n$ in Eq. (\[ernstpot\]) are the roots of the equation $$e(z)+\left[F(z)\right]^2=0$$ and can only be real or complex conjugate pairs. The remaining constants $a_{ij},\, b_i$ and $c_i$ are defined as follows: $$\begin{array}{l} a_{ij} \equiv (-1)^{i+j+1} s_i s_j t_i t_j
(s_i t_j - s_j t_i)
\left|\begin{array}{cc} s_k v_k & s_l v_l \\ t_k u_k & t_l u_l
\end{array}\right|, \\
\mbox{} \\
(i<j; \ k < l; \ k,l\neq i,j; \ \;i,k=1\rightarrow
3;\;j,l=2\rightarrow 4); \\
\mbox{} \\
b_i \equiv (-1)^{i} s_i t_i (s_i - t_i)
\left|\begin{array}{ccc} s^2_k t^2_k & s^2_l t^2_l & s^2_m t^2_m \\
s_k v_k & s_l v_l & s_m v_m \\ t_k u_k & t_l u_l & t_m u_m
\end{array}\right|, \\
\mbox{} \\
(k<l<m; \ k,l,m \neq i;\ \;i=1\rightarrow 4;\;k=1,2; \;l=2,3;\;m=3,4);
\\
\mbox{} \\
c_i \equiv (-1)^{i+1} s_i t_i (s_i - t_i)(K_3 G_i + K_4 H_i),
\end{array}$$ $$\label{ernstpotconst}
G_i \equiv \left|\begin{array}{ccc} s_k t^2_k & s_l t^2_l & s_m t^2_m \\
s_k v_k & s_l v_l & s_m v_m \\ t_k u_k & t_l u_l & t_m u_m
\end{array}\right|,
\ \ \ \
H_i \equiv \left|\begin{array}{ccc} s^2_k t_k & s^2_l t_l & s^2_m t_m \\
s_k v_k & s_l v_l & s_m v_m \\ t_k u_k & t_l u_l & t_m u_m
\end{array}\right|,$$ $$(k<l<m; \ k,l,m \neq i;\ \;i=1\rightarrow 4;\;k=1,2; \;l=2,3;\;m=3,4);
\\$$ $$s_i \equiv \beta_1-\alpha_i, \ \ t_i \equiv \beta_2-\alpha_i,$$ $$u_i \equiv K_1 s_i t_i + K_3^2 t_i + K_3 K_4 s_i, \ \
v_i \equiv K_2 s_i t_i + K_4^2 s_i + K_3 K_4 t_i ,$$ $$K_1 \equiv \frac{m_1 z_2+m_2 z_1 + (m_1 +m_2) \beta_1}{\beta_1-\beta_2}
,
\ \
K_2 \equiv \frac{m_1 z_2+m_2 z_1 + (m_1 +m_2) \beta_2}{\beta_2-\beta_1},$$ $$K_3 \equiv \frac{q_1 z_2+q_2 z_1 + (q_1 +q_2) \beta_1}{\beta_1-\beta_2}
,
\ \
K_4 \equiv \frac{q_1 z_2+q_2 z_1 + (q_1 +q_2) \beta_2}{\beta_2-\beta_1},$$ $$\beta_1 \equiv -\frac{1}{2} \left( z_1 + m_1+z_2+m_2 - \sqrt{ \left(
z_1-z_2 +m_1-m_2\right)^2 + 4 q_1 q_2} \right),$$ $$\beta_2 \equiv -\frac{1}{2} \left( z_1 + m_1+z_2+m_2 + \sqrt{ \left(
z_1-z_2 +m_1-m_2\right)^2 + 4 q_1 q_2} \right),$$ where all of the subsequent quantities introduced are constants ultimately defined in terms of $m_i$, $q_i$, $z_i$, $i=1,2$, which specify the character and locations of the sources in the Weyl-class limit only.
The expressions for $\cal E$ and $\Psi$ are in Kinnersley’s [@kinnersley] form and this permits one to write the corresponding metric functions as $$e^w = \frac{A \bar{A} - B\bar{B}+C\bar{C}}{(A+B)(\bar{A}+\bar{B})},
\ \
e^v = \frac{A \bar{A} - B\bar{B}+C\bar{C}}{K_0 r_1 r_2 r_3 r_4},$$ where $$K_0 = \left(\sum^4_{i<j} a_{ij}\right)\left(\sum^4_{i<j}
\bar{a}_{ij}\right)$$ and a bar denotes complex conjugation. For a static metric, the electrostatic potential $\Phi$ is equal to the Ernst potential $\Psi$ and this completes the solution.
With the knowledge of the full Ernst potentials and the metric functions, the next step would be to evaluate the true mass and charge integrals in terms of the parameters $m_1,m_2,q_1,q_2,z_1,z_2$. It is to be stressed that outside of the Weyl-class, these parameters no longer carry the suggested physical meaning. For the metric (\[metric\]), the integrals (\[mass\]) and (\[charge\]) can be written as relations in flat 3-space ($i=1,2$): $$\label{flatmass}
M_i = \frac{1}{8\pi} \oint_{\sigma_i} \!\! e^{-w} {\cal E}_{,\alpha}
\, n^{\alpha} \, dA$$ $$\label{flatcharge}
Q_i = -\frac{1}{4\pi} \oint_{\sigma_i} \!\! e^{-w} {\Phi}_{,\alpha}
\, n^{\alpha} \, dA \, ,$$ where $n^{\alpha}$ ($\alpha$ runs from 1 to 3) is the unit vector orthogonal to the surface and $dA$ denotes the invariant (flat) surface element (see also Ref. and references therein).
We can extend the Weyl-class definitions of the coordinate positions of the bodies to the non-Weyl-class solution. There are three distinct types of sources of interest. They are characterized by the transition between a source with an event horizon to one without an event horizon. As mentioned previously, the constants $\alpha_n,
n=1\rightarrow 4$ in Eq. (\[ernstpot\]) are either real or complex conjugate pairs. By definition, we choose $\alpha_1 \geq \alpha_2 >
\alpha_3 \ge \alpha_4$. A Reissner-Nordström “black hole” is characterized by real pairs of $\alpha_n$. Fig. \[fig1\] shows that, in the Weyl canonical coordinate system, $\alpha_n$ indicates the end points of a “Weyl rod”, which itself is the event horizon surface. A “superextreme” object [@manko] or “naked singularity” is characterized by a complex conjugate pair of $\alpha_n$. Body 2 of Fig. \[fig2\] illustrates the manifestation of a “superextreme” body in the space-time. An “extreme” object, for example, would be characterized by real $\alpha_n$ for which $\alpha_1 = \alpha_2$. Therefore we have the following definitions for the coordinate positions of the sources:
[*i*]{})
: For a Reissner-Nordström “black hole”, we define $-Z_i$ to be the coordinate position of the center of the “Weyl rod”. For example, the coordinate position of body 1 of Fig. (\[fig2\]) is $$-Z_1 = \frac{1}{2}\left(\alpha_1+\alpha_2\right).$$
[*ii*]{})
: For a “superextreme” object, we define $-Z_i$ to be the coordinate position of the real part of $\alpha_n$. For example, body 2 of Fig. (\[fig2\]) is a “superextreme” object. Therefore its coordinate position is $$-Z_2 = \mbox{Re}(\alpha_3) = \mbox{Re}(\alpha_4).$$
(One could consider the imaginary part of $\alpha_n$ as the end points of a “complex Weyl rod” with the coordinate position of this “complex rod” being defined as its intersection with the real axis ($z$-axis).)
[*iii*]{})
: For an “extreme object”, we define $-Z_i$ to be the coordinate position of the point locating the zero “length” Weyl “rod”. For example, if body 1 was an “extreme” object, then $\alpha_1 = \alpha_2$ and $-Z_1 = \alpha_1$.
We also define $$\mbox{Re}(\alpha_2) > \mbox{Re}(\alpha_3)$$ as the condition for having two separated bodies irrespective of the type of object.
With the above integrals and the coordinate positions as defined above evaluated in terms of $m_1,m_2,q_1,q_2,z_1,z_2$, it would then be possible, in principle, to invert these equations and hence write the solution (\[ernstpot\]–\[ernstpotconst\]) in terms of the true physical parameters $M_i,Q_i$ and the coordinate positions $Z_i,\;
i=1,2$. Ideally, the coordinate positions of the sources should be replaced with the proper separation of the sources. The complexity of the above Ernst potentials makes the analytic evaluation of the integrals (\[flatmass\]-\[flatcharge\]) and the proper separation difficult. As a consequence this goal has not yet been achieved. However, it is possible to numerically integrate Eqs. (\[mass\]–\[charge\]) for a given set $\{m_1,m_2,q_1,q_2,z_1,z_2\}$. This will prove to be useful in studying balance conditions without a strut in Section \[equilibsec\].
Although the numerical evaluation of the physical mass and charge can be achieved from the parameterizations of paper I or paper II, it was hoped that the parameterization proposed in this paper, based on the Weyl-class solution, would facilitate the analytic evaluation of the integrals. It is not difficult to show that the parameterizations in papers I or II do not correctly identify the individual masses and charges of each source. We stated earlier that our parameterization $\{m_1,m_2,q_1,q_2,z_1,z_2\}$ only represents the physical masses and charges and coordinate positions of each source when the Weyl-class condition (Eq. (\[weylclass\]) or (\[weylcondition\])) is imposed (i.e., $\{M_1=m_1,M_2=m_2,Q_1=q_1,Q_2=q_2,Z_1=z_1,Z_2=z_2\}$) . We can best demonstrate the problems with the parameterizations of papers I and II by comparing the representation of a properly parameterized Weyl-class solution with each of the other parameterizations. Let the set $\{m_1,m_2,q_1,q_2,z_1,z_2\}$ represent the physical Weyl-class parameters under the condition $m_1 q_2 = m_2 q_1$. Then the relationships between the three parameterizations is found by solving the set of equations (setting the spin parameters found in papers I and II to zero) $$\label{eqnsystem}
\begin{tabular}{ccccc}
\multicolumn{1}{c}{\underline{Weyl-class}} & &
\multicolumn{1}{c}{\underline{Paper~I}} &
& \multicolumn{1}{c}{\underline{Paper~II}} \\
$m_1 + m_2$ & = & $\tilde{m}_1 + \tilde{m}_2$ & = & $\hat{m}_1 + \hat{m}_2 $\\
$q_1 + q_2$ & = & $\tilde{q}_1 + \tilde{q}_2$ & = & $\hat{q}_1 + \hat{q}_2 $\\
$z_1 + z_2$ & = & $\tilde{z}_1 + \tilde{z}_2$ & = & $\hat{z}_1 + \hat{z}_2 $\\
$m_1 z_2 + m_2 z_1$ & = & $\tilde{m}_1\tilde{z}_2 +
\tilde{m}_2\tilde{z}_1$ & = & $\hat{m}_1\hat{z}_2 + \hat{m}_2\hat{z}_1 +
2 \hat{m}_1\hat{m}_2 $\\
$q_1 z_2 + q_2 z_1 $& = & $\tilde{q}_1\tilde{z}_2 +
\tilde{q}_2\tilde{z}_1$ & = &
$\hat{q}_1\hat{z}_2 + \hat{q}_2\hat{z}_1 + \hat{q}_1\hat{m}_2 +
\hat{q}_2\hat{m}_1$ \\
$z_1 z_2 + m_1 m_2 - q_1 q_2$ & = & $\tilde{z}_1 \tilde{z}_2 +
\tilde{m}_1 \tilde{m}_2$ & = &
$\hat{z}_1 \hat{z}_2 - \hat{m}_1\hat{m}_2 $ .
\end{tabular}$$ The tilded and careted parameters are the parameterizations of papers I and II respectively. Table \[table1\] summarizes the results of solving the system (\[eqnsystem\]) given the values shown in column 1. The solution represents two Weyl-class Reissner-Nordström “critically charged” bodies without an intervening line singularity. It is clear that none of the parameter values in the latter two columns match the physical Weyl-class values. In fact one has to assign negative values to $\tilde{m}_2,
\tilde{q}_2$ in order to obtain a [*positive*]{} physical mass and charge for source 2. Thus, apart from one special case, neither the paper I nor the paper II parameterizations can be interpreted in as the invariant physical parameters. The only exception is for identical bodies (with or without a line singularity) in the parameterization of paper I. In this very special case of the Weyl-class, the parameters $\tilde{m}_1 =\tilde{m}_2,\;\tilde{q}_1 =\tilde{q}_2$ are the physical masses and charges. However, $\tilde{z}_1$ and $\tilde{z}_2$ do not identify the coordinate positions of the bodies as defined earlier. The paper II parameterization is not physical even for identical bodies.
It is the demand for the inclusion of the Weyl-class solution in Ref. which led to our form of $e(z)$ and $F(z)$. It should be emphasized that our parameterization contains as a special case, the simplest clearly individually spherical two-body balance solution of two critically charged bodies. This can be best illustrated by examining the Simon [@simon; @hoen] relativistic multipole moments of each parameterization. The first five Simon relativistic mass and charge multipole moments for our parameterization are $$\label{massmultipoles}
\begin{array}{ccl}
{\cal M}_0 & = & m_1+m_2, \\
{\cal M}_1 & = & \mbox{} - m_1 z_1 - m_2 z_2, \\
{\cal M}_2 & = & m_1 z_1^2 + m_2 z_2^2 - \left(m_1 m_2 - q_1 q_2\right)
\left(m_1 + m_2\right), \\
{\cal M}_3 & = & \mbox{} - m_1 z_1^3 - m_2 z_2^3 + \left(m_1 m_2 - q_1
q_2\right)
\left(2 m_1 z_1 + 2 m_2 z_2 + z_1 m_2 + z_2 m_1 \right), \\
{\cal M}_4 & = & m_1 z_1^4 + m_2 z_2^4 - \left(m_1 m_2 - q_1 q_2\right)
\Big[ \left(m_1 +m_2 \right) \left(q_1 q_2 - m_1 m_2\right) \\
\mbox{} & \mbox{} & \mbox{}
+ 2 \left(m_1 z_1^2 + m_2 z_2^2\right)
+ \left(m_1+m_2\right)
\left(z_1+z_2\right)^2 \\
\mbox{} & \mbox{} & \mbox{}
+ \left. \frac{1}{7} \left(m_1+m_2\right)\left(\left(q_1+q_2\right)^2-
\left(m_1+m_2\right)^2\right)\right] \\
\mbox{} & \mbox{} & \mbox{}
-\frac{1}{210}\left(z_1-z_2\right)\left[16\left(z_1-z_2\right)
\left(m_1+m_2\right)\left(m_1
q_2 - m_2 q_1\right)^2 \right.
\\ \mbox{} & \mbox{} & \mbox{}
+ z_1\left(30 m_1 \left(m_1 m_2 + m_2^2 - q_2^2\right) -3 q_1\left( 3
m_2 q_1 + 7 q_2 m_1\right)\right) \\
\mbox{} & \mbox{} & \mbox{}
- z_2\left(30 m_2 \left(m_1 m_2 + m_1^2 - q_1^2\right) -3 q_2\left( 3
m_1 q_2 + 7 q_1 m_2\right)\right) \Big]
\end{array}$$ and $$\label{chargemultipoles}
\begin{array}{ccl}
{\cal Q}_0 & = & q_1+q_2, \\
{\cal Q}_1 & = & \mbox{} - q_1 z_1 - q_2 z_2 , \\
{\cal Q}_2 & = & q_1 z_1^2 + q_2 z_2^2 - \left(m_1 m_2 - q_1 q_2\right)
\left(q_1 + q_2\right), \\
{\cal Q}_3 & = & \mbox{} - q_1 z_1^3 - q_2 z_2^3 + \left(m_1 m_2 - q_1
q_2\right)
\left(2 q_1 z_1 + 2 q_2 z_2 + z_1 q_2 + z_2 q_1 \right). \\
{\cal Q}_4 & = & q_1 z_1^4 + q_2 z_2^4 - \left(m_1 m_2 - q_1 q_2\right)
\Big[ \left(q_1 +q_2 \right) \left(q_1 q_2 - m_1 m_2\right) \\
\mbox{} & \mbox{} & \mbox{}
+ 2 \left(q_1 z_1^2 + q_2 z_2^2\right) + \left(q_1+q_2\right)
\left(z_1+z_2\right)^2 \\
\mbox{} & \mbox{} & \mbox{}
+ \frac{1}{7} \left. \left(q_1+q_2\right)\left(\left(q_1+q_2\right)^2-
\left(m_1+m_2\right)^2\right)\right] \\
\mbox{} & \mbox{} & \mbox{}
-\frac{1}{210}\left(z_1-z_2\right)\left[16\left(z_1-z_2\right)
\left(q_1+q_2\right)\left(m_1
q_2 - m_2 q_1\right)^2 \right.
\\ \mbox{} & \mbox{} & \mbox{}
- z_1\left(30 q_2 \left(q_1 q_2 - m_1 m_2 + q_1^2\right) -3 m_1\left( 13
m_1 q_2 - 3 m_2 q_1\right)\right) \\
\mbox{} & \mbox{} & \mbox{}
+ z_2\left(30 q_1 \left(q_1 q_2 - m_1 m_2 + q_1^2\right) -3 m_2\left( 13
m_2 q_1 - 3 m_1 q_2\right)\right)\Big]
\end{array}$$ respectively. In Newtonian physics, a system of two monopoles at positions $z_1,\ z_2$ has multipole moments $$\label{newtpoles}
{\cal M}_n = m_1 z_1^n + m_2 z_2^n, \ \ \ \ {\cal Q}_n = q_1 z_1^n +
q_2 z_2^n.$$ It is interesting to observe that this is also the relativistic multipole structure for two Weyl-class critically charged bodies, at least up to ${\cal M}_4,\ {\cal Q} _4$. There is an inherent asphericity imposed upon each, since the two bodies are interacting in a line. For non-linearly interacting sources in a line, one would not expect to realize perfect sphericity of the individual sources. (It is yet to be explained why the sphericity is maintained in the Weyl-class, at least up to ${\cal M}_4,\ {\cal Q} _4$.) Once the solution is written analytically in terms of the physically meaningful constants $M_i,Q_i$ and the coordinate positions $Z_i,\; i=1,2$, one will be able to examine the general multipole structure of non-linearly interacting spherical bodies.
For comparison, the first four Simon relativistic mass and charge multipole moments for the parameterization of paper I (with their spin parameters $a_i = 0, i=1,2$) are $$\label{mankomassmultipoles}
\begin{array}{ccl}
{\cal M}_0 & = & \tilde{m}_1+\tilde{m}_2, \\
{\cal M}_1 & = & \mbox{} - \tilde{m}_1 \tilde{z}_1 - \tilde{m}_2
\tilde{z}_2, \\
{\cal M}_2 & = & \tilde{m}_1 \tilde{z}_1^2 + \tilde{m}_2 \tilde{z}_2^2
- \tilde{m}_1 \tilde{m}_2
\left(\tilde{m}_1 + \tilde{m}_2\right), \\
{\cal M}_3 & = & \mbox{} - \tilde{m}_1 \tilde{z}_1^3 - \tilde{m}_2
\tilde{z}_2^3 + \tilde{m}_1 \tilde{m}_2
\left(2 \tilde{m}_1 \tilde{z}_1 + 2 \tilde{m}_2 \tilde{z}_2 +
\tilde{z}_1 \tilde{m}_2 + \tilde{z}_2 \tilde{m}_1 \right)
\end{array}$$ and $$\label{mankochargemultipoles}
\begin{array}{ccl}
{\cal Q}_0 & = & \tilde{q}_1+\tilde{q}_2, \\
{\cal Q}_1 & = & \mbox{} - \tilde{q}_1 \tilde{z}_1 - \tilde{q}_2
\tilde{z}_2 , \\
{\cal Q}_2 & = & \tilde{q}_1 \tilde{z}_1^2 + \tilde{q}_2 \tilde{z}_2^2
- \tilde{m}_1 \tilde{m}_2
\left(\tilde{q}_1 + \tilde{q}_2\right), \\
{\cal Q}_3 & = & \mbox{} - \tilde{q}_1 \tilde{z}_1^3 - \tilde{q}_2
\tilde{z}_2^3 + \tilde{m}_1 \tilde{m}_2
\left(2 \tilde{q}_1 \tilde{z}_1 + 2 \tilde{q}_2 \tilde{z}_2 +
\tilde{z}_1 \tilde{q}_2 + \tilde{z}_2 \tilde{q}_1 \right).
\end{array}$$ The first four Simon relativistic mass and charge multipole moments for the parameterization of paper II (with their spin parameters $a_i = 0, i=1,2$) are $$\label{chamorromassmultipoles}
\begin{array}{ccl}
{\cal M}_0 & = & \hat{m}_1+\hat{m}_2, \\
{\cal M}_1 & = & \mbox{} - \hat{m}_1 \hat{z}_1 - \hat{m}_2 \hat{z}_2 +
2 \hat{m}_1 \hat{m}_2, \\
{\cal M}_2 & = & \hat{m}_1 \hat{z}_1^2 + \hat{m}_2 \hat{z}_2^2 +
\hat{m}_1 \hat{m}_2
\left(\hat{m}_1 + \hat{m}_2 -2 \hat{z}_1 -2 \hat{z}_2\right), \\
{\cal M}_3 & = & \mbox{} - \hat{m}_1 \hat{z}_1^3 - \hat{m}_2
\hat{z}_2^3 \\
\mbox{} & \mbox{} & \mbox{} + \hat{m}_1 \hat{m}_2
\left(2 \hat{m}_1 \hat{m}_2 + 2 \hat{z}_1 \hat{z}_2 + 2 \hat{z}_1^2 +2
\hat{z}_2^2 - \hat{m}_1 \hat{z}_2 - \hat{m}_2 \hat{z}_1 - 2
\hat{m}_1 \hat{z}_1 - 2 \hat{m}_2 \hat{z}_2 \right)
\end{array}$$ and $$\label{chammorochargemultipoles}
\begin{array}{ccl}
{\cal Q}_0 & = & \hat{q}_1+\hat{q}_2, \\
{\cal Q}_1 & = & \mbox{} - \hat{q}_1 \hat{z}_1 - \hat{q}_2 \hat{z}_2
+\hat{m}_1 \hat{q}_2 + \hat{m}_2 \hat{q}_1, \\
{\cal Q}_2 & = & \hat{q}_1 \hat{z}_1^2 + \hat{q}_2 \hat{z}_2^2 +
\hat{m}_1 \hat{m}_2 \left(\hat{q}_1 +
\hat{q}_2\right) -\left(\hat{q}_1 \hat{m}_2 + \hat{q}_2
\hat{m}_1\right)\left(\hat{z}_1 + \hat{z}_2\right), \\
{\cal Q}_3 & = & \mbox{} - \hat{q}_1 \hat{z}_1^3 - \hat{q}_2
\hat{z}_2^3 - \hat{m}_1 \hat{m}_2
\left(2 \hat{q}_1 \hat{z}_1 + 2 \hat{q}_2 \hat{z}_2 + \hat{z}_1
\hat{q}_2 + \hat{z}_2 \hat{q}_1 \right) \\
\mbox{} & \mbox{} & \mbox{} + \left(\hat{q}_1 \hat{m}_2 + \hat{q}_2
\hat{m}_1\right) \left(\hat{m}_1
\hat{m}_2 + \hat{z}_1 \hat{z}_2 + \hat{z}_1^2 +\hat{z}_2^2\right) .
\end{array}$$ If the above parameterizations did represent the physical mass and charge, it is evident that the multipole structure would not be that of Newtonian spherical bodies even for critically charged bodies. As stated earlier, it should be noted that in the parameterization of paper I, it can be shown that only in the case of identical bodies, the parameters $\tilde{m}_1=\tilde{m}_2, \tilde{q}_1=\tilde{q}_2$ are the physical mass and charge. However, in this case the multipoles still do not have the form of Eq. (\[newtpoles\]) since the parameters $\tilde{z}_1$ and $\tilde{z}_2$ do not identify the positions of the bodies as defined earlier. A simple transformation would correct the multipoles in this case.
The Equilibrium Condition {#equilibsec}
=========================
In order for the space-time to be regular on the $z$-axis between the sources (removal of the Weyl line singularity or imposition of the condition for elementary flatness [@synge]), it is required that the metric function $$\label{bc1}
v(z,\rho =0) = 0$$ between the sources. If the origin of the coordinate system is located between the sources (i.e., Re($\alpha_2$) $> 0$, Re($\alpha_3$) $< 0$), then application of Eq. (\[bc1\]), after some simplification, yields the balance equation $$\label{K6eqn}
K \equiv \frac{a_{12}\left(\bar{a}_{13}+\bar{a}_{14}\right)+\bar{a}_{12}
\left(a_{13}+a_{14}\right)}{K_0} = 0.$$ Three cases were examined: $i$) Two Reissner-Nordström black holes, $ii$) Two Reissner-Nordström superextreme bodies and $iii$) One black hole and one superextreme body.
The procedure for testing for equilibrium without an intervening strut or tension will be as follows:
1. Assign numerical values to five of the six parameters from the unphysical set $\{m_1,m_2,q_1,q_2,z_1,z_2\}$.
2. Solve Eq. (\[K6eqn\]) for the unknown variable.
3. If a real root of Eq. (\[K6eqn\]) exists, then evaluate Eqs. (\[mass\]) and (\[charge\]) to determine the physical mass and charge parameters.
The results for each of the three cases are as follows:
Two Reissner-Nordström Black Holes
----------------------------------
Numerous sets of the parameters $\{m_1,m_2,q_1,q_2,z_1,z_2\}$, such that the constants $\alpha_n, n=1\rightarrow 4$ are real, were investigated. No roots were found of Eq. (\[K6eqn\]). For example, choosing $m_1=9.0,\;\; q_1=3.0,\;\; z_1=-15.0, \;\; m_2=8.0, \;\;
q_2=2.0$, no balance for $0\leq z_2 \leq 10^{10}$ was found. These findings are consistent with other results [@tomimatsu; @perry; @dietz] that two Reissner-Nordström black holes cannot be found in equilibrium without an intervening strut or tension.
Two Reissner-Nordström Superextreme Bodies
------------------------------------------
Numerous sets of the parameters $\{m_1,m_2,q_1,q_2,z_1,z_2\}$, such that the constants $\alpha_n, n=1\rightarrow 4$ are complex conjugate pairs, were investigated. No roots were found of Eq. (\[K6eqn\]). For example, in choosing $m_1=3.0,\;\; q_1=9.0,\;\;
z_1=-15.0, \;\; m_2=2.0, \;\; q_2=8.0$, no balance for $0\leq
z_2 \leq 10^{10}$ was found. These findings suggest that two Reissner-Nordström superextreme bodies cannot be found in equilibrium without a strut or tension.
One Black Hole and One Superextreme Body
----------------------------------------
The following three different cases were found for which Eq. (\[K6eqn\]) has a real root. Each case has the configuration illustrated in Fig. \[fig2\].
Case A)
: For $m_1=6.0,\;\; q_1=2.0, \;\; z_1=-5.0, \;\;
m_2=-0.7,\;\; q_2=4.0,\;\; $ balance at approximately [@footnote0] $z_2=2.08$ was found. The values of $\alpha_n$ are $\alpha_1 = 10.3,\;\; \alpha_2 = 1.74,\;\; \alpha_3 =
-3.11+i4.30,\;\; \alpha_4 = -3.11-i4.30$. Using Eqs. (\[flatmass\]-\[flatcharge\]), the physical masses and charges are $M_1=3.95,\;\; Q_1=-0.887, \;\; M_2=1.35,\;\;
Q_2=6.89$. Using the definitions of coordinate positions described in Section \[non-weyl\], it was found that $Z_1=-6.03$ and $Z_2 =
3.11$. Thus balance has occurred for $M_1 M_2 > Q_1 Q_2$, $Q_1 Q_2
< 0$ at a coordinate separation of ${\cal S} \equiv Z_2-Z_1 = 9.13\;
$. Note that the parameter $m_2$ is negative but both physical masses are positive. The parameterizations of papers I and II yield respectively
--------------- --- --------- -- ------------- --- ---------
$\tilde{m}_1$ = $4.96$ $\hat{m}_1$ = $4.36$
$\tilde{q}_1$ = $2.31$ $\hat{q}_1$ = $-1.05$
$\tilde{m}_2$ = $0.34$ $\hat{m}_2$ = $0.94$
$\tilde{q}_2$ = $3.69$ $\hat{q}_2$ = $7.05$
$\tilde{z}_1$ = $-6.60$ $\hat{z}_1$ = $-6.00$
$\tilde{z}_2$ = $3.68$ $\hat{z}_2$ = $3.08$
--------------- --- --------- -- ------------- --- ---------
which do not agree with the integrated values of Eqs. (\[mass\]) and (\[charge\]). This demonstrates that in general none of the analytic parameterizations proposed, including our own, are suitable choices for the individual masses and charges of the sources.
Case B)
: For $m_1=9.0,\;\; q_1=3.0, \;\; z_1=-40.0, \;\;
m_2=2.5,\;\; q_2=8.0,\;\; $ balance was found at approximately $z_2=34.6$. The values of $\alpha_n$ are $\alpha_1 = 48.4,\;\;
\alpha_2 = 31.61,\;\; \alpha_3 = -34.62+i7.65,\;\; \alpha_4 =
-34.62-i7.65$. The physical masses and charges are $M_1=8.87,\;\;
Q_1=2.00, \;\; M_2=2.63,\;\; Q_2=9.00$. The coordinate positions are $ -Z_1 = 40.01,\ -Z_2 = -34.6$ Thus balance has occurred for $M_1
M_2 > Q_1 Q_2$, $Q_1 Q_2 > 0$ at a coordinate separation of ${\cal
S} = 74.6\; .$
Case C)
: For $m_1=900.0,\;\; q_1=300.0, \;\; z_1=-865.0, \;\;
m_2=0.025,\;\; q_2=0.080,\;\; $ balance was found at approximately $z_2=21.581$. The values of $\alpha_n$ are $\alpha_1 = 1713.5,\;\;
\alpha_2 = 16.474,\;\; \alpha_3 = -21.582+i0.26226,\;\; \alpha_4 =
-21.582-i0.26226$. The physical masses and charges are $M_1=899.71,\;\; Q_1=298.25, \;\; M_2=0.31897,\;\; Q_2=1.8254$. The coordinate positions are $ -Z_1 = 865.00,\ -Z_2 = -21.582$ Thus balance has occurred for $Q_1 Q_2 > M_1 M_2 $, $Q_1 Q_2 > 0$ at a coordinate separation of ${\cal S} = 886.58\; .$
Comparison with Test Particle Analysis
--------------------------------------
Bonnor’s [@bonnor1] examination of a test particle in the field of a Reissner-Nordström source yielded a wide variety of balance conditions. The following cases for separation-independent equilibrium were examined (note: $M$, $Q$ characterize the Reissner-Nordström space-time and $m$, $q$ are the test body parameters):
Case 1)
: For $q=\epsilon m,\; Q=\eta M,\;\; \epsilon,\eta = \pm
1$, balance occurs if $\epsilon=\eta$.
Case 2)
: If $m=|q|,\;\; M\neq\left| Q\right|,$ or $m\neq
|q|,\;\; M=\left| Q\right|,$ no equilibrium is possible.
Case 3)
: If $mM=qQ$ but $m\neq |q|$, then no equilibrium is possible.
Since the exact solution under study contains the Weyl-class solution as a special case, we also find Bonnor’s case 1 as a separation-independent equilibrium condition. Case 2 or 3 cannot be tested readily by our numerical procedure. In order to do so, one would have to have the good fortune of correctly choosing the set $\{m_1,m_2,q_1,q_2,z_1,z_2\}$ such that the physical masses and charges satisfy the given conditions (i.e. $M_1=|Q_1|$ etc.). Then, to test the dependence on separation, one would need to choose a new set of unphysical parameters such that the proper separation changes while the physical masses and charges remain the same.
The following separation-dependent cases were also found in Ref. :
Case 4)
: If $\left|Q\right| > M,\;\; mM=-qQ$ and $m^2\neq q^2$ with $qQ <0$, then an equilibrium exists at $$r=\frac{Q^2}{2M}.$$
Case 5)
: If $\left|Q\right| > M,\;\; |q| < m, \;\; qQ<0$ or
Case 6)
: if $\left|Q\right| > M,\;\; |q| < m, \;\; qQ>0,\;\;
qQ<mM$ then an equilibrium position exists at $$r = \frac{Q^2\left(
M\left(m^2-q^2\right) + q
\sqrt{\left(m^2-q^2\right)\left(Q^2-M^2\right)}
\right)}{m^2M^2-q^2Q^2} .$$
Case 7)
: If $\left|Q\right| < M,\;\; |q| > m, \;\; qQ>0,\;\;
qQ>mM$ then an equilibrium position exists at $$r = \frac{Q^2\left(
M\left(m^2-q^2\right) - q
\sqrt{\left(m^2-q^2\right)\left(Q^2-M^2\right)}
\right)}{m^2M^2-q^2Q^2} .$$
Thus we have found a direct correspondence between cases A–C of the exact solution and cases 5–7 of Bonnor’s test particle analysis. The separation dependence of cases 4–7 cannot be studied in the exact solution using the present methods for the same reasons cases 2–3 cannot be studied. Since the separation dependence cannot be tested using the present methods, there is little value in numerically calculating the proper separation of the sources in cases A–C.
The physical parameters in case C could approximate a test body in a strong gravitational field. Using these values in case 7 and transforming from spherical coordinates to cylindrical coordinates for a single Reissner-Nordström body using the transformation (with $\theta =0$) $$\begin{aligned}
z&=&\left(r-M\right) \cos\theta, \nonumber \\
\rho &=&\sqrt{r^2-2 M r + Q^2}\, \sin\theta ,\end{aligned}$$ Bonnor’s method yields a coordinate separation of ${\cal S} = 1465.5$. Since the separation of the bodies from these two methods are not consistent, it would appear that case C does not sufficiently approximate a test body.
Discussion
==========
The essential departure in the present paper from previous work is the attempt to parameterize the solution in terms of true physical constants of the space-time. For a static axially symmetric solution of the Einstein-Maxwell equations, the integrals of Eq. (\[mass\]) and (\[charge\]) provide the invariant parameters required for meaningful analysis of the properties of the solution.
There are three cases of the exact solution which have not been examined. They are an extreme body with respectively a Reissner-Nordström black hole, a superextreme body, and another extreme body for which the solution is not of the Weyl-class. Knowledge of the solution analytically in terms of the physical parameters is required to analyze these cases adequately.
Ref. defines the terms “undercharged”, “overcharged” and “critically charged” as follows ($i=1,2$): $$\begin{aligned}
M_i^2 & > & Q_i^2 \ \ \ \ \ \mbox{``undercharged''} \\
M_i^2 & < & Q_i^2 \ \ \ \ \ \mbox{``overcharged''} \\
M_i^2 & = & Q_i^2 \ \ \ \ \ \mbox{``critically charged''}
\label{criticaldiscussion} \end{aligned}$$ For the Weyl-class, the “lengths” of the Weyl rods are [@coopcruz] $2l_i=2\sqrt{M_i^2-Q_i^2}\, ,\; i=1,2$. If body 1 is “critically charged” [@footnote1], then $\alpha_1=\alpha_2\;
(=d)$ since $l_1=0$ (see Fig. \[fig1\]). This implies that the terminology “critically charged” body and “extreme” body may be used interchangeably for Weyl-class solutions. If body 1 is “undercharged”, $\alpha_1\;(= d + 2l_1)$ and $\alpha_2\;(=d)$ are real quantities. Thus “undercharged body” and “black hole” are synonymous terms in the Weyl-class. Finally, if body 1 is “overcharged”, $\alpha_1\;(= d + l_1)$ and $\alpha_2\;(=d
+\bar{l_1})$ are complex conjugates. Thus the terms “overcharged” and “superextreme” are equivalent descriptions in the Weyl-class. Unlike the Weyl-class solutions where the “lengths” of the Weyl rods (real or complex) depend only upon the mass and charge of that source, it is strongly suggested from the analysis of Section \[equilibsec\] that for the general (non-Weyl-class) solution, the “lengths” of the rods also depend on the mass and charge of the other source and the distance separating the bodies as well. It would thus be possible to have a “critically” charged body (according to Eq. (\[criticaldiscussion\])) for which the “rod” is either of non-zero “length” or “complex”. This is important in terms of nomenclature for describing the physics of the space-time. Since the transition of a pair (eg. ($\alpha_1,\alpha_2$)) from real values to a complex conjugate pair in Sibgatullin’s [@sib] method defines a differentiation of an object with a horizon to one without, it would seem that the appropriate description would be respectively, a black hole (horizon), “extreme” body (zero “length” Weyl rod) and “superextreme” body (no horizon or naked singularity) as described in paper I. The descriptions “under”, “over” and “critically” charged body should be reserved for the relations $M_i^2 > Q_i^2$, $M_i^2 < Q_i^2$ and $M_i^2 = Q_i^2$ respectively between the individual masses and charges. This classification scheme would describe equilibrium conditions more precisely once all are identified. The appropriateness of such a scheme would become apparent when the analytic physical parameterization of the solution is known.
Bonnor’s [@bonnor1] test particle analysis has been modified [@aguirr] in such a way that the equilibrium conditions of a charged test particle in the field of a Kerr-Newman source can be studied. The generalization of the mathematical solution to two spinning sources (Kerr-Newman sources) is already known [@manko; @chamorro]. One is able to invariantly define angular momentum for a stationary space-time in a manner similar to Eqs. (\[mass\]–\[charge\]) because of the presence of a spacelike Killing vector (rotational symmetry) (see Ref. and references therein for definitions of mass and angular momentum of stationary vacuum fields). It is unknown how the subsequent analysis of two identical spinning bodies in paper I based on the invariant definitions will affect their results, if at all. However, it is clear that the parameterization given is inadequate for the physical analysis of the general case (non-identical bodies).
Conclusions {#conclusion}
===========
The solution derived in papers I, II and this paper is a generalization of the Weyl-class double Reissner-Nordström solution. However, the analytic parameterizations presented in papers I, II and this paper cannot in all cases be interpreted as the true physical constants of the spacetime. The invariant physical charge for each source is found by direct integration of Maxwell’s equations. The physical mass is invariantly defined [@kramer] in a manner similar to which the charge was found. Numerical methods were used to evaluate the invariant individual masses and charges for the axially symmetric superposition of two Reissner-Nordström bodies. It was found that neither the Newtonian balance condition nor critically charged bodies are necessary for electrostatic equilibrium. The dependence of the balance condition on the separation of the bodies is not yet known due to the complexities involved in expressing the solution analytically in terms of the true physical set of parameters. However, all the balance conditions found are consistent with Bonnor’s test particle analysis. This suggests that there exist equilibrium conditions which depend on the separation of the sources. The parameterization of this paper is manifestly physical in the Weyl-class limit.
This research was supported, in part, by a grant from the Natural Sciences and Engineering Research Council of Canada and a Natural Sciences and Engineering Research Council Postgraduate Scholarship (GPP).
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In cases A–C, Eq. (\[K6eqn\]) has been solved to a precision of $|K|<10^{-50}$ using highly refined values of $z_2$.
It should be noted that having identical roots $\alpha_1 = \alpha_2$ is not sufficient for identifying critically charged bodies even in the Weyl-class. The Curzon particle is such an object with $\alpha_1 = \alpha_2$ but it is not necessarily critically charged. See Refs. and .
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------------ ----------------------- --------------------
Weyl-Class Paper I Paper II
$m_1 = 8$ $\tilde{m}_1 = 14.17$ $\hat{m}_1 = 3.58$
$q_1 = 8$ $\tilde{q}_1 = 14.17$ $\hat{q}_1 = 3.58$
$m_2 = 3$ $\tilde{m}_2 = -3.17$ $\hat{m}_2 = 7.42$
$q_2 = 3$ $\tilde{q}_2 = -3.17$ $\hat{q}_2 = 7.42$
$z_1 = -7$ $\tilde{z}_1 = -2.02$ $\hat{z}_1 = -4.7$
$z_2 = 7$ $\tilde{z}_2 = 2.02$ $\hat{z}_2 = 4.7$
------------ ----------------------- --------------------
: The values of the parameters in the parameterizations of papers I and II is shown given the Weyl-Class values.
\[table1\]
|
---
abstract: 'We report a Lattice-Boltzmann scheme that accounts for adsorption and desorption in the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. Lattice Boltzmann simulations made it possible to solve numerically the coupled Navier-Stokes equations of fluid dynamics and Nernst-Planck equations of electrokinetics in complex, heterogeneous media. Associated to the moment propagation scheme, it became possible to extract the effective diffusion and dispersion coefficients of tracers, or solutes, of any charge, *e.g.* in porous media. Nevertheless, the dynamical properties of tracers depend on the tracer-surface affinity, which is not purely electrostatic, but also includes a species-specific contribution. In order to capture this important feature, we introduce specific adsorption and desorption processes in a Lattice-Boltzmann scheme through a modified moment propagation algorithm, in which tracers may adsorb and desorb from surfaces through kinetic reaction rates. The method is validated on exact results for pure diffusion and diffusion-advection in Poiseuille flows in a simple geometry. We finally illustrate the importance of taking such processes into account on the time-dependent diffusion coefficient in a more complex porous medium.'
author:
- Maximilien Levesque
- Magali Duvail
- Ignacio Pagonabarraga
- Daan Frenkel
- Benjamin Rotenberg
title: Accounting for adsorption and desorption in Lattice Boltzmann simulations
---
Introduction
============
The dynamical properties of fluids in heterogeneous materials offers a great challenge and have implications in many technological and environmental contexts. Inherently multi-scale in time and space, mesoscale properties such as the diffusion or dispersion coefficient reflect the nano-to-microscopic geometry of the media and the inter-atomic interactions between flowing particles, or tracers, and surface atoms. Experimentally, information about the microstructure of porous media can be extracted from diffusion measurements by pulsed gradient spin echo nuclear magnetic resonance (PGSE-NMR) [@mitra_diffusion_1997; @sen_time-dependent_2003; @sen_time-dependent_2004]. At short times, the dynamics of a pulse of tracers is connected to the geometry of the porous medium at the pore scale. At longer times, macroscale properties such as porosity and tortuosity come into play. Theoretically, stochastic approaches have been an important support to the understanding of the underlying phenomena [@sen_time-dependent_2003; @sen_time-dependent_2004; @dudko_time-dependent_2005], and have been used recently to show that adsorption and desorption processes may strongly modify the short and long time dynamics of the tracers [@levesque_taylor_2012; @levesque_note_2013].
In numerous if not all practical situations involving particle diffusion and advection, the carrier fluid is in contact with confining walls where adsorption may occur. These processes depend on the chemical nature of the solute, which explain why particles with the same charge may diffuse in the same medium with different effective diffusion coefficients. This species-dependent affinity is at the heart of all chromatographic techniques used in analytic and separation chemistry [@tallarek_study_1998]. It also plays a crucial role in the dissemination of toxic or radioactive pollutants in the environment, and conversely in remediation strategies. Recently, the great interest for nanofluidic devices and for the transport in heterogeneous porous media has also raised the issue of the relevance of models which do not take into account these sorption processes. Moreover, it was recently shown that stochastic resonance between these processes and some external field may be of practical importance, e.g. for molecular sorting [@alcor_molecular_2004; @levesque_taylor_2012].
At the mesoscale, the dynamics of particles in a fluid can be described by the continuity equation $$\partial_{t}\rho\left(\mathbf{r},t\right)=-\mathbf{\nabla}\cdot\mathbf{J}\left(\mathbf{r},t\right),\label{eq:advection-diffusion}$$ where $\rho(\mathbf{r},t)$ is the one-particle density at position $\mathbf{r}$ and time $t$, $\partial_t \equiv \partial / \partial t$ and $\mathbf{J}$ is the particle flux, which is a function of the velocity field of the carrier fluid, the bulk diffusion coefficient $D_{b}$ of the particles and, if any, their charge and the local electric field arising from their environment. If adsorption is taken into account, solid-liquid interfaces located at $\mathbf{r}$ have a surface concentration $\Gamma\left(\mathbf{r},t\right)$ (length$^{-2}$) that evolves with time according to: $$\partial_{t}\Gamma\left(\mathbf{r},t\right)=-k_{d}\Gamma\left(\mathbf{r},t\right)+k_{a}\rho\left(\mathbf{r},t\right),\label{eq:surface_concentration}$$ where $k_{a}$ (length$\cdot$time$^{-1}$) and $k_{d}$ (time$^{-1}$) are kinetic adsorption and desorption rates. For molecules, the rates can vary widely. As an example, the dissociation rate of DNA double strands on a surface grafted with single-strand DNA ranges from $10^{-5}$ to $10^{-3}$ s$^{-1}$ for a few tens of base pairs [@gunnarsson_DNAsorption_2007]. Their adsorption rate can be adjusted by changing the grafting density. Finally, we assume that the tracers (the solutes) neither diffuse into the solid phase (even though that process can easily be accounted for using our algorithm) nor dissolve the surfaces [@Ladd_erosion_PRE_2002].
The time-dependent diffusion coefficient $D(t)$ and the dispersion coefficient $K$ of the tracers can be investigated by following the spreading of a tracer pulse in the fluid. This would amount to solving Eqs. \[eq:advection-diffusion\] and \[eq:surface\_concentration\], *e.g.* with a finite element method, for all possible initial conditions, which is computationally intractable for complex systems such as heterogeneous porous media. An alternative is to deduce $K$ and $D(t)$ from the tracer velocity auto-correlation function (VACF) following [@lowe_super_1995; @Lowe_Frenkel_1996_EPL1; @Capuani_Frenkel_2003_EPL2; @rotenberg_faraday_2010]: $$\begin{aligned}
D_{\gamma}\left(t\right) & = & \int_{0}^{t}Z_{\gamma}\left(t^{\prime}\right)\mbox{d}t^{\prime},\label{eq:D_t_from_vacf}\\
K_{\gamma} & = & \int_{0}^{\infty}\left(Z_{\gamma}\left(t\right)-Z_{\gamma}\left(\infty\right)\right)\mbox{d}t,\label{eq:K_from_vacf}\end{aligned}$$ where the VACF in the direction $\gamma\in\left\{ x,y,z\right\}$ is $$Z_{\gamma}\left(t\right)=\left\langle
v_{\gamma}(0)v_{\gamma}\left(t\right)\right\rangle.
\label{eq:def_vacf}$$ At long times $Z_{\gamma}\left(\infty\right)=\bar{v}_{\gamma}^{2}$ with $\bar{v}_{\gamma}$ the average velocity of the flow. The issue of averaging over initial conditions in Eq. \[eq:def\_vacf\] can be handled elegantly and efficiently using the moment propagation method [@Lowe_Frenkel_1996_EPL1; @Hoef_Frenkel_1990_EPL3; @Merks_2002_EPL4], which was recently extended to charged tracers [@Rotenberg_EPL_2008; @pagonabarraga_PCCP_2010; @rotenberg_faraday_2010; @wang2009electrokinetic; @Wang2010electrokinetic; @Wang2010electroosmosis].
In order to compute the VACF of the tracers from Eq. \[eq:def\_vacf\], one has to keep track of their velocity. For this purpose, we use the underlying dynamics of the fluid given by Eq. \[eq:advection-diffusion\], which does not rely on the velocity of individual particles, but on the one-particle solvent density. Moreover, the simulation of heterogeneous multi-scale media requires a numerically efficient method. The lattice-Boltzmann (LB) method [@frisch_lattice-gas_1986; @mcnamara_use_1988; @ladd_theoretical_1994; @ladd_numerical_1994; @succi_lb; @aidun_lattice-boltzmann_review_2010; @chen_lattice_1998] offers a convenient framework to deal with such situations. In the LB approach, the fundamental quantity is a one-particle velocity distribution function $f_{i}\left(\mathbf{r},t\right)$ that describes the density of particles with velocity $\mathbf{c}_{i}$, typically discretized over 19 values for three-dimensional LB, at a node $\mathbf{r}$, either fluid or solid, of a lattice of spacing $\Delta x$, and at a time $t$ discretized by steps of $\Delta t$. The dynamics of the fluid are governed by transition probabilities of a particle moving in the fluid from one node to the neighboring ones: $$f_{i}\left(\mathbf{r}+\mathbf{c}_{i}\Delta t,t+\Delta t\right)=f_{i}\left(\mathbf{r},t\right)+\Delta_{i}\left(\mathbf{r},t\right),\label{eq:lb_evolution}$$ where $\Delta_{i}$, the so-called collision operator, is the change in $f_{i}$ due to collisions at lattice nodes. This LB equation recovers the fluid dynamics of a liquid and the moments of the distribution function are related to the relevant hydrodynamic variables. The reader is referred to Refs. [@succi_lb] and [@chen_lattice_1998] for reviews of the method.
Algorithm
=========
In order to compute the dynamical properties of tracers evolving in a fluid described by the LB algorithm, *i.e.* to solve Eqs. \[eq:D\_t\_from\_vacf\]–\[eq:def\_vacf\], we use the moment propagation (MP) method [@lowe_super_1995; @Lowe_Frenkel_1996_EPL1]. Other methods could have been used, such as the numerical resolution of the macroscopic equations or Brownian Dynamics to simulate the random walk of tracers biased by the LB flow [@maier_pore-scale_1998]. The latter method has often been successfully used, e.g. by Boek and Venturoli [@Boek20102305]. Nevertheless, the MP method offers many advantages. First, MP relies on the same ground as LB. It is based on the propagation of a position and velocity distribution function [@Lowe_Frenkel_1996_EPL1; @Hoef_Frenkel_1990_EPL3], therefore offering an elegant unified approach. Secondly, MP allows for the propagation of *any* moment of the distribution function $f_{i}\left(\mathbf{r},t\right)$, which offers great opportunities. For example, Lowe, Frenkel and van der Hoef exploited these higher moments to compute self-dynamic structure factors [@lowe_1997_EPL6]. The LB-MP method has been thoroughly validated by Merks for low Péclet and Reynolds numbers [@Merks_2002_EPL4].
In the moment propagation algorithm, any quantity $P\left(\mathbf{r},t\right)$ can be propagated between fluid nodes. This quantity will be modified by adsorption and desorption processes. In their absence, $P\left(\mathbf{r},t+\Delta t\right)=P^{\star}\left(\mathbf{r},t+\Delta t\right)$ with:
$$\begin{aligned}
P^{\star}\left(\mathbf{r},t+\Delta t\right) & = & \sum_{i}\left[P\left(\mathbf{r}-\mathbf{c}_{i}\Delta t,t\right)p_{i}\left(\mathbf{r}-\mathbf{c}_{i}\Delta t,t\right)\right]\nonumber \\
& & +P\left(\mathbf{r},t\right)\left(1-\sum_{i}p_{i}\left(\mathbf{r},t\right)\right),\label{eq:propagated_quantity}\end{aligned}$$
where the first sum runs over all discrete velocities connecting adjacent nodes. The probability of leaving node $\mathbf{r}$ along the direction $\mathbf{c}_{i}$ is noted $p_{i}\left(\mathbf{r},t\right)$. The last term in Eq. \[eq:propagated\_quantity\] represents the fraction of particles that did not move from $\mathbf{r}$ at the previous time-step. The expression for $p_{i}$, which is central in the algorithm, depends on the nature of the tracers. It is given by: $$p_{i}\left(\mathbf{r},t\right)=\frac{f_{i}\left(\mathbf{r},t\right)}{\rho\left(\mathbf{r},t\right)}-\omega_{i}+\frac{\omega_{i}\lambda}{2}Q,$$ where the first two terms account for advection and are obtained by coupling the tracer dynamics to that of the fluid evolving according to the LB scheme. The weigths $\omega_{i}$ are constants depending upon the underlying LB lattice. The last term describes diffusive mass transfers. The dimensionless parameter $\lambda$ determines the bulk diffusion coefficient $D_{b}=\lambda c_{s}^{2}\Delta t/4$, with $c_{s}=\sqrt{k_{B}T/m}$ the sound velocity in the fluid. It also determines the mobility of tracers under the influence of chemical potential gradients (including the electrostatic contribution) which are accounted for in the $Q$ term as described in Ref. [@Rotenberg_EPL_2008]. For neutral tracers, $Q=1$.
We now introduce a new propagation scheme in order to account for adsorption and desorption at the solid-liquid interface. While Eq. \[eq:propagated\_quantity\] still holds for nodes $\mathbf{r}$ which are in the fluid but not at the interface, for the fluid interfacial nodes we define a new propagated quantity $P_{\textrm{ads}}\left(\mathbf{r},t\right)$ associated with adsorbed particles: $$\begin{aligned}
P_{\textrm{\mbox{ads}}}\left(\mathbf{r},t+\Delta t\right) & = & P\left(\mathbf{r},t\right)p_{a}+P_{\textrm{\mbox{ads}}}\left(\mathbf{r},t\right)\left(1-p_{d}\right),\label{eq:Pads}\end{aligned}$$ where $p_{a}=k_{a}\Delta t/\Delta x$ is the probability for a tracer lying at an interface to adsorb, and $p_{d}=k_{d}\Delta t$ is the probability for an adsorbed, immobile tracer to desorb. There is no restriction in the definition of $k_{a}$ and $k_{d}$ so that they may depend on geometrical considerations and on the local tracer density. Finally, the evolution of the propagated quantity associated with free tracers now includes a term accounting for the desorption of adsorbed particles: $$\begin{aligned}
P\left(\mathbf{r},t+\Delta t\right) & = & P^{\star}\left(\mathbf{r},t+\Delta t\right)+P_{\textrm{\mbox{ads}}}\left(\mathbf{r},t\right)p_{d}.\label{eq:newP}\end{aligned}$$ where $P^{\star}$ is still given by Eq. \[eq:propagated\_quantity\]. In order to compute the VACF of the tracers, one propagates as $P$ the probability to arrive at position **$\mathbf{r}$** at time $t$, weighted by the initial velocity of tracers. Thus one needs to initialize, for each direction $\gamma$, a propagated quantity according to the Maxwell-Boltzmann distribution. The Boltzmann weights for solid ($\mathcal{S}$), fluid ($\mathcal{F}$) and interfacial ($\mathcal{I}\subset{\cal F}$) nodes read: $$\begin{cases}
0 & \mbox{for }\mathbf{r}\in{\cal S},\\
e^{-\beta\mu^{ex}\left(\mathbf{r}\right)}/\mathcal{Z} & \mbox{for }\mathbf{r}\in{\cal F\setminus I},\\
e^{-\beta\mu^{ex}\left(\mathbf{r}\right)}\left(1+e^{-\beta\Delta\mu^{ads}\left(\mathbf{r}\right)}\right)/\mathcal{Z} & \mbox{for }\mathbf{r}\in{\cal I},
\end{cases}$$ where $e^{-\beta\Delta\mu^{ads}\left(\mathbf{r}\right)}=k_{a}/\left(k_{d}\Delta x\right)$ corresponds to the sorption free energy for interfacial tracers, $\beta\equiv1/k_{B}T$ with $k_{B}$ the Boltzmann’s constant and $T$ the temperature, and $\mathcal{Z}$ is the partition function of the tracers. The excess chemical potential $\mu^{ex}\left(\mathbf{r}\right)$ includes in the case of tracers with charge $q$ a mean-field electrostatic contribution $q\psi\left(\mathbf{r}\right)$ with $\psi$ the local electrostatic potential. The VACF is then simply given as in the no sorption case [@Rotenberg_EPL_2008] by: $$Z_{\gamma}(t)=\sum_{\mathbf{r}}P\left(\mathbf{r},t\right)\times\left(\sum_{i}p_{i}\left(\mathbf{r},t\right)c_{i\gamma}\right).$$
Validation
==========
In order to validate this new scheme, we compare numerical results and exact theoretical solutions of Eqs. \[eq:advection-diffusion\] and \[eq:surface\_concentration\]. This allows to assess the validity of our method independently of experimental results. We consider the diffusion and dispersion of tracers in a slit pore, *i.e.* between two walls at positions $x=0$ and $x=L$. The time-dependent diffusion coefficient $D\left(t\right)$ in the direction normal to the wall is given by [@levesque_note_2013]:
$$\begin{aligned}
\frac{D(t)}{D_{b}} & = & \frac{1}{\left(2k_{a}+k_{d}L\right)}\times\mathcal{L}^{-1}\left[\frac{k_{d}L}{\chi^{2}}\right.\nonumber \\
& - & \left.\frac{2k_{d}\left(k_{d}+s\right)\sinh\kappa}{\chi^{3}\left(\left(k_{d}+s\right)\cosh\kappa+k_{a}\chi\sinh\kappa\right)}\right],\label{eq:exact_D}\end{aligned}$$
where $s$ is the Laplace conjugate of time $t$, $\mathcal{L}^{-1}$ is the inverse Laplace transform, $\chi=\sqrt{s/D_{b}}$ and $\kappa=\chi L/2$.
![(Color online) The time-dependent diffusion coefficient, $D(t)$, normalized by the bulk diffusion coefficient, $D_{b}$, of neutral tracers in a slit pore, as extracted from Lattice-Boltzmann simulations using our new scheme (symbols) and from the reference exact solution of Eq. \[eq:exact\_D\] (lines). Several fractions of adsorbed tracers, or sorption strength, $f_a$ defined in Eq. \[eq:fa\], are presented: black circles, 0%; red squares, 16%; green upward triangles, 66%; and blue downward triangles, 95%.\[fig:Doft\]](fig1.eps){width="\columnwidth"}
In Fig. \[fig:Doft\], we compare the exact time-dependent diffusion coefficient $D(t)$ of Eq. \[eq:exact\_D\] with the one extracted from Lattice-Boltzmann simulations for different sorption strength. This last quantity is defined by the fraction of adsorbed tracers. Unless otherwise stated, all simulations are performed within a slit pore of width $L=100$ $\Delta x$ and a bulk diffusion coefficient $D_{b}=10^{-2}$ $\Delta x^{2}/\Delta t$. These values are chosen very conservative, since the LB method is known to be efficient even in narrow slits (even for $L<10$ $\Delta x$) and for a wide range of magnitudes in the diffusion coefficients [@Merks_2002_EPL4; @ladd_numerical_1994]. $L$ and $D_b$ therefore account for a negligible part of the difference with exact results. We can thus purposely assess the effect of the new algorithm only. In Fig. \[fig:Doft\], we report time-dependent diffusion coefficients calculated by our method for a fixed sorption rate $k_a=10^{-1}~\Delta x/\Delta t$ and decreasing desorption rates $k_d \Delta t=10^{-2}$, $10^{-3}$ and $10^{-4}$ resulting in an increasing fraction of adsorbed tracers $f_a$ of approximately 16 %, 66 % and 95 %. This fraction is given by [@levesque_note_2013]: $$\label{eq:fa}
f_a=\left(1+\frac{k_d L}{2k_a}\right)^{-1}.$$ Excellent agreement is found between exact solutions and our numerical results for all fractions of adsorbed tracers, *i.e.* all “sorption strengths”.
At the initial time, $D(t=0)$ is given by the fraction of mobile tracers. At intermediate times, sorption/desorption processes significantly decrease the slope of $D(t)$, as the partial immobilization at the surface slows down the exploration of the pore. We have shown recently that for this range of parameters, this slope is given by $k_d/(1+k_aL/2D_b)$ [@levesque_note_2013]. In this illustration of diffusion between two walls, the confinement is total so that for sufficiently long times, the effective diffusion coefficient decreases exponentially with time and tends to zero.
![(Color online) Contour plots of the relative error of our Lattice Boltzmann scheme with respect to exact results, given by Eq. \[eq:exact\_D\], on the slope (top) and origin (bottom) of the linear regression of the time-dependent diffusion coefficient of neutral tracers in a slit pore, as a function of the adsorption and desorption rates $k_{a}$ and $k_{d}$.\[fig:rel\_err\_Doft\]](fig2a.eps "fig:"){width="8cm"} ![(Color online) Contour plots of the relative error of our Lattice Boltzmann scheme with respect to exact results, given by Eq. \[eq:exact\_D\], on the slope (top) and origin (bottom) of the linear regression of the time-dependent diffusion coefficient of neutral tracers in a slit pore, as a function of the adsorption and desorption rates $k_{a}$ and $k_{d}$.\[fig:rel\_err\_Doft\]](fig2b.eps "fig:"){width="8cm"}
In order to quantify the error on $D(t)$ with respect to the exact result of Eq. \[eq:exact\_D\], we plot in figure \[fig:rel\_err\_Doft\] the relative error on the slope and origin of the linear fit of $\log_{10}D(t)$ for long times, *i.e.* for $D_{b}t/L^{2}>0.5$, as a function of $k_{a}$ and $k_{d}$. In the whole range of $k_{a}$ and $k_{d}$, the relative errors on the slope and origin remain under 5 % and 2 %, respectively, which is highly satisfactory.
The effect of a pressure gradient has also been studied on the same system. The resulting Poiseuille flow induces Taylor-Aris dispersion [@taylor_dispersion_1953; @aris_dispersion_1956] of the tracers with a dispersion coefficient $K$, which is known exactly in the presence of adsorption and desorption in the simple slit geometry [@levesque_taylor_2012]: $$\begin{aligned}
\frac{K}{D_{b}} & = &
1+P_{e}^{2}\left[\frac{102y^{2}+18y+1}{210\left(1+2y\right)^{3}}
+\frac{D_{b}}{L^2k_d}\frac{2y}{\left(1+2y\right)^{3}}\right], \label{eq:exact_K}\end{aligned}$$ where $P_{e}=L\bar{v}/D_{b}$ is the Péclet number and $y=k_a/k_dL$.
![(Color online) Dispersion coefficient of neutral tracers in a slit pore in the direction of the flux, normalized by the bulk diffusion coefficient, as a function of the Péclet number, as extracted from our Lattice-Boltzmann scheme (symbols) and from the exact results (lines). Several fractions of adsorbed tracers, or sorption strength, $f_a$ defined in Eq. \[eq:fa\], are presented: black squares, 0%; red circles, 16%; green upward triangles, 66%; and blue downward triangles, 95%.\[fig:dispersion\]](fig3.eps){width="\columnwidth"}
In Fig. \[fig:dispersion\], we compare the dispersion coefficient as calculated by LB with the exact results of Eq. \[eq:exact\_K\] for various sorption strengths, as a function of the Péclet number. Adsorption significantly increases the dispersion, as it slows down part of the tracers. The agreement between our scheme and exact results is excellent, even for strong adsorption and Péclet numbers above 100.
As mentioned above, the time-dependence of the diffusion coefficient is a signature of the intrinsic geometric properties of a porous medium. The simplest model of such media consists of a compact face centered cubic (fcc) lattice of spheres of radius $R$ [@ladd_dissolution_2001]. The porosity, *i.e.* the fraction of empty (or fluid) space, is $1-\pi/\left(3\sqrt{2}\right)\approx26$%. The unit cell contains 4 octahedral cavities of radius $\approx0.41R$ connected by 8 smaller tetrahedral cavities of radius $\approx0.22R$ by a small channels of radius $\approx0.15R$. This fcc lattice is illustrated in Fig. \[fig:bcc\] for a lattice parameter $L=100\Delta x$.
![(Color online) Perspective view of a unit cell of a face-centered cubic packing of spheres, of lattice parameter 100 $\Delta x$. Solid nodes are colored in blue. Interfacial fluid nodes, where adsorption processes may occur, are colored in red. Non-interfacial fluid nodes are in white.\[fig:bcc\]](fig4.eps){width="0.5\columnwidth"}
We report the time-dependent diffusion coefficient for this model porous medium in Fig. \[fig:diffusion\_bcc\]. At $t=0$, the diffusion coefficient is again given by the fraction of free particles times their bulk diffusion coefficient. After a reduced time $D_bt/R^2=0.5$, tracers have explored the whole porosity and the diffusion coefficient tends toward the effective diffusion coefficient. The time dependence is strongly influenced by adsorption/desorption, as in the slit pore case. It is thus essential to consider these phenomena when interpreting experimental measurement of effective and time-dependent diffusion coefficients.
![(Color online) Diffusion coefficient $D(t)$ of a neutral tracer, normalized by the bulk diffusion coefficient $D_{b}$ in the face-centered cubic packing of spheres with radius $R$ illustrated in Fig. \[fig:bcc\], as a function of the reduced time. Several fractions of adsorbed tracers, or sorption strength, $f_a$ defined in Eq. \[eq:fa\], are presented: black circles, 0%, *i.e.*, without adsorption; red squares, 30%; green upward triangles, 79%; and blue downward triangles, 97%.\[fig:diffusion\_bcc\]](fig5.eps){width="\columnwidth"}
Conclusion
==========
In summary, we have proposed a new scheme that accounts for adsorption and desorption in a generic Lattice-Boltzmann scheme, allowing for the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. These processes are modelled by kinetic rates of adsorption and desorption taking place at interfacial, fluid nodes. The algorithm has been validated over a wide range of adsorption and desorption rates and Péclet numbers in the slit pore geometry where exact results are available. Finally, we have shown on a more complex porous medium that adsorption and desorption processes may not be neglected, as they strongly modify the short, intermediate and long time behaviors of the diffusion coefficient as well as the dispersion coefficient. In turn, this demonstrates that neglecting interactions with the surface in the interpretation of $D(t)$ as a probe of the geometry of the porous medium, as measured experimentally *e.g.* by PSGE-NMR, may lead to incorrect conclusions. This scheme may now be used in two ways. First, one could predict the effective diffusion coefficient in complex heterogeneous media for species with known adsorption and desorption rates (from experiments or molecular simulations). Conversely, from reference measurements of the time-dependent diffusion coefficient in controlled geometries, one could extract the adsorption and desorption rates $k_{a}$ and $k_{d}$. Moreover, in the case of diffusion in the solid stationary phase, the method would allow us to relate the relevant diffusion constant to the shape of the elution profile.
While the present method is very general and also applies in principle to the case of irreversible adsorption ($k_d = 0$, i.e. $p_d= 0$ in Eq. \[eq:newP\]), such a situation is of interest only outside of equilibrium. Indeed, in that case at equilibrium all the solute is adsorbed on the surface and its VACF corresponds to the sorbed species only. The propagation scheme (Eqs. \[eq:propagated\_quantity\]-\[eq:newP\]) could nevertheless be used to investigate irreversible adsorption out of equilibrium by considering the density as the propagated quantity $P$ (instead of the one described here to compute the VACF), as was done e.g. by Warren to simulate electrokinetic phenomena [@warren_electroviscous_1997]. As an example of practical application where (possibly irreversible) sorption is coupled to electrokinetic phenomena, we can for example mention the case of ion adsorption onto charged minerals such as clays.
BR and ML acknowledge financial support from the French Agence Nationale de la Recherche under grant ANR-09-SYSC-012. MD acknowledges the French Agence Nationale pour la Gestion des Déchets Radioactifs (ANDRA) for financial support. IP acknowledges Spanish MINECO (Project No. FIS2011-22603) and DURSI (SGR2009-634) for financial support. DF acknowledges a Wolfson Merit Award of the Royal Society of London.
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abstract: 'Magnetic skyrmions in bulk crystals are line-like topologically protected spin textures. They allow for the propagation of magnons along the skyrmion line but are localized inside the skyrmion line. Analogous to the vortex line, these propagating modes are the Kelvin modes of a skyrmion line. In crystals without an inversion center, it is known that the magnon dispersion in the ferromagnetic state is asymmetric in the wavevector. It is natural to expect that the dispersion of the Kelvin modes is also asymmetric with respect to the wavevector. We study the Kelvin modes of a skyrmion line in the ferromagnetic background. In contrast, we find that the lowest Kelvin mode is symmetric in the wavevector in the low energy region despite the inversion symmetry breaking. Other Kelvin modes below the magnon continuum are asymmetric, and most of them have a positive group velocity. Our results suggest that a skyrmion line can function as a one-way waveguide for magnons.'
author:
- 'Shi-Zeng Lin'
- 'Jian-Xin Zhu'
- Avadh Saxena
bibliography:
- 'references.bib'
title: Kelvin modes of a skyrmion line in chiral magnets and the associated magnon transport
---
Lord Kelvin calculated stable propagating wave modes along a straight vortex tube of uniform vorticity in a classical fluid about 130 years ago. [@Kelvin1880] These modes were later called Kelvin modes. The Kelvin modes in quantized vortex lines were subsequently studied [@Pitaevskii1961; @PhysRev.162.143] and observed experimentally [@RevModPhys.59.87] in quantum superfluids. The spectrum of the Kelvin modes in the long wavelength limit is $\hbar\omega=\frac{\hbar^2 k^2}{2m} \ln(1/k\xi)$, where $m$ is the mass of a particle in the superfluid, $k$ is the momentum, and $\xi$ is the healing length.
Recently, a vortex-line like topological spin texture, known as skyrmion, has been observed in magnets by experiments.[@Muhlbauer2009; @Yu2010a] A large family of skyrmion-hosting materials has been identified. A skyrmion has several intrinsic properties, such as polarization, vorticity and helicity, and these properties are determined by the symmetries of the material and/or external magnetic fields. For skyrmions in materials without inversion symmetry, the Dzyaloshinskii-Moriya interaction [@Dzyaloshinsky1958; @Moriya60; @Moriya60b] (DMI) is responsible for the stabilization of a skyrmion lattice. [@Bogdanov89] Skyrmions can also exist in systems with inversion symmetry, where competing magnetic interactions stabilize the skyrmions. [@PhysRevLett.108.017206; @leonov_multiply_2015; @PhysRevB.93.064430; @PhysRevB.93.184413; @hirschberger_skyrmion_2018; @kurumaji_skyrmion_2018] In thin films, skyrmions appear as disk-like excitations and in bulk materials, skyrmions are line-like excitations. Skyrmions can be manipulated by various external drives, such as electric current, electric field, thermal gradient, etc. [@Jonietz2010; @Yu2012; @Schulz2012; @White2012; @PhysRevLett.113.107203; @Kong2013; @Lin2014PRL; @Mochizuki2014] Remarkably, skyrmions can be driven into motion by a small current density of the order of $10^6\ \mathrm{A/m^2}$, which is $5$ to $6$ orders of magnitude smaller than that for magnetic domain walls. [@Jonietz2010; @Yu2012; @Schulz2012] For their superior properties including compact size, high mobility and stability, skyrmions have attracted tremendous attention recently and are deemed as promising candidates for applications in the next generation spintronic devices. [@Fert2013; @nagaosa_topological_2013]
A skyrmion line can also support propagating Kelvin modes inside the line, see Fig. \[f1\] (a). In centrosymmetric systems, the Kelvin mode is symmetric in the propagating wavevector. [@PhysRevD.90.025010] In systems without inversion symmetry, the question is whether the dispersion of the Kelvin modes is asymmetric with respect to the wavevector. If the answer is positive, this would imply a one-way propagation of magnons inside a skyrmion line, and therefore the skyrmion line can work as a one-way magnon waveguide. This is the question we will address in this work.
First, we consider the magnon dispersion in the fully spin polarized state. We begin with a phenomenological description of the magnetization in a chiral magnet. The Hamiltonian of the system in terms of the magnetization field $\mathbf{n}(\mathbf{r})$ with $|\mathbf{n}|=1$ is [@bak_theory_1980; @Bogdanov89] $$\label{eq1}
\mathcal{H} =\int dr^3\left[ \frac{J}{2} \sum_{\mu = x,y} {\left( {{\partial _\mu }{\bf{n}}} \right)^2} + D{\bf{n}}\cdot\nabla \times {\bf{n}} -B_z n_z-\frac{A}{2} n_z^2\right],$$ which successfully captures many experimental observations in chiral magnets. Here $J$ is the exchange interaction, $D$ is the DMI [@Dzyaloshinsky1958; @Moriya60; @Moriya60b] and $B_z$ is the external magnetic field. We have introduced an easy axis anisotropy $A>0$. For B20 compounds with cubic symmetry, this term is not allowed, [@bak_theory_1980] but this anisotropy can be generated by uniaxial stress. It can also exist in other crystals with a layered structure. We have neglected the weak dipolar interaction. Note that the skyrmion size is much bigger than the spin lattice constant, and this justifies the continuum approximation in Eq. . For the field value above the saturation field, a ferromagnetic state is stabilized, where $\mathbf{n}=\hat{z}$ with $\hat{z}$ being a unit vector in the $z$ direction. [@Butenko2010; @leonov_properties_2016] The dynamics of $\mathbf{n}$ is determined by the Berry phase contribution to the action $S_B=\frac{S\hbar}{2a^3}\int dr^3 dt \partial_t \varphi ( \cos\theta +1)$, where $S$ is the total spin of the ion (in the material) and $a$ is the crystal lattice parameter. Here $\varphi$ and $\theta$ are the spherical angles of $\mathbf{n}$, i.e. $\mathbf{n}=(\sin\theta\cos\varphi,\ \sin\theta\sin\varphi,\ \cos\theta)$. The magnon dispersion is $$\label{eq2}
\frac{S\hbar}{a^3}\omega_{\mathrm{FM}}=J \mathbf{k}^2+2D k_z+B_z+A.$$ The magnon dispersion is asymmetric with respect to $k_z$ consistent with the inversion symmetry breaking. This asymmetric magnon dispersion has been observed in experiments, [@PhysRevB.97.224403; @PhysRevLett.120.037203] and provides a useful way to determine the strength of the DMI. Here the ferromagnetic (FM) state is stable for the field above $B_{c}=D^2/J-A$. The asymmetry gives rise to unconventional magnon propagation such as the modified Snell’s law. [@PhysRevB.94.140410] Note that the asymmetry only appears in $k_z$ along the field direction, while the dispersion with respect to $k_x$ and $k_y$ remain symmetric. Therefore for a thin film with a normal magnetic field, the magnon dispersion is symmetric with respect to the in-plane wave vectors, even though the inversion symmetry is broken.
A skyrmion line can exist as a metastable state in the background of the FM state. The skyrmion line provides a centrosymmetric potential for magnon excitations, and it allows for the existence of localized magnons. This was calculated in thin films. [@PhysRevB.90.094423; @Lin_internal_2014] In clean systems, the magnon modes can be labeled by angular momentum $m$ and wavevector $k_z$. In the following discussion, we call these modes the Kelvin modes with quantum numbers $m$ and $k_z$. Let us consider the lowest mode associated with the translation of the skyrmion line. The translation of the whole straight skyrmion line does not cost any energy and it is a Goldstone mode of the system. The bending of the skyrmion line costs energy and results in the dispersion of the corresponding Kelvin mode. The displacement of a rigid skyrmion line can be described by $\mathbf{n}_s[\mathbf{r}-\mathbf{u}(z)]$, where $\mathbf{u}(z)=[u_x(z),\ u_y(z)]$ is the displacement vector. The $z$ independent displacement of a skyrmion line does not cost energy, i.e. $\mathcal{H}[\mathbf{n}_s(\mathbf{r}-\mathbf{u}_0)]=\mathcal{H}[\mathbf{n}_s(\mathbf{r})]$. For a long wavelength distortion, the energy functional can be expanded in the basis of $\partial_z\mathbf{u}$, $\mathcal{F}(\mathbf{u})\propto \int dz (\partial_z\mathbf{u})^2+\cdots$. The first order term $\partial_z\mathbf{u}$ appears as a surface term upon integration. It vanishes when the two ends of the skyrmion line are fixed.
The energy cost to distort a skyrmion line can also be obtained directly from Eq. and is $$E_z=\frac{1}{2} J (\partial_z\mathbf{n})^2 +D\left(n_y\partial_zn_x-n_x\partial_z n_y\right).$$ The contribution from the DMI vanishes as obtained by straightforward calculations. Therefore the distorted skyrmion line has an energy cost $$E_z=\frac{1}{2} \eta J \int dz \left[(\partial_z u_x)^2+(\partial_z u_y)^2\right],$$ with $\eta\equiv\int dr^2(\partial_x \mathbf{n}_s)=\int dr^2(\partial_y \mathbf{n}_s)$. The stiffness of the skyrmion line is $\eta J$ and is independent of the DMI, which is consistent with the fact that $J$ is the largest energy scale of the problem.
In terms of $\mathbf{u}(z)$, the Berrry phase part of the action becomes $$S_B=S_B(\mathbf{u}=0)+\frac{S \hbar \pi }{a^3}\left(u_x {\partial_t u_y}-u_y {\partial_t u_x}\right).$$ The total action associated with the distortion of a skyrmion line is $S_T=S_B-E_z$. The equation of motion for $\mathbf{u}(z)$ is $$\begin{aligned}
\frac{2\pi\hbar S}{a^3} \partial_t u_y-J\eta \partial _z^2u_x=0,\\
-\frac{2\pi\hbar S}{a^3} \partial_t u_x-J\eta \partial _z^2u_y=0.\end{aligned}$$ Therefore the dispersion of this Kelvin mode is (as will be shown below, this Kelvin mode has $m=\pm 1$) $$\label{eqKM8}
\omega =\frac{a^3 \eta J }{2\pi\hbar S}k_z^2.$$ This Kelvin mode is symmetric with respect to $k_z$. This is different from the magnon mode in the ferromagnetic state, where the magnon dispersion is asymmetric due to the presence of DMI. The mode is gapless. A gap exists when there is a local pinning potential or geometric confinement in a small system. The gap can be introduced into the action $S_T$ by adding a mass term $M \mathbf{u}^2/2$ for a straight skyrmion line. Nevertheless, the dispersion of other Kelvin modes is asymmetric with respect to $k_z$ as will be shown below.
![(a) Schematic view of the Kelvin modes of a skyrmion line. (b) Definition of the local spin coordinate $\mathbf{L}$ and its relation to the spin $\mathbf{n}$ in the lab frame.[]{data-label="f1"}](fig1.pdf){width="\columnwidth"}
To go beyond the analysis of the skyrmion displacement field, we calculate the magnon spectrum in the presence of a straight skyrmion line embedded in the ferromagnetic background. For a disk-like skyrmion in thin films, the spectrum was calculated in in Refs. . Here we extend the method used in Ref. to three dimensions. First, we find the stationary solution of a straight skyrmion line. The symmetry of the problem allows us to use cylindrical coordinates $\mathbf{r}=(r,\ \phi,\ z)$. The skyrmion line solution in Eq. has the form $\varphi=\phi+\pi/2$ (skyrmion helicity is $\pi/2$ determined by the DMI) and $\theta(r)$ with $\theta$ changing from $\theta=-\pi$ at the skyrmion center $r=0$ to $\theta=0$ at $r=\infty$. We obtain the equation for $\theta(r)$ by minimizing $\mathcal{H}$ $$\begin{aligned}
\frac{J}{{2r}}\sin \left( {2\theta } \right) + D\cos \left( {2\theta } \right) + {B_z}r\sin\theta + \frac{A}{2}r\sin \left( {2\theta } \right)
\nonumber\\
- \left( {J{\partial _r}\theta + D} \right) - r J\partial _r^2\theta = 0,\end{aligned}$$ from which $\theta(r)$ can be found numerically.
We then introduce a local coordinate system with the local $z$ axis along the spin direction $\mathbf{n}_s(\mathbf{r})$. The spin representation in the lab coordinate and the local coordinate is sketched in Fig. \[f1\] (b). The local coordinate is obtained by the subsequent rotation operations in the lab frame: rotation along the $z$ axis by $\phi_0=\pi/2$, rotation along the $y$ axis by $\theta$ and rotation along the $z$ axis by $\varphi$. Then the spin in the lab frame $\mathbf{n}$ can be obtained from the local coordinate $\mathbf{L}=(L_X,\ L_Y,\ L_Z)$ according to $\mathbf{n}=\hat{O}\mathbf{L}$, with $$\label{eq4}
\hat{O}= \left(
\begin{array}{ccc}
-\text{sin$\varphi $} & -\text{cos$\varphi $} \text{cos$\theta $} & \text{cos$\varphi $} \text{sin$\theta $} \\
\text{cos$\varphi $} & -\text{sin$\varphi $} \text{cos$\theta $} & \text{sin$\varphi $} \text{sin$\theta $} \\
0 & \text{sin$\theta $} & \text{cos$\theta $} \\
\end{array}
\right).$$ The small deviations $\mathbf{L}$ from the skyrmion line solution $\bar{L}_{X}=\bar{L}_Y=0$ and $\bar{L}_Z=1$ are described by the complex magnon fields $$\label{eq5}
\psi=\frac{L_X+i L_Y}{\sqrt{2}}, \ \ \ \psi^*=\frac{L_X-i L_Y}{\sqrt{2}},$$ and $L_Z=1-\psi\psi^*$ with $|\psi|\ll 1$. Expanding the Hamiltonian to second order in $\psi$, we obtain $$\label{eq6}
\mathcal{H}_\psi=\frac{1}{2}\hat{\psi}^\dagger{H}_\psi\hat{\psi}, ~~~ \hat{\psi}^\dagger=(\psi^*,\ \ \psi),$$ $$\begin{aligned}
\label{eq7}
{H}_\psi=(-J \nabla^2+V_0)\sigma_0+V_1\sigma_x\nonumber\\
-2\sigma_z\left[\left(J\frac{\cos\theta}{r^2}-D\frac{\sin\theta}{r}\right)i\partial_\phi-i D\cos\theta\partial_z\right],\end{aligned}$$ with $\sigma_i$ ($i=x,\ y,\ z$) being the Pauli matrices and $\sigma_0$ is the unit matrix. Here $$\begin{aligned}
\label{eq9}
{V_0} = J\frac{{1 + 3\cos \left( {2\theta } \right)}}{{4{r^2}}} -D \frac{{3\sin \left( {2\theta } \right)}}{{2r}} + {B_z}\cos\theta - D {\partial _r}\theta \nonumber\\
- \frac{J}{2}{\left( {{\partial _r}\theta } \right)^2}-\frac{A}{2}\left(1-3\cos^2\theta\right),\end{aligned}$$ $$\label{eq10}
{V_1} = J \frac{{{{\sin }^2}\theta }}{{2{r^2}}} + D \frac{{\sin \left( {2\theta } \right)}}{{2r}} - D {\partial _r}\theta - \frac{J}{2}{\left( {{\partial _r}\theta } \right)^2}+\frac{A}{2}\sin^2\theta.$$ The presence of skyrmion gives rise to an emergent magnetic field acting on the magnons. This can be seen explicitly by introducing an effective vector potential $$\mathbf{a}=-\hat{\phi } \left(\frac{\cos\theta }{r }-\frac{D \sin\theta}{J}\right)+\frac{ D \cos\theta }{J}\hat{z},$$ with $\hat{\phi}$ being the unit vector in the $\phi$ direction. Using $\nabla\cdot \mathbf{a}=0$, $H_\psi$ can be written in a compact form $$H_\psi=J{\left( { - {{i}}\nabla - {\sigma _3}{\bf{a}}} \right)^2} + {\sigma _0}\left( {{V_0} - J{{\bf{a}}^2}} \right) + {\sigma _1}{V_1}.$$ The emergent vector potential $\mathbf{a}$ couples to the magnons and induces a screw scattering of the extended magnons by skyrmions.[@PhysRevB.89.064412; @PhysRevB.90.094423]
![Dispersion of the Kelvin modes with angular momentum $m$. The gray region is the mangon continuum.[]{data-label="f2"}](fig2.pdf){width="\columnwidth"}
The eigenmodes are determined by the equation $$\label{eq11}
-i\frac{S\hbar}{a^3}\sigma_z\partial_t\hat{\psi}={H}_\psi\hat{\psi}.$$ This equation has the form of the Schrödinger equation describing the magnon wave function in a centrosymmetric potential. We can introduce an angular momentum $m$ and wavevector $k_z$ with $\psi=\psi_m(r,t)\exp(i m\phi+i k_z z)$ to label the eigenmodes. The two components of $\hat{\psi}$ are related by complex conjugation because the magnetic moment $\mathbf{n}$ is real. This indicates that the matrix equation, Eq. , is redundant. Indeed ${H}_\psi$ has particle-hole symmetry, ${H}_\psi=\sigma_x K {H}_\psi K \sigma_x$ with $K$ being the complex conjugate operator. This means that if $\exp[i(\omega t+m\phi+k_z z)]\hat{\eta}_m$, with $\hat{\eta}_m^\dagger\equiv (\eta_1^*, \ \ \eta_2^*)$, solves Eq. , then $\exp[-i(\omega t+m\phi+k_z z)]\sigma_x K\hat{\eta}_m$ also solves Eq. . We therefore only take the magnon branch with $\omega\ge 0$. Then $\hat{\psi}$ can be obtained by a linear superposition of the two symmetry-related solutions $$\nonumber
\hat{\psi}_m=b \exp[i(\omega t+m\phi+k_z z)]\hat{\eta}_m+b^*\exp[-i(\omega t+m\phi+k_z z)]\sigma_x K\hat{\eta}_m.$$ The two components of $\hat{\psi}_m$ are complex conjugate to each other. Here $\hat{\eta}$ is determined by the eigenvalue problem $$\label{eq13}
\frac{S\hbar}{a^3}\omega_m\sigma_z\hat{\eta}_m={H}_\psi\hat{\eta}_m.$$ When the frequency is much larger than the magnon gap of the FM state, $\frac{S\hbar}{a^3}\omega_{g}=B_z+A-D^2/J$, i.e. $\omega\gg \omega_g$, the magnon dispersion reduces to that in Eq. and the eigenmodes are $\hat{\eta}_m^\dagger=(1,\ 0)J_m(k r)$. We represent the matrix ${H}_\psi$ using the Bessel function $J_m(k r)$ as an orthogonal basis. [@PhysRevB.96.014407] The basis functions are $$\begin{aligned}
\nonumber
|{p_{m,i}}\rangle = \frac{{\sqrt 2 }}{{{R_c}{J_m}\left( {{k_{m - 1,i}}} \right)}}{J_{m - 1}}\left( {{k_{m - 1,i}}\frac{r}{{{R_c}}}} \right)\exp \left( {{{i}}m\phi+i k_z z } \right)\left( {\begin{array}{*{20}{c}}
1\\
0
\end{array}} \right),\end{aligned}$$ $$\begin{aligned}
\nonumber
|{h_{m,i}}\rangle = \frac{{\sqrt 2 }}{{{R_c}{J_{m + 2}}\left( {{k_{m + 1,i}}} \right)}}{J_{m + 1}}\left( {{k_{m + 1,i}}\frac{r}{{{R_c}}}} \right)\exp \left( {{{i}}m\phi+i k_z z } \right)\left( {\begin{array}{*{20}{c}}
0\\
1
\end{array}} \right),\end{aligned}$$ where we have used the box normalization with $R_c$ being the radius of the box and $k_{m,i}$ is the $i$-th zero of the Bessel function $J_m(kr)$. Then the matrix elements of ${H}_\psi$ are $$\begin{aligned}
\hat{{H}}_{11;ij}^{(m)}=\langle p_{m,i} | {H}_\psi |{p_{m,j}}\rangle, \ \ \hat{{H}}_{12;ij}^{(m)}=\langle p_{m,i} | {H}_\psi |{h_{m,j}}\rangle, \nonumber\\
\hat{{H}}_{21;ij}^{(m)}=\langle h_{m,i} | {H}_\psi |{p_{m,j}}\rangle, \ \ \hat{{H}}_{22;ij}^{(m)}=\langle h_{m,i} | {H}_\psi |{h_{m,j}}\rangle.\end{aligned}$$ By diagonalizing the matrix $\sigma_z {H}_\psi$, we obtain the eigenfrequencies and eigenmodes. We take $R_c=20$ and truncate the Bessel series at $i_{\mathrm{max}}=20$.
![The same as Fig. \[f2\], but with an easy axis anisotropy, $A=1.0\ D^2/J$ and $B_z=0.4\ D^2/J$.[]{data-label="f3"}](fig3.pdf){width="\columnwidth"}
The calculated dispersion of the Kelvin modes with different $m$ is shown in Fig. \[f2\]. There are only two Kelvin modes below the magnon continuum when $B_z=1.4 D^2/J$ in Fig. \[f2\] (a), and as a consequence, these modes are radially localized inside the skyrmion. The other modes mix with the magnon continuum and can easily decay into the extended magnon modes. The Kelvin mode with $m=-1$ corresponds to the distortion of a rigid skyrmion line discussed above. It is symmetric with respect to $k_z$ in the low energy region consistent with that in Eq. . The mode with $m=0$ corresponds to the uniform radial breathing of the skyrmion line. The group velocity $v_{g}=d\omega/dk_z$ is always positive, indicating a one-way propagation of this Kelvin mode. At a lower field, $B=1.05D^2/J$ in Fig. \[f2\] (b), the Kelvin mode with $m=-2$ also appears below the magnon continuum. In the presence of an easy axis anisotropy, there appear more Kelvin modes below the magnon continuum, see Fig. \[f3\]. The Kelvin mode with $m=-1$ in Figs. \[f2\] and \[f3\] has a very small gap originating from the numerical discretization in the calculations, which breaks the translation symmetry. The left branch of the Kelvin mode with $m=-1$ at high energy merges into the magnon continuum, and therefore is strongly damped. It only allows magnons with a positive group velocity to propagate in this region. Here all the Kelvin modes with $m\neq -1$ are gapped, which guarantee the meta-stability of the skyrmion line in the ferromagnetic background.
To use the skyrmion line as a one-way magnonic waveguide, it is required to excite the Kelvin mode with $m\neq -1$. This can be achieved by choosing the angular momentum of the source field. Recently, the propagation of magnons both with symmetric and asymmetric dispersion in a skyrmion line in prismatic geometry was demonstrated using micromagnetic simulations. [@xing_skyrmion_2019] The propagation of linear magnon wave and nonlinear solitary wave excitations along a skyrmion line in chiral magnets was considered in Ref. . The results on the linear magnon wave are consistent with ours. The unidirectional propagation of magnon in the skyrmion line crystal in $\mathrm{Cu_2OSeO_3}$ was investigated both experimentally and theoretically. [@seki_propagating_2019]
We have focused on a system with DMI form in Eq. , which can be realized in crystals having $D_n$ or $C_n$ symmetry, [@Bogdanov89] for example B20 chiral magnets including FeGe and MnSi. For crystals with $C_{nv}$ or $D_{2d}$ symmetry, such as Mn-Pt-Sn Heusler materials, [@nayak_magnetic_2017] $\mathrm{GaV_4S_8}$, [@kezsmarki_neel-type_2015] $\mathrm{GaV_4Se_8}$, [@PhysRevB.95.180410] and $\mathrm{VOSe_2O_5}$ [@PhysRevLett.119.237201] no spatial derivative along the crystal $c$ axis is allowed in the DMI. [@PhysRevB.96.214413] In these systems, all the Kelvin modes are symmetric with respect to $k_z$.
It is possible that skyrmion lines that do not percolate the whole system are stabilized. [@Milde2013; @yokouchi_current-induced_2018; @SZLinMonopole2016; @PhysRevB.94.174428; @PhysRevB.98.054404] At the ends of the lines, there appear emergent magnetic monopoles or antimonopoles. When both ends of a skyrmion line are terminated by a monopole and an antimonopole, the skyrmion line serves as a magnonic cavity for the Kelvin modes because these localized modes cannot penetrate into the ferromagnetic state. In imperfect systems, the skyrmion line is distorted in order to accommodate the pinning potential. [@Blatter94] The pinning opens a gap for the symmetric lowest Kelvin mode. The bent skyrmion line can still guide the Kelvin modes to propagate along the line.
To summarize, we have studied the Kelvin modes of a straight skyrmion line in chiral magnets. There exist several Kelvin modes below the magnon continuum, and these modes are radially localized in the skyrmion line. The Kelvin mode with angular momentum $m=-1$ is symmetric with respect to the wavevector along the skyrmion line in the low energy region. The Kelvin modes with other $m$ are asymmetric. Our results suggest that the skyrmion lines can function as a one-way magnonic waveguide.
SZL would like to thank Congjun Wu and Daniel P. Arovas for motivating the present study. The authors thank Markus Garst for useful discussions and for sharing their results prior to publication. This work was carried out under the auspices of the U.S. DOE NNSA under contract No. 89233218CNA000001 through the LDRD Program and the U.S. DOE Office of Basic Energy Sciences Program E3B5 (SZL and JXZ).
|
---
address: |
Department of Physics & Astronomy,\
Arizona State University,\
Tempe, AZ 85287-1504, USA\
E-mail: richard.lebed@asu.edu
author:
- 'Richard F. Lebed'
title: 'Baryon Resonances in the $1/{\rm N_c}$ Expansion'
---
Introduction
============
About 150 of the 1100 pages in the 2004 [*Review of Particle Properties*]{}[@PDG] catalogue measured properties of baryons; and of these, about 100 describe resonances unstable against strong decay, with lifetimes so short as to appear only as features in partial wave analyses. Such states have resisted a model-independent description for decades. To date there exists no convincing explanation for why QCD produces [*any*]{} baryon resonances, much less for their peculiar observed spectroscopy, mass spacings, and decay widths. Even the unambiguous existence of numerous resonances remains open to debate, as evidenced by the infamous 1- to 4-star classification system.[@PDG]
Baryon resonances are exceptionally difficult to study precisely because they [*are*]{} resonances rather than stable states. For example, treating baryon resonances as Hamiltonian eigenstates in quark potential models is questionable, because such models are strictly speaking valid only when vacuum $q \bar q$ pair production and annihilation is suppressed (to ensure a Hermitian Hamiltonian). It is just this mechanism, however, that provides the means by which baryon resonances occur in scattering from ground-state baryons.
Even so, one of the most natural descriptions of excited baryons in large $N_c$ remains an $N_c$ valence quark picture. The inspiration for this choice is that the ground-state baryon multiplets ($J^P
\! = \! {\frac 1 2}^+ \!$, ${\frac 3 2}^+$ for $N_c \! = \! 3$) neatly fill a single multiplet completely symmetric under combined spin-flavor symmetry \[the SU(6) [**56**]{}, for 3 light flavors\], so that one may suppose the ground state of $N_c$ quarks is also completely spin-flavor symmetric. Indeed, the SU(6) spin-flavor symmetry for ground-state baryons is shown to become exact in the large $N_c$ limit.[@DM] Then, in analogy to the nuclear shell model, excited states are formed by promoting a small number \[$O(N_c^0)$\] of quarks into orbitally or radially excited orbitals. For example, the generalization of the SU(6)$\times$O(3) multiplet $({\bf 70}, 1^- )$ consists of $N_c - 1$ quarks in the ground state and one in a relative $\ell \! = \! 1$ state. One may then analyze observables such as masses and axial-vector couplings by constructing a Hamiltonian whose terms possess definite transformation properties under the spin-flavor symmetry and are accompanied by known powers of $N_c$. By means of the Wigner-Eckart theorem, one then relates observables for different states in each multiplet. This approach has been extensively studied[@CGKM; @Goity; @PY; @CCGL; @GSS; @CC] (see Ref. for a short review), but it falls short in two important respects:
First, a Hamiltonian formalism is not entirely appropriate to unstable particles, since it refers to matrix elements between asymptotic external states. Indeed, a resonance is properly represented by a complex-valued pole in a scattering amplitude, its real and imaginary parts indicating mass and width, respectively. Moreover, a naive Hamiltonian does not recognize the essential nature of resonances as excitations of ground-state baryons.
Second, even a Hamiltonian constructed to respect the instability of the resonances would not necessarily give states in the simple quark-shell baryon multiplets as its eigenstates. Just as in the nuclear shell model, the possibility of [*configuration mixing*]{} suggests that the true eigenstates might consist of mixtures of states with 1, 2, or several excited quarks.
In contrast to quark potential models, chiral soliton models naturally accommodate baryon resonances as excitations resulting from scattering of mesons off ground-state baryons. Such models are consistent with the large $N_c$ limit because the solitons are heavy, semiclassical objects compared to the mesons. As has been known for many years,[@ANW] a number of predictions following from the Skyrme and other chiral soliton models are independent of the details of the soliton structure, and may be interpreted as group-theoretical, model-independent large $N_c$ results. Indeed, the equivalence of group-theoretical results for ground-state baryons in the Skyrme and quark models in the large $N_c$ limit was demonstrated[@Manohar] long ago. Compared to quark models, chiral soliton models tend to fall short in providing detailed spectroscopy and decay parameters for baryon resonances, particularly at higher energies. It is therefore gratifying that large $N_c$ provides a point of reference where both pictures share common ground.
In the remainder of this talk I discuss how the chiral soliton [*picture*]{} (no specific model) may be used to study baryon resonances as well as the full scattering amplitudes in which they appear, and also its relation to the quark [*picture*]{} (again, no specific model). It summarizes a series of papers written in collaboration with Tom Cohen (and more recently our students),[@CL1; @CLcompat; @CDLN1; @CLpent; @CDLN2; @CLSU3; @CDLM] and updates an earlier version[@UMN] of this talk.
Amplitude Relations {#amp}
===================
In the mid-1980’s a series of papers[@HEHW; @MK; @MP; @Mat3; @MM] uncovered a number of linear relations between meson-baryon scattering amplitudes in chiral soliton models. The fundamentally group-theoretical nature of these results, as was pointed out, suggested consistency with the large $N_c$ limit.
Standard $N_c$ counting[@Witten] shows that ground-state baryons have masses of $O(N_c^1)$, but meson-baryon scattering amplitudes are $O(N_c^0)$. Therefore, the characteristic resonant energy of excitation above the ground state and resonance widths are both generically expected to be $O(N_c^0)$. To say that two baryon resonances are degenerate to leading order in $1/N_c$ thus actually means equal masses at both the $O(N_c^1)$ and $O(N_c^0)$ levels.
A prototype of these linear relations was first derived in Ref. . For a ground-state ($N$ or $\Delta$) baryon of spin = isospin $R$ scattering with a meson (indicated by the superscript) of relative orbital angular momentum $L$ (and primes for analogous final-state quantum numbers) through a combined channel of isospin $I$ and spin $J$, the full scattering amplitudes $S$ may be expanded in terms of a smaller set of “reduced” scattering amplitudes $s$: $$\begin{aligned}
S_{LL^\prime R R^\prime IJ}^\pi & \! = & (-1)^{R^\prime \! \!
- R} \sqrt{[R][R^\prime]} \sum_K [K]
\left\{ \begin{array}{ccc} K &
I & J\\ R^\prime & L^\prime & 1 \end{array} \right\} \left\{
\begin{array}{ccc} K & I & J \\ R & L & 1 \end{array} \right\}
s_{K L^\prime L}^\pi \ , \ \label{MPeqn1} \\
S_{L R J}^\eta & = & \sum_K
\delta_{KL} \, \delta (L R J) \, s_{K}^\eta \ ,\label{MPeqn2}\end{aligned}$$ where $[X] \equiv 2X \! + \! 1$, and $\delta(j_1 j_2 j_3)$ indicates the angular momentum addition triangle rule. Both are consequences of a more general formula[@Mat9j] involving $9j$ symbols that holds for mesons of arbitrary spin and isospin, which for brevity we do not reproduce here. The basic feature inherited from chiral soliton models is the quantum number $K$ ([*grand spin*]{}) with [**K**]{}$\,\equiv\,$[**I**]{}$\,$+$\,$[**J**]{}, conserved by the underlying hedgehog configuration, which breaks $I$ and $J$ separately. The physical baryon state is a linear combination of $K$ eigenstates that is an eigenstate of both $I$ and $J$ but no longer $K$. $K$ is thus a good (albeit hidden) quantum number that labels the reduced amplitudes $s$. The dynamical content of relations such as Eqs. (\[MPeqn1\])–(\[MPeqn2\]) lies in the $s$ amplitudes, which are independent for each value of $K$ allowed by $\delta (IJK)$.
In fact, $K$ conservation turns out to be equivalent to the large $N_c$ limit. The proof[@CL1] begins with the observation that the leading-order (in $1/N_c$) $t$-channel exchanges have $I_t \! = \!
J_t,$[@KapSavMan] which in turn is proved using large $N_c$ [*consistency conditions*]{}[@DJM]—essentially, unitarity order-by-order in $1/N_c$ in meson-baryon scattering processes. However, ($s$-channel) $K$ conservation was found—years earlier—to be equivalent to the ($t$-channel) $I_t \! = \! J_t$ rule,[@MM] due to the famous Biedenharn-Elliott sum rule,[@Edmonds] an SU(2) identity.
The significance of Eqs. (\[MPeqn1\])–(\[MPeqn2\]) lies in the fact that there exist more full observable scattering amplitudes $S$ than reduced amplitudes $s$. Therefore, one obtains a number of linear relations among the measured amplitudes holding at leading \[$O(N_c^0)$\] order. In particular, a resonant pole appearing in one of the physical amplitudes must appear in at least one reduced amplitude; but this same reduced amplitude contributes to a number of other physical amplitudes, implying a degeneracy between the masses and widths of resonances in several channels.[@CL1] For example, we apply Eqs. (\[MPeqn1\])–(\[MPeqn2\]) to negative-parity[^1] $I \! = \! \frac 1 2$, $J \! = \frac
1 2$ and $\frac 3 2$ states (called $N_{1/2}$, $N_{3/2}$) in Table \[I\]. Noting that neither the orbital angular momenta $L ,
L^\prime$ nor the mesons $\pi , \eta$ that comprise the asymptotic states can affect the compound state except by limiting available total quantum numbers ($I$, $J$, $K$), one concludes that a resonance in the $S_{11}^{\pi NN}$ channel ($K \! = \! 1$) implies a degenerate pole in $D_{13}^{\pi NN}$, because the latter contains a $K \! = \!
1$ amplitude.
One thus obtains towers of degenerate negative-parity resonance multiplets labeled by $K$: $$\begin{aligned}
N_{1/2} , \; \Delta_{3/2} , \; \cdots \; &~& (K \! = \! 0 \! : \,
s_{0}^\eta) \; , \nonumber \\
N_{1/2} , \; \Delta_{1/2} , \; N_{3/2} , \; \Delta_{3/2} , \;
\Delta_{5/2} , \; \cdots \; &~& (K \! = \! 1 \! :
\, s_{1 0 0}^\pi , \, s_{1 2 2}^\pi) \; , \nonumber \\
\Delta_{1/2} , \; N_{3/2} , \; \Delta_{3/2} , \; N_{5/2} , \;
\Delta_{5/2} , \; \Delta_{7/2} , \; \cdots \;
&~& (K \! = \! 2 \! : \, s_{2 2 2}^\pi, \, s_{2}^\eta ) \; .
\label{towers}\end{aligned}$$ It is now fruitful to consider the quark-shell picture large $N_c$ analogue of the first excited negative-parity multiplet \[the $({\bf
70}, 1^- )$\]. Just as for $N_c \! = \! 3$, there are two $N_{1/2}$ and two $N_{3/2}$ states. If one computes the masses to $O(N_c^0)$ for the entire multiplet in which these states appear, one finds only three distinct eigenvalues,[@CCGL; @CL1; @PS] which are labeled $m_0$, $m_1$, and $m_2$ and listed in Table \[I\]. Upon examining an analogous table containing all the states in this multiplet,[@CL1] one quickly concludes that exactly the required resonant poles are obtained if each $K$ amplitude, $K \! = \! 0,1,2$, contains precisely one pole, which is located at the value $m_K$. The lowest quark-shell multiplet of negative-parity excited baryons is found to be [*compatible*]{} with, [*i.e.*]{}, consist of a complete set of, multiplets classified by $K$. But the quark-shell masses are [*real*]{} Hamiltonian eigenvalues, and therefore present a result less general than that obtained from the $K$ amplitude analysis.
One can prove[@CLcompat] this compatibility for all nonstrange baryon multiplets in the SU(6)$\times$O(3) shell picture.[^2] It is important to note that compatibility does not imply SU(6) is an exact symmetry at large $N_c$ for resonances as it is for ground states.[@DM] Instead, it says that SU(6)$\times$O(3) multiplets are [*reducible*]{} multiplets at large $N_c$. In the example given above, $m_{0,1,2}$ each lie only $O(N_c^0)$ above the ground state, but are separated by $O(N_c^0)$ intervals.
We emphasize that large $N_c$ by itself does not mandate the existence of any resonances at all; rather, it merely tells us that if even one exists, it must be a member of a well-defined multiplet. Although the soliton and quark pictures both have well-defined large $N_c$ limits, compatibility is a remarkable feature that combines them in a particularly elegant fashion.
Phenomenology {#phenom}
=============
Confronting these formal large $N_c$ results with experiment poses two significant challenges, both of which originate from neglecting $O(1/N_c)$ corrections. First, the lowest multiplet of nonstrange negative-parity states covers quite a small mass range (only 1535–1700 MeV), while $O(1/N_c)$ mass splittings can generically be as large as $O$(100 MeV). Any claims that two such states are degenerate while two others are not must be carefully scrutinized. Second, the number of states in each multiplet increases with $N_c$, meaning that a number of large $N_c$ states are spurious in $N_c \! =
\! 3$ phenomenology. For example, for $N_c \! \ge \! 7$ the analogue of the [**70**]{} contains three $\Delta_{3/2}$ states, but only one \[$\Delta(1700)$\] when $N_c \! = \! 3$. As $N_c$ is tuned down from large values toward 3, the spurious states must decouple through the appearance of factors such as $(1-3/N_c)$, which in turn requires one to understand simultaneously leading and subleading terms in the $1/N_c$ expansion.
Nevertheless, it is possible to obtain testable predictions for the decay channels, even using just the leading-order results. For example, note from Table \[I\] that the $K \! = \! 0(1)$ $N_{1/2}$ resonance couples only to $\eta$($\pi$). Indeed, the $N(1535)$ resonance decays to $\eta N$ 30–55$\%$ of the time despite lying barely above that threshold, while the $N(1650)$ decays to $\eta N$ only 3–10$\%$ of the time despite having much more comparable phase space to $\pi N$ and $\eta N$. This pattern clearly suggests that the $\pi$-phobic $N(1535)$ should be identified with $K \! = \! 0$ and the $\eta$-phobic $N(1650)$ with $K \! = \! 1$, the first fully field theory-based explanation for these phenomenological facts.
Configuration Mixing
====================
As mentioned above, one does not expect quark-shell baryon states with a fixed number of excited quarks to be eigenstates of the full QCD Hamiltonian. Rather, configuration mixing likely clouds the situation. Consider, for example, the expectation that baryon resonances have generically broad \[$O(N_c^0)$\] widths. One may ask whether some states escape this restriction and turn out to be narrow in the large $N_c$ limit. Indeed, some of the first work[@PY] on excited baryons combined large $N_c$ consistency conditions and a quark description of excited baryon states to predict that baryons in the [**70**]{}-analogue have widths of $O(1/N_c)$, while states in an excited negative-parity spin-flavor symmetric multiplet ([**56$^\prime$**]{}) have $O(N_c^0)$ widths.
In fact there arise, even in the quark-shell picture, operators inducing configuration mixing between these multiplets.[@CDLN1] The spin-orbit and spin-flavor tensor operators (respectively $\ell s$ and $\ell^{(2)} g \, G_c$ in the notation of Refs. ,$\,$,$\,$), which appear at $O(N_c^0)$ and are responsible for splitting the eigenvalues $m_0$, $m_1$, and $m_2$, give nonvanishing transition matrix elements between the [**70**]{} and [**56$^\prime$**]{}. Since states in the latter multiplet are broad, configuration mixing forces at least some states in the former multiplet to be broad as well. One concludes that the possible existence of any excited baryon state narrow in the large $N_c$ limit requires a fortuitous absence of significant configuration mixing.
Pentaquarks
===========
The possible existence of a narrow isosinglet, strangeness +1 (and therefore exotic) baryon state $\Theta^+ (1540)$, claimed to be observed by numerous experimental groups (but not seen by several others), remains an issue of great dispute. Although the jury remains out on this important question, one may nevertheless use the large $N_c$ method described above to determine the quantum numbers of its degenerate partners.[@CLpent] For example, if one imposes the theoretical prejudice $J_{\Theta} \! = \! \frac 1 2$, then there must also be pentaquark states with $I \! = \! 1$, $J \! = \! \frac 1 2,
\frac 3 2$ and $I \! = \! 2$, $J \! = \! \frac 3 2, \frac 5 2$, with masses and widths equal that of the $\Theta^+$, up to $O(1/N_c)$ corrections.
The large $N_c$ analogue of the “pentaquark” actually carries the quantum numbers of $N_c \! + \! 2$ quarks, consisting of $(N_c \! +
\! 1)/2$ spin-singlet, isosinglet $ud$ pairs and an $\bar s$ quark. The quark operator picture, for example, shows the partner states we predict to belong to SU(3) multiplets [**27**]{} ($I \! = \! 1$) and [**35**]{} ($I \! = \! 2$).[@JMpent] However, the existence of partners does not depend upon any particular picture for the resonance or any assumptions regarding configuration mixing. Since the generic width for such baryon resonances remains $O(N_c^0)$, the surprisingly small reported width ($<$10 MeV) does not appear to be explicable by large $N_c$ considerations alone, but may be a convergence of small phase space and a small nonexotic-exotic-pion coupling.
$1/N_c$ Corrections
===================
All the results exhibited thus far hold at the leading nontrivial order ($N_c^0$) in the $1/N_c$ expansion. We saw in Sec. \[phenom\] that $1/N_c$ corrections are essential not only to explain the sizes of effects apparent in the data, but in the very enumeration of physical states. Clearly, if this analysis is to carry real phenomenological weight, one must demonstrate a clear path to characterize $1/N_c$ corrections to the scattering amplitudes. Fortunately, such a generalization is possible: As discussed in Sec. \[amp\], the constraints on scattering amplitudes obtained from the large $N_c$ limit are equivalent to the $t$-channel requirement $I_t \! = \! J_t$. In fact, Refs. showed not only that the large $N_c$ limit imposes this constraint, but also that exchanges with $|I_t \! - \! J_t| \! = \! n$ are suppressed by a relative factor $1/N_c^n$.
This result permits one to obtain relations for the scattering amplitudes incorporating all effects up to and including $O(1/N_c)$: $$\begin{aligned}
\! \! \! \! S_{LL^\prime R R^\prime I_s J_s} \! \! \! \! & = &
\sum_{\mathcal J} \left[
\begin{array}{ccc} 1 & R^\prime & I_s \\ R & 1 & I_t \! = \!
{\mathcal J}
\end{array} \right] \left[
\begin{array}{ccc} L^\prime & R^\prime & J_s \\ R & L & J_t \! = \!
{\mathcal J}
\end{array}
\right] s_{{\mathcal J} L L^\prime}^t \nonumber \\ &
-\frac{1}{N_c} & \sum_{\mathcal J} \left[ \begin{array}{ccc} 1 &
R^\prime & I_s\\ R & 1 & I_t \! = \! {\mathcal J}
\end{array} \right] \left[
\begin{array}{ccc} L^\prime & R^\prime & J_s \\ R & L &
J_t \! = \! {{\mathcal J} \! + \! 1}
\end{array}
\right] s_{{\mathcal J} L L^\prime}^{t(+)} \nonumber \\ &
- \frac{1}{N_c} & \sum_{\mathcal J} \left[
\begin{array}{ccc} 1 & R^\prime & I_s\\ R & 1 & I_t \! = \!
{\mathcal J}
\end{array} \right] \left[
\begin{array}{ccc} L^\prime & R^\prime & J_s \\ R & L &
J_t \! = \! {{\mathcal J} \! - \! 1}
\end{array}
\right] s_{{\mathcal J} L L^\prime}^{t(-)} +
O(\mbox{\small{$\frac{1}{N_c^2}$}}) ,
\label{MPplus}\end{aligned}$$ One obtains this expression by first rewriting $s$-channel expressions such as Eqs. (\[MPeqn1\])–(\[MPeqn2\]) in terms of $t$-channel amplitudes. The $6j$ symbols in this case contain $I_t$ and $J_t$ as arguments (which for the leading term are equal). One then introduces[@CDLN2] new $O(1/N_c)$-suppressed amplitudes $s^{t(\pm)}$, for which $J_t - I_t = \pm 1$. The square-bracketed $6j$ symbols in Eq. (\[MPplus\]) differ from the usual ones only through normalization factors, and in particular obey the same triangle rules.
Relations between observable amplitudes that incorporate the larger set $s^t$, $s^{t(+)}$, and $s^{t(-)}$ are expected to be a factor of $N_c \! = \! 3$ better than those merely including the leading $O(N_c^0)$ results. Indeed, this is dramatically evident in $\pi N \!
\to \! \pi \Delta$, where sufficient numbers of amplitudes are measured (Fig. \[inter\]). For example, (c) and (d) in the first four insets give the imaginary and real parts, respectively, of partial wave data for $SD_{31}$ ([$\circ$]{}) and $(1/\sqrt{5})
DS_{13}$ ($\square$), which are equal up to $O(1/N_c)$ corrections; in (c) and (d) of the second four insets, the [$\circ$]{} points again are $SD_{31}$ data, while $\lozenge$ represent $-\sqrt{2}
DS_{33}$, and by Eq. (\[MPplus\]) these are equal up to $O(1/N_c^2)$ corrections.
=4.1in
=4.1in
Pion Photoproduction
====================
Meson-baryon scattering is not the only process that can be considered in the soliton-inspired picture. As long as one knows the isospin and spin quantum numbers of the field coupling to the baryon along with the corresponding $N_c$ power suppression of each coupling, one may carry out precisely the same sort of analysis as described above.
The processes we have in mind are those involving real or virtual photons (photoproduction,[@CDLM] electroproduction, real or virtual Compton scattering). One minor complication is that the electromagnetic interaction breaks isospin, in that the photon is a mixture of isoscalar ($I \! = \! 0$) and isovector ($I \! = \! 1$) sources. The former is suppressed by a factor $1/N_c$ compared to the latter since baryon couplings carrying both a spin index (coupling to the photon polarization vector) and an isospin index are larger than those carrying just a spin index by a factor $N_c$.[@JJM]
Moreover, electromagnetic processes are typically parametrized in terms of multipole amplitudes, which combine the intrinsic photon spin with its relative orbital angular momentum; in fact, this is very convenient, because then the photon can be treated effectively as a spinless field whose effective orbital angular momentum is the order of the multipole. Note that this makes processes with virtual photons just as simple as those with real photons, even though the former can carry not only spin-1 but spin-0 amplitudes as well. With these caveats in mind, carrying out an analysis of pion photoproduction amplitudes, including $1/N_c$ corrections (leading plus subleading $I
\! = \! 1$ amplitudes and leading $I \! = \! 0$ amplitudes), is straightforward.[@CDLM]
For example, a relationship receiving only $O(1/N_c^2)$ corrections reads $$M^{\textrm{m},\,p(\pi^+)n}_{L,L,-}
=
M^{\textrm{m},\,n(\pi^-)p}_{L,L,-}
-\left(\frac{L+1}{L}\right)
\left[
M^{\textrm{m},\,p(\pi^+)n}_{L,L,+}-M^{\textrm{m},\,n(\pi^-)p}_{L,L,+}
\right] , \label{NLO}$$ where the superscript m means magnetic multipoles, $N (\pi^a)
N^\prime$ means the process $N \gamma \! \to \! N^\prime \pi^a$, and the subscripts $L, L, \pm$ mean that an electromagnetic multipole of order $L$ creates a pion in the $L^{\rm th}$ partial wave, with total $J \! = \! L \! \pm \! \frac 1 2$. Including just the first term on the right-hand side (r.h.s.) gives a relation valid up to $O(1/N_c)$ corrections, and the quality of both this relation and its extension to next-to-leading order may be assessed.
A sample result appears in Fig. \[MAID\], where the left-hand side (l.h.s.) is a solid line, the $O(1/N_c)$ result is dotted, and the $O(1/N_c^2)$ is dashed. While the agreement at first glance may not seem impressive, some very heartening features may be discerned. First, the agreement in the region below the appearance of resonances is quite good, and indeed improves at $O(1/N_c^2)$. Second, unlike the solid line \[containing D$_{13}$(1520)\], the dotted line gives no hint of a resonance but the dashed line does \[D$_{15}$(1675)\]; and the fact that their positions do not precisely match should not alarm us, as one expects them to differ by an amount of $O(\Lambda_{\rm
QCD}/N_c) \! \sim \! 100$ MeV. One may in fact use the helicity amplitudes compiled[@PDG] for these two resonances and relate them directly to the amplitudes appearing in Eq. (\[NLO\]). In order to obtain dimensionless and scale-independent results, one divides the linear combination of helicity amplitudes corresponding to Eq. (\[NLO\]) by the same expression with all signs made positive. The $O(1/N_c)$ and $O(1/N_c^2)$ combinations give[@CDLM] $-0.38 \!
\pm \! 0.06$ and $-0.13 \pm 0.06$, respectively, showing that the $1/N_c$ expansion works beautifully—even better than one might expect.
2.2 in 2.2 in\
-1.0ex
Conclusions: The Way Forward
============================
There now exist reliable and convincing calculational techniques using the $1/N_c$ expansion of QCD that handle not only long-lived ground-state baryons, but also unstable baryon resonances and the scattering amplitudes in which they appear. The approach, originally noted in chiral soliton models but eventually shown to be a true consequence of large $N_c$ QCD, is found to have phenomenological consequences \[such as the large $\eta$ branching fraction of the $N(1535)$\] that compare favorably with real data.
The first steps of obtaining $1/N_c$ corrections to the leading-order results, absolutely essential to make comparisons with the full data set, are complete. The measured scattering amplitudes appear to obey the constraints placed by these corrections, and more work along these lines is forthcoming. For example, the means by which the spurious extra resonances of $N_c \! > \! 3$ decouple as one takes the limit $N_c \! \to \! 3$ is crucial and not yet understood.
The explicit results presented here, as mentioned in Sec. \[amp\], have used only relations among states of fixed strangeness. Moving beyond this limitation means using flavor SU(3) group theory, which is rather more complicated than isospin SU(2) group theory. Nevertheless, this is merely a technical complication, and existing work shows that it can be overcome.[@GSS; @CLSU3]
At the time of this writing, all of the essential tools appear to be in place to commence a full-scale analysis of baryon scattering and resonance parameters. One may envision a sort of resonance calculation factory, which I have previously dubbed [*Baryons $IN_C$*]{}.[@UMN] Given sufficient time and researchers, the whole baryon resonance spectrum can be disentangled using a solid, field-theoretical approach based upon a well-defined limit of QCD.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank my co-organizers for continuing the tradition of large $N_c$ meetings and for assembling a excellent program. I also appreciate the fine assistance of the ECT$^*$ staff. Finally, I would like to thank Korova Coffee Bar, San Diego, whose superb French Roast and free wireless internet access aided the timely writing of this paper. The work described here was supported in part by the National Science Foundation under Grant No. PHY-0140362.
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[^1]: Parity enters by restricting allowed values of $L,L^\prime$.[@CLcompat]
[^2]: Studies to extend these results to flavor SU(3) are underway[@CLSU3]; while the group theory is more complicated, it remains tractable.
|
---
abstract: |
The method of finding the minimal distance between smooth non crossing submanifolds in N-dimensional Euclidean space are presented. It based on solution of the equations that describe the dynamics of the pair of material points. The dynamical system can be presented as a natural mechanical system determined by Riemannian geometry on the manifold and chosen potential energy. Such an approach makes it possible to find Lyapunov function of the considered system and to formulates the requirements on the form of potential energy that brings to the convergence of the method.
[***Keywords:*** minimal distance, geometrical mechanics, optimization, stability]{}.
author:
- |
Stanislav S. Zub\
Faculty of Computer Science and Cybernetics,\
Taras Shevchenko National University of Kyiv,\
Glushkov boul., 2, corps 6.,\
Kyiv, Ukraine 03680\
`stah@univ.kiev.ua`\
\
Sergiy I. Zub\
Institute of Metrology,\
Mironositskaya st., 42,\
Kharkiv, Ukraine 61002\
`sergii.zub@gmail.com`\
\
Vladimir V. Semenov\
Faculty of Computer Science and Cybernetics,\
Taras Shevchenko National University of Kyiv,\
Glushkov boul., 2, corps 6.,\
Kyiv, Ukraine 03680\
`semenov.volodya@gmail.com`
title: '**Application of Lagrangian mechanics equations for finding of the minimum distance between smooth submanifolds in N-dimensional Euclidean space – Part II**'
---
Introduction
==============
Application of Lagrange mechanics equations (see section \[secLagrangeForm\]) for finding out the minimal distance between smooth non crossing surfaces in N-dimensions Euclidean space are presented.
Method base on solving of equations that describe the dynamics of the material point pair that moving under the action of a mutual force of attraction between them. Each point on the own surface is held back by holonomic constraints. The potential energy of points interaction depends only from the distance (see article [@GOPM]). For stopping-down in position that corresponds to minimum distance we add dissipation through Rayleigh function. The correspondent equations follow from d’Alembert principle and represented in section \[secDalamber\].
Because of every of the surfaces are Riemannian manifold, so their direct product is also a Riemannian manifold (see section \[RimanGeom\]) and our dynamical system can be represented as a natural mechanical system [@SteveSmale67] that is defined of Riemannian geometry on manifold and potential energy [@MarRat98 (1.1.7),p.3].
Such an approach makes it possible to find out Lyapunov function of considered system and formulates the requirements on the form of potential energy that gives the method convergence (see section \[Lyapunov\]).
Set up of the problem and notations
=====================================
Let $\mathfrak{N}$, $\mathfrak{M}$ are smooth non crossing surfaces in Euclidean space $E^N$.
In local coordinate systems the points $\xi$,$\eta$ of surfaces are described by radius-vector $$\label{Rvectors}
\begin{cases}
\vec{x}(\xi^1,\cdots,\xi^n)\in\mathfrak{N},\quad n<N;\\
\vec{y}(\eta^1,\cdots,\eta^m)\in\mathfrak{M},\quad m<N.
\end{cases}$$
Suppose that the kinetic energy of material points has the form $$\label{Rvectors}
\begin{cases}
T^{(1)}=\frac{m^{(1)}\dot{\vec{x}}^2}{2}=\frac{m^{(1)}}{2}({\partial_a\vec{x}}\cdot{\partial_b\vec{x}})\dot{\xi}^a\dot{\xi}^b;\\
T^{(2)}=\frac{m^{(2)}\dot{\vec{y}}^2}{2}=\frac{m^{(2)}}{2}({\partial_q\vec{y}}\cdot{\partial_p\vec{y}})\dot{\eta}^q\dot{\eta}^p,
\end{cases}$$ where “$\cdot$” is a scalar product in $E^N$; $a,b,c,d=1\ldots n$ are the indices of the internal variables of the 1-st surfaces; $q,p,r,s=1\ldots m$ are the indices of the internal variables of the 2-st surfaces.
Potential energy depends only on the distance between the points and is a monotonically increasing function $$\label{vecr}
\begin{cases}
U=U(r),\quad r=|\vec{r}|;\\
\vec{r}=\vec{y}(\eta)-\vec{x}(\xi)=\vec{y}(\eta^1,\cdots,\eta^m)-\vec{x}(\xi^1,\cdots,\xi^n);\\
|\vec{r}|=\sqrt{\vec{r}^{\ 2}}=\sqrt{r^I r_I}=\sqrt{\sum_{I=1}^N (r^I)^2};\\
\frac{\partial}{\partial\xi_a}(U(|\vec{r}|))=\frac{\partial U}{\partial r}\frac{\partial r}{\partial\xi_a}=-\frac{1}{r}\frac{\partial U}{\partial r}(\vec{y}-\vec{x})\cdot{\partial_a\vec{x}};\\
\frac{\partial}{\partial\eta_s}(U(|\vec{r}|))=\frac{\partial U}{\partial r}\frac{\partial r}{\partial\eta_s}=\frac{1}{r}\frac{\partial U}{\partial r}(\vec{y}-\vec{x})\cdot{\partial_s\vec{y}},
\end{cases}$$ where $I,J=1\ldots N$ indices enumerate the components of the radius-vectors in Euclidean space.
Then Lagrange function of generalized coordinates ($\xi^1,\cdots,\xi^n,\eta^1,\cdots,\eta^m$) and generalized velocities ($\dot{\xi}^1,\cdots,\dot{\xi}^n,\dot{\eta}^1,\cdots,\dot{\eta}^m$) has the form $$\label{L0}
\mathcal{L}_0=T^{(1)}+T^{(2)}-U=$$ $$=\frac{m^{(1)}}{2}({\partial_a\vec{x}}\cdot{\partial_b\vec{x}})\dot{\xi}^a\dot{\xi}^b
+\frac{m^{(2)}}{2}({\partial_q\vec{y}}\cdot{\partial_p\vec{y}})\dot{\eta}^q\dot{\eta}^p
-U(|\vec{y}(\eta)-\vec{x}(\xi)|).$$
At the surfaces $\mathfrak{N}$, $\mathfrak{M}$ insymbol $g^{(1)},g^{(2)}$ of Riemannian metrics using fundamental quadratic forms of surfaces [@MarRat98 (1.1.7),p.3] $$\label{Tensors}
\begin{cases}
g^{(1)}_{ab}=
m^{(1)}(\partial_{a}\vec{x}\cdot\partial_{b}\vec{x})=m^{(1)}\sum_{I=1}^N \partial_{a} x^I \partial_{b} x_I,\\
g^{(2)}_{st}=
m^{(2)}(\partial_{s}\vec{y}\cdot\partial_{t}\vec{y})=m^{(2)}\sum_{J=1}^N \partial_{s} y^J \partial_{t} y_J.
\end{cases}$$
The Lagrangian formalism {#secLagrangeForm}
==========================
Taking into account (\[Tensors\]) Lagrange function (\[L0\]) has the form: $$\label{L0}
\mathcal{L}(\xi^1,\ldots,\xi^n,\eta^1,\ldots,\eta^m;\dot{\xi}^1,\ldots,\dot{\xi}^n,\dot{\eta}^1,\ldots,\dot{\eta}^m)=$$ $$=\frac12 g^{(1)}_{ab}\dot{\xi}^a\dot{\xi}^b
+\frac12 g^{(2)}_{st}\dot{\eta}^s\dot{\eta}^t
-U(|\vec{y}(\eta)-\vec{x}(\xi)|).$$
Let’s write down Lagrange equations of our natural mechanical system $$\label{LEq}
\begin{cases}
\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial\dot{\xi}^a}\right)
-\frac{\partial \mathcal{L}}{\partial \xi^a}=0;\\
\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial\dot{\eta}^q}\right)
-\frac{\partial \mathcal{L}}{\partial \eta^q}=0.
\end{cases}$$
For the first equation of (\[LEq\]) we have $$\label{LEq_Comp}
\begin{cases}
\frac{\partial\mathcal{L}}{\partial\xi_a}=m^{(1)}({\partial_b\vec{x}}\cdot\partial^2_{ac}\vec{x})\dot{\xi}^c\dot{\xi}^b
+\partial_a(U(|\vec{y}(\eta)-\vec{x}(\xi)|));\\
\frac{\partial \mathcal{L}}{\partial\dot{\xi_a}}=m^{(1)}({\partial_a\vec{x}}\cdot{\partial_b\vec{x}})\dot{\xi}^b;\\
\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial\dot{\xi_a}}\right)=m^{(1)}(({\partial_b\vec{x}}\cdot{\partial^2_{ac}\vec{x}}+{\partial_a\vec{x}}\cdot{\partial^2_{bc}\vec{x}})\dot{\xi^c}\dot{\xi^b}
+({\partial_a\vec{x}}\cdot{\partial_b\vec{x}})\ddot{\xi}^b).
\end{cases}$$
Similarly, for the second equation in (\[LEq\]).
Substituding (\[LEq\_Comp\]) in (\[LEq\]) we get $$\label{LagrangeEq7}
\begin{cases}
g^{(1)}_{ab}\ddot{\xi}^b+
m^{(1)}({\partial_a\vec{x}}\cdot{\partial^2_{bc}\vec{x}})\dot{\xi^c}\dot{\xi}^b
+\partial_a(U(|\vec{r}|))=0; \\
g^{(2)}_{st}\ddot{\eta}^t+
m^{(2)}({\partial_s\vec{x}}\cdot{\partial^2_{tu}\vec{x}})\dot{\eta}^u\dot{\eta}^t
+\partial_s(U(|\vec{r}|))=0.
\end{cases}
$$
Lagrangian dynamics on Riemannian manifold {#RimanGeom}
============================================
Let’s show that our problem reduces to the dynamics of a point on Riemannian manifold that is the direct product of two Riemannian manifolds (the original surfaces in Euclidean space).
Insymbol $$\label{coord_impals}
\begin{cases}
q=(\xi^1,\ldots,\xi^n,\eta^1,\ldots,\eta^m); \\
\dot{q}=(\dot{\xi}^1,\ldots,\dot{\xi}^n,\dot{\eta}^1,\ldots,\dot{\eta}^m); \\
p=(\mu_1,\ldots,\mu_n,\nu_1,\ldots,\nu_m).
\end{cases}
$$
$$\label{metr_matrix}
g=\begin{bmatrix}
g^{(1)}\circ pr_1 & 0 \\
0 & g^{(2)}\circ pr_2
\end{bmatrix},
$$
$$\label{projection}
\begin{cases}
pr_1: q=(\xi^1,\ldots,\xi^n,\eta^1,\ldots,\eta^m)\longrightarrow\xi=(\xi^1,\ldots,\xi^n); \\
pr_2: q=(\xi^1,\ldots,\xi^n,\eta^1,\ldots,\eta^m)\longrightarrow\eta=(\eta^1,\ldots,\eta^m).
\end{cases}$$
Using (\[coord\_impals\]),(\[metr\_matrix\]) lets write (\[L0\]) in the form $$\label{L0_metr}
\mathcal{L}(q,\dot{q})=\frac12 g_{ik}\dot{q}^i\dot{q}^k-U(|\vec{r}(q)|),$$ or $$\label{L0_Marsden}
\mathcal{L}(q,v)=\frac12\langle v,v\rangle-U(|\vec{r}(q)|),
\quad v^i=\dot{q}^i, \quad i,j,k,l=1,\dots,n+m.$$
Since our configuration space is Riemannian space with metric (\[metr\_matrix\]) that is given by the kinetic energy of the system and Lagrangian has the form (\[L0\_Marsden\]) (see [@MarRat98 (7.7.2)–(7.7.3), p.198]) then our problem can be reduced to the theory of natural mechanical system with kinetic energy that is determined by the metric of Riemannian manifold [@MarRat98 §7.7].
Let’s transform the equations (\[LagrangeEq7\]) to the form $$\label{LagrangeEq8}
\begin{cases}
g^{(1)}_{ab}\ddot{\xi}^b+
m^{(1)}({\partial_a\vec{x}}\cdot{\partial^2_{bc}\vec{x}})\dot{\xi^c}\dot{\xi}^b
=-\partial_a(U(|\vec{r}|)); \\
g^{(2)}_{st}\ddot{\eta}^t+
m^{(2)}({\partial_s\vec{x}}\cdot{\partial^2_{tu}\vec{x}})\dot{\eta}^u\dot{\eta}^t
=-\partial_s(U(|\vec{r}|)).
\end{cases}$$
Then the left-hand sides can be expressed in terms of Riemannian geometry (metric, connection coefficient).
Indeed $$\label{LagrangeInvForm}
g\ddot{q}=\begin{pmatrix}
g^{(1)}\circ pr_1 & 0 \\
\\
\\
0 & g^{(2)}\circ pr_2
\end{pmatrix}
\begin{pmatrix}
\ddot{\xi}^{(1)}\\
\vdots\\
\ddot{\xi}^{(n)}\\
\ddot{\eta}^{(1)}\\
\vdots\\
\ddot{\eta}^{(m)}
\end{pmatrix}
$$ and $$\label{secondin7}
\begin{split}
m^{(1)}({\partial_a\vec{x}}\cdot\partial^2_{bc}\vec{x})
&=\frac12(\partial_b g_{ac}+\partial_c g_{ab})-\\
&-\frac12(\partial^2_{ab}\vec{x}\cdot\partial_c\vec{x}+\partial^2_{ac}\vec{x}\cdot\partial_b\vec{x})=\\
&=\frac12(\partial_b g_{ac}+\partial_c g_{ab})-\frac12\partial_a g_{bc}=\\
&=\frac12(\partial_b g_{ac}+\partial_c g_{ab}-\partial_a g_{bc})=\Gamma_{b,ac};
\end{split}$$ so far as $$\label{LagrangeEq5}
\begin{cases}
\partial_a g_{bc}=\partial^2_{ab}\vec{x}\cdot\partial_c\vec{x}+\partial_b\vec{x}\cdot\partial^2_{ac}\vec{x};\\
\partial_b g_{ac}=\partial^2_{ab}\vec{x}\cdot\partial_c\vec{x}+\partial_a\vec{x}\cdot\partial^2_{bc}\vec{x};\\
\partial_c g_{ab}=\partial^2_{ac}\vec{x}\cdot\partial_b\vec{x}+\partial_a\vec{x}\cdot\partial^2_{bc}\vec{x}.
\end{cases}
$$
Then the equations (\[LagrangeEq8\]) take the form $$\label{LagrangeEq9}
\begin{cases}
g^{(1)}_{ab}\ddot{\xi}^b+
\Gamma_{b,ac}\dot{\xi^c}\dot{\xi}^b
=-\partial_a(U(|\vec{r}|)); \\
g^{(2)}_{st}\ddot{\eta}^t+
\Gamma_{t,su}\dot{\eta}^t\dot{\eta}^u
=-\partial_s(U(|\vec{r}|));\\
\Gamma_{b,ac}=\frac12(\partial_b g_{ac}+\partial_c g_{ab}-\partial_a g_{bc});\\
\Gamma_{t,su}=\frac12(\partial_t g_{su}+\partial_u g_{st}-\partial_s g_{tu}).
\end{cases}$$
Let’s use the well-known relation for the Christoffel symbol $$\label{SymCryst1}
\Gamma^a_{bc}=g^{ad}\Gamma_{b,dc},$$ where $g^{ad}$ is contravariant metric tensor given by an inverse matrix with respect to a covariant matrix. I.e. $$\label{MetricRev}
\begin{cases}
g_{(1)}^{ab}g^{(1)}_{bc}=\delta^a_c;\\
g_{(2)}^{st}g^{(2)}_{tu}=\delta^s_u;\\
g_{ik}g^{kl}=\delta^l_i,
\end{cases}$$ where $g$ in last relation is taken from (\[metr\_matrix\]) with taking into account the block structure of this matrix.
Then the Lagrange equations of motion take the form $$\label{LagrangeEq6}
\begin{cases}
\ddot{\xi}^a+(\Gamma^{(1)})^a_{bc}\dot{\xi}^b\dot{\xi}^c=-g_{(1)}^{ac}\partial_c(U(|\vec{r}|));\\
\ddot{\eta}^s+(\Gamma^{(2)})^s_{tu}\dot{\eta}^t\dot{\eta}^u=-g_{(2)}^{st}\partial_t(U(|\vec{r}|)),
\end{cases}
$$ i.e. $$\label{LagrangeSymCryst3}
\ddot{q}^i+\Gamma^i_{jk}\dot{q}^j\dot{q}^k+g^{ik}\partial_k(U(|\vec{r}|))=0.$$
The Hamiltonian equations and dissipative term {#secDalamber}
================================================
Let’s represent the equation (\[LagrangeSymCryst3\]) in the form $$\label{LagrangeEqCryst4}
\ddot{q}^i=\gamma(q,\dot{q})^i-(\nabla U)^i,$$ where $$\label{Grad}
\begin{cases}
\gamma(q,\dot{q})^i=-\Gamma^i_{jk}\dot{q}^j\dot{q}^k;\\
(\nabla U)^i=g^{ik}\frac{\partial U}{\partial q^k}.
\end{cases}$$
Equation (\[LagrangeEqCryst4\]) fully coincides with [@MarRat98 (7.7.3),p.198].
The transition to the first-order Hamiltonian equations is realized with the help of Legendre transformations $$\label{LegandreTransf1}
\begin{cases}
p_i=v_i=g_{ij}v^j=g_{ij}\dot{q}^j;\\
\mathcal{H}(q,p)=E(v)=A(v)-\mathcal{L}(v)=\\
=\langle v,v\rangle_q -\mathcal{L}(v)=\frac12\langle v,v\rangle_q+U(q).
\end{cases}$$ $$\label{LegandreTransf2}
\begin{cases}
\dot{q}=v;\\
\dot{v}=\gamma(q,v)-\nabla U(q).
\end{cases}$$
Strictly speaking the Hamiltonian equations must be written with respect to the covariant vector $p_i$ rather than a contravariant vector $v^i$. However, the equations of the first order (\[LegandreTransf2\]) are completely equivalent to the Hamiltonian equations that can easily be obtained from (\[LegandreTransf2\]) by using covariant differentiation.
Indeed $$\label{CovarDeriv0}
\begin{cases}
\frac{d v}{dt}-\gamma(q,v)=\frac{D v}{dt};\\
\frac{D v}{dt} = -\nabla U;\\
g^\flat\frac{D v}{dt} = -d U;\\
\frac{D g}{dt} = 0;\\
g^\flat\frac{D v}{dt} = \frac{D (g^\flat v)}{dt} = \frac{D p}{dt};\\
\frac{D p}{dt} = - d U,
\end{cases}$$ where $\frac{D}{dt}$ is a symbol of the covariant derivative along the trajectory.
As it shown in [@MarRat98 с.205] the dissipation can be described via the Rayleigh function and the generalized Lagrange equation takes the form $$\label{Rayleigh1}
\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}^i}\right)-\frac{\partial \mathcal{L}}{\partial q^i}=-\frac{\partial \mathcal{R}}{\partial \dot{q}^i},
$$ where $$\mathcal{R}=\frac12 R_{ij}(q)v^i v^j.$$ Then the equations (\[LegandreTransf2\]) taking into account the dissipation can be written $$\label{Rayleigh2}
\begin{cases}
\dot{q}=v;\\
\dot{v}=\gamma(q,v)-\nabla U(q)-F_\mathcal{R}(q,v);\\
F_\mathcal{R}^i=g^{ik}\frac{\partial \mathcal{R}}{\partial v^k}
\end{cases}$$
First the Lagrange equations (\[LagrangeEq7\]) for our system were written in the local coordinate system. However, the principle of action for the Lagrange equation (\[LegandreTransf2\]) and d’Alembert principle [@MarRat98 p.202–203] from that follows the equations (\[Rayleigh2\]) that formulated on Riemannian manifold, independently of local coordinate systems. Thus, the equations (\[Rayleigh2\]) define a global vector field on the Riemannian manifold of our natural mechanical system.
Liapounov function {#Lyapunov}
====================
The point $(q_0,0)$ where $\nabla U(q_0)=0$ and $v=0$ is an equilibrium point for the (\[Rayleigh2\]).
This is obvious from the form of the right-hand sides of the equations (\[Rayleigh2\]).
The function $L(q,v)=E(q,v)-U(q_0)$ is the Lyapunov function for the field (\[Rayleigh2\]).
- $L(q_0,0)=0$ by construction.
- For provide $L(q,v)>0$ for anyone $(q,v)\in U\backslash \{(q_0,0)\}$ it is necessary that $U(q)>U(q_0)$ on a whole neighborhood area (as known that the kinetic energy is positive definite).
- Let’s show that $\dot{L}(q,v):=X[L](q,v)<0$ anywhere $(q,v)\in U\backslash \{(q_0,0)\}$.
$$\label{Energy1}
\begin{split}
\frac{dE}{dt}=\frac{DE}{dt}=\frac{D}{dt}\left(\frac12\langle v,v\rangle_q+U(q)\right)
&=\left\langle v,\frac{Dv}{dt}\right\rangle+\frac{dU}{dt}=\\
&=-\langle v,\nabla U\rangle-\langle v,\nabla^{(v)}\mathcal{R}\rangle+\langle\nabla U, v\rangle=\\
&=-\langle v,\nabla^{(v)}\mathcal{R}\rangle=\\
&=-v^i\frac{\partial \mathcal{R}}{\partial v^i}=-2\mathcal{R}<0,
\end{split}$$
where $E(q,v)$ is the total energy of the Lagrangian system without dissipation.
In last relation we use the fact that Rayleigh function is a quadratic positive-definite form of velocities, i.e. homogeneous velocity function of 2-degree.
Then from Lyapunov theorem (the second method)[@OrtegaLect; @ModContr] take place the asymptotic stability of the equations (\[Rayleigh2\]) at the equilibrium point.
Let [*note*]{} that the potential energy of type $U=k r^2$, where $k>0$ is suitable case.
[20]{} S.S. Zub, N.I. Lyashko , V.V. Semenov, Dynamic systems for finding best approximation pairs relative to two smooth curves in euclidean space – Part I, arXiv \[math-ph\] S. Smale, Differentiable dynamical systems. Bull. – Amer. Math. Soc., 1967. – Vol.73, 747–817 pp. J. Marsden, T. Ratiu, Introduction to mechanics and symmetry. – New York: Springer, 1999. – 553 p. J.-P. Ortega, Symmetric Hamiltonian Systems. Stability Methods and Applications. – La Cristalera, 2014. – 181 p. J. Hsu, A. Meyer, Modern Control Principles and Applications. – McGraw-Hill, 1968. – 769 p.
|
---
abstract: |
As a result of a new improved fit to old bubble chamber data of the dominant axial $C_5^A$ nucleon-to-Delta form factor, and due to the relevance of this form factor for neutrino induced coherent pion production, we re-evaluate our model predictions in Phys. Rev. D [**79**]{}, 013002 (2009) for different observables of the latter reaction. Central values for the total cross sections increase by 20%$\sim$30%, while differential cross sections do not change their shape appreciably. Furthermore, we also compute the uncertainties on total, differential and flux averaged cross sections induced by the errors in the determination of $C_5^A$. Our new results turn out to be compatible within about $1\sigma$ with the former ones. Finally, we stress the existing tension between the recent experimental determination of the $\frac{\sigma({\rm CC coh}\pi^+)}{\sigma({\rm NC
coh}\pi^0)}$ ratio by the SciBooNE Collaboration and the theoretical predictions.
author:
- 'E. Hernández'
- 'J. Nieves'
- 'M. Valverde'
title: Coherent pion production off nuclei at T2K and MiniBooNE energies revisited
---
Introduction
============
Experimental analyses of neutrino induced coherent pion production generally rely on the Rein–Sehgal (RS) model [@rein; @rein2] which is based on the partial conservation of the axial current (PCAC) hypothesis. In the RS model the pion-nucleus coherent cross section is written in terms of the pion-nucleon elastic cross section by means of approximations that are valid for high neutrino energies and small values of the nucleus momentum transfer square and of the lepton momentum transfer square ($q^2$). As pointed out in Refs. [@Amaro:2008hd; @Hernandez:2009vm], those approximations are less reliable for neutrino energies below/around 1 GeV, light nuclei, like carbon or oxygen, and finite values of $q^2$. These are the energies and targets used in present and forthcoming neutrino oscillation experiments. There are other approaches to coherent production that do not rely on PCAC but on microscopic models for pion production at the nucleon level [@Amaro:2008hd; @Fogli:1979cz; @kelkar; @Sato:2003rq; @singh; @luis1; @luis2; @Nakamura:2009iq; @Nakamura:2009kr; @Martini:2009uj]. The dominant contribution to the elementary amplitude at low energies is given by the $\Delta$-pole mechanism ($\Delta$ excitation and its subsequent decay into $\pi
N$). Medium effects on the $\Delta$ mass and width, final pion distortion, as well as nonlocalities in the pion momentum, are very important and are taken into account in microscopic calculations. Similarly to PCAC models, the process is dominated by the axial part of the weak current and it is thus very sensitive to nucleon-to-Delta axial form factors.
Very recently, the role of nonlocalities in the $\Delta$ momentum has also been investigated [@Leitner:2009ph; @Praet:2009zz; @Nakamura:2009iq; @Nakamura:2009kr]. In Ref. [@Leitner:2009ph] it is claimed that their neglect, the so called, local approximation, leads to an overestimate of the coherent production cross section that can be as large as a factor of 2 for neutrino energies of 500MeV. Similar results were obtained in Ref. [@Praet:2009zz]. Final pion distortion and in medium modifications of the $\Delta$ properties were not considered, and it was not clear whether those approximations could not affect the results. Final pion distortion and in medium modification of the $\Delta$ properties were included in Refs. [@Nakamura:2009iq; @Nakamura:2009kr], were nonlocal effects on the $\Delta$ momentum were incorporated in the $\Delta$ self-energy in the first-order approximation. They also observe a large reduction in the total cross section due to the nonlocal aspects of the $\Delta$ propagation in the medium. However, as claimed by the authors of Ref. [@Nakamura:2009iq], this that not mean that earlier microscopic calculations [@Amaro:2008hd; @Fogli:1979cz; @kelkar; @Sato:2003rq; @singh; @luis1; @luis2; @Martini:2009uj] are wrong, as there $\Delta$ nonlocal effects are taken into account in an effective way through the in medium modification of the $\Delta$ properties which were fitted to observables.
In the model we developed in Ref. [@Amaro:2008hd], the $\Delta$ was treated in the local approximation. However, the modifications of the $\Delta$ in medium properties are such that similar models give a good reproduction of pionic atoms and $\pi-$nucleus scattering [@GarciaRecio:1989xa; @Nieves:1993ev], pion photoproduction [@Carrasco:1989vq], pion electroproduction [@Gil:1997bm], ($^3$He,t) [@FernandezdeCordoba:1992df] and elastic $\alpha-$proton [@FernandezdeCordoba:1993az] reactions. We share the claim of the authors of Ref. [@Nakamura:2009iq] and we believe this treatment of the $\Delta$, where certainly non-local effects are being effectively (partially) taken into account, is also adequate for neutrino induced reactions. Nevertheless, this interesting issue deserves future investigations.
Our model in Ref. [@Amaro:2008hd] is based on a microscopic model at the nucleon level, described in detail in Ref. [@Hernandez:2007qq], that, besides the dominant $\Delta$ pole contribution, takes into account background terms required by chiral symmetry. As a result of the inclusion of background terms, we had to re adjust the strength of the dominant $\Delta$ pole contribution. The least known ingredients of the model are the axial nucleon-to-$\Delta$ transition form factors of which $C_5^A$, not only gives the largest contribution, but it also controls all other axial form factors if one assumes Adler’s model [@Adler:1968tw] that gives $C_4^A(q^2) = -\frac{C_5^A(q^2)}{4}, \
C_3^A(q^2) = 0 $, and PCAC is used to obtain $C_6^A(q^2) = C_5^A(q^2)
\frac{M^2}{m_\pi^2-q^2}$. This strongly suggested to us the readjustment of $C_5^A$ to the experimental data.
Information on pion production off the nucleon comes mainly from two bubble chamber experiments, ANL [@anlviejo; @anl; @anl-cam73] and BNL [@bnlviejo; @bnl]. Assuming, as proposed in Ref. [@Pa04], the $q^2$ dependence $
C_5^A(q^2) = \frac{C_5^A(0)}{(1-q^2/M^2_{A\Delta})^2}\,(
1-\frac{q^2}{3M_{A\Delta}^2})^{-1}
$, we fitted in Ref. [@Hernandez:2007qq] the flux-averaged $\nu_\mu
p\to \mu^- p \pi^+$ ANL $q^2$-differential cross section for pion-nucleon invariant masses $W < 1.4$ GeV [@anl; @anlviejo] obtaining $C_5^A(0) = 0.867 \pm 0.075$ and $M_{A\Delta}=0.985\pm 0.082\,{\rm}$, with a Gaussian correlation coefficient $r=-0.85$ and a $\chi^2/dof=0.4$. The fitted axial mass was in good agreement with the estimates of about 0.95 GeV and 0.84 GeV given in Refs. [@anl; @Pa05]. On the other hand, the $C_5^A(0)$ value is some 30% smaller than the prediction obtained from the off diagonal Goldberger-Treiman relation (GTR) that gives $\left.C_5^A(0)\right|_{GTR}=1.2$. $C_5^A(0)$ is not constrained by chiral perturbation theory ($\chi$PT) and lattice calculations are still not conclusive about the size of possible violations of the GTR. For instance, though values for $C_5^A(0)$ as low as 0.9 can be inferred in the chiral limit from the results of Ref. [@negele07-prd], they also predict $C_5^A(0)/\left.C_5^A(0)\right|_{GTR}$ to be greater than one.
Recently, two re-analysis have been carried out trying to make compatible the GTR prediction for $C_5^A(0)$ and ANL data. In Ref. [@Leitner:2008ue], $C_5^A(0)$ is kept to its GTR value and three additional parameters, that control the $C_5^A(q^2)$ fall off, are fitted to the ANL data. Although ANL data are well reproduced, we find the outcome in [@Leitner:2008ue] to be unphysical, as it provides a quite pronounced $q^2-$dependence giving rise to a too large axial transition radius of around 1.4 fm (further details are discussed in [@Hernandez:2009zg]).
A second re-analysis [@Graczyk:2009qm] brings in the discussion two interesting points. First that both ANL and BNL data were measured in deuterium, and second, the uncertainties in the neutrino flux normalization. It is claimed in Ref. [@Graczyk:2009qm], that the latter could be responsible for BNL total cross sections being systematically larger than ANL ones. In Ref. [@Graczyk:2009qm] the authors do a combined best fit to the ANL and BNL data including deuteron effects, which they evaluate as in Ref. [@AlvarezRuso:1998hi], and flux normalization uncertainties, treated as systematic errors, and taken to be 20% for ANL data and 10% for BNL data. With a pure dipole dependence for $C_5^A$, they found $C_5^A(0)=1.19 \pm 0.08$, in agreement with the GTR estimate.
The works in Refs. [@Leitner:2008ue; @Graczyk:2009qm] consider only the $\Delta $ pole mechanism but ignore the sizable non-resonant contributions which are of special relevance for neutrino energies below 1 GeV. When background terms are considered, the tension between ANL data and the GTR prediction for $C_5^A(0)$ substantially increases as the results in Ref. [@Hernandez:2007qq] clearly shows.
In our work in Ref. [@Hernandez:2010bx] we have performed a fit to both ANL and BNL data in which: [*i)*]{} We have included the BNL total $\nu_\mu p \to \mu^- p \pi^+$ cross section measurements of Ref. [@bnlviejo]. We have just included the three lowest neutrino energies: 0.65, 0.9 and 1.1 GeV, since there is no cut in the outgoing pion-nucleon invariant mass in the BNL data, and we want to avoid heavier resonances from playing a significant role. We have not used the BNL measurement of the $q^2-$differential cross section, since it lacked an absolute normalization. [*ii)*]{} We have taken into account deuteron effects, [*iii)*]{} the uncertainties in the neutrino flux normalizations, 20% for ANL and 10% for BNL data, are treated as fully correlated systematic errors, improving thus the simpler treatment adopted in Ref. [@Graczyk:2009qm], and finally [*iv)*]{} in some fits, we have relaxed Adler’s model constraints, in order to extract some direct information on $C_{3,4}^A(0)$. For simplicity we took $C_5^A(q^2) = \frac{C_5^A(0)}{(1-q^2/M^2_{A\Delta})^2}$. As in Ref. [@Graczyk:2009qm], the consideration of BNL data and flux uncertainties increased the value of $C_5^A(0)$ by about 9%, while strongly reduced the statistical correlations between $C_5^A(0)$ and $M_{A\Delta}$. The inclusion of background terms reduced $C_5^A(0)$ by about 13%, and deuteron effects increased it by about 5%, consistently with the results of [@Hernandez:2007qq] and [@AlvarezRuso:1998hi; @Graczyk:2009qm], respectively. Fitted data was quite insensitive to $C_{3,4}^A(0)$.
In our most robust fit in Ref. [@Hernandez:2010bx] we used Adler’s constraints, and we obtained $C_5^A(0) = 1.00 \pm 0.11, \ M_{A\Delta}=0.93\pm 0.07\,{\rm
GeV}$, with a small Gaussian correlation coefficient $r=-0.06$ and a $\chi^2/dof=0.32$. This violation of the GTR is about 15%, and it is smaller than that suggested in Ref. [@Hernandez:2007qq], though it is definitely greater than that claimed in Ref. [@Graczyk:2009qm], mostly because in Ref. [@Graczyk:2009qm] background terms were not considered. However, the GTR value and the $C_5^A(0)$ above differ in less than $2\sigma$, and the discrepancy is even smaller if Adler’s constraints are removed.
These new results are quite relevant for the neutrino induced coherent pion production in nuclei which is a low $q^2$ dominated reaction. Background term contributions to coherent production largely cancel for symmetric nuclei [@Amaro:2008hd] making the $\Delta$ pole mechanism the unique contribution. As the process is dominated by the axial part of the weak current, it is very sensitive to $C_5^A(0)$. Thus, we would expect the results in Ref. [@Amaro:2008hd], based in the determination of $C_5^A(0)$ of Ref. [@Hernandez:2007qq], to underestimate cross sections by some 30%. In this work we re-evaluate different pion coherent production observables using our model of Ref. [@Amaro:2008hd] but with the new parameterization and results for $C_5^A$ obtained in Ref. [@Hernandez:2010bx]. As the correlation coefficient is small in this case, we shall treat the theoretical errors that derive from the uncertainties in $C_5^A(0)$ and $M_{A\Delta}$ as independent and we shall add them in quadratures.
New results {#sec:results}
===========
We start by showing in the left panel of Fig. \[fig:flux-cross\] the differential cross section with respect to the $E_\pi(1-\cos\theta_\pi)$ variable for neutral current (NC) coherent $\pi^0$ production on carbon. $E_\pi$ is the pion energy in the laboratory frame (LAB) while $\theta_\pi$ is the LAB angle between the pion and the incoming neutrino. The shapes of the distributions are completely similar to the ones we obtained in Ref. [@Amaro:2008hd], but the absolute values increase by some 20%$\sim$ 30% depending on the neutrino energy. This is generally true for other differential cross sections that we do not show here. For the distribution convoluted with the MiniBooNE flux we find an increase in the total cross section of about 29%.
------------------------------- ------------ ---------------------- ---------------------------- --------------------- ---------------------------------------------------- -----------------------------------------------------
Reaction Experiment $\bar \sigma$ $\sigma_{\rm exp}$ $E_{\rm max}^i $ $\int_{E_{\rm low}^i}^{E_{\rm max}^i} dE \phi^i(E) $\int_{E_{\rm low}^i}^{E_{\rm max}^i} dE \phi^i(E)$
\sigma(E)$
\[$10^{-40}$cm$^2$\] \[$10^{-40}$cm$^2$\] \[GeV\] \[$10^{-40}$cm$^2$\]
CC $\nu_\mu + ^{12}$C K2K $6.1\pm1.3$ $<7.7 $ [@Hasegawa:2005td] 1.80 $5.0\pm1.0$ 0.82
CC $\nu_\mu + ^{12}$C MiniBooNE $3.8\pm0.8$ 1.45 $3.5\pm0.7$ 0.93
CC $\nu_\mu + ^{12}$C T2K $3.2\pm0.6$ 1.45 $2.9\pm0.6$ 0.91
CC $\nu_\mu + ^{16}$O T2K $3.8\pm0.8$ 1.45 $3.4\pm0.7$ 0.91
NC $\nu_{{\mu}} + ^{12}$C MiniBooNE $2.6\pm0.5$ $7.7\pm1.6\pm3.6$ [@Raaf] 1.34 $2.2\pm0.5$ 0.89
NC $\nu_{{\mu}} + ^{12}$C T2K $2.3\pm0.5$ 1.34 $2.1\pm0.5$ 0.90
NC $\nu_{{\mu}} + ^{16}$O T2K $2.9\pm0.6$ 1.35 $2.6\pm0.6$ 0.90
CC $\bar\nu_\mu + ^{12}$C T2K $2.6\pm0.6$ 1.45 $1.8\pm0.4$ 0.67
NC $\bar\nu_{{\mu}} + ^{12}$C T2K $2.0\pm0.4$ 1.34 $1.3\pm0.3$ 0.64
------------------------------- ------------ ---------------------- ---------------------------- --------------------- ---------------------------------------------------- -----------------------------------------------------
: NC/CC $\nu_\mu$ and $\bar\nu_\mu$ coherent pion production total cross sections, with errors, for K2K, MiniBooNE and T2K experiments. In the case of CC K2K, the experimental threshold for the muon momentum $|\vec{k}_\mu|>$ 450 MeV is taken into account. Details on the flux convolution are compiled in the last three columns. []{data-label="tab:res"}
In Table \[tab:res\] we show our new predictions for, both NC and charged current (CC) processes, for the K2K [@Hasegawa:2005td] and MiniBooNE [@AguilarArevalo:2008xs] flux averaged cross sections as well as for the future T2K experiment. In the middle and right panels of Fig. \[fig:flux-cross\], we show some results for T2K and MiniBooNE experiments. In all cases, the flux $\phi$ is normalized to one. As in Ref. [@Amaro:2008hd], and since we neglect all resonances above the $\Delta(1232)$, we have set up a maximum neutrino energy ($E_{\rm max}^i$) in the flux convolution, approximating the convoluted cross section by $\bar \sigma \approx {\int_{E_{\rm low}^i}^{E_{\rm max}^i} dE \phi^i(E)
\sigma(E)}{ /}{\int_{E_{\rm low}^i}^{E_{\rm max}^i} dE \phi^i(E)} $, where we fixed the upper limit in the integration to $E_{\rm max}=1.45\,$GeV and 1.34 GeV for CC and NC $\nu_\mu/\bar\nu_\mu$ driven processes, respectively. $E_{\rm low}^i$ is the lower flux limit. For the K2K case a threshold of 450 MeV for muon momentum is also implemented [@Hasegawa:2005td] and we can to go up to $E_{\rm max}^{\rm
CC,K2K}$=1.8 GeV. We cover about 90% of the total flux in most of the cases. For the T2K antineutrino flux, we cover just about 65%, and therefore our results are less reliable.
Our central value cross sections increase by some 23%$\sim$30%, while the errors associated to the uncertainties in the $C_5^A(0)$ and $M_{A\Delta}$ determination are of the order of 21%. Our new results are thus compatible with former ones in Ref. [@Amaro:2008hd] within $1\sigma$. Our prediction for the K2K experiment lies more than 1$\sigma$ below the K2K upper bound, while we still predict an NC MiniBooNE cross section notably smaller than that given in the PhD thesis of J.L. Raaf [@Raaf]. Note however, that the MiniBooNE Collaboration has not given an official value for the total coherent cross section yet, and only the ratio coherent/(coherent+incoherent) has been presented [@AguilarArevalo:2008xs]. For the future T2K experiment, we now get cross sections of the order 2.4$-$3.2$\times10^{-40}$cm$^2$ in carbon and about 2.9$-$3.8 $\times
10^{-40}$cm$^2$ in oxygen.
In Fig. \[fig:cross\] we show new $\nu_\mu/\bar\nu$ CC and NC coherent pion production total cross sections off carbon and oxygen targets. As in Ref. [@Amaro:2008hd], we observe sizable corrections to the approximate relation $\sigma_{\rm CC} \approx 2
\sigma_{\rm NC}$ for these two isoscalar nuclei in the whole range of $\nu/\bar\nu$ energies examined. As pointed out in Refs. [@rein2; @Berger:2007rq], this is greatly due to the finite muon mass, and thus the deviations are dramatic at low neutrino energies. In any case, these corrections can not account for the apparent incompatibility among the CC K2K cross section and the NC value quoted in Ref. [@Raaf].
The SciBooNE Collaboration has just reported a measurement of NC $\pi^0$ production on carbon by a $\nu_\mu$ beam with average energy 0.8GeV [@sciboone]. Based on previous measurements of CC coherent $\pi^+$ production [@sciboone2], they conclude that $\left.\frac{\sigma({\rm CC coh}\pi^+)}{\sigma({\rm NC coh}\pi^0)}\right|_{\rm
SciBooNE}=0.14^{+0.30}_{-0.28}$. This result can not be accommodated within our model, or any other present theoretical model, neither microscopic [@luis1; @luis2; @Martini:2009uj; @Nakamura:2009iq] nor PCAC based [@rein; @rein2; @Paschos:2009ag; @Berger:2008xs]. Theoretically, this ratio cannot be much smaller than 1.4-1.6. For instance, for a carbon target and for a neutrino energy of 0.8GeV we find a value of $1.45\pm 0.03$ for that ratio, ten times bigger that the value given by the SciBooNE Collaboration. From the $\nu_\mu+^{12}C$ CC and NC MiniBooNE convoluted results shown in Table \[tab:res\], we obtain $1.46 \pm 0.03$. We believe this huge discrepancy with the SciBooNE result stems form the use in Ref. [@sciboone] of the RS model to estimate the ratio between NC coherent $\pi^0$ production and the total CC pion production. As clearly shown in Refs. [@Amaro:2008hd; @Hernandez:2009vm], the RS model is not appropriate to describe coherent pion production in the low energy regime of interest for the SciBooNE experiment.
\
\
Work supported by DGI and FEDER funds, contracts FIS2008-01143/FIS, FIS2006-03438, FPA2007-65748, CSD2007-00042, by JCyL, contracts SA016A07 and GR12, by GV, contract PROMETEO/2009-0090 and by the EU HadronPhysics2 project, contract 227431.
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abstract: 'Using upper $\ell_p$-estimates for normalized weakly null sequence images, we describe a new family of operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ with parameters $1\leq p\leq\infty$ and $1\leq\xi\leq\omega_1$. These classes contain the completely continuous operators, and are distinct for all choices $1\leq p\leq\infty$ and, when $p\neq 1$, for all choices $\xi\neq\omega_1$. For the case $\xi=1$, there exists an ideal norm ${\lVert\cdot\rVert}_{(p,1)}$ on the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ under which it forms a Banach ideal.'
address: 'Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115'
author:
- Ben Wallis
title: 'Constructing Banach ideals using upper $\ell_p$-estimates'
---
Introduction
============
The roots of the theory of operator ideals extend at least as far back as 1941 when J.W. Calkin observed that if $\mathcal{H}$ is a Hilbert space, then the subspaces of finite-rank operators, compact operators, and Hilbert-Schmidt operators all form multiplicative ideals in the space $\mathcal{L}(\mathcal{H})$ of continuous linear operators on $\mathcal{H}$ ([@Ca41]). However, the concept of an ideal as a class of operators between arbitrary Banach spaces developed more recently, with the first thorough treatment of the subject, a monograph by Albrecht Pietsch, appearing in 1978 ([@Pi78]). In this paper we define and study a new family of operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ with parameters $1\leq p\leq\infty$ and $1\leq\xi\leq\omega_1$, where $\omega_1$ denotes the first uncountable ordinal. For any fixed value of $\xi$, these ideals are distinct for all choices of $p$, which is to say that for any $1\leq p<q\leq\infty$ there exist Banach spaces $X$ and $Y$ for which the components satisfy $\mathcal{WD}_{\ell_p}^{(\xi,\infty)}(X,Y)\neq\mathcal{WD}_{\ell_q}^{(\xi,\infty)}(X,Y)$. It remains an open question whether, for fixed $1<p\leq\infty$, the ideals are distinct for all choices of $\xi$. However, we do obtain a partial positive answer by establishing $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}\neq\mathcal{WD}_{\ell_p}^{(\infty,\omega_1)}$ whenever $\xi\neq\omega_1$ and $p\neq 1$. We shall also see that $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ always strictly includes the ideal of completely continuous operators $\mathcal{V}$, which shows that they are distinct from some other notable families of operator ideals with a parameter related to the $\ell_p$ spaces. For instance, let $\mathcal{N}_p$, $\mathcal{I}_p$, and $\Pi_p$ denote the ideals of $p$-nuclear, $p$-integral, and absolutely $p$-summing operators, respectively. Then $\mathcal{N}_p\subsetneq\mathcal{I}_p\subsetneq\Pi_p\subsetneq\mathcal{V}\subsetneq\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ (cf., e.g., Proposition 22 in [@Wo91] together with Theorem 2.17 in [@DJT95]).
Of special interest are the those operator ideals whose components are always norm-closed. For instance, given arbitrary Banach spaces $X$ and $Y$, the compact, weakly compact, and completely continuous operators from $X$ into $Y$ are always norm-closed in $\mathcal{L}(X,Y)$, whereas the finite-rank operators are not. We shall see that, when $p\neq 1$, there always exist separable spaces $X$ for which $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X)$ fails to be norm-closed in $\mathcal{L}(X)$, and when $p\neq\infty$, we can choose $X$ to be reflexive. Nevertheless, in the case $\xi=1$, we can construct an ideal norm ${\lVert\cdot\rVert}_{(p,1)}$ for the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ so that it forms a Banach ideal, that is, a “nice” norm assignment for each component space $\mathcal{WD}_{\ell_p}^{(\infty,1)}(X,Y)$ under which it becomes a Banach space.
The ideas for the construction of this family originate with [@BF11] and [@ADST09]. In [@BF11], the authors defined the subset $\mathcal{WS}(X,Y)$ of [**$\boldsymbol{(w_n)}$-singular**]{} operators in $\mathcal{L}(X,Y)$ as those operators $T$ for which, given any normalized basic sequence $(x_n)$ in $X$, the image sequence $(Tx_n)$ fails to dominate $(w_n)$. Here, $(w_n)$ is taken to be some normalized 1-spreading basis for some fixed Banach space $W$. They showed that when $(w_n)$ is the summing basis for $c_0$, the unit vector basis for $\ell_1$, or the unit vector basis for $c_0$, the resulting classes $\mathcal{WS}$ are the norm-closed ideals of weakly compact, Rosenthal, or compact operators, respectively. Meanwhile, in [@ADST09] the authors constructed and studied classes of operators based on Schreier family support. In particular, they defined $\mathcal{SS}_\xi$, the [**$\boldsymbol{\mathcal{S}_\xi}$-strictly singular**]{} operators, as the class of all continuous linear Banach space operators $T$ for which if $(x_n)$ is any normalized basic sequence in the domain space, for any $\epsilon>0$ there exists some $z\in[x_n]$ with support lying in the $\xi$th Schreier family $\mathcal{S}_\xi$, and satisfying ${\lVertTz\rVert}<\epsilon{\lVertz\rVert}$.
In this paper, we use similar ideas to produce operator ideals with certain nice properties. However, whereas classes $\mathcal{WS}$ and $\mathcal{SS}_\xi$ were constructed using normalized basic sequences and singular estimates on their images, for the classes $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ we instead use normalized weakly null sequences and uniform upper estimates. Since continuous linear operators preserve weak convergence, the choice of weakly null sequences in place of basic sequences allows us to show that the classes $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ are indeed multiplicative ideals between arbitrary Banach spaces. The choice of uniform upper estimates instead of singular estimates then gives us a natural way to show that each class $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ is closed under addition.
Now we shall take a moment to recall some essential definitions and basic facts relevant to our project. Let $\mathcal{J}$ be a subclass of the class $\mathcal{L}$ of all continuous linear operators between Banach spaces, such that for each pair of Banach spaces $X$ and $Y$, $\mathcal{J}(X,Y):=\mathcal{L}(X,Y)\cap\mathcal{J}$ is a linear subspace containing all the finite-rank operators from $X$ into $Y$. We call $\mathcal{J}$ an **operator ideal** if whevenever $W,X,Y,Z$ are Banach spaces and $T\in\mathcal{J}(X,Y)$, then for all operators $A\in\mathcal{L}(W,X)$ and $B\in\mathcal{L}(Y,Z)$ we have $BTA\in\mathcal{J}(W,Z)$. An [**ideal norm**]{} with respect to an operator ideal $\mathcal{J}$ is a rule $\rho$ that assigns to every $T\in\mathcal{J}(X,Y)$, a nonnegative real value $\rho(T)$, and satisfying the following conditions for all Banach spaces $W$, $X$, $Y$, and $Z$. First, $\rho(x^*\otimes y)={\lVertx^*\rVert}{\lVerty\rVert}$ for all $x^*\in X^*$ and $y\in Y$, where $x^*\otimes y$ is viewed as the 1-dimensional operator $x\mapsto x^*(x)y$ lying in $\mathcal{J}(X,Y)$; second, $\rho(S+T)\leq\rho(S)+\rho(T)$ for all $S,T\in\mathcal{J}(X,Y)$; and third, $\rho(BTA)\leq{\lVertB\rVert}\rho(T){\lVertA\rVert}$ for all $T\in\mathcal{J}(X,Y)$, $A\in\mathcal{L}(W,X)$, and $B\in\mathcal{L}(Y,Z)$. A [**Banach ideal**]{} is then an operator ideal $\mathcal{J}$ equipped with an an ideal norm $\rho$ such that all components $\mathcal{J}(X,Y)$ are complete with respect to the norm on that space induced by $\rho$.
We will also need to use the **Schreier families**. These are denoted $\mathcal{S}_\xi$ for each countable ordinal $0\leq\xi<\omega_1$, and we must define them as follows. Put $\mathcal{S}_0:=\{\{n\}:n\in\mathbb{N}\}\cup\{\emptyset\}$ and $\mathcal{S}_1:=\{F\subset\mathbb{N}:\# F\leq\min F\}\cup\{\emptyset\}$. Now fix a countable ordinal $1\leq\xi<\omega_1$. In case $\xi=\zeta+1$ for some countable ordinal $1\leq\zeta<\omega_1$ we define $\mathcal{S}_\xi$ as the set containing $\emptyset$ together with all $F\subset\mathbb{N}$ such that there exist $n\in\mathbb{N}$ and $F_1<\cdots<F_n\in\mathcal{S}_\zeta$ satisfying $\{\min F_k\}_{k=1}^n\in\mathcal{S}_1$ and $F=\bigcup_{k=1}^nF_k$. In case $\xi$ is a limit ordinal we fix a strictly increasing sequence $(\zeta_n)$ of non-limit-ordinals satisfying $\sup_n\zeta_n=\xi$, and define $\mathcal{S}_\xi:=\bigcup_{n=1}^\infty\{F\in\mathcal{S}_{\zeta_n}:n\leq F\}$.
Usually in the literature, the family of finite subsets of natural numbers is denoted $[\mathbb{N}]^{<\infty}$, or $\mathcal{P}_{<\infty}(\mathbb{N})$. However, for convenience, let us abuse our notation and write this family as if it were the “$\omega_1$th Schreier family.” In other words, we set $\mathcal{S}_{\omega_1}:=\{F\subset\mathbb{N}:\#F<\infty\}$. This will greatly simplify the writing.
The sets $\mathcal{S}_\xi$ ($1\leq\xi\leq\omega_1$) have some very nice properties, most especially that each is [**spreading**]{}. This means that if $\{m_1<\cdots<m_k\}\in\mathcal{S}_\xi$ and $\{n_1<\cdots<n_k\}$ satisfies $m_i\leq n_i$ for all $1\leq i\leq k$, then $\{n_1<\cdots<n_k\}\in\mathcal{S}_\xi$ also holds. They are also [**hereditary**]{}, which means that if $E\in\mathcal{S}_\xi$ and $F\subseteq E$ then $F\in\mathcal{S}_\xi$. Contrary to what we might expect, though, the Schreier families are [*not*]{} increasing under the inclusion relation. However, it is easily seen that, for all $1\leq\xi\leq\omega_1$, we have $\mathcal{S}_1\subseteq\mathcal{S}_\xi$, and in particular we have $\{k\}\in\mathcal{S}_\xi$ for all $k\in\mathbb{N}$. Moreover, the Schreier families do behave somewhat nicely under the inclusion relation in the sense that, if $1\leq\zeta<\xi\leq\omega_1$ are ordinals, then there is $d=d(\zeta,\xi)\in\mathbb{N}$ such that for all $F\in\mathcal{S}_\zeta$ satisfying $d\leq\min F$ we have $F\in\mathcal{S}_\xi$.
We will appeal several times to the Bessaga-Pełczyński Selection Principle. However, the version that we need is slightly stronger than typically stated in the literature. More specifically, we need a small uniform bound on the equivalence constant. The proof is practically identical to the standard small perturbations and gliding hump arguments found, for instance, in Theorem 1.3.9 and Proposition 1.3.10 from [@AK06].
Suppose $X$ is a Banach space with a basis $(e_i)$, and corresponding coefficient functionals $(e_i^*)\subset X^*$. Let $(x_n)\subset X$ be a sequence satisfying $\lim_{n\to\infty}{\lVertx_n\rVert}=1$ and $\lim_{n\to\infty}e_i^*(x_n)=0$ for all $i\in\mathbb{N}$. Then for any $\epsilon>0$, there exists a subsequence $(x_{n_k})$ which is $(1+\epsilon)$-congruent to a normalized block basis of $(e_i)$.
We divide the remainder of this paper into sections 2 and 3. In section 2 we define the classes $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$, and establish that, for the nontrivial case $p\neq 1$, they fail to be norm-closed, but as long as $\xi=1$ they form Banach ideals. Then, in section 3 we discuss the significance of the parameters $p$ and $\xi$.
The operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$
==========================================================
Let us state formally the definition of classes $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$.
Let $X$ and $Y$ be Banach spaces, and fix some constants $0\leq C<\infty$ and $1\leq p\leq\infty$, and some countable ordinal $1\leq\xi<\omega_1$. Put $\mathcal{A}_\xi:=\{(\alpha_k)\in c_{00}:\text{supp}(\alpha_k)\in\mathcal{S}_\xi\}$, the set of all scalar sequences with support in the $\xi$th Schreier family. Then we denote by $\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$ the set of all operators $T\in\mathcal{L}(X,Y)$ for which, given $\epsilon>0$, each normalized weakly null sequence $(x_n)\subset X$ admits a subsequence $(x_{n_k})$ such that for all $(\alpha_k)\in\mathcal{A}_\xi$, the estimate ${\lVert\sum\alpha_kTx_{n_k}\rVert}\leq(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$ holds. Then we set $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y):=\bigcup_{C\geq 0}\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$.
Immediate from the definitions and the inequality ${\lVert(\alpha_k)\rVert}_{\ell_p}\leq{\lVert(\alpha_k)\rVert}_{\ell_q}$ for all $(\alpha_k)\in c_{00}$ and $1\leq q\leq p\leq\infty$, we get the following relations.
\[inclusion\]Let $X$ and $Y$ be Banach spaces, and fix some constants $0\leq C\leq D\leq\infty$ and $1\leq q\leq p\leq\infty$, and some ordinal $1\leq\xi\leq\omega_1$. Then $\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)\subseteq\mathcal{WD}_{\ell_q}^{(C,\xi)}(X,Y)$ and $\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)\subseteq\mathcal{WD}_{\ell_p}^{(D,\xi)}(X,Y)$.
When checking that an operator satisfies the definition of $\mathcal{WD}_{\ell_p}^{(C,\xi)}$, the following Propositions will come in handy.
\[seminormalized-tends-to-1\]Let $X$ and $Y$ be Banach spaces, and fix constants $0\leq C<\infty$ and $1\leq p\leq\infty$, and some ordinal $1\leq\xi\leq\omega_1$. Then $T\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$ if and only if for all $\epsilon>0$ and every seminormalized weakly null sequence $(x_n)\subset X$ which admits a subsequence $(x_{n_k})$ satisfying ${\lVertx_{n_k}\rVert}\to 1$, there exists a further subsequence $(x'_{n_k})$ such that for all $(\alpha_k)\in\mathcal{A}_\xi$, the estimate ${\lVert\sum\alpha_kTx'_{n_k}\rVert}\leq(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$ holds.
We need only prove the “only if” part since the “if” part is obvious. Suppose $T\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$. Let $(x_n)$ be a seminormalized weakly null sequence with a subsequence tending to 1 in norm, and pick $\epsilon>0$. Let $1<\delta<1+\frac{\epsilon}{2C}$, which gives us $C\delta+\frac{\epsilon}{2}<C+\epsilon$, and pass to a further subsequence so that ${\lVertx_{n_k}\rVert}\leq\delta$ for all $k$. By definition of $T\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$ we can pass to yet a further subsequence so that $(\frac{x_{n_k}}{{\lVertx_{n_k}\rVert}})$ satisfies ${\lVert\sum\alpha_kT\frac{x_{n_k}}{{\lVertx_{n_k}\rVert}}\rVert}\leq(C+\frac{\epsilon}{2\delta}){\lVert(\alpha_k)\rVert}_{\ell_p}$ for all $(\alpha_k)\in\mathcal{A}_\xi$. Since also $({\lVertx_{n_k}\rVert}\alpha_k)\in\mathcal{A}_\xi$ for each $(\alpha_k)\in\mathcal{A}_\xi$, we get
$$\begin{gathered}
{\lVert\sum\alpha_kTx_{n_k}\rVert}={\lVert\sum{\lVertx_{n_k}\rVert}\alpha_kT\frac{x_{n_k}}{{\lVertx_{n_k}\rVert}}\rVert}\leq\left(C+\frac{\epsilon}{2\delta}\right){\lVert({\lVertx_{n_k}\rVert}\alpha_k)\rVert}_{\ell_p}\\\\\leq\left(C\delta+\frac{\epsilon}{2}\right){\lVert(\alpha_k)\rVert}_{\ell_p}\leq(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}.\end{gathered}$$
\[norm-null\]Let $X$ and $Y$ be Banach spaces, and fix constants $0\leq C<\infty$ and $1\leq p\leq\infty$. If $(x_n)\subset X$ is a sequence for which $(Tx_n)$ has a norm-null subsequence, then given $\epsilon>0$, there exists a further subsequence $(x_{n_k})$ for which the estimate ${\lVert\sum\alpha_kTx_{n_k}\rVert}\leq(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$ holds for all $(\alpha_k)\in c_{00}$.
Pick a subsequence so that ${\lVertTx_{n_k}\rVert}\leq\epsilon 2^{-k}$ and hence, by Hölder, if $q$ is conjugate to $p$ so that $\frac{1}{p}+\frac{1}{q}=1$, ${\lVert\sum\alpha_kTx_{n_k}\rVert}\leq\epsilon{\lVert(2^{-k}\alpha_k)\rVert}_{\ell_1}\leq\epsilon{\lVert(2^{-k})\rVert}_{\ell_q}{\lVert(\alpha_k)\rVert}_{\ell_p}\leq(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$ for any sequence $(\alpha_k)\in c_{00}$.
Recall that a linear operator between Banach spaces $X$ and $Y$ is called [**completely continuous**]{} just in case it always sends weakly null sequences into norm-null ones. We write $\mathcal{V}(X,Y)$ for the space of these completely continuous operators. (As mentioned previously, $\mathcal{V}$ is a norm-closed operator ideal.) Thus, Proposition \[norm-null\] yields the following.
\[VinWD\]Let $X$ and $Y$ be Banach spaces, and let $1\leq p\leq\infty$, $0\leq C\leq\infty$, and $1\leq\xi<\omega_1$. Then $\mathcal{V}(X,Y)\subseteq\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$.
Let us observe, via several steps, that the class $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ is indeed an operator ideal.
\[multideal\]Let $W$, $X$, $Y$, and $Z$ be Banach spaces, and fix constants $1\leq p\leq\infty$ and $0\leq C<\infty$, and some ordinal $1\leq\xi\leq\omega_1$. Suppose $T\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$ with $A\in\mathcal{L}(W,X)$ and $B\in\mathcal{L}(Y,Z)$. Then $TA\in\mathcal{WD}_{\ell_p}^{(C{\lVertA\rVert},\xi)}(W,Y)$ and $BT\in\mathcal{WD}_{\ell_p}^{(C{\lVertB\rVert},\xi)}(X,Z)$.
Let’s first show that $TA\in\mathcal{WD}_{\ell_p}^{(C{\lVertA\rVert},\xi)}(W,Y)$. Recall that an operator is weak-to-weak continuous if and only if it is norm-to-norm continuous. Thus if $(w_n)$ is a normalized weakly null sequence in $W$, we get that $(Aw_n)$ is weakly null in $X$. If it contains a norm-null subsequence then so does $TAw_n$, and so by Proposition \[norm-null\] we are done. Otherwise, we can pass to a subsequence if necessary so that ${\lVertAw_n\rVert}\to\delta$ for some $0<\delta\leq{\lVertA\rVert}$. Hence ${\lVert\delta^{-1}Aw_n\rVert}\to 1$, and by Proposition \[seminormalized-tends-to-1\] we get, for any $\epsilon>0$, a subsequence $(n_k)$ satisfying ${\lVert\sum\alpha_kT\delta^{-1}Aw_{n_k}\rVert}\leq(C+\frac{\epsilon}{\delta}){\lVert(\alpha_k)\rVert}_{\ell_p}$ and hence ${\lVert\sum\alpha_kTAw_{n_k}\rVert}\leq(C\delta+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}\leq(C{\lVertA\rVert}+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$ for all $(\alpha_k)\in\mathcal{A}_\xi$.
Next, we show that $BT\in\mathcal{WD}_{\ell_p}^{(C{\lVertB\rVert},\xi)}(X,Z)$. Fix a normalized weakly null sequence $(x_n)\subset X$, and let $\epsilon>0$. To make things nontrivial, we may assume $B\neq 0$. Then we can find a subsequence $(x_{n_k})$ such that for all $(\alpha_k)\in\mathcal{A}_\xi$ we get ${\lVert\sum\alpha_kTx_{n_k}\rVert}\leq(C+\frac{\epsilon}{{\lVertB\rVert}}){\lVert(\alpha_k)\rVert}_{\ell_p}$ and hence ${\lVert\sum\alpha_kBTx_{n_k}\rVert}\leq{\lVertB\rVert}{\lVert\sum\alpha_kTx_{n_k}\rVert}\leq(C{\lVertB\rVert}+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$.
By “pushing out” a scalar sequence $(\alpha_k)\in\mathcal{A}_\xi$, and using the spreading property of $\mathcal{S}_\xi$, we obtain the following obvious Lemma.
\[subsequence-hereditary\]Let $Y$ be a Banach space, and fix an ordinal $1\leq\xi\leq\omega_1$. Suppose $(y_n)$ and $(y'_k)$ are sequences in $Y$ such that $(y'_k)_{k\geq m}$ is a subsequence of $(y_n)_{n\geq m}$ for some $m\in\mathbb{N}$. If $(\alpha_k)\in\mathcal{A}_\xi$ satisfies $\min\text{supp}(\alpha_k)\geq m$ then there exists $(\beta_n)\in\mathcal{A}_\xi$ which satisfies $\sum\alpha_ky'_k=\sum\beta_ny_n$ and ${\lVert(\alpha_k)\rVert}_{\ell_p}={\lVert(\beta_n)\rVert}_{\ell_p}$ for all $1\leq p\leq\infty$.
\[addition\]Let $X$ and $Y$ be Banach spaces, and fix constants $0\leq C,D<\infty$ and $1\leq p\leq\infty$. Then for any $S\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$ and $T\in\mathcal{WD}_{\ell_p}^{(D,\xi)}(X,Y)$ we have $S+T\in\mathcal{WD}_{\ell_p}^{(C+D,\xi)}(X,Y)$.
Let $\epsilon>0$ and pick a normalized weakly null sequence $(x_n)\subset X$. By definition of $S\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$ applied to $\frac{\epsilon}{2}>0$ and $(x_n)$ we get a subsequence $(x_{n_k})$ such that for all $(\alpha_k)\in\mathcal{A}_\xi$, the estimate ${\lVert\sum\alpha_kSx_{n_k}\rVert}\leq(C+\frac{\epsilon}{2}){\lVert(\alpha_k)\rVert}_{\ell_p}$ holds. Next, apply the definition of $T\in\mathcal{WD}_{\ell_p}^{(D,\xi)}(X,Y)$ to $\frac{\epsilon}{2}>0$ and $(x_{n_k})$ to find a further subsequence $(k_i)$ such that for all $(\alpha_i)\in\mathcal{A}_\xi$ we get ${\lVert\sum\alpha_iTx_{n_{k_i}}\rVert}\leq(D+\frac{\epsilon}{2}){\lVert(\alpha_i)\rVert}_{\ell_p}$. Notice that since $(x_{n_{k_i}})$ is a subsequence of $(x_{n_k})$, then for each scalar sequence $(\alpha_i)\in\mathcal{A}_\xi$, by Lemma \[subsequence-hereditary\], ${\lVert\sum\alpha_i(S+T)x_{n_{k_i}}\rVert}\leq{\lVert\sum\alpha_iSx_{n_{k_i}}\rVert}+{\lVert\sum\alpha_iTx_{n_{k_i}}\rVert}\leq(C+D+\epsilon){\lVert(\alpha_i)\rVert}_{\ell_p}$.
From Propositions \[VinWD\], \[multideal\], and \[addition\], it now follows immediately that $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ is an operator ideal. In fact, the same combination of Propositions shows that $\mathcal{WD}_{\ell_p}^{(0,\xi)}$ is an operator ideal, but it turns out that, using Proposition \[VinWD\] along with the fact that every family $\mathcal{S}_\xi$ contains all the singletons, regardless of our choice of $p$ or $\xi$ we always get $\mathcal{WD}_{\ell_p}^{(0,\xi)}=\mathcal{V}$, the completely continuous operators.
Let us now construct two important examples.
\[compact\]Let $X$ be a Banach space which fails to contain a copy of $\ell_1$. (This is true in particular if $X$ is reflexive.) Fix constants $1\leq q<p\leq\infty$ and $0\leq C\leq\infty$, and some countable ordinal $1\leq\xi<\omega_1$. Then $\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,\ell_q)=\mathcal{K}(X,\ell_q)$.
Assume $0\leq C<\infty$. By Proposition \[VinWD\] we already have $\mathcal{K}(X,\ell_q)\subseteq\mathcal{V}(X,\ell_q)\subseteq\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,\ell_q)$, and so it suffices to prove $\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,\ell_q)$ contains only compact operators. For suppose towards a contradiction that there exists $T\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,\ell_q)$ which is not compact. Then we can find a seminormalized sequence $(x_n)\subset X$ for which $(Tx_n)$ fails to have a convergent subsequence. Since $X$ fails to contain a copy of $\ell_1$, we can apply Rosenthal’s $\ell_1$ Theorem to pass to a subsequence so that $(x_n)$ is weak Cauchy. Hence we can pass to a further subsequence if necessary so that $(x_{2n}-x_{2n+1})$ and $(Tx_{2n}-Tx_{2n+1})$ are both weakly null and seminormalized. This means the sequence $(x'_n)$ defined by $x'_n:=(x_{2n}-x_{2n+1})/{\lVertx_{2n}-x_{2n+1}\rVert}$ is normalized and weakly null, whereas the sequence $(Tx'_n)$ is seminormalized and weakly null. By passing to yet another subsequence if necessary, by Proposition 2.1.3 in [@AK06] we can assume $(Tx'_n)$ is $K$-equivalent, $K\geq 1$, to the unit vector basis of $\ell_q$. Thus, by this equivalence together with the definition of $T\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$, for any $\epsilon>0$ we can find a subsequence $(n_k)$ such that ${\lVert(\alpha_k)\rVert}_{\ell_q}\leq K{\lVert\sum\alpha_kTx'_{n_k}\rVert}\leq K(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$ for all $(\alpha_k)\in\mathcal{A}_\xi$. Due to $\mathcal{S}_1\subseteq\mathcal{S}_\xi$, the above inequality holds also for all $(\alpha_k)\in\mathcal{A}_1$. Notice that every $(\beta_k)\in c_{00}$ induces a corresponding “spread out” sequence $(\alpha_k)\in\mathcal{A}_1$ satisfying ${\lVert(\alpha_k)\rVert}_{\ell_r}={\lVert(\beta_k)\rVert}_{\ell_r}$ for all $1\leq r\leq\infty$. Thus we obtain the impossible estimate ${\lVert(\beta_k)\rVert}_{\ell_q}={\lVert(\alpha_k)\rVert}_{\ell_q}\leq K(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}=K(C+\epsilon){\lVert(\beta_k)\rVert}_{\ell_p}$ for all $(\beta_k)\in c_{00}$.
\[whole-space\]Let $X$ and $Y$ be Banach spaces, and fix numbers $1\leq p\leq q<\infty$ and an ordinal $1\leq\xi\leq\omega_1$. Suppose $T\in\mathcal{L}(X,Y)$ is an operator such that $TX$ has a $K$-embedding, $K\geq 1$, into $\ell_q$. (In other words, suppose there is an operator $Q\in\mathcal{L}(TX,\ell_q)$ which satisfies $K^{-1}{\lVerty\rVert}\leq{\lVertQy\rVert}\leq K{\lVerty\rVert}$ for all $y\in TX$.) Then $T\in\mathcal{WD}_{\ell_p}^{(K^2{\lVertT\rVert},\xi)}(X,Y)$, and the same result holds if $1\leq p\leq\infty$ and $TX$ has a $K$-embedding into $c_0$. Thus, for $1\leq p\leq q<\infty$ we have $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,\ell_q)=\mathcal{L}(X,\ell_q)$, and for $1\leq p\leq\infty$ we have $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,c_0)=\mathcal{L}(X,c_0)$.
Fix a normalized weakly null sequence $(x_n)\subset X$, and denote by $Q\in\mathcal{L}(TX,\ell_q)$ (resp. $Q\in\mathcal{L}(TX,c_0)$) the $K$-embedding. If $(Tx_n)$ contains a norm-null subsequence then we are done by Proposition \[norm-null\]. Otherwise let $\epsilon>0$, and find a subsequence so that ${\lVertQTx_{n_k}\rVert}\to r$ with $0<r\leq K{\lVertT\rVert}$, and quickly enough so that by the uniform version of Bessaga-Pełczyński combined with Lemma 2.1.1 in [@AK06], we can pass to a further subsequence if necessary so that $(\frac{1}{r}QTx_{n_k})$ is $(1+\frac{\epsilon}{Kr})$-equivalent to the unit vector basis of $\ell_q$ (resp. $c_0$). This gives us, in the $\ell_q$ case,
$$\begin{gathered}
{\lVert\sum\alpha_kTx_{n_k}\rVert}\leq K{\lVert\sum\alpha_kQTx_{n_k}\rVert}=Kr{\lVert\sum\alpha_k\frac{1}{r}QTx_{n_k}\rVert}\\\\\leq Kr\left(1+\frac{\epsilon}{Kr}\right){\lVert(\alpha_k)\rVert}_{\ell_q}\leq(K^2{\lVertT\rVert}+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_q}\leq(K^2{\lVertT\rVert}+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}\end{gathered}$$
for all $(\alpha_k)\in c_{00}$, and a similar inequality holds in the $c_0$ case.
We must lay some groundwork aimed at showing that, in case $\xi=1$, the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ forms a Banach ideal. We begin by defining a seminorm on the linear space $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$.
Let $X$ and $Y$ be Banach spaces, and fix a constant $1\leq p\leq\infty$ and an ordinal $1\leq\xi\leq\omega_1$. For each $T\in\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$, we define $C_{(p,\xi)}(T):=\inf\{0\leq C<\infty:T\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)\}$.
\[seminorm\]Let $X$ and $Y$ be Banach spaces, and fix a constant $1\leq p\leq\infty$ and an ordinal $1\leq\xi\leq\omega_1$. If $T\in\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$ then $T\in\mathcal{WD}_{\ell_p}^{( C_{(p,\xi)}(T),\xi)}(X,Y)$. Furthermore, $T\mapsto C_{(p,\xi)}(T)$ defines a seminorm on the linear space $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$.
The first part of the Proposition is clear from applying the definition of $T\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$ for $ C_{(p,\xi)}(T)<C<\infty$, and absolute homogeneity is similarly obvious. The only thing nontrivial to show is that the triangle inequality holds. Indeed, let $S,T\in\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$, and suppose $(x_n)$ is a normalized weakly null sequence. Let $\epsilon>0$. Then we can apply the definition of $S\in\mathcal{WD}_{\ell_p}^{( C_{(p,\xi)}(S),\xi)}(X,Y)$ to $(x_n)$ and $\frac{\epsilon}{2}>0$ to find a subsequence $(n_k)$ satisfying ${\lVert\sum\alpha_kSx_{n_k}\rVert}\leq( C_{(p,\xi)}(S)+\frac{\epsilon}{2}){\lVert(\alpha_k)\rVert}_{\ell_p}$ for all $(\alpha_k)\in\mathcal{A}_\xi$. Then, we successively apply the definition of $T\in\mathcal{WD}_{\ell_p}^{( C_{(p,\xi)}(T),\xi)}(X,Y)$ to $(x_{n_k})$ and $\frac{\epsilon}{2}>0$ to find to a further subsequence $(n_{k_j})$ so that ${\lVert\sum\alpha_jTx_{n_{k_j}}\rVert}\leq( C_{(p,\xi)}(T)+\frac{\epsilon}{2}){\lVert(\alpha_j)\rVert}_{\ell_p}$ for all $(\alpha_j)\in\mathcal{A}_\xi$. Thus, by these facts together with Lemma \[subsequence-hereditary\], $$\begin{gathered}
{\lVert\sum\alpha_j(S+T)x_{n_{k_j}}\rVert}\leq{\lVert\sum\alpha_jSx_{n_{k_j}}\rVert}+{\lVert\sum\alpha_jTx_{n_{k_j}}\rVert}
\\\\=\left( C_{(p,\xi)}(S)+\frac{\epsilon}{2}\right){\lVert(\alpha_j)\rVert}_{\ell_p}+\left( C_{(p,\xi)}(T)+\frac{\epsilon}{2}\right){\lVert(\alpha_j)\rVert}_{\ell_p}\\\\=( C_{(p,\xi)}(S)+ C_{(p,\xi)}(T)+\epsilon){\lVert(\alpha_j)\rVert}_{\ell_p}.\end{gathered}$$
Thus, $C_{(p,\xi)}(S+T)\leq C_{(p,\xi)}(S)+ C_{(p,\xi)}(T)$, and we are done.
Next we show that $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ fails to be norm-closed (as a class) whenever $p\neq 1$. The main idea toward this end proceeds from the following Lemma.
\[nonclosed-space\]Fix constants $1<p\leq\infty$ and $1\leq q<\infty$, and an ordinal $1\leq\xi\leq\omega_1$. Let $(X_m)$ and $(Y_m)$ be sequences of Banach spaces, and for each $m\in\mathbb{N}$, let $T_m\in\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X_m,Y_m)$ be an operator satisfying ${\lVertT_m\rVert}=1$. If $ C_{(p,\xi)}(T_m)\to\infty$ then there exists a subsequence $(m_j)$ and a sequence of operators $S_j\in\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$ for which $S_j\to S\in\mathcal{L}(X,Y)$ but $S\notin\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$, where we define $X:=(\bigoplus_{j=1}^\infty X_{m_j})_{\ell_q}$ and $Y:=(\bigoplus_{j=1}^\infty Y_{m_j})_{\ell_q}$.
Define the subsequence by letting $(m_j)$ be an increasing sequence satisfying $C_j:= C_{(p,\xi)}(T_{m_j})>j2^j$ for every $j$. Next, set $S:=\bigoplus_{j=1}^\infty 2^{-j}T_{m_j}\in\mathcal{L}(X,Y)$. For each $i$, let $P_i\in\mathcal{L}(Y)$ denote the continuous linear projection onto the first $i$ coordinates of $Y$, and set $S_i:=P_iS$. It’s easy to see that $S_i\to S$. Next, we claim that each $S_i\in\mathcal{WD}_{\ell_p}^{(M_i,\xi)}(X,Y)\subseteq\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$, where we set $M_i:={\lVert(2^{-j}C_j)_{j=1}^i\rVert}_{\ell_q}$. Indeed fix any $i\in\mathbb{N}$, and let $(x_n)$ be a normalized weakly null sequence in $X$. Pick any $\epsilon>0$. For each $j$, let $\widetilde{X}_j$ be the obvious isometrically isomorphic copy of $X_j$ contained in $X$, and let $U_j:\widetilde{X}_j\to X_j$ be the corresponding isometric isomorphism. For each $n$, write $x_n=(x_{n,j})_j\in X$. Then $(x_{n,j})_n$ is a sequence in $X_j$ which is bounded by 1. If $(x_{n,j})_n$ has a norm-null subsequence, then by Proposition \[norm-null\] we can find a subsequence $(n_k)$ such that, for all $(\alpha_k)\in c_{00}$,
$$\label{inequality}{\lVert\sum_k\alpha_kT_{m_j}x_{n_k,j}\rVert}\leq\left(C_j+\frac{\epsilon 2^j}{i^{1/q}}\right){\lVert(x_{n_k,j})_k\rVert}_{\ell_p}$$
Otherwise we can find a subsequence $(n_k)$ so that ${\lVertx_{n_k,j}\rVert}_{X_j}\to r$ as $k\to\infty$ for some $0<r\leq 1$. Clearly, $(x_{n,j})_n$ is weakly null in $X_j$, and so by Propositions \[seminormalized-tends-to-1\] and \[seminorm\], we can pass to a further subsequence if necessary so that, for all $(\alpha_k)\in\mathcal{A}_\xi$,
$$\begin{gathered}
{\lVert\sum_k\alpha_kT_{m_j}x_{n_k,j}\rVert}=r{\lVert\sum_k\alpha_kT_{m_j}\frac{x_{n_k,j}}{r}\rVert}\\\\\leq r\left(C_j+\frac{\epsilon 2^j}{i^{1/q}r}\right){\lVert(x_{n_k,j})_k\rVert}_{\ell_p}\leq\left(C_j+\frac{\epsilon 2^j}{i^{1/q}}\right){\lVert(x_{n_k,j})_k\rVert}_{\ell_p}.\end{gathered}$$
In either case, for each $j$ and any subsequence of $(x_{n,j})_n$, we can pass to a further subsequence so that the inequality holds for all $(\alpha_k)\in\mathcal{A}_\xi$.
Thus, by successively passing to further subsequences for $j=1,\cdots,i$, due to Lemma \[subsequence-hereditary\], we get a subsequence $(n_k)$ such that holds for all $j=1,\cdots,i$ and all $(\alpha_k)\in\mathcal{A}_\xi$. In particular, this means
$$\begin{gathered}
{\lVert\sum_k\alpha_kS_ix_{n_k}\rVert}=\left(\sum_{j=1}^i2^{-jq}{\lVert\sum_k\alpha_kT_{m_j}x_{n_k,j}\rVert}^q\right)^{1/q}\\\\\leq\left(\sum_{j=1}^i2^{-jq}\left(C_j+\frac{\epsilon 2^j}{i^{1/q}}\right)^q{\lVert(\alpha_k)\rVert}_{\ell_p}^q\right)^{1/q}={\lVert\left(2^{-j}C_j+\frac{\epsilon}{i^{1/q}}\right)_{j=1}^i\rVert}_{\ell_q}{\lVert(\alpha_k)\rVert}_{\ell_p}\\\\\leq\left({\lVert\left(2^{-j}C_j\right)_{j=1}^i\rVert}_{\ell_q}+{\lVert\left(\frac{\epsilon}{i^{1/q}}\right)_{j=1}^i\rVert}_{\ell_q}\right){\lVert(\alpha_k)\rVert}_{\ell_p}=(M_i+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p},\end{gathered}$$
which proves the claim that $S_i\in\mathcal{WD}_{\ell_p}^{(M_i,\xi)}(X,Y)\subseteq\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$.
However, it cannot be that $S\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$ for any $0\leq C<\infty$. To show why not, fix $i\in\mathbb{N}$, and let $(x_n)$ be a normalized weakly null sequence in $X_i$. Then let $\epsilon>0$ be such that, for any subsequence $(n_k)$, there exists $(\alpha_k)\in\mathcal{A}_\xi$ with ${\lVert\sum_k\alpha_kT_{m_i}x_{n_k}\rVert}>(i2^i+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$. Let $Q_i:X_i\to X$ be the canonical embedding of $X_i$ into $X$, and observe that $(Q_ix_n)_n$ is a normalized weakly null sequence in $X$. However, for every subsequence $(n_k)$ there exists $(\alpha_k)\in\mathcal{A}_\xi$ with
$$\begin{gathered}
{\lVert\sum_k\alpha_kSQ_ix_{n_k}\rVert}=2^{-i}{\lVert\sum_k\alpha_kT_{m_i}x_{n_k}\rVert}\\>2^{-i}(i2^i+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}\geq(i+2^{-i}\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}.\end{gathered}$$
It follows that $S\notin\mathcal{WD}_{\ell_p}^{(i,\xi)}(X,Y)$ for any $i$, and hence $S\notin\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$.
\[not-norm-closed-example\]Fix a constant $1<p\leq\infty$ and an ordinal $1\leq\xi\leq\omega_1$. There exists a Banach space $X$ for which $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X)$ is not norm-closed. If $p\neq\infty$, then we can choose $X$ to be reflexive.
For convenience in writing, let us consider the case where $p\neq\infty$. The case where $p=\infty$ uses $c_0$ in place of $\ell_p$, and the resulting proof is nearly identical, except that the resulting space $X$ is not reflexive.
Let $(e_n)$ denote the unit vector basis of $\ell_p$. For each finite $E\subset\mathbb{N}$, define the functional $f_E\in\ell_p^*$ by the rule $f_E(e_n)=1$ if $n\in E$ and $f_E(e_n)=0$ otherwise. Now, fix $m\in\mathbb{N}$, and define the norming set $\mathcal{B}_m:=B_{\ell_p^*}\cup\{f_E:E\subset\mathbb{N},\#E=m\}$, where $B_{\ell_p^*}$ denotes the closed unit ball of $\ell_p^*=\ell_q$. Notice that for every $E\subset\mathbb{N}$ of size $m$, we have $|f_E(\sum\alpha_ke_k)|\leq{\lVert(\alpha_k)_{k\in E}\rVert}_{\ell_1}\leq m^{1-\frac{1}{p}}{\lVert(\alpha_k)\rVert}_{\ell_p}$ so that ${\lVertf_E\rVert}_{\ell_p^*}\leq m^{1-\frac{1}{p}}$. So $\mathcal{B}_m$ is a bounded subset of $\ell_p^*$ containing $B_{\ell_p^*}$. Due to the identity ${\lVertx\rVert}_{\ell_p}=\sup_{f\in B_{\ell_p^*}}|f(x)|$, we can now define an equivalent norm ${\lVert\cdot\rVert}_m$ on $\ell_p$ by the rule ${\lVertx\rVert}_m:=\sup_{f\in\mathcal{B}_m}|f(x)|$. Put $X_m:=(\ell_p,{\lVert\cdot\rVert}_m)$, and notice that for all $n$ and $E$, we have $|f_E(e_n)|\leq 1$. Hence $(e_n)$ is still normalized in $X_m$, and weakly null since $X_m$ is isomorphic to $\ell_p$. Furthermore, this isomorphism also means the identity map $I_m\in\mathcal{L}(X_m)$ is a norm-1 operator which lies in $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X_m)$ by Example \[whole-space\]. However, we will show that $ C_{(p,\xi)}(I_m)\geq m^{1-\frac{1}{p}}$.
Suppose $C<m^{1-\frac{1}{p}}$, and let $0<\epsilon<m^{1-\frac{1}{p}}-C$. Then, let $(e_{n_k})$ be any subsequence of $(e_n)$, which we have previously observed is normalized and weakly null in $X_m$. Pick $E=(m+1<m+2<\cdots<2m)\in\mathcal{S}_1\subseteq\mathcal{S}_\xi$ of size $m$, and define $F:=(n_{m+1}<n_{m+2}<\cdots<n_{2m})$. Since $\mathcal{S}_\xi$ is spreading, we have $F\in\mathcal{S}_\xi$, and also of size $m$. Next, define $(\alpha_k)\in\mathcal{A}_\xi$ by letting $\alpha_k=1$ for all $k\in E$ and $\alpha_k=0$ otherwise. Then
$$\begin{gathered}
{\lVert\sum\alpha_ke_{n_k}\rVert}_m\geq\left|f_F\left(\sum\alpha_ke_{n_k}\right)\right|=\left|f_F\left(\sum_{n\in F}e_n\right)\right|=m\\\\=m^{1-\frac{1}{p}}{\lVert(\alpha_k)\rVert}_{\ell_p}>(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}.\end{gathered}$$
Thus, the identity map $I_m$ does not lie in $\mathcal{WD}_{\ell_p}^{(C,\xi)}(X_m)$, and $ C_{(p,\xi)}(I_m)\geq m^{1-\frac{1}{p}}$ as claimed.
We have therefore constructed a sequence $(X_m)$ of Banach spaces and a corresponding sequence $I_m\in\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X_m)$ of norm-1 operators with $C_{(p,\infty)}(I_m)\to\infty$. By Lemma \[nonclosed-space\], there exists a space $X$ for which $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X)$ fails to be norm-closed, and in case $p\neq\infty$, we can choose it to be reflexive.
Even though $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ is not a norm-closed operator ideal, when $\xi=1$ its components are $F_\sigma$-subsets of $\mathcal{L}(X,Y)$, as the following Proposition shows.
\[norm-closed\]Let $X$ and $Y$ be Banach spaces, and fix constants $0\leq C<\infty$ and $1\leq p\leq\infty$. We consider the case $\xi=1$. Then $\mathcal{WD}_{\ell_p}^{(C,1)}(X,Y)$ is a norm-closed subset of $\mathcal{L}(X,Y)$.
Let $(T_j)$ be a sequence in $\mathcal{WD}_{\ell_p}^{(C,1)}(X,Y)$ converging to some $T\in\mathcal{L}(X,Y)$. Fix any $\epsilon>0$, and let $(x_n)\subset X$ be a normalized weakly null sequence in $X$. Without loss of generality we may assume ${\lVertT-T_j\rVert}<\epsilon/(2j^{1-\frac{1}{p}})$ for all $j$. Let $(x_{n_k})$ be a subsequence formed by a diagonal argument using the $T_j$’s with $\frac{\epsilon}{2}>0$. In other words, begin with a subsequence $(x_{n_{1,k}})$ given by the definition of $T_1\in\mathcal{WD}_{\ell_p}^{(C,1)}(X,Y)$, corresponding to $\frac{\epsilon}{2}>0$. Then find a *further* subsequence $(x_{n_{2,k}})\subset(x_{n_{1,k}})$ given by the definition of $T_2\in\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$, also corresponding to $\frac{\epsilon}{2}>0$, and so on. Finally, for each $k$, put $n_k:=n_{k,k}$.
Let $(\alpha_k)\in\mathcal{A}_1$, and set $m:=\min\text{supp}(\alpha_k)\leq\#\text{supp}(\alpha_k)$. Notice that $(x_{n_k})_{k\geq m}$ is a subsequence of $(x_{n_{m,i}})_{i\geq m}$ so that by Lemma \[subsequence-hereditary\], $$\begin{gathered}
{\lVert\sum\alpha_kTx_{n_k}\rVert}\leq{\lVert\sum_k\alpha_kT_mx_{n_k}\rVert}+{\lVertT-T_m\rVert}{\lVert\sum\alpha_kx_{n_k}\rVert}
\\\\<\left(C+\frac{\epsilon}{2}\right){\lVert(\alpha_k)\rVert}_{\ell_p}+m^{1-\frac{1}{p}}\left(\frac{\epsilon}{2m^{1-\frac{1}{p}}}\right){\lVert(\alpha_k)\rVert}_{\ell_p}=(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}\end{gathered}$$
Fix Banach spaces $X$ and $Y$, along with a constant $1\leq p\leq\infty$ and an ordinal $1\leq\xi\leq\omega_1$. We define the norm ${\lVert\cdot\rVert}_{(p,\xi)}$ on the space $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$ by the rule ${\lVertT\rVert}_{(p,\xi)}:={\lVertT\rVert}_{\mathcal{L}(X,Y)}+ C_{(p,\xi)}(T)$.
Notice that ${\lVert\cdot\rVert}_{(p,\xi)}$ is indeed a norm on $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$, as it is the sum of a norm and a seminorm.
\[banach-ideal\]Fix $1\leq p\leq\infty$. In case $\xi=1$, the rule ${\lVert\cdot\rVert}_{(p,1)}$ is an ideal norm which makes $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ into a Banach ideal.
As was observed earlier, that $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ is an operator ideal follows from Propositions \[VinWD\], \[multideal\], and \[addition\]. To show that ${\lVert\cdot\rVert}_{(p,1)}$ induces a complete norm on each component space $\mathcal{WD}_{\ell_p}^{(\infty,1)}(X,Y)$, suppose $(T_n)$ is a ${\lVert\cdot\rVert}_{(p,1)}$-Cauchy sequence. Then it is ${\lVert\cdot\rVert}_{(p,1)}$-bounded and hence $C_{(p,1)}$-bounded, say by $M>0$. It is also Cauchy in the operator norm so that $T_n\to T$ for some $T\in\mathcal{L}(X,Y)$. By Proposition \[seminorm\], every $T_n$ lies in the set $\mathcal{WD}_{\ell_p}^{(M,1)}(X,Y)$, which is closed under the operator norm by Proposition \[norm-closed\]. Hence, $T\in\mathcal{WD}_{\ell_p}^{(\infty,1)}(X,Y)$ as well, and it remains only to show that ${\lVert\cdot\rVert}_{(p,1)}$ is indeed an ideal norm.
Since any element of the form $x^*\otimes y$ is rank-1, it is completely continuous. By Proposition \[VinWD\], this means $C_{(p,1)}(x^*\otimes y)=0$ and hence ${\lVertx^*\otimes y\rVert}_{(p,1)}={\lVertx^*\otimes y\rVert}_{\mathcal{L}(X,Y)}=\sup_{x\in S_X}{\lVert(x^*\otimes y)(x)\rVert}_Y=\sup_{x\in S_X}|x^*(x)|{\lVerty\rVert}_Y={\lVertx^*\rVert}_{X^*}{\lVerty\rVert}_Y$. The triangle inequality follows from the fact that ${\lVert\cdot\rVert}_{(p,1)}$ is a norm on each component space $\mathcal{WD}_{\ell_p}^{(\infty,1)}(X,Y)$. That ${\lVertBTA\rVert}_{(p,1)}\leq{\lVertB\rVert}_{\mathcal{L}(Y,Z)}{\lVertT\rVert}_{(p,1)}{\lVertA\rVert}_{\mathcal{L}(W,X)}$ for all $T\in\mathcal{J}(X,Y)$, $A\in\mathcal{L}(W,X)$, and $B\in\mathcal{L}(Y,Z)$, follows naturally from Propositions \[multideal\] and \[seminorm\].
Significance of parameters
==========================
Let $1<q<p<\infty$. By Example \[compact\] we get $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(\ell_q)=\mathcal{K}(\ell_q)$, whereas by Example \[whole-space\] we get $\mathcal{WD}_{\ell_q}^{(\infty,\xi)}(\ell_q)=\mathcal{L}(\ell_q)$. Applying Proposition \[inclusion\] therefore gives us the following.
\[p-distinct\]Fix any ordinal $1\leq\xi\leq\omega_1$. For $1\leq q<p\leq\infty$, the norm-closed operator ideals $\overline{\mathcal{WD}}_{\ell_p}^{(\infty,\xi)}$ and $\overline{\mathcal{WD}}_{\ell_q}^{(\infty,\xi)}$ are distinct (as classes).
However, it is natural to also ask whether the classes $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ are distinct as $\xi$ ranges over $1\leq\xi\leq\omega_1$. Obviously, this is not the case for the trivial ideal $\mathcal{WD}_{\ell_1}^{(\infty,\xi)}=\mathcal{L}$. The question remains open in general for $1<p\leq\infty$, but in this section we do give a [*partial*]{} answer by showing that $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}\neq\mathcal{WD}_{\ell_p}^{(\infty,\omega_1)}$ for all countable ordinals $1\leq\xi<\omega_1$. Our task requires a few preliminaries, given below.
\[ordinal-inclusion\]Let $X$ and $Y$ be Banach spaces, and fix a constant $1\leq p\leq\infty$. Let $1\leq\xi<\zeta\leq\omega_1$ be ordinals, and $0\leq C\leq\infty$. Then $\mathcal{WD}_{\ell_p}^{(C,\zeta)}(X,Y)\subseteq\mathcal{WD}_{\ell_p}^{(C,\xi)}(X,Y)$.
We may assume $C\neq\infty$. Suppose $T\in\mathcal{WD}_{\ell_p}^{(C,\zeta)}(X,Y)$, and let $(x_n)$ be a normalized weakly null sequence in $X$, and $\epsilon>0$. Then there exists a subsquence $(n_k)$ such that ${\lVert\sum\alpha_kTx_{n_k}\rVert}\leq(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$ for all $(\alpha_k)\in\mathcal{A}_\zeta$. Let $d=d(\xi,\zeta)$ be such that if $E\in\mathcal{S}_\xi$ with $\min E\geq d$ then $E\in\mathcal{S}_\zeta$. Now, let $(\alpha_k)\in\mathcal{S}_\xi$, and define the scalar sequence $(\beta_k)$ as $\beta_k:=\alpha_{k-d}$ for $k\geq d$ and $\beta_k:=0$ for $k<d$. By the spreading property of $\mathcal{S}_\xi$ we have $(\beta_k)\in\mathcal{A}_\xi$, and since also $\min\text{supp}(\beta_k)\geq d$ we have $(\beta_k)\in\mathcal{A}_\zeta$. Thus, ${\lVert\sum\alpha_kTx_{n_{k+d}}\rVert}={\lVert\sum\beta_kTx_{n_k}\rVert}\leq(C+\epsilon){\lVert(\beta_k)\rVert}_{\ell_p}=(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$.
Let $1\leq\xi<\omega_1$ be a countable ordinal. A finite sequence $(E_i)_{i=1}^j$ of finite subsets of $\mathbb{N}$ is called [**$\boldsymbol{\mathcal{S}_\xi}$-admissible**]{} whenever $E_1<\cdots<E_j$ and $\{\min E_i\}_{i=1}^j\in\mathcal{S}_\xi$. Then the [**Tsirelson-type space**]{} $T[\frac{1}{2},\mathcal{S}_\xi]$ is the completion of $c_{00}$ under the norm ${\lVert\cdot\rVert}_T$ uniquely defined by the implicit equation ${\lVertx\rVert}_T=\max\{{\lVertx\rVert}_{\ell_\infty},\frac{1}{2}\sup\sum_i{\lVertE_ix\rVert}_T\}$, where the “sup” is taken over all $j\in\mathbb{N}$ and all $\mathcal{S}_\xi$-admissible families $(E_i)_{i=1}^j$. Here we use the notation $E_ix:=\sum_{n\in E_i}\alpha_ne_n$ for $x:=\sum\alpha_ne_n\in c_{00}$, where $(e_n)$ are the canonical basis vectors in $c_{00}$. We also use the abbreviation $T=T[\frac{1}{2},\mathcal{S}_\xi]$ when the ordinal $\xi$ is understood from context.
It is easily seen that the canonical unit vectors in $c_{00}$ form a normalized 1-unconditional basis for $T$. For $1\leq q<\infty$, its [**$\boldsymbol{q}$-convexification**]{} $T_q[\frac{1}{2},\mathcal{S}_\xi]$, where we again use the abbreviation $T_q=T_q[\frac{1}{2},\mathcal{S}_\xi]$, is then usually defined in the literature by setting $T_q=T_q[\textstyle\frac{1}{2},\mathcal{S}_\xi]=\{(\alpha_n):(|\alpha_n|^q)\in T=T[\frac{1}{2},\mathcal{S}_\xi]\}$, which is a Banach space under the norm ${\lVert(\alpha_n)\rVert}_{T_q}:={\lVert(|\alpha_n|^q)\rVert}_T^{1/q}$, and with the canonical unit vectors in $c_{00}$ again forming a normalized 1-unconditional basis. (Notice also that if $q=1$ then we get $T_1=T$.) However, it will serve our purposes much better to use instead the following equivalent construction (cf. [@JL03], p1062). We inductively define a sequence $({\lVert\cdot\rVert}_n)$ of norms on $c_{00}$. Set ${\lVert\cdot\rVert}_0:={\lVert\cdot\rVert}_{\ell_\infty}$ and define each successive ${\lVert\cdot\rVert}_{n+1}$ by the rule ${\lVertx\rVert}_{n+1}=\max\{{\lVertx\rVert}_{\ell_\infty},2^{-1/q}\sup(\sum_i{\lVertE_ix\rVert}_n^q)^{1/q}\}$, where the “sup” is taken over all $j\in\mathbb{N}$ and all $\mathcal{S}_\xi$-admissible families $(E_i)_{i=1}^j$. Then ${\lVertx\rVert}_{T_q}:=\lim_{n\to\infty}{\lVertx\rVert}_n$ defines a norm on $c_{00}$. In fact, it is easily seen (cf., e.g., the kind of argument used in the proof to Theorem 10.3.2 in [@AK06]) that ${\lVert\cdot\rVert}_{T_q}$ is the unique norm on $c_{00}$ satisfying the implicit equation ${\lVertx\rVert}_{T_q}=\max\{{\lVertx\rVert}_{\ell_\infty},2^{-1/q}\sup(\sum_i{\lVertE_ix\rVert}_{T_q}^q)^{1/q}\}$. The space $T_q$ is just the completion of $c_{00}$ under this norm.
Due to this construction, ${\lVertx\rVert}_{T_q}\leq{\lVertx\rVert}_{\ell_q}$ for each $x\in c_{00}$. Furthermore, $T_q$ is known to be a reflexive Banach space which contains no copy of $\ell_q$. When $q=1$ this follows from Proposition 5.1 in [@AA92]. In case $1<q<\infty$, Remark T.1 on p1062 of [@JL03] tells us that $T_q$ is an asymptotic $\ell_q$ space which contains no copy of $\ell_q$, and thus by Remark 6.3 in [@MT93] it is also reflexive. Therefore each dual space $T_q^*$ is a reflexive space which fails to contain any copy of $\ell_p$, $\frac{1}{p}+\frac{1}{q}=1$. Notice that $T_q^*$ can also be viewed as a completion of $c_{00}$ under some norm ${\lVert\cdot\rVert}_{T_q^*}$, with the usual action $f(x)=\sum\alpha_n\beta_n$ for $f=(\alpha_n)\in T_q^*$ and $x=(\beta_n)\in T_q$.
In [@OA13] was given an implicit formula for the norm of $T_1[\frac{1}{2},\mathcal{S}_1]^*$. It is natural to conjecture that a similar formula always holds for the norm of $T_q[\frac{1}{2},\mathcal{S}_\xi]^*$, but for our purposes we only need a crude estimate.
\[convexified-estimate\]Let $1<p\leq\infty$ and $1\leq q<\infty$ be conjugate, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Set $1\leq\xi<\omega_1$ and $T_q=T_q[\frac{1}{2},\mathcal{S}_\xi]$. Then ${\lVertx\rVert}_{T_q^*}\leq 2^{1/q}{\lVert({\lVertE_ix\rVert}_{T_q^*})\rVert}_{\ell_p}$ for all $x\in c_{00}$ and $\mathcal{S}_\xi$-admissible families $(E_i)_{i=1}^j$ satisfying $x=\sum_{i=1}^jE_ix$.
Let $y\in T_q$. Since $x=\sum_{i=1}^jE_ix$ we have $x(y)=\sum_{i=1}^j(E_ix)(E_iy)$. Then by this fact together with Hölder and the relation $2^{-1/q}(\sum_{i=1}^j{\lVertE_iy\rVert}_{T_q}^q)^{1/q}\leq{\lVerty\rVert}_{T_q}$ (which follows from the construction of $T_q$), we have
$$\begin{gathered}
|x(y)|=|\sum_{i=1}^j(E_ix)(E_iy)|\leq\sum_{i=1}^j|(E_ix)(E_iy)|\leq\sum_{i=1}^j{\lVertE_ix\rVert}_{T_q^*}{\lVertE_iy\rVert}_{T_q}\\\\\leq{\lVert({\lVertE_ix\rVert}_{T_q^*})\rVert}_{\ell_p}\left(\sum_{i=1}^j{\lVertE_iy\rVert}_{T_q}^q\right)^{1/q}\leq 2^{1/q}{\lVert({\lVertE_ix\rVert}_{T_q^*})\rVert}_{\ell_p}{\lVerty\rVert}_{T_q}.\end{gathered}$$
\[Tsirelson-block-estimate\]Let $T_q=T_q[\frac{1}{2},\mathcal{S}_\xi]$, $1\leq q<\infty$ and $1\leq\xi<\omega_1$, and let $(u_k)$ be any normalized block basic sequence in the dual space $T_q^*$ (with respect to the canonical unit vectors in $c_{00}$). Then for every $(\alpha_k)\in\mathcal{A}_\xi$ we have ${\lVert\sum\alpha_ku_k\rVert}_{T_q^*}\leq 2^{1/q}{\lVert(\alpha_k)\rVert}_{\ell_p}$, where $1<p\leq\infty$ is conjugate to $q$, that is, $\frac{1}{p}+\frac{1}{q}=1$.
Write $\text{supp}(\alpha_k)=:\{k_1,\cdots,k_j\}\in\mathcal{S}_\xi$, and set $E_i:=\text{supp}\{u_{k_i}\}$ for each $1\leq i\leq j$. Then $x:=\sum\alpha_ku_k=\sum_{i=1}^jE_ix$, where ${\lVertE_ix\rVert}_{T_q^*}={\lVert\alpha_{k_i}u_{k_i}\rVert}_{T_q^*}=|\alpha_{k_i}|$ for each $1\leq i\leq j$. Furthermore, due to $k_i\leq\min\text{supp}\{u_{k_i}\}=\min E_i$ together with $\{k_1,\cdots,k_j\}\in\mathcal{S}_\xi$ and the spreading property of Schreier families, we see that $(E_i)_{i=1}^j$ is $\mathcal{S}_\xi$-admissible. All of this together with Lemma \[convexified-estimate\] means ${\lVert\sum\alpha_ku_k\rVert}_{T_q^*}\leq 2^{1/q}{\lVert({\lVertE_ix\rVert}_{T_q^*})_{i=1}^j\rVert}_{\ell_p}=2^{1/q}{\lVert(\alpha_{k_i})_{i=1}^j\rVert}_{\ell_p}=2^{1/q}{\lVert(\alpha_k)\rVert}_{\ell_p}$.
\[convexified-lower-estimate\]Set $T_q=T_q[\frac{1}{2},\mathcal{S}_\xi]$, $1\leq q<\infty$ and $1\leq\xi<\omega_1$. Let $1<p\leq\infty$ denote the conjugate of $q$, that is, $\frac{1}{p}+\frac{1}{q}=1$. Then ${\lVertx^*\rVert}_{\ell_p}\leq{\lVertx^*\rVert}_{T_q^*}$ for all $x^*\in c_{00}$.
Since $c_{00}\subseteq\ell_p=\ell_q^*$ with $c_{00}$ dense in $\ell_q$, for each $\epsilon>0$ we can find $x\in c_{00}$ such that $|x^*(x)|\geq({\lVertx^*\rVert}_{\ell_p}-\epsilon){\lVertx\rVert}_{\ell_q}$. Combining this with the relation ${\lVertx\rVert}_{T_q}\leq{\lVertx\rVert}_{\ell_q}$ (which follows from the construction of $T_q$) we get $|x^*(x)|\geq({\lVertx^*\rVert}_{\ell_p}-\epsilon){\lVertx\rVert}_{\ell_q}\geq({\lVertx^*\rVert}_{\ell_p}-\epsilon){\lVertx\rVert}_{T_q}$ and hence ${\lVertx^*\rVert}_{T_q^*}\geq{\lVertx^*\rVert}_{\ell_p}-\epsilon$. Letting $\epsilon\to 0$ completes the proof.
\[ordinal-distinct\]Set $T_q=T_q[\frac{1}{2},\mathcal{S}_\xi]$, $1\leq q<\infty$ and $1\leq\xi<\omega_1$, and let $T_q^*$ denote its dual. Let $1<p\leq\infty$ be conjugate to $q$, that is, $\frac{1}{p}+\frac{1}{q}=1$. Then $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(T_q^*)=\mathcal{L}(T_q^*)$, whereas $\mathcal{WD}_{\ell_p}^{(\infty,\omega_1)}(T_q^*)\neq\mathcal{L}(T_q^*)$.
Consider the identity operator $I:T_q^*\to T_q^*$. We claim that $I\in\mathcal{WS}_{\ell_p}^{(2^{1/q},\xi)}(T_q^*)$. Indeed, let $(x_n)$ be a normalized weakly null sequence in $T_q^*$, and let $\epsilon>0$. By the uniform version of the Bessaga-Pełczyński Selection Principle, there exists a subsquence $(x_{n_k})$ which is $(1+2^{-1/q}\epsilon)$-equivalent to a normalized block basic sequence $(u_k)$ of the unit vector basis. Thus, by Lemma \[Tsirelson-block-estimate\], for every $(\alpha_k)\in\mathcal{A}_\xi$ we have ${\lVert\sum\alpha_kx_{n_k}\rVert}_{T_q^*}\leq(1+2^{-1/q}\epsilon){\lVert\sum\alpha_ku_k\rVert}_{T_q^*}\leq(2^{1/q}+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$, and the claim is proved.
On the other hand, we also claim $I\notin\mathcal{WD}_{\ell_p}^{(\infty,\omega_1)}(T_q^*)$. Let $(e_n)$ be the unit vector basis of $T_q^*$, which is also weakly null since $T_q^*$ is reflexive. Recall from Lemma \[convexified-lower-estimate\] that ${\lVert(\alpha_n)\rVert}_{\ell_p}\leq{\lVert(\alpha_n)\rVert}_{T_q^*}$ for all $(\alpha_n)\in c_{00}$. Hence, for any subsequence $(n_k)$ we have ${\lVert\sum\alpha_ke_{n_k}\rVert}_{T_q^*}\geq{\lVert(\alpha_k)\rVert}_{\ell_p}$. Since $T_q^*$ fails to contain a copy of $\ell_p$, then for any $C\geq 0$ and $\epsilon>0$ we must now be able to find some $(\alpha_k)\in c_{00}$ with ${\lVert\sum\alpha_ke_{n_k}\rVert}_{T_q^*}\geq(C+\epsilon){\lVert(\alpha_k)\rVert}_{\ell_p}$.
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abstract: |
Let $\Omega$ be a non-empty open proper and connected subset of $\mathbb R^{n}$. Consider the elliptic Schrödinger type operator $L_{E}u=$ $A_{E}u+Vu=$ $-\Sigma_{ij}a_{ij}(x)$ $u_ {x_i x_j}+Vu$ in $\Omega$, and the linear parabolic operator $L_{P}u=A_{P}u+Vu=$ $u_{t}-\Sigma a_{ij}(x,t)u_{x_{i}x_{j}}+Vu$ in $\Omega_{T}=\Omega\times ( 0,T )$, where the coefficients $a_{ij}\in VMO$ and the potential $V$ satisfies a reverse-Hölder condition. The aim of this paper is to obtain a priori estimates for the operators $L_{E}$ and $L_{P}$ in weighted Sobolev spaces involving the distance to the boundary and weights in a local- $A_{p}$ class.
[**Mathematics Subject Classification (2010):**]{} Primary: 35J10; Secondary: 35B45; 42B35.
address:
- |
Departamento de Matemática\
Fac. de Cs. Exactas, Ingeniería y Agrimensura\
Universidad Nacional de Rosario\
Pellegrini 250, 2000 Rosario
- |
Facultad de Ciencias Exactas\
Universidad Nacional del Centro de la Provincia de Buenos Aires\
Pinto 399, 7000 Tandil
- |
Instituto de Matemática Aplicada del Litoral\
CONICET- Universidad Nacional del Litoral\
IMAL-CCT CONICET Santa FE\
Colectora Ruta Nac. No 168, Paraje El Pozo 3000 Santa Fe
author:
- 'Isolda Cardoso – Pablo Viola – Beatriz Viviani'
title: 'Interior $L^p$ - estimates for elliptic and parabolic Schrödinger type operators and local $A_p$ -weights'
---
Introduction {#Intro}
============
Let $\Omega$ be a non-empty open proper and connected subset of ${\mathbb{R}}^{n}$. We are going to consider the following two operators: the elliptic Schrödinger type operator
$$L_{E}u=A_{E}u+Vu=-\sum_{ij}a_{ij}(x)u_ {x_i x_j}+Vu$$
in $\Omega$, and the linear parabolic operator
$$L_{P}u=A_{P}u+Vu=u_{t}-\sum a_{ij}(x' ,t)u_{x_{i}x_{j}}+Vu$$
in $\Omega_{T}=\Omega\times ( 0,T )$, with $T>0$, under the following assumptions:
- $a_{ij}=a_{ji}$, and $$\frac{1}{C} |\xi|^{2} \leq \sum _{ij} a_{ij}(.)\xi_{i}\bar{\xi_{j}}\leq C|\xi|^{2}$$ for a.e. $x\in \Omega$ or $x=(x',t)\in \Omega_{T}$, respectively;
- $a_{ij}\in L^{\infty}\cap VMO({\mathbb{R}}^{n})$. Here we have the space of functions of vanishing mean oscillation defined as
$$VMO({\mathbb{R}}^{n})=\big\{g\in BMO({\mathbb{R}}^{n}): \eta(r)\to 0, r\to 0^{+}\big\},$$
where $$\eta(r)=\sup _{\rho\leq r} \sup _{x\in{\mathbb{R}}^{n}}\Bigg(\dfrac{1}{|B_{\rho}(x)|}\int _{B_{\rho}(x)}\big|g(y)-g_{B_{\rho}}\big|dy\Bigg).$$ Here $g_{B_\rho }=|B(\rho(x) )|^{-1}\int_{B_\rho (x)} g(y)\, dy$. The parabolic $VMO({\mathbb{R}}^{n+1})$ is defined in the same way, except this time we take the supremum over the parabolic balls (see section \[prelim:parabolicsetting\]);
- The potential $V\geq 0$ satisfies a reverse Hölder condition of order $q$, shortly $V\in RH_{q}$, which means that $$\label{RHq}
\Big(\dfrac{1}{|B|}\int _{B} V^q dx\Big)^{1/q} \leq \dfrac{1}{|B|}\int _{B} Vdx,$$ where the ball $B$ is in ${\mathbb{R}}^{n}$.
Sometimes we will use $A$ for either the operators $A_{E}$ or $A_{P}$, and $\Lambda$ for either the subset $\Omega$ or $\Omega_{T}$.
When the coefficients $a_{ij}$ are at least uniformly continuous, existence and uniqueness results together with a-priori $W^{2,p}$ estimates are well known (see e.g. [@GT]). The theory for operators with discontinuous coefficients, in the sense of $VMO$, goes back to the 90’s with the works of Chiarenza-Frasca-Longo in [@CFL1] and [@CFL2] for elliptic operators and Bramanti-Cerutti in [@BC] for the parabolic case. Since then, many authors have considered this problem in different situations and contexts. The Schrödinger operator when $A$ is the Laplacian and the potential $V$ satisfies the reverse-Hölder condition (3), was studied by Shen in [@Sh2] and related results when $V(x) =|x|$ (Hermite operator) have been proved by Thangavelu in [@T]. For the elliptic type Schrödinger operator under consideration, a global $W^{2,p}({\mathbb{R}}^{n})$ estimate and the existence and uniqueness results deduced from them were obtained in [@BBHV]. We are interested in obtaining a priori interior estimates in weighted Sobolev spaces for the operator $L$, where $L$ is either the elliptic Schrödinger type operator $L_{E}$ or the parabolic operator $L_{P}$, defined in a non necessarily bounded domain. We follow the strategy adopted in [@BBHV]. First we get a weighted version of the a priori estimates obtained in [@CFL1] and in [@BC] for the principal operator $A_E$ and $ A_P$ respectively. Thanks to these estimates we are reduced to prove a weighted $L^p$ bound on $Vu$ in terms on $Lu$. Then, we give a representation formula for $Vu$ by means of the fundamental solution of a constant coefficient operator of the type $A_0 + V$, for which a global estimate was proved by Dziubanski in [@D] for $L_{E}$ and by Kurata in [@K] for $L_{P}$. These representation formulas involve suitable integral operators with positive kernel, applied to $Lu$, and their positive conmutators, applied to the second order derivatives of $u$.
In order to prove that these operators are bounded on weighted $L^p$, we use local maximal functions, $M_{\text{loc}}f$ (see section \[prelim\]), defined in a proper open set imbedded in a metric space. This maximal operator and the classes of weight involved $A_{p,\text{loc}}$ (see below), were first studied by Nowak and Stempak in [@NS] when $\Omega=(0,\infty)$ and by Lin and Stempak in [@LS] for $\Omega= {\mathbb{R}}^{n}\setminus \{0\}$. In a general setting, that is in metric spaces, this maximal operator and the corresponding classes of weights were considered by Harboure, Salinas and Viviani in [@HSV] and by Lin, Stempak and Wan in [@LSW].
We consider the local weights class $A_{p,\text{loc}}$ defined as follows: let $(X,d)$ be a metric space and let $\Lambda$ be a nonempty open proper subset of $X$, if $0< \beta < 1$ we define the family of balls
$$F_{\beta} =\big\{ B=B(x_{B},r_{B}): x_{B}\in\Gamma, r_{B}< \beta d(x_{B},\Lambda^{C}) \big\},$$
where $d(x_{B},\Lambda^{C})$ denotes the distance from the center $x_B$ of the ball $B$ to the complementary set of $\Lambda$. Given a Borel measure $\mu$ defined on $\Lambda$, for $1< p< \infty$, we define $$\label{apbeta}
\mbox{ $w\in A^{\beta}_{p,\text{loc}}(\Lambda)$\ \ iff \ \ }
\sup _{B\in\mathcal{F}_{\beta} }\frac{1}{\mu(B)}
\Big(\int _{B} w d\mu\Big)^{1/p}
\Big(\int _{B} w^{-p/p'}d\mu\Big)^{1/p'} < \infty.$$ We remark that the classes $ A^{\beta}_{p,\text{loc}}(\Lambda)$ are independent of $\beta$, as was shown in [@HSV]. In view of this fact, we shall refer to theses weights as $ A_{p,\text{loc}}(\Lambda)$. We also consider the following weighted Sobolev spaces, defined in ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{n+1}$, respectively:
$$W^{2,p}_{\delta ,w}(\Omega )=
\Big\{u \in L^{1}_{\text{loc}}(\Omega ): \|u\|_{W^{2,p}_{\delta ,w}(\Omega )}=\sum _{|\gamma|\leq 2} \|\delta ^{|\gamma |} D^{\gamma} u\|_{L^{p}_{w}(\Omega )}< \infty\Big\},$$
and $$W^{2,p}_{\delta ,w}(\Omega_{T} \! )
\! = \!
\Big\{ \! u \! \in \!
L^{1}_{\text{loc}} \! (\Omega_{T} )
\! : \!
\|u\|_{W^{2,p}_{\delta ,w}(\Omega_{T} )}
\!\! = \!\!\!
\sum _{|\gamma |\leq 2}
\!\!
\|\delta ^{|\gamma|} D_{x}^{\gamma} u\|_{L^{p}_{w}(\Omega_{T} )} + \|\delta^{2}D_{t}u\|_{L^{p}_{w}(\Omega_{T})}
\!\! < \! \infty \!
\Big\} \! ,$$ where $\delta(x)=\min \{ 1 , d(x,\Lambda^{C})\}$, with either $\Lambda=\Omega$ or $\Omega_{T}$, and $d$ denotes the corresponding distance.
We will prove the following results:
\[thm:principal\] Let $\Omega$ be a nonempty, proper, open and connected subset of ${\mathbb{R}}^{n}$. Let $p\in(1,q]$ and $w\in A_{p,\text{loc}}(\Omega )$. If $u\in W^{2,p}_{\delta ,w}(\Omega )$ is a solution of $$Lu = Au+Vu =-\sum_{i,j}a_{ij}u_{x_ix_j}+Vu=f\qquad\text{in $\Omega $},$$ under the assumptions (1), (2) and (3), then $$\|u\|_{W^{2,p}_{\delta ,w}(\Omega )} + \|\delta ^2Vu\|_{L^p_{w}(\Omega )}\leq C\big[\|\delta ^2 f\|_{L^p_w(\Omega )}+\|u\|_{L^p_w(\Omega )}\big],$$ where $\delta (x)=\min\{1,d(x,\Omega^{C})\}, \,x\in {\mathbb{R}}^{n}$.
The parabolic version of this theorem goes as follows:
\[thm:principalP\] Let $\Omega$ be a nonempty, proper, open and connected subset of ${\mathbb{R}}^{n}$. For $T>0$ define $\Omega_{T}=\Omega\times \big(0,T\big)$. Let $p\in(1,q]$ and $w\in A_{p,\text{loc}}(\Omega_{T} )$. If $u\in W^{2,p}_{\delta ,w}(\Omega_{T})$ is a solution of $$Lu = Au+Vu =u_{t}-\sum_{i,j}a_{ij}u_{x_i x_j}+Vu=f\qquad\text{in $\Omega_{T} $},$$ under the assumptions (1), (2) and (3), then $$\|u\|_{W^{2,p}_{\delta ,w}(\Omega_{T} )} + \|\delta ^2Vu\|_{L^p_{w}(\Omega_{T} )}\leq C\big[\|\delta ^2 f\|_{L^p_w(\Omega_{T} )}+\|u\|_{L^p_w(\Omega_{T} )}\big],$$ where $\delta (x',t)=\min\{1,d((x',t),\Omega^{C}_{T} )\}$ .
We note that, as it is easy to check, $w(x)= \delta^\alpha(x)$ belongs to $A_{p,\text{loc}}$ for any exponent $\alpha\in\mathbb{R}$. Therefore the data function $f$ appearing on the right hand side of Theorem \[thm:principal\] and Theorem \[thm:principalP\] could increase polynomially when approaching the boundary of $\Omega$ or $\Omega_{T}$ and still we might have some control for the derivatives of the solution up to the order $2$.
The paper is organized as follows: in Section \[prelim\] we put together the preliminary definitions and results, and prove some useful lemmas; in Section \[previous\] we prove some results that will build the proof of the Main Theorem for the operator $L_{E}$, and in Section \[previousP\] we show similar results for the operator $L_{P}$. Finally, in [Section \[mains\]]{} we end up proving the main results stated above: Theorems \[thm:principal\] and \[thm:principalP\].
Acknowledgments {#acknowledgments .unnumbered}
---------------
The first author is partially supported by Universidad Nacional de Rosario and a grant from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). The second author is partially supported by Núcleo Consolidado de Matemática Pura y Aplicada, Universidad Nacional del Centro de la Provincia de Buenos Aires and by CONICET. The third author is partially supported by grants from CONICET and Universidad Nacional del Litoral.
Preliminaries {#prelim}
=============
Definition and notations {#prelim:definitions}
------------------------
### The parabolic setting {#prelim:parabolicsetting}
The parabolic setting we are considering consists of ${\mathbb{R}}^{n+1}$ endowed with the following parabolic metric $$d(x,y)=(|x'-y'|^{2}+|t-s|)^{\frac12},$$ where we write $x=(x',t), y=(y',s)\in{\mathbb{R}}^{n+1}$, with $x',y'\in{\mathbb{R}}^{n}$ and $t,s\in{\mathbb{R}}^+$. We denote the parabolic balls as usual: $$B(x,r)=\{ y\in{\mathbb{R}}^{n+1}: d(x,y)<r\}.$$ and its Lebesgue measure by $|B(x,r)|= c_n r^{n+2}$.
### The local maximal operator {#prelim:localmaximal}
In this subsection we will denote by $X$ a metric space satisfying the weak homogeneity property, that is, there exists a fix number $N$ such that for any ball $B(x,r)$ there are no more than $N$ points in the ball whose distance from each other is greater than $r/2$. Also $\Lambda$ will mean any open proper and non empty subset of $X$ such that all balls contained in $\Lambda$ are connected sets and $\mu$ will be a Borel measure defined on $\Lambda$ which satisfies a doubling condition on $F_{\beta}$, that is, there is some constant $C_\beta$ such that for any ball $B\in F_{\beta} $ $$\mu(B)\leq C_\beta \mu(\tfrac{1}{2} B)$$ with $0< \mu(B)<\infty$ for any ball $B\in\mathcal{F}=\bigcup_{0<\alpha<1}\mathcal{F}_\alpha$.
We shall use the following local maximal operator associated to $\mathcal{F}_{\beta}$: given $0<\beta<1$ and $\mu$ as above $$\label{maxi}
M_{\mu,\beta}f(x)=\sup_{x\in B\in \mathcal{F}_{\beta}}\frac{1}{\mu(B)}\int_B|f|\; d\mu$$ for any $f\in L^1_{\text{loc}}(\Lambda, d\mu)$ and $x\in \Lambda$. When $\mu$ is the Lebesgue measure we denote $M_{\mu,\beta}f$ by $M_{\beta,\text{loc}}f$.
The boundness property for $M_{\mu,\beta}f$ is contained in the next Theorem:
\[thm:acot.mxml.loc\] Let $X$ and $\Lambda$ as above. Let $0<\beta<1$ and $\mu$ a Borel measure satisfying the doubling property on $\mathcal{F}_{\beta}$. Then, for $1<p<\infty$, $M_{\mu,\beta}f$ is bounded on $L^{p}_{w}(\Lambda,w d\mu)$ if and only if $w\in A^{\beta}_{p,\text{loc}}(\Lambda)$.
### The properties of the potential $V$ {#prelim:propertiesofV}
The potential $V$ satisfies assumption (3) and, as it is remarked in [@BBHV], the condition $V\in RH_{q}$ implies that for some $\epsilon >0$ we have also that $V\in RH_{q+\epsilon}$, where the $RH_{q+\epsilon}$ constant of $V$ is controlled in terms of the $RH_{q}$ constant of $V$. They also remark the useful fact that the measure $V(y)dy$ is doubling.
Associated to the function $V\in RH_{q}$ there is a function $\rho(x)$, called *critical radious*, defined by Shen in [@Sh2]: $$\label{rho}
\rho(x)= \sup\bigg\{ r>0: {\dfrac{r^{2}}{|B(x,r)|}}\int _{B(x,r)} V(y)dy\leq 1\bigg\},$$ which, under our assumptions on $V$, is finite almost everywhere. We note that by definition of $\rho$, we have that $$\label{14} {1\over{\rho(x)^{n-2}}} \int _{B(x,\rho(x))}V(y)dy\leq 1.$$
Shen also proved that the following inequalities hold: $$\begin{aligned}
\label{rhoxrhoy} & C\Big( 1+ \frac{|x-y|}{\rho(y)}\Big)^{1\over{k_{0}}}\leq 1+ \frac{|x-y|}{\rho(x)} \leq C \Big( 1+ \frac{|x-y|}{\rho(y)}\Big)^{1\over{k_{0}}},\end{aligned}$$ for some $k_{0}\in{\mathbb{N}}$ and any $x,y\in{\mathbb{R}}^{n}$ and $$\label{13}
{1\over{r^{n}}}\int _{B(x,r)}V(y)dy\leq C \bigg( {R\over r} \bigg)^{n\over q} {1\over{R^{n}}} \int _{B(x,R)} V(y),$$ for any $0<r<R<\infty$.
### Bounds for the fundamental solutions of the constant coefficient operators $L_{0}$ {#prelim:fundamentalsolutions}
Let us now consider the operator $A$, which denotes either $A_{E}$ or $A_{P}$. For fixed $x_{0}\in\Lambda$, where $\Lambda$ denotes $\Omega$ or $\Omega_{T}$, respectively, freeze the coefficients $a_{ij}(x_{0})$ and denote $L_{0}$ the operator $L$ with these constant coefficients.
Dziubanski in [@D] proved that the elliptic operator $L_{0}$ has a fundamental solution $\Gamma(x_{0};x,y)$ which satisfies that for any $k\in{\mathbb{N}}$ there exists a constant $c_{k}$ (independent of $x_{0}$) such that $$\begin{aligned}
\label{ellipticfundsolbound}
\Gamma(x_{0};x,y)
& \leq c_{k}{1\over \big({1+{|x-y|\over{\rho(x)}}}\big)^{k}} {1\over{|x-y|^{n-2}}},\end{aligned}$$ for any $x,y\in{\mathbb{R}}^{n}$, $x\neq y$. Here $\rho$ is the critical radious associated to $V$ defined in \[rho\]. We remark that the kernel
$$W(x,y) = V(y) {1\over {\big( 1+{{|x-y|}\over{\rho(x)}} \big)^{k} }} {1\over{|x-y|^{n-2}}},$$
satisfies *Hörmander’s condition of order $q$*, briefly *condition $H_{1}(q)$*, in the first variable (see Proposition 12 in [@BBHV]). This means that there exists a constant $C>0$ such that for any $r>0$ and any $x,x_{0}\in{\mathbb{R}}^{n}$ with $|x-x_{0}|<r$, the following inequality holds: $$\label{H1qcondition}
\sum _{j=1}^{\infty} j |B(x_0, 2^jr)|^\frac{1}{q'}
\Big( \int _{2^{j}r\leq |x_{0}-y|\leq 2^{j+1}r} |W(x,y)-W(x_{0},y)|^{q} dy \Big)^{1\over q} \leq C.$$ Also, observe that from inequalities \[rhoxrhoy\] we can replace $\rho(y)$ with $\rho(x)$ in the kernel $W$, possibly changing the integer $k$.
For the parabolic operator $L_{0}$, Kurata showed in Corollary 1 of [@K] that it has a fundamental solution $\Gamma(x_{0};x,y)$ which satisfies that for each $k\in{\mathbb{N}}$ there exists constants $c_{k}$ and $c_{0}$ (independents of $x_{0}$) such that
$$\Gamma(x_{0};x,y) \leq c_{k} {1\over{\big(1+{{d(x,y)}\over{\rho(x')}} \big)^{k}}} {1\over{|t-s|^{n/2}}} e^{-c_{0}{{|x'-y'|^{2}}\over{|t-s|}}},$$
where $d$ is the parabolic distance given in \[prelim:parabolicsetting\]. Thus, $$\begin{aligned}
\label{parabolicfundsolbound}
\Gamma(x_{0};x,y) & \leq c_{k} {1\over{\big(1+{{d(x,y)}\over{\rho(x')}} \big)^{k}}} {1\over{d(x,y)^{n}}}.\end{aligned}$$ The parabolic kernel, appearing on the right hand side [of \[parabolicfundsolbound\]]{}, also satisfies condition $H_{1}(q)$, as we prove in the next subsection.
Previous Lemmas {#prelim:lemmas}
---------------
\[lemm:parabolickernelh1q\] The kernel $$W(x,y)= V(y') {1\over{ \big( 1+ {{d(x,y)}\over{\rho(y')}} \big)^{k}}} {1\over{d(x,y)^{n}}}$$ satisfies condition $H_{1}(q)$ for $k$ large enough, that is, there exists a constant $C>0$ such that for every $r>0$, $x,x_{0}\in{\mathbb{R}}^{n+1}$ with $d(x,x_{0})<r$, $$\sum _{j=1}^{\infty} j (2^{j}r)^{{n+2}\over{q'}} \Big( \int _{2^{j}r<d(x_{0},y)\leq 2^{j+1}r} |W(x,y)-W(x_{0},y)|^{q}dy \Big)^{1\over q}\leq C.$$
We follow the lines of Proposition 12 of [@BBHV]. As usual, we may assume $q>{n\over 2}$. Let $x,x_{0},y\in\Omega_{T}$ be such that $d(x,x_{0})\leq r$ and $d(y,x_{0})\ge 2r$, so that in particular $d(x_{0},y)\simeq d(x,y)$.
The first step is to compute $$\begin{aligned}
|W(x,y) - & W(x_{0},y)|
\leq V(y') \Bigg( {1\over{ \big( 1 + {{d(x_{0},y)}\over{\rho(y')}}\big)^{k} }} \bigg| {1\over{d(x,y)^{n}}}-{1\over{d(x_{0},y)^{n}}} \bigg| + \\
& + {1\over{d(x,y)^{n}}} \bigg| {1\over{ \big( 1 + {{d(x,y)}\over{\rho(y')}}\big)^{k} }}-{1\over{ \big( 1 + {{d(x_{0},y)}\over{\rho(y')}}\big)^{k} }} \bigg| \Bigg) = A+B.\end{aligned}$$
We note that by the mean value Theorem
$$\bigg| {1\over{d(x,y)^{n}}}-{1\over{d(x_{0},y)^{n}}} \bigg| \leq C {{d(x,x_0)}\over{d(x_{0},y)^{n+1}}},$$
Also
$$\begin{aligned}
\bigg| {1\over{ \big( 1 + {{d(x,y)}\over{\rho(y')}}\big)^{k} }}-{1\over{ \big( 1 + {{d(x_{0},y)}\over{\rho(y')}}\big)^{k} }} \bigg|
& \leq C {k\over{\rho(y')}} {{d(x,x_{0})}\over{ \big( 1+{{d(x_{0},y)}\over{\rho(y')}} \big)^{k+1}}}\\
& \leq C d(x_{0},y)^{-1} {{d(x,x_{0})}\over{ \big( 1+{{d(x_{0},y)}\over{\rho(y')}} \big)^{k}}},\end{aligned}$$
which we obtain from applying again the mean value Theorem.
Thus, by using the fact that $d(x_{0},y)\simeq d(x,y)$, we obtain that $A$ and $B$ are [bounded]{} by $$\begin{aligned}
C V(y') {1\over{ \big( 1 + {{d(x_{0},y)}\over{\rho(y')}}\big)^{k} }} {{d(x,x_{0})}\over{d(x_{0},y)^{n+1}}}.\end{aligned}$$
The second step is to consider the balls $B_{j}=B(x_{0},2^{j}r)$, the annuli $C_{j}=\{y:2^{j}r<d(y,x_{0})\leq 2^{j+1}r\}=\overline{B_{j+1}}\backslash\overline{B_{j}}$ and the rectangles $B'_{j}\times I_{j}$, where $B'_{j}=\{y'\in{\mathbb{R}}^{n}:|y'- x_0' |\leq 2^{j}r \}$ and $I_{j}=\{s\in{\mathbb{R}}:|s-t_0|\leq (2^{j}r)^2\}$. Thus, $C_{j}\subset B'_{j+1}\times I_{j+1}$.\
In view of \[rhoxrhoy\] replacing $\rho(y')$ with $ \rho(x')$ (possibly with a change of the integer $k$), we have that $$\begin{aligned}
\Big( \int _{C_{j}} A^{q} dy \Big)^{1\over q}
& \leq C {1\over{\big( 1 + {{2^{j}r}\over{\rho(x')}}\big)^{k}}} {r\over{(2^{j}r)^{n+1}}}
\Big( \int _{C_{j}} V(y')^{q} dy \Big)^{1\over q} \\
& \leq C {1\over{\big( 1 + {{2^{j}r}\over{\rho(x')}}\big)^{k}}} {r\over{(2^{j}r)^{n+1}}}
\Big( \int _{I_{j+1}} ds \int _{B'_{j+1}} V^{q}(y') dy' \Big)^{1\over q} \\
& \leq C {1\over{\big( 1 + {{2^{j}r}\over{\rho(x')}}\big)^{k}}} {r\over{(2^{j}r)^{n+1}}} (2^{j+1}r)^{{n+2}\over q} \big( {1\over{|B'_{j+1}|}} \int _{B'_{j+1}} V^{q}(y') dy' \big)^{1\over q} \\
&\leq C {1\over{\big( 1 + {{2^{j}r}\over{\rho(x')}}\big)^{k}}} {r\over{(2^{j}r)^{n+1}}} (2^{j}r)^{{n+2}\over q} {1\over{(2^{j}r)^{n}}} \int _{B'_{j+1}} V(y') dy',\end{aligned}$$ where in the last inequality we used the reverse Hölder condition on the potential $V$.
The third step is to add up and split, as follows: $$\begin{aligned}
\sum _{j=0}^{\infty} j (2^{j}r)^{{n+2}\over{q'}}
&
\Big(\int _{C_{j}} A^{q} dy \Big)^{1\over q}\\
& \leq C \sum _{j=0}^{\infty} j (2^{j}r)^{n+2} {1\over{\big( 1 + {{2^{j}r}\over{\rho(x')}} \big)^{k}}} {r\over{(2^{j}r)^{n+1}}} {1\over{(2^{j}r)^{n}}} \int _{B'_{j+1}} V(y')dy' \\
& \leq C \sum _{j:2^{j}r<\rho(x')} \big( \dots \big)
+ C \sum _{j:2^{j}r\ge\rho(x')} \big( \dots \big) = A_{I}+A_{II}.\end{aligned}$$
Therefore, $$\begin{aligned}
A_{I}
& \le
C \sum _{j:2^{j}r<\rho(x')} j (2^{j}r)^{n+2} {r\over{(2^{j}r)^{n+1}}} {1\over{(2^{j+1}r)^{n}}} \int _{B'_{j+1}} V(y')dy' \\
& \leq C \sum _{j:2^{j}r<\rho(x')} j (2^{j}r)^{n+2} {r\over{(2^{j}r)^{n+1}}} \Big( {{\rho(x')}\over{2^{j}r}} \Big)^{n\over q} {1\over{\rho(x')^{n}}} \int _{B(x',\rho(x'))} V(y')dy',\end{aligned}$$ because of equation \[13\]. Finally, by definition of $\rho$ (see \[rho\]) and since $q>{n\over 2}$ we conclude that $A_{I}$ is finite: $$\begin{aligned}
A_{I} & \leq C \sum _{j:2^{j}r<\rho(x')} {j\over{2^{j}}} \Big( {{\rho(x')}\over{2^{j}r}} \Big)^{{n\over q}-2}
\leq C \sum _{j:2^{j}r<\rho(x')} {j\over{2^{j}}}.\end{aligned}$$ Similarly, by using the doubling property of the measure $V(y')dy'$, equation \[13\] and definition of $\rho$, we have that $$\begin{aligned}
A_{II} &\leq C \sum _{j:2^{j}r\ge\rho(x')} {j\over 2^{j}} (2^{j}r)^{2} \Big({{\rho(x')}\over{2^{j}r}} \Big)^{k} {1\over{(2^{j}r)^{n}}} \int _{B'_{j+1}} V(y')dy' \\
&\leq C \sum _{j:2^{j}r\ge\rho(x')} {j\over 2^{j}} (2^{j}r)^{2} \Big({{\rho(x')}\over{2^{j}r}} \Big)^{k} {1\over{(2^{j}r)^{n}}} \Big({{2^{j}r}\over{\rho(x')}} \Big)^{\alpha} \int _{B(x',\rho(x'))} V(y')dy' \\
&\leq C \sum _{j:2^{j}r\ge\rho(x')} {j\over 2^{j}} \Big({{\rho(x')}\over{2^{j}r}} \Big)^{k-\alpha+n-2},\end{aligned}$$ which is finite for $k$ large enough, and the proof of the Lemma follows.
\[lemm:covering.Omega\] Let $(X,d)$ be a metric space with the weak homogeneity property (hence separable) and let $\Lambda$ be a nonempty open proper subset of $X$. Let $0 <r_0 < \beta /10$. Then, there exist two families of balls, denoted by $\mathcal G_r, \tilde{\mathcal G}_r$, such that $$\mathcal W_{r_0 }=\mathcal G_{r_0 }\cup \tilde{\mathcal G}_{r_0 }=\{B_i\}$$ is a covering of $\Lambda$ by balls of $\mathcal F_\beta $ with the following properties:
1. If $B=B(x_B,s_B)\in \tilde{\mathcal G}_{r_{0}}$, then $10B\in \mathcal{F}_\beta $, $d(x_B, \Lambda^{C} )\leq 1 $ and $\tfrac{1}{2} r_0 d(x_B, \Lambda^{C} )$ $\leq s_B\leq r_0 d(x_B, \Lambda^{C} )$.
2. If $B\in\mathcal G_{r_0 }$, then $B \equiv B(x_B,r_0 )$ , $10B\in \mathcal F_\beta $ and $d(x_B, \Lambda^{C} )> 1 $.
3. If $B,B'\in \mathcal W_{r_0 }$ and $B\cap B'\neq \emptyset $, then: $B\subset 5B'$ and $B'\subset 5B$.
4. There exists $ M> 0$ such that $\sum _{B\in\mathcal W_{r_0 }} \chi_B(x)\leq M$.
Let $ r_0<\beta /10$ and define $$\Lambda _k =\{x\in \Lambda : 2^{-k}<d(x, \Lambda^{C})\leq 2^{-k+1}\}$$ for $k>0$, and $$\Lambda _0=\{x\in\Lambda : 1<d(x,\Lambda^{C}) )\}.$$ We have that $\Lambda =\bigcup _{i=0}^\infty \Lambda_k$. For each $k\geq 0$ let us choose a maximal family of points $\{x_{ik}\}_{i=1}^\infty$ in $ \Lambda_k$ such that $ d(x_{ik},x_{ij})> r_0 2^{ -k}.$ For each $k\geq 0$ let us consider the family of balls $\{B(x_{ik},r_0 2^{-k})\}$. This family clearly verifies that $ \Lambda_k \subset \bigcup _{i=1}^\infty B(x_{ik}, r_0 2^{-k})$, and
$$\Lambda = \bigcup _{k=0}^\infty \bigcup _{i=1}^\infty B(x_{ik}, r_0 2^{-k}).$$
Let us consider for each $k\geq1$ a ball $ B_{ik}= B(x_{ik}, r_{B_{ik}})$ such that $r_{B_{ i_k}}=r_{0} 2^{-k}$. We can easily see that $\{B_{ik}\}$ is a covering of $\Lambda \setminus \Lambda _0$ such that $10B_{ik}\in \mathcal F_\beta$ [and]{} $$\dfrac 12 r_0 d(x_{ik},\Lambda^{C} ) < r_{B_{ik}} \leq r_0 d(x_{ik}, \Lambda^{C}).$$
For $k=0$ let us consider the family $\{B_{i0}\}=\{B(x_{i0},r_{0})\}_{i=1}^\infty$. We have that $B_{i0}\in \mathcal F_\beta $ and $10 B_{i0}\in\mathcal F_\beta $. If $B_{ik}\cap B_{jl}\neq \emptyset $,with $k,l\geq 0$, then: $$B_{jl}\subset 5B_{ik}.$$ Indeed, if $z\in B_{ik}\cap B_{jl}$, then $$2^{-k} \leq d(x_{ik},\Lambda^{C}) \leq d(x_{jl},\Lambda^{C}) + d(x_{jl},z) + d(z,x_{ik})\leq 2^{-l+1} + r_{0}2^{-l}+r_{0}2^{k},$$ from where $2^{-k+l}\leq {{2+r_{0}}\over{1-r_{0}}} < 3$, and by simmetry, also $2^{-l+k}< 3$, which leads us to $|k-l|\leq 1$. The worst possible situation is $k=l+1$. Let us consider $y\in B_{jl}$, then $$d(y,x_{ik})\leq d(y,x_{jl})+d(x_{jl},z)+d(z,x_{ik}) < r_{0}2^{-l}+r_{0}2^{-l}+r_{0}^{-k}=5r_{ik}.$$
Thus, from the above computations, we can conclude that property 3 holds and $x_{jl}$ is in the same band $\Lambda_{k}$ or in a neighbour band $\Lambda_{j}$. Hence, the sets $\{x_{jl}\in\Lambda_{j}: B_{ik}\cap B_{jl}\neq \emptyset \}$, with $|k-j|\leq 1$, have at most finite cardinal which does not depend on $B_{ik}$. Then, there exists $M$, independent of $r_{0}$ and $\beta$, such that $$\sum_{k=0}^\infty\sum_{i=1}^\infty \chi_{B_{ik}}(x)\leq M.$$
Let us state the following Lemma, which is often used in the paper without mentioning it.
\[lemm:tecnico\] Let $(X,d)$ be a metric space and let $\Lambda$ be a nonempty open proper subset of $X$. Let $0<\beta<1$ and $\alpha >1$. Given $B_{0}=B(z_{0},r_{0})$ such that $\alpha B_{0}\in\mathcal{F}_{\beta}$ and any $x\in B_{0}$ we have that $r_{0} < \frac{\beta}{\alpha -\beta} d(x,\Lambda^{C}) $ and $B\big(x,(\alpha - \beta)r_0 \big)\in\mathcal{F}_{\beta}$.
Since $\alpha B_{0}\in\mathcal{F}_{\beta}$, we have that $$r_{0} <{\frac\beta\alpha} d\big(z_{0},\Lambda^{C}\big)<{\frac\beta\alpha}(d(x,z_{0})+d(x,\Lambda^{C}))<{\frac\beta\alpha}r_{0}+{\frac\beta\alpha}d(x,\Lambda^{C}),$$
therefore $\big( 1-{\frac\beta\alpha} \big)r_{0}<{\frac\beta\alpha}d(x,\Lambda^{C})$, and finally $$(\alpha -\beta) r_{0} < \beta d(x,\Lambda^{C}).$$
We also need the following version of the Fefferman-Stein inequality on spaces of homogeneous type:
\[lemm:FS-PS.tipo.homog\] Let $(X,d,\mu)$ be a space of homogeneous type regular in measure, such that $\mu(X)<\infty$. Let $f$ be a positive function in $L^{\infty}$ with bounded support and $w\in A_{\infty}$. Then, for every $p$, $1<p<\infty$, there exists a positive constant $C=C([w]_{A_{\infty}})$ such that if $\|M_{X}f\|_{L^{p}(w)}<+\infty$, then $$\|M_{X}f\|_{L^{p}(w)}^{p} \leq C \|M^{\sharp}_{X}f\|_{L^{p}(w)}^{p},$$ where $$\begin{aligned}
M_{X}f(x)
&=\sup _{x\in P \in {F( X)} } {1\over{\mu(P\cap X|)}}\int _{P \cap X}|f(y)| d\mu(y), \\
M^{\sharp}_{X}f(x)
&=\sup _{x\in P \in {F( B)} } {1\over{\mu(P \cap X)}}\int _{P \cap X}|f(y) - f_{P\cap X}| d\mu(y)+ \frac{1}{\mu(X)}\int_{X} f(y) d\mu(y),\end{aligned}$$ with $${F( B)}= \{B(x_{B},r_{B}) : x_B \in X, r_{B} >0 \}.$$
Previous results for the proof of the Theorem \[thm:principal\] {#previous}
===============================================================
In order to prove Theorem \[thm:principal\] we will need the following results:
\[thm:bound.D2u\] Under assumptions (1) and (2), for any $p\in(1,\infty)$ and $w\in A_{p,\text{loc}}(\Omega )$, there exist $C$ and $r_0> 0$ such that for any ball $B_{0}= B(x_0, r_0)$ in $\Omega$ with $10 B_{0}\in\mathcal{F}_{\beta}$ and any $u\in W^{2,p}_0(B_0)$ the following inequality holds $$\|D^2 u\|_{L_w^p(B_0)}\leq C\|Au\|_{L^p_w(B_0)}.$$
The proof follows the same lines of the proof of Lemma 4.1 in [@CFL1], which makes use of expansion into spherical harmonics on the unit sphere in ${\mathbb{R}}^{n}$. After that, all is reduced to obtain $L^p$- boundedness ofón-Zygmund operator $T$ and its conmutator on a ball $B$ contained in $\Omega$ (see Theorems 2.10, 2.11 and the representation formula (3.1) in this paper). We can look at the operator $T$ and its conmutator $[T,b]$ acting on functions defined over the space of homogeneous type $B$ equipped with the Euclidean metric and the restriction of Lebesgue measure. Also, it is easy to check that the weight $w\chi_{B}$ is in $A_p(B)$, provided $w$ belongs to $A_{p,\textit{loc}}(\Omega)$, since $B$ has been chosen such that $10B\in\mathcal{F}_{\beta}$. By the weighted theory of singular integrals and conmutators on spaces of homogeneous type (see for instance [@PS]), applied to our operator the result follows.
\[thm:bound.epsilon\] Let $1<p<\infty$ and $w\in A_{p,\text{loc}}(\Omega)$. For any function $u\in W^{k,p}_{\delta,w}(\Omega)$, and any $j$, $1\leq j\leq k-1$, and $\gamma$ such that $|\gamma|=j$, we have $$\|\delta ^jD^\gamma u\|_{L^p_w(\Omega )}\leq C(\epsilon ^{-j}\|u\|_{L^p_w(\Omega )}+\epsilon ^{k-j}\|\delta ^k D^ku\|_{L^p_w(\Omega )}),$$ for any $0<\epsilon<1$ and $C$ independent of $u$ and $\epsilon$, with $\delta(x)=\min\{1 , d(x,\Omega)^{C}\}$.
The main Theorem of this section is the following:
\[thm:potencial\] Let $a_{ij}\in VMO$, for $i,j=1,\dots,n$, $V\in RH_q$ with $1< p\leq q$, and $w\in A_{{{q-1}\over{q-p}}p,\text{loc}}$. Then there exist positive constants $C$ and $r_{0}$ such that for any ball $B_{0}= B(z_0, r_0)$ in $\Omega$ with $10 B_{0}\in\mathcal{F}_{\beta}$ and any $u\in C^{\infty}_{0}(B_0)$, we have that $$\|Vu\|_{{L^p_w}(B_0)} \leq C\|Lu\|_{L^p_w(B_0)}.$$
For $z_{0}\in\Omega$ pick a ball $B_{0}:=B(z_{0},r_{0})$ with $r_{0}$ to be chosen later. We follow the argument from [@BBHV]: let $x_{0}\in B_{0}$ and fix the coefficients of $A$ at $x_{0}$, namely $a_{ij}(x_{0})$, to obtain the operator
$$L_{0}u=-\sum _{i,j=1}^{n} a_{ij}(x_{0})
u_{x_{i} x_{j}} + Vu = A_{0}u+Vu.$$
Rewrite the operator $L_{0}$ in divergence form:
$$L_{0}u=-\bigg(\sum _{i,j=1}^{n} a_{ij}(x_{0}) u_{x_{i}}\bigg)_{x_{j}} + Vu.$$
From proposition 4.9 of [@D] we know that the operator $L_{0}$ has a fundamental solution $\Gamma(x_{0};x,y)$ which satisfies that for every positive integer $k$ there exists a constant $C_{k}$, independent of $x_{0}$, such that $$\begin{aligned}
\label{cota.Gamma} \Gamma(x_{0};x,y) & \leq C_{k} {1\over {\big( 1+{{|x-y|}\over{\rho(x)}} \big)^{k} }} {1\over{|x-y|^{n-2}}},\end{aligned}$$ where $\rho(x)$ is the critical radius (recall section \[prelim:definitions\]).
Thus, for any $u\in C^{\infty}_{0}(B_{0})$, $x\in B_{0}$, $$\begin{aligned}
u(x) = & \int \Gamma(x_{0};x,y) L_{0}u(y)dy= \\
= & \int \Gamma(x_{0};x,y)Lu(y)dy + \int \Gamma(x_{0};x,y) [A_{0}u(y)-Au(y)] dy.\\\end{aligned}$$
Now if we let $x_{0}=x$, we obtain $$\begin{aligned}
\label{u.defreeze}
u(x) = & \int \Gamma(x;x,y) Lu(y) dy + \sum _{i,j=1}^{n} \int \Gamma(x;x,y) [a_{ij}(y)-a_{ij}(x)] u_{x_{i}x_{j}}(y) dy.\end{aligned}$$ Then the following pointwise bound holds for all $k\in{\mathbb{N}}$, $x\in B_{0}$, $$\begin{aligned}
\label{Vu}
|V(x)u(x)|
& \leq C_{k} V(x) \int _{B_{0}} {1\over {\big( 1+{{|x-y|}\over{\rho(x)}} \big)^{k} }} {1\over{|x-y|^{n-2}}}
\Big( |Lu(y)| + \\
\notag
& \qquad \qquad \qquad \qquad \qquad
+ \sum _{i,j=1}^{n} |a_{ij}(y)-a_{ij}(x)|
|u_{x_{i}x_{j}}(y)| \Big) dy.\end{aligned}$$
Next let us rewrite (\[Vu\]) as $$\begin{aligned}
\label{9}
|V(x)u(x)| \leq C_{k} S_{k}(|Lu|)(x)+\sum _{i,j=1}^{n} S_{k,a_{ij}}(|u_{x_{i}x_{j}}|)(x),\end{aligned}$$ where $S_{k}$ and $S_{k,a}$ are the integral operators defined as $$\begin{aligned}
\label{Sk}
S_{k}f(x) = & V(x) \int {1\over {\big( 1+{{|x-y|}\over{\rho(x)}} \big)^{k} }} {1\over{|x-y|^{n-2}}} f(y) dy,\end{aligned}$$ and $$\begin{aligned}
\label{Ska}
S_{k,a}f(x) = & V(x) \int {1\over {\big( 1+{{|x-y|}\over{\rho(x)}} \big)^{k} }} {1\over{|x-y|^{n-2}}} |a(y)-a(x)| f(y) dy,\end{aligned}$$ with $a\in L^{\infty}\cap VMO({\mathbb{R}}^{n})$, $k\in{\mathbb{N}}$.
We will prove in Theorem \[thm:Sk.sin.conmutador\] below that for all $p\in (1,q]$ and $k$ large enough, $$\begin{aligned}
\label{10}
\|S_{k}f\|_{L^{p}_{w}(B_{0})} \leq C \|f\|_{L^{p}_{w}(B_{0})}.\end{aligned}$$ Also, we will prove in Theorem \[thm:Ska.con.conmutador\] below that for each $\epsilon >0$ there exists $r_{0}>0$ depending on the VMO-modulus of the function $a$ such that $$\begin{aligned}
\label{11}
\|S_{k,a}f\|_{L^{p}_{w}(B_{0})} \leq \epsilon \|f\|_{L^{p}_{w}(B_{0})}.\end{aligned}$$
Then, by (\[9\]), (\[10\]), (\[11\]) and Theorem \[thm:bound.D2u\] we have that for any $u\in C_{0}^{\infty}(B_{0})$ with $r_{0}$ small enough, $$\begin{aligned}
\|Vu\|_{L^{p}_{w}(B_{0})} & \leq C \|Lu\|_{L^{p}_{w}(B_{0})} + \epsilon \|u_{x_{i}x_{j}}\|_{L^{p}_{w}(B_{0})} \leq C \|Lu\|_{L^{p}_{w}(B_{0})} + C \epsilon \|Au\|_{L^{p}_{w}(B_{0})} \\
& \leq (C+C\epsilon) \|Lu\|_{L^{p}_{w}(B_{0})} + C \epsilon \|Vu\|_{L^{p}_{w}(B_{0})},\end{aligned}$$ and Theorem \[thm:potencial\] follows.
Statement and proof of Theorems \[thm:Sk.sin.conmutador\] and \[thm:Ska.con.conmutador\]:
-----------------------------------------------------------------------------------------
Following the lines of [@BBHV], let us also consider the operators defined in $\Omega $
$$\begin{aligned}
S^{\ast}_{k}f(x)
&=\int {{V(y)}\over{\big( 1+ {{|x-y|}\over{\rho(y)}}\big)^{k}}} {1\over{|x-y|^{n-2}}} f(y)dy, \qquad x\in\Omega. \qquad \mbox{ and}\\
S^{\ast}_{k,a}f(x) &= \int {{V(y)}\over{\big( 1+ {{|x-y|}\over{\rho(y)}}\big)^{k}}} {1\over{|x-y|^{n-2}}} |a(y)-a(x)| f(y)dy,\end{aligned}$$
for each positive integer $k$ and $a\in VMO$. These operators are the adjoint of the integral operator $S_{k}$ and $S_{k,a}$, given in and respectively.
\[thm:Sk.sin.conmutador\] Let $B_{0}$ be a ball in $\mathcal{F}_{\beta}$ such that $10 B_{0}\in\mathcal{F}_{\beta}$. Then for $k$ large enough and $p \in [1,q]$, the operator $S_{k}$ is bounded on $L^{p}_{w}(B_{0})$, with $w\in A_{{{q-1}\over{q-p}}p ,\text{loc}}(\Omega)$.
It is enough to prove that the adjoint operator $S^{\ast}_{k}$ is bounded on $L^{p'}_{v}(B_{r_{0}})$, with $v = w^{-1/p-1}\in A_{p'/q',\text{loc}}(\Omega)$ for $p' \in [q',\infty]$, since $p'\over q'$ and ${{q-1}\over{q-p}}p$ are conjugate exponents. As we pointed out in section \[prelim:fundamentalsolutions\], we may replace $\rho(y)$ by $\rho(x)$ in the kernel of the operator $S^{\ast}_{k}$ (and maybe changing the integer $k$). Assume, without loss of generality, that $f\ge 0$. Also assume that $q>{n\over 2}$, which can be done because of the fact that if $V$ satisfies the $RH_{q}$ property, then $V$ satisfies the $RH_{q+\epsilon}$ property for some $\epsilon >0$.
We will prove the pointwise bound $$S^{\ast}_{k}f(x)\leq C (M_{\beta,\text{loc}}(|f|^{q'})(x))^{{1\over{q'}}} =:M_{q',\text{loc}},$$ for $x\in B_{0}$, $f\in L^{p}_{w}(B_{0})$ and $f\ge 0$. If $p>q'$ the theorem then follows by the boundedness of the local-maximal function (Theorem \[thm:acot.mxml.loc\]), and if $p=q'$ the theorem follows from the fact that $V$ satisfies the $RH_{q+\epsilon}$ property for some $\epsilon >0$.
We have that $$\begin{aligned}
S^{\ast}_{k}f(x)
& \leq C \int _{|x-y|<\rho(x)}{{V(y)}\over{\big( 1+ {{|x-y|}\over{\rho(x)}}\big)^{k}}} {1\over{|x-y|^{n-2}}} \chi_{B_{0}}(y)f(y) dy \, + \\
& \qquad
+ C \int _{|x-y|\ge\rho(x)}{{V(y)}\over{\big( 1+ {{|x-y|}\over{\rho(x)}}\big)^{k}}} {1\over{|x-y|^{n-2}}} \chi_{B_{0}}(y)f(y) dy \\
& \leq C \int _{|x-y|<\rho(x)}{{V(y)}\over{|x-y|^{n-2}}} \chi_{B_{0}}(y)f(y) dy \, + \\
& \qquad
+ C \int _{|x-y|\ge\rho(x)} \Big({{\rho(x)}\over{|x-y|}} \Big)^{k} {{V(y)}\over{|x-y|^{n-2}}} \chi_{B_{0}}(y)f(y) dy = \mathbf{A}(x)+\mathbf{B}(x).\\\end{aligned}$$
Let $x\in B_{0}=B(z_{0},r_{0})$.
Let us first study $\mathbf{A}(x)$. Denote by $B_{j}$ the balls $B_{j}=B(x,2^{-j}\rho(x))$ and by $C_{j}$ the annuli defined as $C_{j}=\{ y: 2^{-(j+1)}\rho(x)<|x-y|\leq 2^{-j}\rho(x) \}=\overline{B_{j}}\backslash \overline{B_{j+1}}$, $j\in{\mathbb{N}}_{0}$.
If $\rho(x)\leq r_{0}$ then, by the Lemma \[lemm:tecnico\] we have that $\rho(x)\leq r_{0}<{\beta\over{10-\beta}}d(x,\Omega^{C})$. Then $B(x,\rho(x))\in\mathcal{F}_{\beta}$ and we proceed as in [@BBHV]. That is, $$\begin{aligned}
\mathbf{A}(x) & \leq C \sum _{j=0}^{\infty} {1\over{(2^{-j}\rho(x))^{n-2}}}
\int _{C_{j}} V(y)f(y) dy \leq \\
& \leq C \sum _{j=0}^{\infty} (2^{-j}\rho(x))^{2}
\bigg( {1\over{|B_{j}|}} \int _{B_{j}} V(y)^{q} dy \bigg)^{1\over q}
\bigg( {1\over{|B_{j}|}} \int _{B_{j}} f(y)^{q'}dy \bigg)^{1\over q'} \\
&\leq C M_{q',\text{loc}}(f)(x)\sum _{j=0}^{\infty} (2^{-j}\rho(x))^{2}
\bigg( {1\over{|B_{j}|}} \int _{B_{j}} V(y) dy \bigg),\end{aligned}$$ by Hölder inequality, $RH_{q}$ condition and the definition of local Maximal function of exponent $q'$.
A slight modification of the argument is needed in the case $\rho(x) > r_{0}$: there exists $j_{0}\in{\mathbb{N}}_{0}$ such that $2^{-(j_{0}+1)}\rho(x)< r_{0} \leq 2^{-j_{0}}\rho(x)$. Let $y\in C_{j}$, for $j\leq j_{0}-2$. Then, $$2^{-(j+1)}\rho(x)<|x-y|\leq 2^{-j}\rho(x),$$ and also $$2r_{0}< 2^{-j_{0}+1}\rho(x)\leq 2^{-(j+1)}\rho(x),$$ from where $$2r_{0}\leq 2^{-(j+1)}\rho(x)<|x-y|\leq |x-z_{0}|+|z_{0}-y|<r_{0} + |z_{0}-y|.$$ Therefore $|z_{0}-y|>r_{0}$, and thus $B_{0}\cap C_{j}=\emptyset$ if $j\leq j_{0}-2$. Then, $$\begin{aligned}
\mathbf{A}(x)
& \leq C \sum _{j=j_{0}-1}^{\infty} {1\over{(2^{-j}\rho(x))^{n-2}}} \int _{B_{0}\cap C_{j}} V(y)f(y) \, dy \\
&\leq C \sum _{j=j_{0}-1}^{\infty} (2^{-j}\rho(x))^{2} \big( {1\over{|B_{j}|}} \int _{B_{j}} V(y)^{q} dy \big)^{1\over q} \bigg( {1\over{|B_{j}|}} \int _{B_{j}} f(y)^{q'} \, dy \bigg)^{1\over q'},\end{aligned}$$ by Hölder inequality and the fact that $C_{j}\subset \overline{B_{j}}$. Since $B_{j}=B(x,2^{-j}\rho(x))\subset B(x,4r_{0}) \subset B(z_{0},5r_{0})$ and $B(z_{0},10r_{0})\in\mathcal{F}_{\beta}$, we have that $B_{j}\in\mathcal{F}_{\beta}$, $j\ge j_{0}-1$, in view of Lemma \[lemm:tecnico\].
Then, applying the $RH_{q}$ condition on $V$, we obtain $$\mathbf{A}(x)\leq C M_{q',\text{loc}}(f)(x)\sum _{j=j_{0}-1}^{\infty} (2^{-j}\rho(x))^{2}\bigg( {1\over{|B_{j}|}} \int _{B_{j}} V(y) dy \bigg).$$ Finally, we follow the same steps as in [@BBHV] to conclude that $$\mathbf{A}(x)\leq C M_{q',\text{loc}}(f)(x),$$ namely, choose $R=\rho(x)$ and $r=2^{-j}\rho(x)$ in \[13\], and use \[14\] from section \[prelim:propertiesofV\], when needed.
Next we study $\mathbf{B}(x)$.
This time, if $\rho(x) >2r_{0}$ we have that $\mathbf{B}(x)=0$.
The other case goes as follows: now consider the balls $B_{j}=B(x,2^{j}\rho(x))$ and the annuli $C_{j}=\{y:2^{j-1}\rho(x)<|x-y|\leq 2^{j}\rho(x)\} \subset \overline{B_{j}}\backslash\overline{B_{j-1}}$, for $j\in{\mathbb{N}}_{0}$. There exists $j_{0}\in{\mathbb{N}}_{0}$ such that $2^{j_{0}-1}\rho(x)<r_{0}\leq 2^{j_{0}}\rho(x)$. Consider $y\in C_{j}$ for $j\ge j_{0}+2$. Then, $$2^{j-1}\rho(x)<|x-y|\leq 2^{j}\rho(x),$$ and since $2r_{0}\leq 2^{j_{0}+1}\rho(x)\leq 2^{j-1}\rho(x)$, we have that $$2r_{0}<|x-y|\leq |x-z_{0}|+|z_{0}-y|<r_{0}+|z_{0}-y|.$$ Therefore, $|z_{0}-y|>r_{0}$ and we conclude that $B_{0}\cap C_{j}=\emptyset$, for $j\ge j_{0}+2$. Then, $$\begin{aligned}
\mathbf{B}(x) &
\leq C \sum _{j=0}^{j_{0}+1} {{2^{-jk}}\over{(2^{j}\rho(x))^{n-2}}} \int _{B_{0}\cap C_{j}} V(y)f(y) dy \\
& \leq C \sum _{j=0}^{j_{0}+1} {{(2^{j}\rho(x))^{2}}\over{2^{jk}}} \Big( {1\over{|B_{j}|}} \int _{B_{j}} V(y)^{q}dy\Big)^{1\over q} \Big( {1\over{|B_{j}|}} \int _{B_{j}} f(y)^{q'}dy\Big)^{1\over q'},\end{aligned}$$ by Hölder inequality and the fact that $C_{j}\subset B_{j}$. Then, for $0\leq j\leq j_{0}+1$, we have that $B(x,2^{j}\rho(x))\subset B(x,4r_{0}) \subset B(z_{0},5r_{0})$. Again, since $ B(z_{0},10r_{0})\in\mathcal{F}_{\beta}$, we get $B_{j}\in\mathcal{F}_{\beta}$. Thus, from the $RH_{q}$ condition $$\mathbf{B}(x)\leq C M_{q',\text{loc}}(f)(x) \sum _{j=0}^{j_{0}+1} {{(2^{j}\rho(x))^{2}}\over{2^{jk}}} \Big( {1\over{|B_{j}|}} \int _{B_{j}} V(y)dy\Big).$$
Now we continue the proof given in [@BBHV], that is, use again \[13\] and \[14\], to conclude that
$$\mathbf{B}(x)\leq C M_{q',\text{loc}}(f)(x).$$
\[thm:Ska.con.conmutador\] Let $p \in (1,q]$ and $w\in A_{{{q-1}\over{q-p}}p,\text{loc}}(\Omega)$. Then, given $\epsilon >0$ there exist $r_{0}>0$, depending on the $VMO-$modulus of $a$, such that for any ball $B_{0}= B(z_0, r_0)$ in $\Omega$ with $10 B_{0}\in\mathcal{F}_{\beta}$, the inequality $$\begin{aligned}
{\|S_{k,a}f\|_{L^{p}_{w}(B_{0})} \leq \epsilon \|f\|_{L^{p}_{w}(B_ {0})} }\end{aligned}$$ holds for all $f\in L^{p}_{w}(B_{0})$ and $k$ large enough.
Now we can write $$S^{\ast}_{k,a}f(x) = \int |a(y)-a(x)|W(x,y)f(y)dy,$$ where $W(x,y)$ is the kernel given in Lemma \[lemm:parabolickernelh1q\] which satisfies the $H_{1}(q)$ condition, and we deduce Theorem \[thm:Ska.con.conmutador\], from the following abstract result:
\[thm:abstracto\] Let $w\in A_{p/q',\text{loc}}(\Lambda)$ with $q'<p<\infty$ and $\Lambda= \Omega$ or $\Omega_T$. Let $B_0$ be a ball in $\Lambda$ such that $10 B_{0}\in\mathcal{F}_{\beta}$. Assume that $W(x,y)$ is a non-negative kernel satisfying the $H_{1}(q)$ condition on the first variable, for some $q>1$ such that the operator $$Tf(x)=\int W(x,y) f(y) dy$$ is bounded on $L^{p}_{w}(B_{0})$. Then for $b\in BMO({\mathbb{R}}^n) $ or $BMO({\mathbb{R}}^{n+1})$ the operator“positive commutator” $$T_{b}f(x)=\int _{B_{0}} |b(x)-b(y)| W(x,y)f(y)dy$$ is bounded on $L^{p}_{w}(B_{0})$, and $$\|T_{b}f\|_{L^{p}_{w}(B_{0})}\leq C \|b\|_{BMO} \|f\|_{L^{p}_{w}(B_{0})}.$$
In view of Lemma \[lemm:FS-PS.tipo.homog\], we will prove the following pointwise inequality: for $s>q'$ there exists a constant $C>0$ independent of $b$ and $f$ such that $$\begin{aligned}
\label{18}
\ M_{B_0}^{\sharp} (T_{b}f )(x)\leq C \|b\|_{BMO} [ M_{s,\text{loc}}(Tf)(x)+M_{s,\text{loc}}(f)(x)],\end{aligned}$$ for all $x\in B_{0}$, where
$$M^{\sharp}_{B_0}f(x)= \sup _{ x\in B, x_B \in B_0} \inf _{c>0} {1\over{|B\cap B_0|}} \int _{B\cap B_0} |f(y)-c| dy + {1\over{|B_{0}|}}\int _{B_{0}}|f(y)|dy.$$
Fixed $x\in B_0$ and choose $B=B(x_{B },r_B)$ with $x\in B$ and $x_B\in B_0$. Thus $ |B| \simeq |B\cap B_{0}|$. Let $\widetilde{B}=2B=B(x_{B},2r_{B})$. From Lemma \[lemm:tecnico\] it follows that $\widetilde{B}\in \mathcal{F}_{\beta}$. Now for a positive function $f$ let us split it into the sum $f=f_{1}+f_{2}$, where $f_{1}=f\chi_{\widetilde{B}}$ and $f_{2}=f\chi_{\widetilde{B}^{C}}$.
Proceeding as in [@BBHV], we obtain the expression $$\begin{aligned}
|T_{b}f(y) - C_{B}|
& \leq |b(y)-b_{B}| \,
Tf(y) + T(|b-b_{B}|f_{1})(y) \\
& \qquad\qquad\qquad
+ \int _{B_{0}} |W(y,z)-W(x_{B},z)|
|b(z)-b_{B}| f_{2}(z) dz \\
&= \mathbf{A}(y) + \mathbf{B}(y) + \mathbf{C}(y)\end{aligned}$$ for any $y\in B$, where $c_{B}=T(|b-b_{B}|f_{2})(x_{B})=\int _{B_{0}} |b(z)-b_{B}| W(x_{B},z) f_{2}(y)dz$.
Let us first bound $\mathbf{A}(y)$. Taking average over $B\cap B_0$, for $s>q'$, $$\begin{aligned}
Av(\mathbf{A})
& = {1\over{|B\cap B_0|}} \int _{B\cap B_0} |b(y)-b_{B}|Tf(y)dy\\
& \leq C\Big( {1\over{|B|}} \int _{B} |b(y)-b_{B}|^{s'}dy \Big)^{1\over s'}
\Big( {1\over{|B|}} \int _{B} \chi_ {B_0}|Tf(y)|^{s} dy \Big)^{1\over s} \leq \\
& \leq C\|b\|_{BMO} M_{s,\text{loc}}(\chi_ {B_0} Tf)(x),\end{aligned}$$
Choose now $\gamma$ such that $s>\gamma>q'$. The computations for the average of $\mathbf{B}$ from [@BBHV] also hold in our case: $$\begin{aligned}
Av(\mathbf{B})
& \leq {C\over{|B|}}\int _{B}\chi_ {B_0}T(|b-b_{B}|f_{1})(x)dx
\leq C\Big( {1\over{|B|}}\int _{B}T(|b-b_{B}|f_{1})^{\gamma}(x)dx \Big)^{{1\over{\gamma}}} \leq \\
& \leq C \Big( {1\over{|B|}}\int _{\widetilde{B}} |b(x)-b_{B}|^{\gamma} |f_{1}(x)|^{\gamma} dx \Big)^{{1\over{\gamma}}},\end{aligned}$$ since $T$ is bounded on $L^{p}({\mathbb{R}}^n)$(see Theorem 3.1 in [@Sh2] and Theorem 5 in [@BBHV]). Then, by Hölder’s inequality, $$\begin{aligned}
Av(\mathbf{B}) & \leq C \bigg( {1\over{|\widetilde{B}|}} \int _{\widetilde{B}} |f(x)|^{s} dx \bigg)^{1\over s}
\bigg( {1\over{|B|}} \int _{\widetilde{B}} |b(x)-b_{B}|^{\gamma ( {s\over{\gamma}} )'} \bigg)^{1\over{\gamma ( {s\over{\gamma}} )'}} \leq \\
& \leq C \bigg( {1\over{|\widetilde{B}|}} \int _{\widetilde{B}} |f(x)|^{s} dx \bigg)^{1\over s} \bigg[ \bigg( {1\over{|\widetilde{B}|}} \int _{\widetilde{B}} |b(x)-b_{\widetilde{B}}|^{\gamma ( {s\over{\gamma}} )'} \bigg)^{1\over{\gamma ( {s\over{\gamma}} )'}} + |b_{B}-b_{\widetilde{B}}| \bigg] \leq \\
& \leq C \|b\|_{BMO} M_{s,\text{loc}}(f)(x),\end{aligned}$$ because $|b_{B}-b_{\widetilde{B}}|\leq C \|b\|_{BMO}$ and the John-Nirenberg inequality.
Next we choose $\gamma$ such that ${1\over\gamma}+{1\over q} +{1\over s} =1$, and we define the balls $B_{j}=B(x_{B},2^{j}r)$ and the annuli $C_{j}=\{z:2^{j-1}r<|x_{B}-z|\leq 2^{j}r\}$. Like in the proof of theorem \[thm:Sk.sin.conmutador\], there exists $j_{0}\in{\mathbb{N}}_{0}$ such that $C_{j_{0}}\cap B_{0} \neq \emptyset$ and $C_{j_{0}+1}\cap B_{0} = \emptyset$, then by Lemma \[lemm:tecnico\], we have that $B_{j}\in\mathcal{F}_{\beta}$ for $j\leq j_{0}$. Then, for any $y\in B$, we have that $$\begin{aligned}
\mathbf{C}(y)
& = \int _{\widetilde{B}^{C}\cap B_{0}}
|b(z)-b_{B}|
|W(y,z)-W(x_{B},z)| f(z) dz \\
& \leq \sum _{j=2}^{j_{0}}
\int _{C_{j}\cap B_{0}} |b(z)-b_{B}|
|W(y,z)-W(x_{B},z)| f(z)dz \leq \\
& \leq C \sum _{j=2}^{j_{0}}
\bigg( {1\over{|B_{j}|}} \int _{B_{j}} |b(z)-b_{B}|^{\gamma} dz \bigg)^{1\over{\gamma}}
\\
&\qquad\qquad\quad
\bigg( {1\over{|B_{j}|}} \int _{C_{j}} |W(y,z)-W (x_{B},z)|^{q}d \bigg)^{1\over q} \bigg( {1\over{|B_{j}|}} \int _{B_{j}} |\chi_{B_{0}}f(z)|^{s} dz \bigg)^{1\over s} \leq \\
&\leq C \sum _{j=2}^{j_{0}} |B_j| \bigg[ \bigg( {1\over{|B_{j}|}} \int _{B_{j}} |b(z)-b_{B_{j}}|^{\gamma} dz \bigg)^{1\over{\gamma}} + |b_{B}-b_{B_{j}}| \bigg]
\\
&\qquad\qquad\quad
\bigg( {1\over{|B_{j}|}} \int _{C_{j}} |W(y,z)-W(x_{B},z)|^{q}dz \bigg)^{1\over q} M_{s,\text{loc}}(f)(x) \leq \\
& \leq C \|b\|_{BMO} M_{s,\text{loc}}(f)(x) \sum _{j=2}^{\infty} (2^{j}r)^{n\over{q'}}j \bigg( \int _{C_{j}} |W(y,z)-W(x_{B},z)|^{q}dz \bigg)^{1\over q} \leq \\
&\leq C \|b\|_{BMO} M_{s,\text{loc}}(f)(x),\end{aligned}$$ because of the $H_{1}(q)$ condition, the John-Nirenberg inequality and the fact that $|b_{B}-b_{B_{j}}|\leq Cj\|b\|_{BMO}$. Then putting together all the above estimates, we get $$\sup _{\substack{x\in B \\ x_B \in B_0}}
\inf _{c>0}
{1\over{|B\cap B_0|}}
\int _{B\cap B_0} |f(y)-c| \, dy
\leq C \|b\|_{BMO} \Big(M_{s,\text{loc}}(f)(x) + M_{s,\text{loc}}(Tf)(x)\Big).$$ On the other hand, proceeding as above we also have $$\begin{aligned}
{1\over{|B_{0}|}}\int _{B_{0}}|T_{b}f(y)|dy
&\leq {1\over{|B_{0}|}}
\int _{B_{0}}(|b(y)-b_{B_0}|Tf(y) + T(|b-b_{B_0}|f)(y)) \, dy\\
& \leq C \|b\|_{BMO} \Big(M_{s,\text{loc}}(\chi_{B_0} Tf)(x) + M_{s,\text{loc}}(f)(x)\Big)\end{aligned}$$
Thus we obtain \[18\], which together with Lemma \[lemm:FS-PS.tipo.homog\] and Theorem \[thm:acot.mxml.loc\] imply the Theorem.
By duality, we prove the theorem for the adjoint operator $S^{\ast}_{k,a}$ with $v = w^{-1/p-1}\in A_{p'/q',\text{loc}}(\Omega)$ for $p' \in [q',\infty)$.
Applying Theorem \[thm:abstracto\] to the operator $S^{\ast}_{k,a}$ for $k$ large enough we get that if $q'<p'<\infty$, $$\|S^{\ast}_{k,a}f\|_{L^{p'}_{v}(B_{0})} \leq C \|a\|_{BMO} \|f\|_{L^{p'}_{v}(B_{0})},$$ and if $p'=q'$ we use again that $V\in RH_{q+\epsilon}$.
Since $a\in VMO({\mathbb{R}}^{n} )$, there exists a bounded uniformly continuous function $\phi$ in ${\mathbb{R}}^{n}$ such that\
$\|a-\phi\|_{BMO}<\epsilon$. Also, for $z_{0}\in \Omega$ and $r_0 >0$ there exists a uniformly continuous function $\psi$ such that $ \psi = \phi$ in $B _0 =B(z_0,r_0)$ and $$\|\psi\|_{BMO}\leq \omega_\phi (2r_0),$$ where $\omega_\phi (2r_0)$ denote the modulus of continuity of $\phi$ (see [@CFL1]). Choosing $r_0$ small enough, for all $f \in L^{p}_{v}(B_{0})$, we have $$\begin{aligned}
\|S^{\ast}_{k,a}f\|_{L^{p'}_{v}(B_{0})}
&\leq \|S^{\ast}_{k,a-\phi}f\|_{L^{p'}_{v}(B_{0})} + \|S^{\ast}_{k,\phi}f\|_{L^{p'}_{v}(B_{0})}
\\
&=
\|S^{\ast}_{k,a-\phi}f\|_{L^{p'}_{v}(B_{0})} + \|S^{\ast}_{k,\psi}f\|_{L^{p'}_{v}(B_{0})}\\
& \leq C \|a-\phi\|_{BMO} \|f\|_{L^{p'}_{v}(B_{0})} + C \|\psi\|_{BMO} \|f\|_{L^{p'}_{v}(B_{0})} \\
& \leq C \epsilon \|f\|_{L^{p'}_{v}(B_{0})},\end{aligned}$$ thus, the Theorem follows.
Previous results for the proof of the Theorem \[thm:principalP\] {#previousP}
=================================================================
We now present the parabolic-interpolation Theorem, which makes use of the Theorem \[thm:acot.mxml.loc\].
\[thm:bound.epsilonP\] Let $1<p<\infty$ and $w\in A_{p,\text{loc}}(\Omega_T)$. For any function $u\in W^{k,p}_{\delta,w}(\Omega_T)$, any $j$, $1\leq j\leq k-1$, and $\gamma$ such that $|\gamma|=j$, we have that $$\label{interin}
\|\delta ^jD^\gamma u\|_{L^p_w(\Omega_T )}\leq C(\epsilon ^{-j}\|u\|_{L^p_w(\Omega_T )}+\epsilon ^{k-j}\|\delta ^k D^ku\|_{L^p_w(\Omega_T )}).$$ for any $0<\epsilon<1$ and $C$ independent of $u$ and $\epsilon$ with $\delta (x',t)=\min\{1,d((x',t),\Omega_{T}^C )\}$, where $D^\gamma$ denotes the derivative with respect to the first variable.
The proof follows the same lines of the proof of Theorem \[thm:bound.epsilon\] of [@HSV] with appropriate changes. We include it for completeness. We consider the following Sobolev’s integral representation (see [@B]): $$|D^\gamma v(x',s)|\leq C\bigg(\sigma^{-n-j}\int_{B(x',\sigma)}|v(y',s)|\,+\int_{B(x',\sigma)}\frac{|D^k v(y',s)|}{|x'-y'|^{n-k+j}}dy'\bigg),$$ for any $\sigma>0$, $(x',s) \in\mathbb{R}^{n}\times(0,T)$ and $v\in W_{\text{loc}}^{k,1}(\mathbb{R}^{(n+1)})$.
Let us choose a Whitney’ type covering $\mathcal W_{r_0 }$ of $\Omega_T$ with $\beta=1/2$ and $r_0<1/20$. For $P=B(x_P,r_P)\in\mathcal W_{r_0 }$, take a $\mathcal{C}_0^\infty$ function $\eta_P$ such that $\hbox{supp}(\eta_P)\subset 4P\subset\Omega_T$, $0\leq\eta_P\leq 1$, and $\eta_P\equiv 1$ on $2P$.
We apply now the above inequality to $u\eta_P$ which, by our assumptions, belongs to $W_{\text{loc}}^{k,1}(\mathbb{R}^n)$. Observe that for $(x',s)\in P$ and $\sigma\leq r_P$ we have $B((x',s),\sigma)\subset 2P$ and consequently $u\eta_P$ as well as its derivatives coincide with $u$ and its derivatives when integrated over such balls.
Therefore for $(x',s)\in P$ and $\sigma\leq r_P$, we obtain the above inequality with $v$ replaced by $u$, namely $$\begin{aligned}
\label{pwinter}
| D^\gamma u(x',s)|
&=| D^\gamma (u\eta_P)(x',s) |\\
\notag
&\leq C \sigma^{-n-j}\int_{B(x',\sigma)}|u(y',s)|dy'\,+C\int_{B(x',\sigma)}\frac{|D^k u(y',s)|}{|x'-y'|^{n-k+j}}dy' .\end{aligned}$$
Moreover, as is easy to check from the properties of the covering $\mathcal W_{r_0 }$, the balls $B(x,\sqrt{2}\sigma)$, for $x\in P$ and $\sqrt{2}\sigma\leq r_P$, belong to the family ${\mathcal{F}_\beta}$ for $\beta=1/2$. In fact, for $x\in P$, since from properties 1 and 2 of Whitney’s Lemma we get $10 P\in {\mathcal{F}_\beta}$, applying the Lemma \[lemm:tecnico\] we get $$B(x,\sqrt{2}\sigma)\subset B(x,(10-\beta)\sqrt{2}\sigma)\subset B(x,(10-\beta)r_P)\in{\mathcal{F}_\beta}.$$ Let $x=(x',t)\in P.$ Integrating in over $I_\sigma(t)= (t-\sigma^2,t+\sigma^2)$ and noticing that $B(x',\sigma)\times I_\sigma(t)\subset B(x,\sqrt{2}\sigma)\in {\mathcal{F}_\beta}, $ we get $$\begin{aligned}
\sigma^{-2}
\!\!
\int_{I_\sigma(t)}
&
| D^\gamma u(x',s) | \, ds \\
&\leq C \sigma^{-n-2-j}
\!\!\!\!\!
\iint \limits _{B(x',\sigma)\times I_\sigma(t)}
\!\!\!\!\!\!
|u|(y',s) \, dy'ds\,
+C \sigma^{-2}
\!\!\!\!\!
\iint \limits_{B(x',\sigma)\times I_\sigma(t)}
\!\!\!\!\!\!
\frac{|D^k u(y',s)|}{|x'-y'|^{n-k+j}} \, dy'ds \\
&\leq C \sigma^{-j}M_{\beta,\text{loc}}u(x',t)+ C\sigma^{-2}
\!\!\!\!\!
\iint \limits_{{B(x',\sigma)\times I_\sigma(t)}}
\!\!\!\!\!\!
\frac{|D^k u(y',s)|}{|x'-y'|^{n-k+j}}dy'ds\end{aligned}$$ for all $x= (x',t)\in P$ and $\sqrt{2}\sigma\leq r_P$.
As for the second term, splitting the integral dyadically, we obtain that is bounded by $$\label{MaximalD}
\sigma^{k-j}\sum_{i=0}^\infty 2^{i(j-k)}\,\frac{1}{\sigma^2|2^{-i}B(x'\sigma) |}\int_{I_\sigma(t)}\int_{2^{-i}B(x',\sigma)}|D^k u(y',s)|dy'ds.$$ Since for $x\in P$ and $\sqrt{2}\sigma\leq r_P$ all averages involved correspond to balls in $\mathcal{F}_{1/2}$ and $j<k$, the term in is bounded by a constant times $\sigma^{k-j}M_{\beta,\text{loc}}D^ku (x)$ for all $x\in P$.
Putting together both estimates and taking $\sqrt{2}\sigma=\varepsilon r_P$, using that $r_P\simeq \delta(x)$ for $x\in P$ and denoting $$M_{\text{loc}}^2f(x',t)
=\sup_{\substack{s\in I_\sigma(t)\\ \sigma\leq r_P }}
\frac{1}{\sigma^2}\int_{I_\sigma(t)}|f(x',s)|\; ds,$$ we obtain $$\begin{aligned}
\label{cotamloc}
|D^\gamma (u)(x',t)|
&\leq C
M_{\text{loc}}^2 (D^\gamma u)(x',t) \\
\notag
& \leq C\big((\varepsilon\delta(x))^{-j}
M_{\beta,\text{loc}}(u)(x)
+ (\varepsilon \delta(x))^{k-j}M_{\beta,\text{loc}}(D^{k}u(x)\big)\end{aligned}$$ for a.e. $ (x',t) \in P$. Since $\mathcal W_{r_0 }$ is a covering of $\Omega_T$ and the right hand side of no longer depends of $P$, we obtain that (\[cotamloc\]) holds for a.e. $x=(x',t)\in\Omega_T$.
Multiplying both sides by $\delta^j(x)$ and taking the norm in $L^p_w(\Omega_T)$, we arrive to $$\|\delta^j\,D^\gamma u \|_ {L^p_w(\Omega_T)}\leq C\bigl(\varepsilon^{-j}\|M_{\beta,\text{loc}}u\|_{L_w^p(\Omega_T)}+
\varepsilon^{k-j}\|M_{\beta,\text{loc}}(D^{k} u)\|_{L^p_{w\delta^{kp}}(\Omega_T)}\bigr).$$ Next, we observe that if the weight $w$ belongs to $A_{p,\text{loc}}(\Omega_T)$ also does $w\delta^s$, for any real number $s$. In fact, for any ball $B$ in $\mathcal{F}_{1/2}$ we have that $\delta(x)\simeq\delta(x_B)$, for any $x\in B$ so that holds provided it is satisfied by $w$.
Therefore, an application of the continuity results for $M_{\beta,\text{loc}}f$, given in Theorem \[thm:acot.mxml.loc\], leads to the interpolation inequality .
Next we state the parabolic version of Theorem \[thm:bound.D2u\].
\[thm:bound.D2uP\] Under assumptions (1) and (2), for any $p\in(1,\infty)$ and $w\in A_{p,\text{loc}}(\Omega_T )$, there exist $C$ and $r_0> 0$ such that for any ball $B_{0}= B(z_0, r_0)$ in $\Omega_T$ with $10 B_{0}\in\mathcal{F}_{\beta}$ and any $u\in W^{2,p}_0(B_0)$ the following inequalities hold
$$\begin{aligned}
\|u_{x_{i}x_{j}}\|_{L^{p}_{w}(B_{0})}
& \leq C \|A_{P}u\|_{L^{p}_{w}(B_{0})} , \\
\|u_{t}\|_{L^{p}_{w}(B_{0})}
& \leq C \|A_{P}u\|_{L^{p}_{w}(B_{0})}.
\end{aligned}$$
The proof is similar to the elliptic case, as is proved in Corollary 2.13 in [@BC], by using again expansion into spherical harmonics on the unit sphere, this time in ${\mathbb{R}}^{n+1}$. After that, all is reduced to obtain $L^p$- boundedness of a parabolic Calderón-Zygmund operator $T$ and its conmutator on a ball $B$ contained in $\Omega_T$ (see Theorems 2.12 and the representation formula (1.4) in this paper). We can look at the operator $T$ and its conmutator $[T,b]$ acting on functions defined over the space of homogeneous type $B$ equipped with the parabolic metric and the restriction of Lebesgue measure. As before, the weight $w\chi_{B}$ is in $A_p(B)$. By the weighted theory of singular integrals and conmutators on spaces of homogeneous type, (see again [@PS]), applied to our operators the result follows.
Now we focus our attention in the proofs of the main Theorem of this section, that is, the parabolic version of Theorem \[thm:potencial\].
\[thm:potencialP\] Let $a_{ij}\in VMO({\mathbb{R}}^{n+1}) $, for $i,j=1,\dots,n$, $V\in RH_q({\mathbb{R}}^{n}) $ with $1< p\leq q$, and $w\in A_{{{q-1}\over{q-p}}p,\text{loc}}(\Omega_T)$. Then there exist positive constants $C$ and $r_{0}$ such that for any ball $B_{0}= B(z_0, r_0)$ in $\Omega_T$ with $10 B_{0}\in\mathcal{F}_{\beta}$ and any $u\in C^{\infty}_{0}(B_0)$, we have that
$$\|Vu\|_{{L^p_w}(B_0)} \leq C\big\|Lu\|_{L^p_w(B_0)}.$$
For $z_{0}=(z'_{0},\tau)\in\Omega_{T}$ pick a ball $B_{0}:=B(z_{0},r_{0})$ with $r_{0}$ to be chosen later. Again we let $x_{0}\in B_{0}$ and fix the coefficients $a_{ij}(x_{0})$ to obtain the operator
$$L_{0}u=u_{t}-\sum _{i,j=1}^{n} a_{ij}(x_{0})u_{x_{i}}u_{x_{j}} + Vu = A_{0}u+Vu.$$
From [@K] we know that the fundamental solution for this operator is bounded by the expression (see section \[prelim:fundamentalsolutions\]): $$\begin{aligned}
|\Gamma(x_{0},x,y)| &\leq C_{k}{1\over{\big(1+{{d(x,y)}\over{\rho(x')}}\big)^{k}}} {1\over{d(x,y)^{n}}},\end{aligned}$$ for every $x=(x',t), y=(y',s)\in\Omega_{T}$, $t>s$, $k>0$, and for some constants $C_{k},C_{0}$ independent of $x_0$. Here again $\rho(x')$ is the critical radious.
As usual, we defreeze the coefficients to obtain \[u.defreeze\] and again the following pointwise bound holds for all $k\in{\mathbb{N}}$, $x\in B_{0}$,
$$\begin{aligned}
\label{VuP}
|V(x')u(x)|
& \leq C_{k} V(x') \int _{B_{0}} {1\over {\big( 1+{{d(x,y)}\over{\rho(x')}} \big)^{k} }} {1\over{d(x,y)^{n}}} \bigg( |Lu(y)| + \\
\notag
& \qquad\qquad\qquad\qquad
+ \sum _{i,j=1}^{n} |a_{ij}(y)-a_{ij}(x)|
|u_{x_{i}x_{j}}(y)| \bigg) dy,\end{aligned}$$
and rewrite (\[VuP\]) as $$\begin{aligned}
\label{9P}
|V(x')u(x)| \leq C_{k} S_{k}(|Lu|)(x)+\sum _{i,j=1}^{n} S_{k,a_{ij}}(|u_{x_{i}x_{j}}|)(x),\end{aligned}$$ where $S_{k}$ and $S_{k,a}$ are the integral operators defined as $$\begin{aligned}
S_{k}f(x) &= V(x') \int {1\over {\big( 1+{{d(x,y)}\over{\rho(x')}} \big)^{k} }} {1\over{d(x,y)^{n}}} f(y) dy, \qquad \mbox{ and} \\
S_{k,a}f(x)& = V(x') \int {1\over {\big( 1+{{d(x,y)}\over{\rho(x')}} \big)^{k} }} {1\over{d(x,y)^{n}}} |a(y)-a(x)| f(y) dy,\end{aligned}$$ with $a\in L^{\infty}\cap VMO({\mathbb{R}}^{n})$, $k\in{\mathbb{N}}$.
Thus, as in the elliptic case, the Theorem follows from Theorem \[thm:bound.D2uP\] and the next parabolic version of Theorems \[thm:Sk.sin.conmutador\] and \[thm:Ska.con.conmutador\].
Now we need to prove the following parabolic version of Theorem \[thm:Sk.sin.conmutador\]:
\[thm:Sk.sin.conmutadorP\] Let $B_{0}$ be a ball in $\mathcal{F}_{\beta}$ such that $10 B_{0}\in\mathcal{F}_{\beta}$. Then for $k$ large enough and $p \in [1,q]$, the operator $S_{k}$ is bounded on $L^{p}_{w}(B_{r_{0}})$, with $w\in A_{{{q-1}\over{q-p}}p ,\text{loc}}(\Omega_{T})$.
This proof is also done by duality. The remarks we made along the proof of Theorem \[thm:Sk.sin.conmutador\] also hold this time so we won’t mention them.
The adjoint operator of $S_{k}$ is $$\begin{aligned}
S^{\ast}_{k}f(x)&= \int {{V(y')}\over{\big( 1+ {{d(x,y)}\over{\rho(y')}}\big)^{k}}} {1\over{d(x,y)^{n}}} f(y)dy, \qquad x\in\Omega_{T}.\end{aligned}$$
Just like before we can split $$\begin{aligned}
S^{\ast}_{k}f(x) & \leq C \int _{d(x,y)<\rho(x')} {1\over{d(x,y)^{n}}} V(y') \chi_{B_{0}}(y)f(y) dy \, + \\
& \qquad \qquad
+ C \int _{d(x,y)\ge\rho(x')} \Big({{\rho(x')}\over{d(x,y)}} \Big)^{k} {1\over{d(x,y)^{n}}} V(y') \chi_{B_{0}}(y)f(y) dy \\
& = \mathbf{A}(x)+\mathbf{B}(x).\end{aligned}$$ We will prove the pointwise [bound]{} $$S^{\ast}_{k}f(x)\leq C M_{q',\text{loc}}(f)(x).$$
In order to study $\mathbf{A}(x)$, let $x\in B_{0}=B(z_{0},r_{0})$. Denote by $B_{j}$ the balls $B_{j}=B(x,2^{-j}\rho(x'))$, by $C_{j}$ the annuli defined as $C_{j}=\{ y: 2^{-(j+1)}\rho(x')<d(x,y)\leq 2^{-j}\rho(x) \} = \overline{B_{j}}\backslash\overline{B_{j+1}}$, and by $R_{j}$ the rectangles $R_{j}=B'_{j}\times I_{j}$ where $B'_{j}$ denotes the ball in ${\mathbb{R}}^{n}$, $B'_{j}=B(x',2^{-j}\rho(x'))$ and $I_{j}$ denotes the real ball $I_{j}=B(t,(2^{-j}\rho(x')^{2})$, $j\in{\mathbb{N}}_{0}$. We have that $C_{j}\subset B_{j} \subset R_{j}$, and let us remark that the ball measures are $|B_{j}|= c_n(2^{-j}\rho(x))^{n+2}$ and $|B'_{j}|=C_n(2^{-j}\rho(x'))^{n}$. The same steps as before prove that $$\mathbf{A}(x)\leq C M_{q',\text{loc}}(f)(x),$$ for $x\in B_{0}$, $f\in L^{p}_{w}(B_{0})$ and $f\ge 0$, where $M_{q',\text{loc}}$ denotes the local maximal function of exponent $q'$, in the parabolic setting. Indeed, if $\rho(x')\leq r_{0}$ we have that $$\begin{aligned}
\mathbf{A}(x)& \leq C \sum _{j=0}^{\infty} {{|B_{j}|}\over{(2^{-j}\rho(x'))^{n}}} \bigg( {1\over{|B'_{j}|}} \int _{B'_{j}} V(y')^{q}dy' \bigg)^{1\over q} \bigg( {1\over{|B_{j}|}} \int _{B_{j}} f(y)^{q'}dy \bigg)^{1\over{{q'}}} \\
&\leq C M_{q',\text{loc}}(f)(x) \sum _{j=0}^{\infty} (2^{-j}\rho(x'))^{2} \bigg({1\over{|B'_{j}|}}\int _{B'_{j}} V(y')dy'\bigg),\end{aligned}$$ because of the Hölder inequality, the reverse Hölder condition $V$ and the definition of local maximal function. And in the case $\rho(x') >r_{0}$, again there exists $j_{0}\in{\mathbb{N}}_{0}$ such that $C_{j}\cap B_{0}=\emptyset$ for $j\leq j_{0}+2$. The same steps as before show us that $$\mathbf{A}(x)
\leq C M_{q',\text{loc}}(f)(x) \sum _{j=j_{0}-1}^{\infty} (2^{-j}\rho(x'))^{2} \bigg({1\over{|B'_{j}|}}\int _{B'_{j}} V(y')dy'\bigg).$$ Now we use again equations and to conclude that $\mathbf{A}(x)\leq C M_{q',\text{loc}}(f)(x)$.
To study $\mathbf{B}(x)$, we consider the balls $B_{j}=B(x,2^{j}\rho(x'))$, the annuli $C_{j}=\{y:2^{j}\rho(x')<d(x,y)\leq 2^{j+1}\rho(x')\}$, and the rectangles $R_{j}=B'_{j}\times I_{j}=B(x',2^{j}\rho(x')) \times B(t,(2^{j}\rho(x'))^{2}) \subset {\mathbb{R}}^{n}\times {\mathbb{R}}$, for $j\in{\mathbb{N}}_{0}$. We have that $C_{j}\subset B_{j} \subset R_{j}$. Observe that if $\rho(x')>2r_{0}$, then $\mathbf{B}(x)=0$, thus we consider only the case $\rho(x')\leq 2r_{0}$. There exists $j_{0}\in{\mathbb{N}}_{0}$ such that $C_{j}\cap B_{0}=\emptyset$ if $j\ge j_{0}+2$. Thus we have that $$\begin{aligned}
\mathbf{B}(x)& \leq C M_{q',\text{loc}}(f)(x) \sum _{j=0}^{j_{0}+1} {{(2^{j}\rho(x'))^{2}}\over{2^{jk}}} \bigg({1\over{|B'_{j}|}}\int _{B'_{j}} V(y')dy'\bigg),\end{aligned}$$ because of the use of Hölder inequality, the reverse Hölder conditionon $V$ and the definition of local maximal function of the order $q'$. Thus, using again equations \[13\] and \[14\], $\mathbf{B}(x)\leq C M_{q',\text{loc}}(f)(x)$.
\[thm:Sk.sin.conmutadorPsin pesos\] We note that arguing in a similar way as in the proof of Theorem \[thm:Sk.sin.conmutadorP\] it can be show that the operator $S_{k}$ is bounded on $L^{p}({\mathbb{R}}^{(n+1)})$ with $w = 1$ and $p\in [1,q]$. In this case the operator is pointwisely bounded by the maximal Hardy-Littlewood function of order $q'$.
We turn now to the proof of parabolic Theorem \[thm:Ska.con.conmutador\]:
\[thm:Ska.con.conmutadorP\] Let $p \in (1,q]$ and $w\in A_{{{q-1}\over{q-p}}p,\text{loc}}(\Omega_T)$. Then, given $\epsilon >0$ there exist $r_{0}>0$, depending on the $VMO-$modulus of $a$ such that for any ball $B_{0}= B(z_0, r_0)$ in $\Omega_T$ with $10 B_{0}\in\mathcal{F}_{\beta}$ , the inequality $$\begin{aligned}
\|S_{k,a}f\|_{L^{p}_{w}(B_{0})} \leq \epsilon \|f\|_{L^{p}_{w}(B_{0})}.\end{aligned}$$ holds for all $f\in L^{p}_{w}(B_{0})$ and $k$ large enough.
This proof is also done by duality as in the proof of Theorem \[thm:Ska.con.conmutador\], and follows by Theorem \[thm:abstracto\] with $\Lambda= \Omega_T$ and $b\in BMO({\mathbb{R}}^{n+1})$.
Here, $$\begin{aligned}
S^{\ast}_{k,a}f(x) &= \int {{V(y')}\over{\big( 1+ {{d(x,y)}\over{\rho(y')}}\big)^{k}}} {1\over{d(x,y)^{n}}} |a(y)-a(x)| f(y)dy,\end{aligned}$$ for each positive integer $k$ and $a\in VMO$; and the kernel is $$w(x,y)={1\over {\big( 1+{{d(x,y)}\over{\rho(x')}} \big)^{k} }} {1\over{d(x,y)^{n}}},$$ which satisfies the $H_{1}(q)$ condition as shown in section \[prelim:lemmas\] (Lemma\[lemm:parabolickernelh1q\])
Proof of the Main Result {#mains}
========================
We are in position to proof Theorem \[thm:principal\].
Let $\mathcal W_{r_{0}}=\{B_{i}=B(x_{i},r_{i})\}$ be a covering as in Lemma \[lemm:covering.Omega\], with $r_{0}$ as in Theorems \[thm:bound.D2u\] and \[thm:potencial\] and $0<r_0 < \beta /10$. For each $B_i\in W_{r_0 }$ we consider a function $\eta_{i}$ such that the family $\{\eta_i\}_{i=1}^\infty$ satisfies
1. $\eta_i\in\mathcal C_0^\infty(2 B(x_i,r_i))$, $\eta_i\equiv 1$ in $B_i$,
2. $\|\eta_i\|_\infty\leq 1$, $\|D^\alpha \eta_i\|_\infty\leq Cr_i^{-|\alpha |}$ where $ r_i\approx d(x_i, \partial\Omega )$ if $B(x_i,r_i)\in \tilde{\mathcal G}_{r_0 }$ and $ r_i \approx 1$ when $B(x_i,r_i)\in \mathcal G_{r_0 }$,
3. $\sum _{i=1}^\infty \chi_{2B_i}(x)\leq M$.
By using Theorem \[thm:bound.D2u\], for each $i$, we get $$\begin{aligned}
\| \chi_{B_i} D^2 & (u\eta_{i}) \|^p_{L^p_w(2B_i)}\\
& \leq C \| A(u\eta_i)\|^p_{L^p_w(2 B_i)}\\
& \leq C \big(\| Au\|^p_{L^p_w(2B_i)}
+ r_i^{-1}\| Du\|^p_{L^p_w(2B_i)}
+ r_i^{-2}\|u\|^p_{L^p_w(B_i)} \big)^{p}\\
& \leq C \big(\| Au\|_{L^p_w(2B_i )}
+ r_i^{-1}\| Du\|_{L^p_w(2B_i )}
+ r_i^{-2}\|u\|_{L^p(2B_i )}\big)^{p} \\
& \leq C\big(\| Lu\|_{L^p_w( 2B_i)}
+\| Vu\|_{L^p_w(2B_i )}
+ r_i^{-1}\|Du\|_{L^p_w(2B_i )}
+ r_i^{-2}\|u\|_{L^p_w(2B_i )}\big)^p.
\end{aligned}$$
Analogously, using this time Theorem \[thm:potencial\], since $w\in A_{p,\text{loc}}(\Omega )\subset A_{\frac{q-1}{q-p}p,\text{loc}}(\Omega) $ we obtain $$\begin{aligned}
\| \chi_{B_i} V (u\eta_{i})\|^p_{L^p_w(2B_i )}
&\leq C\|L(u\eta_{i})\|^p_{L^p_w(2B_i)} \\
&\leq C \|Lu\|^p_{L^p_w(B_i)}+r_i^{-1} \|Du\|^p_{L^p_w(B_i)}+r_i^{-2} \|u\|^p_{L^p_w(B_i)}.
\end{aligned}$$
Now, we note that for $x\in B_i$ the function $\eta_i u$ coincides with $u$, and also for $x\in 2B_i$, we have $\delta (x_i)\approx r_i $ with $\delta (x_i)=\min\{1,d(x_i,{\partial}\Omega )\}$. Hence, putting together both estimates, multiplying both sides by $\delta^2$, adding over $i$, using de finite overlapping property of the covering $\{2B_i \}$ and taking the $1/p$-th power, we arrive to
$$\begin{aligned}
\|u\|_{W^{2,p}_{\delta ,w}(\Omega )}+\|\delta ^2Vu\|_{L^p_w(\Omega )}
\leq C(\|\delta ^2 Lu\|_{L^p_w(\Omega )} + \|\delta Du\|_{L^p_w(\Omega )} + \|u\|_{L^p_w(\Omega )})
.\\
\intertext{Using the interpolation Theorem \ref{thm:bound.epsilon}}
\leq C(\|\delta ^2 Lu\|_{L^p_w(\Omega )} + \epsilon \|\delta^2 D^2 u\|_{L^p_w(\Omega )})+(C+\epsilon^{-1}) \|u\|_{L^p_w(\Omega )}.
\end{aligned}$$
Finally, choosing $\epsilon $ such that $C\epsilon = 1/2$ and subtracting the term $\|\delta ^2 D^2 u\|_{L^p_w(\Omega )}$, it follows $$\|u\|_{W^{2,p}_{\delta ,w}(\Omega )}
\leq C \{\|Lu\|_{L^p_w(\Omega )}+\|u\|_{L^p_w(\Omega )}\},$$ whence the desired estimate follows.
The proof of Theorem \[thm:principalP\] is obtained by a few changes:
Just like in the previous proof, from Lemma \[lemm:covering.Omega\] applying this time to $\Gamma= \Omega_T$, we consider a covering $\mathcal{W}_{r_{0}}$ and a family $\{\eta_{i}\}$ which satisfies 1 and 3, and the following 2: $\|\eta_{i}\|_{\infty}\leq 1$, $$\begin{aligned}
\|D^{\alpha}_{x}\eta_{i}\|_{\infty} \leq C r_{i}^{-|\alpha|}, \\ \|D_{t}\eta_{i}\|_{\infty}\leq C r_{i}^{-2},
\end{aligned}$$ where $ r_i\approx d(x_i, \partial\Omega )$ if $B(x_i,r_i)\in \tilde{\mathcal G}_{r_0 }$ and $ r_i \approx 1$ when $B(x_i,r_i)\in \mathcal G_{r_0 }$.
Now for each $i$ we use theorems \[thm:bound.D2uP\] and \[thm:potencialP\] to get $$\begin{aligned}
\|\chi _{B_{i}}D_{x}^{2}(u\eta_{i})\|_{L_{w}^{p}(2B_{i})}
& \leq C \big( \| Lu\|_{L^p_w( 2B_i)}
+\| Vu\|_{L^p_w(2B_i )} + \\
& \qquad\qquad\qquad
+ r_i^{-1}\|Du\|_{L^p_w(2B_i )}
+ r_i^{-2}\|u\|_{L^p_w(2B_i )}\big), \\
\|\chi _{B_{i}}D_{t}(u\eta_{i})\|^{p}_{L_{w}^{p}(2B_{i})}
&\leq C \big( \|Lu\|_{L^p_w( 2B_i)}
+\| Vu\|_{L^p_w(2B_i )} + \\
& \qquad\qquad\qquad
+ r_i^{-1}\|Du\|_{L^p_w(2B_i )}+ r_i^{-2}\|u\|_{L^p_w(2B_i )} \big), \\
\|\chi _{B_{i}}Vu\eta_{i}\|^{p}_{L_{w}^{p}(2B_{i})} & \leq C \big( \|Lu\|_{L_{w}^{p}(2B_{i})} + r_{i}^{-1}\|D_{x}u\|_{L_{w}^{p}(2B_{i})}+ r_{i}^{-2}\|u\|_{L_{w}^{p}(2B_{i})}\big),\end{aligned}$$ then, by performing analogous operations to the previous Theorem, we obtain $$\begin{aligned}
\|u\|_{W^{2,p}_{\delta ,w} (\Omega_{T})} + \|\delta^{2}Vu\|_{L^{p}_{w}(\Omega_{T})} &\leq C\big( \|\delta^{2}Lu\|_{L^{p}_{w}(\Omega_{T})} + \|\delta D_{x}u\|_{L^{p}_{w}(\Omega_{T})} +\|u\|_{L^{p}_{w}(\Omega_{T})} \big).
\end{aligned}$$ From the interpolation Theorem \[thm:bound.epsilonP\] we have that $$\begin{aligned}
\|\delta D_{x}u\|_{L^{p}_{w}(\Omega_{T})}
& \leq C \big( \epsilon^{-1}\|u\|_{L^{p}_{w}(\Omega_{T})} + \epsilon \|\delta^{2}D^{2}_{x}u\|_{L^{p}_{w}(\Omega_{T})} \big),
\end{aligned}$$ which finally leads us to $$\begin{aligned}
\|u\|_{W^{2,p}_{\delta ,w} (\Omega_{T})} + \|\delta^{2}Vu\|_{L^{p}_{w}(\Omega_{T})} & \leq C\big( \|\delta ^2 Lu\|_{L^p_w(\Omega_{T} )}+\|u\|_{L^p_w(\Omega_{T} )}\big)
\end{aligned}$$ as we desired.
[99]{}
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|
---
abstract: 'The paper presents a novel instrumentation for rare events selection which was tested in our research of short lived super heavy elements production and detection. The instrumentation includes an active catcher multi elements system and dedicated electronics. The active catcher located in the forward hemisphere is composed of 63 scintillator detection modules. Reaction products of damped collisions between heavy ion projectiles and heavy target nuclei are implanted in the fast plastic scintillators of the active catcher modules. The acquisition system trigger delivered by logical branch of the electronics allows to record the reaction products which decay via the alpha particle emissions or spontaneous fission which take place between beam bursts. One microsecond wave form signal from FADCs contains information on heavy implanted nucleus as well as its decays.'
address: |
$^a$M. Smoluchowski Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348, Kraków, Poland\
$^b$Cyclotron Institute, Texas A&M University, USA\
$^c$University of Silesia, Katowice, Poland\
author:
- 'Z. Majka$^a$[^1], R. Płaneta$^a$, Z. Sosin$^a$, A. Wieloch$^a$[^2], K. Zelga$^a$, M. Adamczyk$^a$, K. Pelczar$^a$, M. Barbui$^b$, S. Wuenschel$^b$, K. Hagel$^b$, X. Cao$^b$, E-J. Kim$^b$, J. Natowitz$^b$, R. Wada$^b$, H. Zheng$^b$, G. Giuliani$^b$, S. Kowalski$^c$'
title: A NOVEL EXPERIMENTAL SETUP FOR RARE EVENTS SELECTION AND ITS POTENTIAL APPLICATION TO SUPER HEAVY ELEMENTS SEARCH
---
Introduction {#intro}
============
A frequent challenge for contemporary researchers in experimental physics is associated with the need to identify rare events out of the huge number of cases that are uninteresting. As examples of such investigations one can mention searches for the Higgs boson [@Aad:12:1] and neutrino-less double beta decay experiments [@Agos:17:1]. We are facing a similar problem in our searches of new super heavy elements (SHE). The question “How heavy can an atomic nucleus be?” is a fundamental problem in nuclear physics. The possible existence of island(s) of stable super heavy nuclei has been an inspiring problem in heavy ion physics for almost four decades [@Armb:00:1]. No stable or long life-times SHE (Z$>$103) has been found either in the natural environment of the Earth or in probes of meteorites or in cosmic rays. All have been produced artificially in complete fusion (CF) reactions between beam and target nuclei. Unfortunately, experimental studies have demonstrated that the cross section for SHE production in CF reactions is decreasing quite rapidly with the increasing atomic number, dropping for the synthesis of $^{277}_{112}$Cn to about 1 pb [@Hofm:96:1] and for a synthesis of element $^{294}_{118}$Og to about 0.5 pb [@Ogan:06:1]. Moreover, half-life times of the SHEs are becoming very short decreasing to 0.7 ms for oganesson ($^{294}_{118}$Og). One of the possible explanations for these results is that the newly produced elements were highly neutron deficient isotopes and they should in fact have quite short lifetimes.
From what was said above two basic conclusions can be drawn. Firstly, the CF experiments dedicated to super heavy nuclei synthesis require a large amount of the accelerator beam time, especially for nuclei with Z$>$118 (one can expect that the SHE production cross section in CF reactions will be in the region of tens of fb). As a consequence, a completely new generation of heavy ion sources is needed to supply the intensity of ion beams as high as 10$^{14}$-10$^{15}$ particles/s. This creates a serious limitation for the CF method being used so far. Secondly, available combinations of stable projectiles and targets cannot be used to produce neutron rich and longer lived SHEs in the predicted island of stability.
In this context, another approach is urgently needed to achieve further progress in super heavy nuclei production. A promising possibility is to utilize multi nucleon transfer reactions occurring in collisions between heavy nuclei. Such reaction mechanisms have been already studied over thirty years ago [@Hild:77:1; @Gagg:80:1; @Jung:78:1; @Frei:79:1; @Reid:79:1; @Krat:86:1; @Gagg:81:1; @Scha:82:1], however in both thin target and thick target irradiation experiments no new elements were observed. Although the cross-sections to produce SHE by multi nucleon transfer reactions occurring in collisions between heavy nuclei predicted theoretically are comparable with the cross-sections to the formation of SHE by a complete fusion method, the multi nucleon transfer processes in near barrier collisions of heavy and very heavy ions seem to be the only reaction mechanism (besides the multiple neutron capture process) allowing one to produce and explore neutron rich heavy nuclei including those located at the SHE island of stability [@Zagr:15:1]. Our research [@Donn:99:1; @Barb:09:1:AW; @Barb:10:1; @Majk:14:1; @Wiel:16:1], which we are conducting since year 1998, indicates that the collision process between heavy nuclei leads to the creation of very heavy systems which disintegrate through the emission of highly energetic alpha particles which are our main signature of the very heavy systems formation. The arguments that we followed when undertaking and continuing this research are shortly summarized in the next section where we briefly outline the multi nucleon transfer concept of SHE creation.
Formation of SHE is a very rare event which should be selected out of the huge number of cases that are uninteresting. In section \[apparatus\] we present a new concept and realization of a detection system and dedicated electronics for registration of rare events in high intensity beam environment. The results of test measurements are shown in section \[testresults\]. Suggestions to further developments of our experimental setup and conclusions are presented in section \[summary\].
SHE production {#production}
==============
Our experimental research of SHE production in collisions between very heavy nuclei was initiated in the late 90s of the last century [@Donn:99:1]. A heavy projectile nucleus (e.g. $^{172}$Yb, $^{197}$Au) at a few MeV/nucleon incident energy goes into contact with a fissile target nucleus (e.g. $^{232}$Th, $^{238}$U). In the initial stage of the collision, a heavy projectile initiates deformation of the target nucleus and nuclear interaction takes place between the objects for a period long enough to transfer a large amount of mass to the projectile nucleus (e.g. by fusion of projectile nucleus with one of the target nucleus fission fragments). If such scenario takes place super heavy nucleus can be produced.
Our early studies have indicated the possibility of forming in these reactions very heavy nuclei that emit high-energy alpha particles \[15-16\]. These results as well as other theoretical analyzes have motivated us to continue this research and to develop an innovative experimental approaches \[17\]. New stabilizing shell structures of very high Z nuclei as well as possible exotic shapes such as toroids and bubbles have been predicted [@Herm:79:1; @Fler:83:1; @Armb:99:1; @Grei:99:1; @Ogan:06:1; @Ogan:15:1; @Dech:99:1; @Bend:01:1; @Wong:73:1; @Najm:15:1]. Model calculations indicate existence of such stabilizing shell structures for nuclei from the islands of stability and predict that the fission barriers of these nuclei reduce the probability of spontaneous fssion [@Bend:13:1; @Kire:12:1; @Stas:13:1; @Bara:15:1; @Agbe:15:1; @Agbe:17:1; @Angh:17:1; @Giul:17:1; @Karp:12:1; @Mart:12:1; @Mark:16:1]. Thus the main modes of decay in and near these islands are predicted to be alpha and beta decay [@Kire:12:1; @Stas:13:1; @Karp:12:1; @Mart:12:1; @Mark:16:1]. Predicted fission barriers and alpha decay energies rely upon model-dependent mass surface extrapolations [@Kire:12:1; @Stas:13:1; @Bara:15:1; @Agbe:15:1; @Agbe:17:1; @Angh:17:1; @Giul:17:1; @Karp:12:1; @Mart:12:1; @Mark:16:1]. The predicted survival of heavy and super-heavy nuclei are extremely sensitive to details of these mass surface extrapolations and the location of closed shells. Uncertainties of 1 MeV in the fission barriers can lead to an order of magnitude change in the fission probabilities due to quantal effects of the barrier penetration [@Bara:15:1]. Uncertainties in level densities, temperature dependencies of fission barriers and details of the fission dynamics further complicate calculations of fission probabilities. While quantitative predictions vary widely, systematic theoretical studies indicate high survival probabilities of nuclei in and near the island of stability [@Kire:12:1; @Stas:13:1; @Bara:15:1; @Angh:17:1; @Giul:17:1; @Karp:12:1; @Mart:12:1; @Mark:16:1]. Notably, recent microscopic fission model results indicate significant increases in fission survivability compared to those of statistical models employing the same fission barriers [@Zhu:17:1; @Xia:11:1] and a strong increase in survivability is already evident in the experimental fusion cross section data for the heaviest elements [@Hami:15:1; @Hofm:15:1; @Utyo:16:1; @Ogan:17:1]. Also some calculations suggest that near the valley of stability, beta decay competes with alpha and spontaneous fission decay and that short-lifetime beta minus decay will be dominant for the more neutron rich isotopes in that region [@Karp:12:1; @Mart:12:1; @Mark:16:1]. This raises the interesting possibility that the production of neutron rich lower Z products can feed higher Z products through $\beta^{-}$ decay, increasing the effective production cross section for such higher Z products near the line of stability. Recent systematic efforts to explore the utility of multi-nucleon transfer reactions for production of new neutron-rich isotopes suggest that the experimental cross sections for projectile (target) like fragment production exceed predicted cross sections by 2-3 orders of magnitude [@Wels:17:1; @Krat:15:1]. It is interesting to ask whether a similar trend exists for heavier elements. The production of alpha particle decaying heavy nuclei produced in massive transfer reaction between heavy nuclei has been explored in our recent research [@Wuen:15:1] using an in-beam detection array composed of YAP scintillators instead fast scintillators used in our work presented in this paper. Heavy nuclei with Z as high as 116, and perhaps higher, are being observed in these reactions what justify our innovative approach to the production of super heavy nuclei. Good experimental data are needed to guide future efforts in heavy element research.
Experimental apparatus {#apparatus}
======================
The construction of the detection system used in the test measurement reported in this paper was based on experience collected during a decade of our experimental searches of SHEs. A picture of the experimental setup is presented in Fig. \[fig:1\]a and its schematic visualization is shown in Fig. \[fig:1\]b. The detection system is composed of two separated units i.e. the forward hemisphere active catcher (AC) detection system composed of 63 scintillator detection modules and a set of ionization chambers equipped with 7 strip position sensitive Si detectors ($\Delta$E-E) placed at backward angles. We focus in this paper on the AC detection system which allows to select candidates for a short lived SHE production out of large number of other uninteresting reaction products. The reaction products of collisions between heavy projectiles and targets are deposited in the AC modules and some of them which are radioactive heavy nuclei will decay by emission of alpha particles and/or by fission. The active catcher detection system is only 10 cm from the target and can detect the creation of a radioactive nucleus with very short, even a few nano-seconds, half-lives. The possibility of discovering the production of such short-lived SHEs was at the basis of the idea of the constructed apparatus.
![The active catcher detection system (the right side in the panel a) located behind the target (a bar in the middle of the panel a) and the backward wall of the gas – Si detectors (the left side in the panel a). A schematic visualization of the detection setup (panel b).[]{data-label="fig:1"}](fig1.eps)
The active catcher detection element presented in Fig. \[fig:2\] consists of fast plastic scintillator of 0.8 mm thickness, an aluminum cylinder with a cavity to accommodate a light guide and a photomultiplier tube (PMT). The light signals generated in the fast scintillator by the implanted reaction product and alpha particles and/or fission fragments emitted from the implanted heavy nucleus are converted by the PMT into electrical pulses which are processed by dedicated electronics.
![Schematic drawing of the active catcher detection module. []{data-label="fig:2"}](fig2.eps)
The PMT signal from each detection module of the active catcher is split and sent into analog and digital logic branches of the electronics (see Fig. \[fig:3\]a). The main trigger produced by the logical branch of the electronics allows the recording of a signal wave form using the CAEN FADC V1742 digitizer module. This module was set to a sampling rate of 1 Gs/s and 1024 points buffer. Therefore each event covers a time window of 1 $\mu$s. In order to manage a very high signal rate caused by a high intensity of reaction products and to record information on the SHE candidate production, the main acquisition trigger is generated by logical electronics presented in Fig. \[fig:3\]b. For this experiment the beam structure of Texas A&M University accelerator consisted of beam bursts of 5 ns width separated by 50 ns. The cyclotron RF signal is used to generate a logical veto to disable event recording during the beam burst (see Fig. \[fig:3\]c). The fast plastic scintillator BC-418 prepared by Saint-Gobain Crystals, used in the active catcher module, generates pulses of 0.5 ns rise time and 1.4 ns decay time. This scintillators are coupled to a small size Hamamatsu R9880U-110 photomultiplier (active window of 8 mm diameter) by a lucite light guide. Each active catcher detection module has a very good time resolution (PMT pulse width is about 5 ns and the rise time is of the order of ns).
![A schematic drawing of the electronics. []{data-label="fig:3"}](fig3.eps)
The PMT signal of the logical branch is sent to a comparator which allows a computer controlled setting of a detection threshold and then a fast logical signal of 2 ns width is generated. The logical signals from all active catcher modules are sent into a logical OR of FPGA card. If the signal from the logical OR of the FPGA card (2 ns width) does not coincide with the beam burst logical signal generated from the cyclotron RF (2 ns width) the main trigger is generated. The trigger signal caused by decays between beam bursts of the reaction products implanted into the active catcher scintillator can occur as fast as a few ns after beam burst ions hit the target (time of flight of the reaction products on a distance of about 10 cm between the target and the active catcher detection module). The main trigger starts recording of the signal wave forms from all active catcher modules. The FADC acquisition time window of 1 $\mu$s is divided into 600 ns and 400 ns intervals which are located before and after the trigger signal time, respectively and the acquisition system records all signals from the active catcher modules 600 ns before and 400 ns following time intervals with respect to the trigger signal time.
Test measurement results {#testresults}
========================
Fig. \[fig:4\] presents two examples of recorded events obtained in a summer 2015 experiment. A beam of $^{197}$Au (15-50 nA) at 7.5 A.MeV was delivered to the $^{232}$Th target of 12 mg/cm$^{2}$ thickness. Fig. \[fig:4\]a shows the event when two signals were detected in only one of the active catcher modules. The pulse located at 602 ns is the triggering signal and represents decays of the implanted reaction product into the active catcher module scintillator which must occur between the beam bursts due to the triggering condition. The second peak at 42.5 ns may represent a signal from the deposition of the reaction product. The time distance between the two peaks is 559.5 ns. If this time interval is divided by 55 ns, i.e. the separation time between the beam bursts, the rest of division is 9.5 ns what is well beyond of the burst duration, Fig. 3c. The peak at 42.5 ns precedes the peak at 559.5 ns and has much higher amplitude which may suggest that it originates from deposition of the reaction product produced during the beam burst. Moreover, we know that the peak at later time was generated by a particle emitted between beam bursts and we can conclude that this event can be a candidate for observation of implantation of the heavy reaction product which decays by the alpha particle emission after 517 ns plus a few ns needed by the heavy nucleus created in the target to travel about 10 cm distance to the active catcher module scintillator. We found about few tens similar cases among 1.5 million recorded events during our test measurements. The time interval between signals assigned as the implantation of the reaction product and the trigger signal assigned as the alpha particle emission from this reaction product covers the full range of the FADC window i.e. 600 ns.
![The wave forms of two recorded events which might indicate on the production and observation of the SHEs. []{data-label="fig:4"}](fig4.eps)
In the collected data we also found a several of three peak events which may represent production and implantation of SHE into the active catcher module scintillator followed by two alpha particle emissions. An example of such a three pulse event is shown in Fig. \[fig:4\]b. Both presented in this work as well as other collected cases for SHE candidates require more advanced analysis to confirm the production of very heavy nuclei in the massive transfer process. Such analysis should allow for a more precise filtering of false signals and for more precise determination of the energy of particles which generate signals in active catcher detectors [@Wiel:17:1].
![Time spectrum of pulses locations with respect to the trigger position. []{data-label="fig:5"}](fig5.eps)
The stability and time resolution determination of the constructed electronics is visualized in Fig. \[fig:5\] which shows a time spectrum of pulses’ positions recorded for all fired channels in one of the FADCs with respect to the trigger location. In order to accommodate sufficient statistics the triggers include also events associated with deposition in the triggering module beam burst reaction products. Observed regular structure of 55 ns period is a result of deposition in other detection modules reaction products associated with another beam bursts. The broadenings of the pulses’ positions are the result of around 5 ns beam burst width. Presented data proves that the electronics were stable and timing was determined with very high accuracy.
Summary and conclusions {#summary}
=======================
The article presents a new concept of the detection apparatus together with dedicated electronics for registering rare events produced in nuclear reactions at high beam intensity. This concept has been applied in our experimental searches for the production and detection of SHEs. The deposition of the reaction product signal in the active catcher detection module as well as the signal of its decay via the alpha particle emission or spontaneous fission which takes place between the beam bursts are recorded. The FADC acquisition time window allows to record up to one microsecond separation between those signals. Preliminary results of the test measurements showed that the new concept and constructed apparatus allow for the selection and recording candidates for short-lived heavy nuclei among other reaction products without overloading the acquisition system. The test run shows that constructed detection system requires improvements to achieve better energy resolution and position determination of deposited reaction products. One possibility is to use diamond detectors (2 mm by 2 mm active area), which have very good energy resolution (better than 10 keV) while preserving their timing characteristics similar to that of the fast plastic scintillators. Authors are very indebted to the Cyclotron Institute crew for their great help and for operation of the accelerator. This work is supported by the National Science Center in Poland, contract no. UMO-2012/04/A/ST2/ 00082, by the U.S. Department of Energy under Grant No. DE-FG03-93ER40773 and by the Robert A. Welch Foundation under Grant A0330.
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[^1]: zbigniew.majka@uj.edu.pl
[^2]: andrzej.wieloch@uj.edu.pl
|
---
abstract: 'This short paper provides further details of the *diveXplore* system (formerly known as CoViSS), which has been used by team ITEC1 for the Video Browser Showdown [@SchoffmannBailer2012] (VBS) 2018. In particular, it gives a short overview of search features and some details of final system changes, not included in the corresponding VBS2018 paper [@primusvbs2018], as well as a basic analysis of how the system has been used for VBS2018 (from a user perspective).'
author:
- 'Klaus Schoeffmann, Bernd M[ü]{}nzer, Jürgen Primus, Andreas Leibetseder'
bibliography:
- 'bib.bib'
title: 'The diveXplore System at the Video Browser Showdown 2018 – Final Notes'
---
System Overview
===============
The *distributed interactive video exploration* (diveXplore) system has been developed at Klagenfurt University for several years (and used for VBS2017 [@ITECUU2017] and VBS2018 [@primusvbs2018]) with the idea to provide a very flexible set of content retrieval features that fit many different search scenarios. Therefore, it provides a number of different components that can be used individually or in combination (see also Figure \[fig:divexplore\]):
- browsing *full feature maps* of the entire dataset (arranged by similar color or similar classification results in CNNs).
- browsing *pre-filtered feature maps* (e.g., keyframes containing *faces*, *texts*, or some other semantic concept) – for this the interface also provides a textual search feature, since there could be several hundreds of maps.
- a *storyboard/shot view* of every video, as well as a *video player* that reveals the shot structure and provides different playback speeds.
- a *color filter* to filter keyframes based on dominant colors.
- a *textual concept search* feature to filter keyframes containing specific visual concepts.
- a *similarity search* feature for query-by-example with a selected keyframe.
- a *color-sketch* feature (more details in [@leibetsedervbs18]).
- a *collaboration feature* in case the interface is used by several users, which allows to see each other’s position in a map, to send hints about shots, etc. The diveXplore system also provides an additional *SpectatorView* that shows the current state of all collaborators (more details in [@primusvbs2018]).
{width="42.00000%"} {width="42.00000%"} {width="42.00000%"} {width="42.00000%"}
[l|c|c|c|]{} & **KIS visual** & **KIS textual** & **AVS**\
&
---------------------------------
concept search or sketch search
(typically in combination
with shot filtering and
video inspection)
---------------------------------
&
----------------
mostly
concept search
----------------
&
--------------------------------------
a combination of
concept search, map search,
map browsing, and similarity search
(typically in combination with
shot filtering and video inspection)
--------------------------------------
\
&
----------------
mostly
concept search
----------------
& - &
------------------------------
mostly concept search
(often with shot filtering),
sometimes map search
and browsing
------------------------------
\
Final System Changes
====================
The system described in our VBS2018 paper [@primusvbs2018] has been further developed and improved after paper submission. Therefore, we describe these changes here in more detail.
First of all, we significantly improved our feature map browsing component in several ways: (i) all feature maps are now arranged by using a self-organizing map algorithm with some similarity feature (e.g., color or semantic concept), (ii) we provide pre-filtered feature maps that contain only keyframes of a specific concept (e.g., faces or texts – see top-left in Figure \[fig:divexplore\]). This includes maps created for concepts that have been recognized in a large enough volume (at least 576 corresponding keyframes) in the dataset by different CNNs. Since this resulted in more than 1200 feature maps, we have also implemented a *textual map search* feature (Figure \[fig:divexplore\]).
Next, the *storyboard* used for VBS2017 [@ITECUU2017] – that consisted of small and uniformly sampled thumbnails – has been replaced by a *shot view* of each video. Similarly to the previous version, it allows to browse over different videos of the dataset quite quickly and easily (and to jump to one specific video).
We introduced a new *similarity* tab in our interface that always displays the results of the previous similarity search. This facilitates jumping between different components of our interface (e.g., looking into the shot view of a video and then going back to the similarity search results). In addition to that we implemented a history feature that allows going back to previously retrieved search results.
Finally, we disabled the *web-example* search component (see [@primusvbs2018]) in our system, because we found it hard to use for the IACC.3 dataset [@2017trecvidawad], due to its low visual quality. Instead we integrated a color-sketch feature in collaboration with our second team [@leibetsedervbs18].
System Usage at VBS2018
=======================
The usage of diveXplore at VBS2018 differs heavily for the two different user groups (*experts* and *novices*), as indicated in Table \[tab:usage\]. While novices mostly used textual concept search, the experts utilized many more features of the system and followed a more diverse search strategy. However, due to the different type of query presentation (visual KIS queries are presented as video clips, while textual KIS and AVS tasks are described by text), the strategy for completing visual tasks is much more visually-oriented (hence, users closely inspected the video and sometimes used sketches). For textual KIS the experts mostly employed simple textual concept search, since no or merely a few clues about the visual representation of the scene were provided. For AVS, however, due to the fact that multiple correct submissions are possible (and their correctness may be immediately confirmed by the VBS Server during the task), users tend to use additional features that could further improve the performance (i.e., allow them to find more instances more easily/quickly), such as map search (followed by browsing filtered maps) and similarity search based on an already found instance [@kletzMMM18].
Conclusions
===========
The usage experience of our system gained at VBS2018 clearly shows that the strategy to build a flexible system with many different retrieval options is the right one. Different type of queries need different search features and interactions means, this is also evident from the many different interfaces already proposed in the literature for video search and interaction [@Schoeffmann2015Survey]. We will continue to improve our interactive video retrieval system diveXplore and try to further improve our performance, which was quite solid in the past two years (2nd place at VBS2017 and VBS2018).
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abstract: 'We describe the automated spectral classification, redshift determination, and parameter measurement pipeline in use for the Baryon Oscillation Spectroscopic Survey (BOSS) of the Sloan Digital Sky Survey III (SDSS-III) as of the survey’s Ninth Data Release (DR9), encompassing 831,000 moderate-resolution optical spectra. We give a review of the algorithms employed, and describe the changes to the pipeline that have been implemented for BOSS relative to previous SDSS-I/II versions, including new sets of stellar, galaxy, and quasar redshift templates. For the color-selected “CMASS” sample of massive galaxies at redshift $0.4 \la z \la 0.8$ targeted by BOSS for the purposes of large-scale cosmological measurements, the pipeline achieves an automated classification success rate of 98.7% and confirms 95.4% of unique CMASS targets as galaxies (with the balance being mostly M stars). Based on visual inspections of a subset of BOSS galaxies, we find that approximately 0.2% of confidently reported CMASS sample classifications and redshifts are incorrect, and about 0.4% of all CMASS spectra are objects unclassified by the current algorithm which are potentially recoverable. The BOSS pipeline confirms that $\sim$51.5% of the quasar targets have quasar spectra, with the balance mainly consisting of stars and low signal-to-noise spectra. Statistical (as opposed to systematic) redshift errors propagated from photon noise are typically a few tens of kms$^{-1}$ for both galaxies and quasars, with a significant tail to a few hundreds of kms$^{-1}$ for quasars. We test the accuracy of these statistical redshift error estimates using repeat observations, finding them underestimated by a factor of 1.19 to 1.34 for galaxies, and by a factor of 2 for quasars. We assess the impact of sky-subtraction quality, signal-to-noise ratio, and other factors on galaxy redshift success. Finally, we document known issues with the BOSS DR9 spectroscopic data set, and describe directions of ongoing development.'
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title: 'Spectral Classification and Redshift Measurement for the SDSS-III Baryon Oscillation Spectroscopic Survey'
---
Introduction
============
The Sloan Digital Sky Survey III (SDSS-III, @Eisenstein11) is the third phase of the SDSS [@York00].[^1] Within the SDSS-III, the Baryon Oscillation Spectroscopic Survey (BOSS, @Dawson12) is currently mapping a larger volume of the universe than any previous spectroscopic survey. The Ninth Data Release of the SDSS-III (DR9, @Nine12, released publicly on 2012 July 31) is the first SDSS-III data release to include BOSS spectroscopic data, and comprises good observations of 831 unique plate-pluggings of 813 unique tilings (plates worth of targets) on the sky. Each plate delivers simultaneous spectroscopic observations of 1000 lines of sight with optical fibers that feed a pair of two-arm spectrographs, giving a total of 831,000 BOSS DR9 spectra.
The main science goal of BOSS is to trace the large-scale mass structure of the universe using massive galaxies and quasar Ly$\alpha$ absorption systems, in order to measure the length scale of the “baryon acoustic oscillation” feature in the spatial correlation function of these objects [e.g., @Eisenstein05], and thereby to constrain the nature of the dark energy that drives the accelerated expansion of the present-day universe. To meet this goal, the BOSS project has specified a series of scientific requirements, including: (1) an RMS galaxy redshift precision better than 300 kms$^{-1}$; (2) a galaxy redshift success rate of at least 94%, including both targeting inefficiency and spectroscopic redshift failure; (3) a catastrophic galaxy redshift error rate of less than 1%; and (4) spectroscopic confirmation of at least 15 quasars at $2.2 < z < 3.5$ per degree$^2$ from among no more than 40 targets per degree$^2$. To satisfy these requirements within such a large survey, automated spectroscopic calibration, extraction, classification, and redshift measurement methods are essential.
This paper, one of a series of technical papers describing SDSS-III DR9 in general and the BOSS data set in particular, presents the automated classification and redshift measurement software for the main galaxy and quasar target samples implemented for the BOSS project. This software is written in the IDL language, and is titled `idlspec2d`. Earlier versions of this code were used to analyze SDSS-I/II data (see @Aihara11), alongside the complementary and independently developed pipeline software `spectro1d` (see @Subbarao02 and @Adelman06); for the BOSS project, the `idlspec2d` software has been adopted as the primary code, due to its robust error estimation methods and its tight integration of redshift measurement and classification with the lower-level operations of raw data calibration and extraction. The code has also been upgraded with new redshift-measurement templates and several new algorithms in order to meet the scientific requirements of the BOSS project. The tagged software version `v5_4_45` was used to process all BOSS spectroscopic data for DR9[^2], and the classification and redshift results delivered by this code have been used for recently published BOSS DR9-sample cosmological analyses [@Anderson12; @Manera12; @Nuza12; @Reid12; @RossA12; @Sanchez12; @Tojeiro12]. An overview of the BOSS project, including experimental design, scientific goals, observational operations, and ancillary programs, is given in @Dawson12. A description of the `idlspec2d` calibration and extraction methods which transform raw CCD pixel data into one-dimensional object spectra will be presented in @Schlegel12.
The organization of this paper is as follows. Section \[sec:data\] presents an overview of the spectroscopic data sample of BOSS DR9. Section \[sec:pipeline\] describes the classification and redshift pipeline algorithms and procedures, including the core redshifting algorithm (§\[subsec:zmeasure\]), special classification handling for the galaxy target samples (§\[subsec:z\_noqso\]), measured spectroscopic parameters (§\[subsec:params\]), and output files (§\[subsec:outfiles\]). Section \[sec:templates\] describes the templates constructed for the automated spectroscopic identification and redshift analysis of BOSS galaxies (§\[subsec:galtemp\]), quasars (§\[subsec:qsotemp\]), and stars (§\[subsec:startemp\]). Section \[sec:performance\] analyzes the completeness, purity, accuracy, and precision of the samples classified and measured by the `idlspec2d` pipeline. Section \[sec:issues\] documents known issues in the DR9 release of BOSS data, and §\[sec:summary\] provides a summary and conclusions.
Data Overview {#sec:data}
=============
The main BOSS survey program consists of two galaxy target samples [@Padmanabhan12] and a quasar target sample including both color-selected candidates and known quasars [@Bovy11; @Kirkpatrick11; @RossN12]. The galaxy samples are designated CMASS (for “constant mass”) and LOWZ (for “low-redshift”). The LOWZ galaxy sample is composed of massive red galaxies spanning the redshift range $0.15 \la z \la 0.4$. The CMASS galaxy sample is composed of massive galaxies spanning the redshift range $0.4 \la z \la 0.7$. Both samples are color-selected to provide near-uniform sampling over the combined volume. The faintest galaxies are at $r=19.5$ for LOWZ and $i=19.9$ for CMASS. Colors and magnitudes for the galaxy selection cuts are corrected for Galactic extinction using @Schlegel98 dust maps. The BOSS quasar sample is selected to recover as many objects as possible in the redshift range $2.2 < z < 3.5$ for the purposes of measuring the 3D structure in the Ly$\alpha$ forest. A variety of selection algorithms are employed to select the quasar sample, which lies close to the color locus of F stars. The faint-end magnitude limits of the quasar target sample are extinction-corrected PSF magnitudes of $g=22$ and $r=21.85$.
A summary of the DR9 BOSS spectroscopic data set (observed between 2009 December and 2011 July) is given in Table \[table:dr9summary\], along with performance metrics that will be discussed in detail further below. Representative BOSS survey spectra are shown in Figure \[fig:mosaic\]. The automated classification and measurement software described here is applied to all spectra obtained by the BOSS spectrographs [@Smee12], including spectra targeted under ancillary programs described in @Dawson12. In this work we focus on the analysis of the main BOSS galaxy and quasar survey targets, since the performance on these samples is the primary scientific driver of the design, development, and verification of the pipeline.
Item Number
------------------------------------------------------------ --------
Plate pluggings 831
Unique plates 819
Unique tiles (plates worth of targets) 813
Spectra 831000
Effective spectra 829073
Unique spectra 763425
CMASS sample spectra 353691
Unique CMASS spectra 324198
Unique CMASS with `ZWARNING_NOQSO == 0` 320031
Unique CMASS that are galaxies 309307
LOWZ sample spectra 111347
Unique LOWZ spectra 103729
Unique LOWZ with `ZWARNING_NOQSO == 0` 103610
Unique LOWZ that are galaxies 102890
CMASS && LOWZ sample spectra 3201
Unique CMASS && LOWZ spectra 2990
Unique CMASS && LOWZ with `ZWARNING_NOQSO == 0` 2976
Unique CMASS && LOWZ that are galaxies 2935
Quasar sample spectra 166034
Unique quasar sample spectra 154433
Unique quasar sample with `ZWARNING == 0` 122488
Unique quasar sample spectra that are quasars 79570
Number of above with $2.2 \le z \le 3.5$ 51903
Unique quasar sample scanned visually 154173
Visual $2.2 \le z \le 3.5$ quasars missed by pipeline 895
Pipeline $2.2 \le z \le 3.5$ QSOs with visual disagreement 327
Sky spectra 78573
Unique sky-spectrum lines of sight 75850
Spectrophotometric standard star spectra 16905
Unique standard star spectra 14915
Ancillary program spectra 32381
Unique ancillary target spectra 28968
Other spectra (commissioning, calibration, etc.) 74620
Unique other spectra 65461
: \[table:dr9summary\] BOSS DR9 summary spectrum totals
For the purposes of this paper, we define the samples of unique LOWZ and CMASS spectra according to the following cuts:
1. Selected by the appropriate sample color cuts (encoded by bit 0 of the `BOSS_TARGET1` mask for the LOWZ sample, and by bit 1 of that mask for the CMASS sample.) The LOWZ and CMASS samples are not mutually exclusive, although they are mostly non-overlapping.
2. \[item:withdata\] Observed with a spectroscopic fiber that is well plugged, successfully mapped to the target object, and not affected by bad CCD columns that remove a large fraction of the wavelength coverage. These conditions are reported via bits 1 and 7 of the `ZWARNING` bitmask described in §\[subsec:zmeasure\].
3. Apparent (not extinction-corrected) $i$-band magnitude less than 21.5 within a 2$\arcsec$-diameter circular aperture, corresponding to the angular size of the BOSS fibers. This criterion excludes low surface-brightness targets for which the spectroscopic signal-to-noise ratio (S/N) becomes unacceptably low for nominal survey exposure times of 60 minutes in good conditions.
4. \[item:specprimary\] Best single observation within the survey data set, for the case of multiply observed spectra. This designation is described in §\[subsec:outfiles\].
The sample of unique BOSS quasar spectra for the current work is defined according to the following cuts:
1. Selected from one of the four categories of known quasars with redshifts optimal for Ly$\alpha$ forest analysis (bit 12 of the `BOSS_TARGET1` mask), quasars selected from the FIRST survey [bit 18, @Becker95], and candidates from the BOSS “Core” and “Bonus” quasar candidate selection algorithms (bits 40 and 41 respectively: see @RossN12).
2. Plugged, mapped, and well-covered in wavelength (as for Item \[item:withdata\] of the previous list for galaxy targets).
3. Best single observation within the survey data set (as for Item \[item:specprimary\] of the previous list).
Pipeline Overview {#sec:pipeline}
=================
Imaging and spectroscopic data for the BOSS Survey are obtained with the 2.5-m Sloan Telescope at Apache Point Observatory [@Gunn06], first with the imaging camera [@Gunn98] and then with an upgraded (relative to SDSS-I/II) spectrograph system capable of obtaining 1000 spectra simultaneously using optical fibers plugged into a drilled aluminum focal plate and feeding two double-arm spectrographs. The characteristics of this instrument are summarized in Table \[table:spectrograph\], and described in detail by @Smee12. The outputs of the fibers feeding each spectrograph are arrayed linearly along a “slit-head” and numbered within the spectroscopic pipeline by the sequential index `FIBERID`, which by convention runs from 1–500 in spectrograph 1 and from 501-1000 in spectrograph 2. A unique physical target plate is specified by the `PLATE` identifier. Since the same plate can be plugged and observed on multiple occasions, with different mappings between fibers and target holes, the modified Julian date of a unique plugging is tracked as well via the `MJD` parameter. Together, the combination of `PLATE`, `MJD`, and `FIBERID` constitute a unique identifier for a BOSS spectrum (as was also the case for SDSS-I/II spectra). Each plugging is observed with multiple exposures which are exactly 15 minutes each in length and can be distributed across more than one night of observation. Typically four to six exposures are required to attain sufficient S/N per pixel at a fiducial magnitude. All good data from a unique plugging are co-added together during the spectroscopic data reduction process. Data from different pluggings are never combined together.
Parameter Value
---------------------- -----------------------------------------
On-sky field of view 3$^{\circ}$ diameter
Fiber aperture 2$\arcsec$ diameter
Multiplex capability 1,000 objects
Wavelength coverage 3,600Å$\la \lambda \la$10,400Å
Spectral resolution $\lambda / \Delta \lambda \approx$2,000
: \[table:spectrograph\] BOSS spectrograph system characteristics
The wavelength calibration, extraction, sky subtraction, flux calibration, and co-addition of BOSS spectra from raw CCD pixel data are described in @Schlegel12. The extraction implementation is a variation of the optimal-extraction algorithm described by @Hewett85 and @Horne86, including a forward-modeling solution that de-blends the cross-talk between neighboring fibers on the CCD. (A similar approach is described in @Sandin10.) The outputs of this “two-dimensional” pipeline software are stored on a plate-by-plate basis for sets of 1000 spectra in the multi-extension “`spPlate`” FITS files, which are the inputs to the “one-dimensional” (1D) pipeline software described in this work. The full contents of the `spPlate` files are described in detail in the online data model[^3]; for the purposes of the redshift measurement and classification pipeline, the most important products are:
1. Wavelength-calibrated, sky-subtracted, flux-calibrated, and co-added object spectra, rebinned onto a uniform baseline of $\Delta \log_{10} \lambda = 10^{-4}$ (about 69kms$^{-1}$) per pixel.
2. Statistical error-estimate vectors for each spectrum (expressed as inverse variance) incorporating contributions from photon noise, CCD read noise, and sky-subtraction error.
3. Mask vectors for each spectrum identifying pixels where warning conditions exist in either any (`ORMASK`) or all (`ANDMASK`) of the spectra contributing to the final co-added spectrum.
Redshift measurement and classification {#subsec:zmeasure}
---------------------------------------
The BOSS spectral classification and redshift-finding analysis is approached as a $\chi^2$ minimization problem. Linear fits are made to each observed spectrum using multiple templates and combinations of templates evaluated for all allowed redshifts, and the global minimum-$\chi^2$ solution is adopted for the output classification and redshift. This approach requires the spectra and their errors to be well-understood, and requires the template spectra to sufficiently span the observed space. Both these conditions are satisfied for the BOSS galaxy and quasar targets, resulting in accurate redshift fits and enabling a quantitative assessment of the confidence of those fits. By performing a statistically objective analysis, confident redshifts are obtained even for data at lower S/N where manual inspection may fail.
The basic outputs of the redshift determination and classification algorithm described in this section are the measured redshift `Z`, its associated 1-sigma statistical error `Z_ERR`, a classification category `CLASS` (either `GALAXY`, `QSO` for quasar, or `STAR`), and a confidence flag `ZWARNING` that is zero for confident measurements and non-zero otherwise. Section \[subsec:z\_noqso\] describes special variations on these outputs that are recommended for use with the BOSS LOWZ and CMASS galaxy sample spectra.
The least-squares minimization is performed by comparison of each spectrum to a full range of templates spanning galaxies, quasars, and stars. A range of redshifts is explored, with trial redshifts spaced every pixel (69kms$^{-1}$) for most templates and spaced by four pixels (276kms$^{-1}$) for quasar templates. At each redshift the spectrum is fit with an error-weighted least-squares linear combination of redshifted template “eigenspectra” in combination with a low-order polynomial. The polynomial terms absorb Galactic extinction, intrinsic extinction, and residual spectro-photometric calibration errors (typically at the 10% level) that are not fully spanned by the eigenspectra. The template basis sets are derived from rest-frame principal-component analyses (PCA) of training samples of galaxies, quasars, and cataclysmic variable stars, and from a set of archetype spectra for other types of stars. CV stars are handled as a separate class from other stars due to their significant range of emission-line strengths. The construction of these basis sets is described in detail in §\[sec:templates\] below. This best-fit combination model gives a $\chi^2$ value for that trial redshift, and these values define a $\chi^2 (z)$ curve when computed across the redshift range under consideration. To facilitate comparison between template classes with differing numbers of basis vectors, these $\chi^2$ values are analyzed in reduced form $\chi^2_r$, i.e., divided by the number of degrees of freedom. In practice this is nearly equivalent to working in terms of $\chi^2$ for any given spectrum, as the number of pixels ($\sim$4500) greatly exceeds the number of free parameters in all fits. The best redshifts for a particular class under consideration are defined by the locations of the lowest minima in the $\chi^2_r$ curve, where that curve is fit by a quadratic function using the five points nearest the minimum (11 points for quasars). The initial quasar fits where the trial redshifts are spaced every four pixels are re-fit near the five best fits with redshifts spaced every pixel. This two-step fitting for the quasars is done for computational efficiency, since most of the computational time is spent evaluating quasar fits which are performed on all targets. The statistical error on the final redshift is evaluated at the location of the minimum of the $\chi^2$ curve as the change in redshift $\pm \delta z$ for which $\chi^2$ increases by one above the minimum value. Noise in the spectra can result in multiple local minima in the neighborhood of the global minimum that are not significant. These are typically separated by a few pixels, or $\sim$200kms$^{-1}$. For all template fits, we collapse minima separated by less than 1000kms$^{-1}$ to a single minimum. Solutions with separations exceeding 1000kms$^{-1}$ must be explicitly evaluated since they represent catastrophic redshift failures for BOSS galaxies if they are statistically indistinguishable from one another (see panel “h” of Figure \[fig:crappo\] further below). The redshift-finding procedure is shown schematically in Figure \[fig:chi2fit\]. (See also @Glazebrook98.)
This core algorithm is applied within the pipeline according to the following sequence:
1. Read the spectrum, error estimates, and mask vectors for a single spectroscopic plate from the `spPlate` file.
2. Mask pixels outside the range 3600Å–10400Å, pixels at wavelengths where the typical sky-subtraction model residuals are more than 3-sigma worse than the errors expected from a Poisson model in any sub-exposure (`BADSKYCHI` set in the `ORMASK` vector output by the reduction software; @Schlegel12), pixels where the sky brightness is in excess of the extracted object flux plus ten times its statistical error in all sub-exposures (`BRIGHTSKY` set in the `ANDMASK`), and pixels with negative flux at greater than 10-$\sigma$ significance.
3. Find the best five galaxy redshifts between $z = -0.01$ and $z = 1.0$, using a rest-frame template basis of four eigenspectra (§\[subsec:galtemp\]).
4. Find the best five quasar redshifts between $z=0.0033$ and $z = 7.0$, using a rest-frame template basis of four eigenspectra (§\[subsec:qsotemp\]).
5. Find the best single redshift for each of 123 sub-classes of star from $-$1200kms$^{-1}$ to $+$1200kms$^{-1}$, using a single rest-frame archetype spectrum for each one (§\[subsec:startemp\]).
6. Find the best single cataclysmic variable star redshift from $-$1000kms$^{-1}$ to $+$1000kms$^{-1}$, using a rest-frame template basis of three eigenspectra (in order to capture emission-line strength variations, §\[subsec:startemp\].)
7. Sort all redshifts and classifications together by ascending $\chi_r^2$, and assign the best spectroscopic redshift and classification from among `GALAXY`, `QSO` (quasar), and `STAR` (including CV) based on the overall minimum $\chi_r^2$ across all classes and redshifts.
8. Compare the change in $\chi_r^2$ between the best classification and redshift and the next-best classification and redshift with a velocity difference greater than 1000km/s$^{-1}$, and assign a low-confidence “`ZWARNING`” flag if this difference is either less than 0.01 in absolute terms or less than 0.01 times the overall minimum $\chi_r^2$ value. For the $\sim$4500 degrees of freedom typical of a BOSS spectrum, the absolute threshold of $\Delta \chi_r^2 = 0.01$ corresponds to $\Delta \chi^2 \approx 45$ (naively interpreted as 6.7-sigma). The relative requirement on $\Delta \chi_r^2$ serves to make the statistical confidence threshold progressively more conservative at higher S/N levels where the redshift templates fits are worse in an absolute sense but nevertheless have greater distinguishing power among multiple hypotheses.
The threshold value of $\Delta \chi_r^2 > 0.01$ used to assign confidence to the classifications is empirically determined. The threshold could be lowered further to recover more redshifts but at the cost of more mis-classifications and incorrect redshifts. Tests on repeat CMASS sample data show that decreasing the threshold to 0.008 (0.005) would increase redshift completeness by 0.3% (0.6%), with 8% (16%) of the added measurements being incorrect. (A full analysis of BOSS galaxy redshift completeness and purity is given in §\[subsec:galcomp\].) An additional test is made possible by the nearly 80,000 blank BOSS sky spectra in DR9. Among these, 2% fit a template with a confidence $\Delta \chi_r^2 > 0.01$, implying they would be assigned a confident redshift in the absence of prior knowledge of their status as blank sky spectra. (Although a small fraction of these are in fact real objects detected spectroscopically in the sky fibers.)
As discussed above, the condition `ZWARNING`$= 0$ designates that the BOSS pipeline has determined a confident classification and redshift for a spectrum. The primary source of `ZWARNING`$\ne 0$ spectra is the $\Delta \chi_r^2$ threshold. However, several other flags are also encoded bit-wise in the `ZWARNING` mask, as documented in Table \[table:zwarning\]. The definitions of the `ZWARNING` mask-bits in BOSS are identical to their definitions in SDSS-I/II. Two of the bits have been disabled for BOSS DR9, and are only retained for historical consistency: (1) the `NEGATIVE_MODEL` bit, which was previously triggered by stellar model fits with negative amplitudes, which are now disallowed entirely; and (2) the `MANY_OUTLIERS` bit, which was found to flag too many good, high-S/N quasar redshifts in BOSS.
Bit Name Definition
----- --------------------- ----------------------------------------------------------------------------------------------------------------
0 `SKY` Sky fiber
1 `LITTLE_COVERAGE` Insufficient wavelength coverage
2 `SMALL_DELTA_CHI2` $\Delta \chi_r^2$ between best and second-best fit is less than 0.01 (or 0.01 $\times$ the minimum $\chi_r^2$)
3 `NEGATIVE_MODEL` Synthetic spectrum negative, **disabled for BOSS DR9**
4 `MANY_OUTLIERS` More than 5% of points above 5-$\sigma$ from synthetic spectrum, **disabled for BOSS DR9**
5 `Z_FITLIMIT` $\chi^2$ minimum for best model is at the edge of the redshift range
6 `NEGATIVE_EMISSION` Negative emission in a quasar line at 3-$\sigma$ significance or greater (see §\[subsec:params\])
7 `UNPLUGGED` Broken or unplugged fiber
Table \[table:templates\] summarizes the number of PCA template and polynomial degrees of freedom associated with each spectroscopic object class, along with the name of the file containing the template basis within the `idlspec2d/v5_4_45` software product. In most cases, the number of PCA templates and number of polynomial terms used in the redshift and classification analysis match the SDSS-I/II `idlspec2d` values. The one exception is the number of CV star templates, which has decreased from four in SDSS-I/II to three in BOSS due to a smaller available training sample at the time the DR9 code version was frozen. For all target classes we have verified that the choices are nearly optimal by performing tests of the completeness and purity of classification and redshift measurement (relative to visually inspected subsets) as a function of the size of the PCA and polynomial basis. As can be expected, increasing the number of PCA and polynomial terms used for the modeling of a particular class increases both the completeness and the *im*purity of the resulting sample for that class. Increases in impurity arise from both catastrophic mis-classification and catastrophic redshift error, with the former decreasing the completeness of other classes. Each spectrum in the survey is fitted with all classes of objects in order to determine a spectroscopic redshift and classification that is independent of photometric data and targeting information (but see §\[subsec:z\_noqso\] below).
------------- ---------------------------- ------------------------------------------
Template $N_{\mathrm{temp}}$, $N_{\mathrm{poly}}$
Class Filename Per Fit
`GALAXY` `spEigenGal-55740.fits` 4, 3
`QSO` `spEigenQSO-55732.fits` 4, 3
`STAR` `spEigenStar-55734.fits` 1, 4
`STAR` (CV) `spEigenCVstar-55734.fits` 3, 3
------------- ---------------------------- ------------------------------------------
: \[table:templates\] Summary of BOSS redshift & classification degrees of freedom
Special galaxy target handling {#subsec:z_noqso}
------------------------------
The implementation of the `idlspec2d` redshift code is designed to meet the BOSS scientific requirements on redshift success rates, as discussed in the Introduction. The original SDSS-I/II code operated on spectra alone, without imposing classification or redshift priors based on photometric data or other targeting information. At the S/N typical of SDSS-I/II spectra, this technique proved highly successful, resulting in a redshift success rate better than 99% for the main galaxy sample and a negligible incidence of catastrophic errors. For the BOSS galaxy samples, however, some prior information from the targeting photometric catalog is needed to achieve the required redshift success rate. Specifically, we have found in practice that the LOWZ and CMASS targets can be galaxies, stars, or superpositions of the two, but are almost never quasars (however, see Item \[item:type2\] in §\[sec:issues\].) Without using any prior information, the redshift code produces an excess of erroneous quasar classifications for CMASS targets due to unphysical quasar basis-plus-polynomial combinations yielding the global minimum-$\chi^2$ (see panel “a” in Figure \[fig:crappo\] further below.)
To remedy this, the adopted BOSS survey values for the redshifts of LOWZ and CMASS galaxy targets are taken from the parameters `Z_NOQSO` and `CLASS_NOQSO`, together with the associated statistical error estimates `Z_ERR_NOQSO` and confidence flags `ZWARNING_NOQSO`, which represent the best-fit redshift and classification determined through the procedure described in §\[subsec:zmeasure\], but *excluding the consideration of quasar template fits.* This effectively imposes the red-color and extended-image priors of the galaxy target sample over the blue-color and point-like image priors of the quasar target sample. The `SMALL_DELTA_CHI2` bit for the `ZWARNING_NOQSO` mask is set only on the absolute criterion of $\Delta \chi_r^2 < 0.01$ relative to the next-best non-quasar model (i.e., with no relative $\Delta \chi_r^2$ cut). We recommend the use of these “`NOQSO`” quantities for statistical analyses of the BOSS galaxy samples. The parameter `Z` and its associated values are also retained and reported for consistency with the original SDSS-I/II approach, representing the global minimum-$\chi^2$ redshift inclusive of all spectral template classes.
Parameter measurements {#subsec:params}
----------------------
The primary outputs of the analysis code described in this work are the classification, redshift, redshift error, and best template-based model fit to each spectrum. The code also measures a number of parameters assuming the best-fit classification and redshift. Specifically: stellar velocity dispersions are measured for objects classified as galaxies; emission-line parameters are measured for all objects; and supplemental stellar sub-classifications and radial velocities are measured for objects classified as stars. We now describe these three measurement procedures in turn.
Stellar velocity dispersions $\sigma_v$ of galaxies are measured using a stellar template basis derived from the ELODIE library [@Prugniel01], covering the rest-frame spectral range 4100Å–6800Å. The high-resolution ELODIE spectra are degraded to the binning scale and approximate resolution of the co-added BOSS spectra, and a PCA of the library is performed. The first 24 principal components are used as a basis for fitting the galaxy spectra. The entire PCA basis is incrementally broadened from 0 to 850kms$^{-1}$ in units of 25kms$^{-1}$, and the set of all broadened PCA components is cached for the analysis of all galaxy spectra. For each galaxy spectrum, the stellar PCA basis is redshifted to match that galaxy’s redshift. At each trial broadening, the galaxy spectrum is fit with an error-weighted least-squares linear combination of the broadened stellar PCA basis plus a quartic polynomial, while masking the regions surrounding common emission lines. This marginalization over stellar-population effects at each trial $\sigma_v$ value serves to absorb some of the systematic errors of “template mismatch” into the statistical velocity dispersion error. The $\chi^2$ goodness-of-fit statistic for the best model is tabulated for each broadening step, to define a $\chi^2 (\sigma_v)$ curve. The minimum-$\chi^2$ velocity-dispersion value (with sub-grid localization) is reported as the measured value `VDISP`, and the error on this measurement `VDISP_ERR` is estimated from the curvature of the $\chi^2$ function at the position of the minimum. Note that this analysis is highly analogous to the redshift measurement procedure described in §\[subsec:zmeasure\]. This velocity dispersion measurement algorithm is unchanged from SDSS-I/II.
Within the BOSS data set, the S/N per pixel in galaxy spectra is often below the threshold commonly adopted as minimally sufficient for accurate point estimation of the stellar velocity dispersion. However, it has been shown by @Shu12 that unbiased measurements of the *distribution* of velocity dispersions within a large sample of galaxies can be made even when the individual spectra are of low S/N, by means of a hierarchical analysis that marginalizes statistically over the likelihoods of all possible velocity-dispersion values for each galaxy. To enable such analyses, we also compute and report the velocity-dispersion likelihood function for each galaxy in the vector-valued column `VDISP_LNL`. This is defined by $- \chi^2(\sigma_v) / 2$ for velocity dispersions $\sigma_v$ from 0 to 850kms$^{-1}$ in steps of 25kms$^{-1}$. The baseline and overall $\chi^2$ computation method are the same as described above for the measurement of the `VDISP` point estimators. However, the `VDISP_LNL` calculation employs only the first five stellar PCA template basis spectra, and also marginalizes over galaxy redshift uncertainties. An additional difference is that while the `VDISP` computations are done only for objects with `CLASS` of galaxy (for consistency with the SDSS-I/II practice), the `VDISP_LNL` calculations are done only for objects with `CLASS_NOQSO` of galaxy (for consistency with BOSS practice).
Emission-line parameters for the 31 transitions listed in Table \[table:emlines\] are computed for all spectra for which those lines fall into the observed BOSS wavelength range. Each line is modeled as a Gaussian, and the amplitudes, centroids, and widths of all lines are optimized non-linearly to obtain a minimum-$\chi^2$ fit to the data. The background continuum spectrum is taken from the best-fit velocity-dispersion model (for galaxies), from the best-fit redshift-pipeline model (for stars), and from a linear fit to the sidebands of each line (for quasars and for ranges of the galaxy spectra that extend beyond the coverage of the ELODIE-based velocity-dispersion templates.) All lines are constrained to have the same redshift within the fit, with the exception of Ly$\alpha$, which is allowed to fit at a different redshift to account for the asymmetric effects of Ly$\alpha$ forest absorption. In addition, groups of lines are constrained to have the same line-width as noted in Table \[table:emlines\], so as to allow robust fits to the strengths of low-S/N emission lines. Hence, the reported line-widths are effectively a strength-weighted average over the group. An initial guess for the line redshifts is taken from the best-fit pipeline redshift. Emission-line redshifts are allowed to depart arbitrarily from this value, but in practice are well-constrained in cases with significant emission in any lines. 96% of the quasars with significant C<span style="font-variant:small-caps;">iv</span> emission have line fits within 6000kms$^{-1}$ of the template redshift, and 96% of the galaxies with significant \[O<span style="font-variant:small-caps;">ii</span>\] emission have fits within 100kms$^{-1}$. Line fluxes, line widths, line redshifts, estimated continuum levels, and observed-frame equivalent widths are reported by the line fitting code, along with associated errors. In the SDSS-I/II implementation of the `idlspec2d` emission-line measurement code, equivalent widths were measured relative to the estimated continuum spectrum at line center, while for BOSS DR9 this has been changed to use a continuum level estimated from the sidebands of the line.
Based on the results of the line-fitting code, galaxy spectra with emission in all four of the lines H$\beta$, \[O<span style="font-variant:small-caps;">iii</span>\] 5007, H$\alpha$, and \[N<span style="font-variant:small-caps;">ii</span>\] 6583 detected at 3-sigma or greater are sub-classified into `AGN`, `STARFORMING`, and `STARBURST` according to the following rules. First, galaxies are sub-classified as `AGN` if $$\log_{10}([O\textsc{iii}]/H\beta) > 1.2 \log_{10}([N\textsc{ii}]/H\alpha) + 0.22$$ [@Baldwin81]. For galaxies falling on the other side of this cut, sub-classification is made based on the equivalent width of H$\alpha$: `STARFORMING` if less than 50Å, and `STARBURST` if greater. Galaxies and quasars may be given an additional sub-classification as `BROADLINE` if they have line widths in excess of 200kms$^{-1}$, with line-width measurement significance of at least 5-sigma, and line-flux measurement significance of at least 10-sigma.
------------ -------------------------------------------------------------- ------------ --------------------------------------------------
Line Line Redshift Width
Wavelength Name Group Group
1215.67 Ly$\alpha$ Ly$\alpha$ Ly$\alpha$
1240.81 N<span style="font-variant:small-caps;">v</span> 1240 emission N<span style="font-variant:small-caps;">v</span>
1549.48 C<span style="font-variant:small-caps;">iv</span> 1549 emission emission
1640.42 He<span style="font-variant:small-caps;">ii</span> 1640 emission emission
1908.734 C<span style="font-variant:small-caps;">iii</span>\] 1908 emission emission
2799.49 Mg<span style="font-variant:small-caps;">ii</span> 2799 emission emission
3726.032 \[O<span style="font-variant:small-caps;">ii</span>\] 3725 emission emission
3728.815 \[O<span style="font-variant:small-caps;">ii</span>\] 3727 emission emission
3868.76 \[Ne<span style="font-variant:small-caps;">iii</span>\] 3868 emission emission
3889.049 H$\epsilon$ emission Balmer
3970.00 \[Ne<span style="font-variant:small-caps;">iii</span>\] 3970 emission emission
4101.734 H$\delta$ emission Balmer
4340.464 H$\gamma$ emission Balmer
4363.209 \[O<span style="font-variant:small-caps;">iii</span>\] 4363 emission emission
4685.68 He<span style="font-variant:small-caps;">ii</span> 4685 emission emission
4861.325 H$\beta$ emission Balmer
4958.911 \[O<span style="font-variant:small-caps;">iii</span>\] 4959 emission emission
5006.843 \[O<span style="font-variant:small-caps;">iii</span>\] 5007 emission emission
5411.52 He<span style="font-variant:small-caps;">ii</span> 5411 emission emission
5577.339 \[O<span style="font-variant:small-caps;">i</span>\] 5577 emission emission
5754.59 \[N<span style="font-variant:small-caps;">ii</span>\] 5755 emission emission
5875.68 He<span style="font-variant:small-caps;">i</span> 5876 emission emission
6300.304 \[O<span style="font-variant:small-caps;">i</span>\] 6300 emission emission
6312.06 \[S<span style="font-variant:small-caps;">iii</span>\] 6312 emission emission
6363.776 \[O<span style="font-variant:small-caps;">i</span>\] 6363 emission emission
6548.05 \[N<span style="font-variant:small-caps;">ii</span>\] 6548 emission emission
6562.801 H$\alpha$ emission Balmer
6583.45 \[N<span style="font-variant:small-caps;">ii</span>\] 6583 emission emission
6716.44 \[S<span style="font-variant:small-caps;">ii</span>\] 6716 emission emission
6730.82 \[S<span style="font-variant:small-caps;">ii</span>\] 6730 emission emission
7135.790 \[Ar<span style="font-variant:small-caps;">iii</span>\] 7135 emission emission
------------ -------------------------------------------------------------- ------------ --------------------------------------------------
: \[table:emlines\] Emission lines measured by the BOSS pipeline
For spectra classified as stars, an additional fitting to the ELODIE stellar library [@Prugniel01] is performed. The ELODIE library contains 709 stars spanning spectral types O to M, luminosity classes V to I, and metallicities \[Fe/H\] from $-3.0$ to $+0.8$. The observed resolution was 42,000 over the wavelength range 4100 to 6800Å. Our fitting makes use of the release of this library at resolution 10,000 that was calibrated to 0.5% in narrow-band spectrophotometric precision and 2.5% in broad-band precision. This library was trimmed from 709 to 610 stars that are not binary or triple stars. The ELODIE spectra are convolved with Gaussian functions to match the resolution of the BOSS spectra. A later release of this library (ELODIE 3.1) was not used due to extensive masking of regions near sky emission that compromises its use for measuring radial velocities to high precision [@Prugniel07]. Each BOSS spectrum classified as a star is re-fit to all spectra in this trimmed ELODIE library with the identical redshift-fitting code used to determine the primary redshift (§\[subsec:zmeasure\]). These fits are limited to the 4100–6800Å wavelength range, include 3 polynomial terms, and span velocities from $-1000$ to $+1000$kms$^{-1}$. The physical parameters of the best-fit ELODIE template are included in the pipeline outputs (`ELODIE_TEFF`, `ELODIE_LOGG`, `ELODIE_FEH`), along with the redshift (`ELODIE_Z`), statistical error of the redshift (`ELODIE_Z_ERR`) and reduced $\chi^2$ of that fit (`ELODIE_RCHI2`). An estimate of the template-mismatch effects on the redshift is provided as the standard deviation in redshift among the best 12 ELODIE template fits (`ELODIE_Z_MODELERR`).
The BOSS pipeline also computes and reports median spectroscopic signal-to-noise ratios per 69kms$^{-1}$ pixel (`SN_MEDIAN`) over the five SDSS broadband wavelength ranges ($ugriz$, @Fukugita96), along with the synthetic broadband fluxes predicted by the spectrum (`SPECTROFLUX`) and the best-fit template model to the spectrum (`SPECTROSYNFLUX`).
As described in @Nine12, DR9 also includes catalogs of alternative parameter measurements for BOSS galaxies, which are documented in other publications. @Chen12 describe PCA-based stellar-population parameter measurements and velocity-dispersion estimates. @Thomas12 have measured stellar velocity dispersions using the pPXF software of @Cappellari04 and emission-line properties using the GANDALF software of @Sarzi06. Finally, @Maraston12 have derived photometric stellar-mass estimates for BOSS galaxies. All these measurements are distributed with DR9, but are separate from the core `idlspec2d` pipeline system described here.
Output files {#subsec:outfiles}
------------
The BOSS `idlspec2d` redshift pipeline generates output files for each plate, along with summary files to aggregate photometric and spectroscopic parameters across the entire BOSS survey data set. These files are listed in Table \[table:files\]; together they contain all the parameters described in this paper. Access to these files on the SDSS-III Science Archive Server (SAS), as well as full data-model documentation of their formats and contents, can be obtained through the SDSS-III DR9 website. The `spAll` summary file from the BOSS pipeline is analogous but not identical in form and content to the `specObj` file loaded by the SDSS-III Catalog Archive Server (CAS), which contains both SDSS-I/II and BOSS data.
--------------------------- -----------------------------------
`spZbest-pppp-mmmmm.fits` Best-fit redshift & class param.s
`spZall-pppp-mmmmm.fits` Parameters for all fits
`spZLine-pppp-mmmmm.fits` Emission-line parameters
`spAll-v5_4_45.fits` Summary param.s for all spectra
`spAllLine-v5_4_45.fits` Line fit param.s for all spectra
--------------------------- -----------------------------------
: \[table:files\] DR9 Redshift and classification pipeline output files
Approximately 8% of BOSS spectra are repeat observations of previously observed targets, due both to re-observations of entire plates and to re-targeting of a number of objects on more than one plate (see @Dawson12). Of particular note within the summary files, the best spectroscopic observation of each object (defined by a 2$^{\prime\prime}$ positional match) in the survey is defined according to the following rules:
1. Prefer spectra with positive median S/N per spectroscopic pixel within the $r$-band wavelength range over other observations.
2. Prefer spectra with `ZWARNING` $ = 0$ over other spectra (or `ZWARNING_NOQSO` $ = 0$ for galaxy-sample targets.)
3. Prefer spectra with higher median S/N per spectroscopic pixel within the $r$-band wavelength range.
The best observation for each object is designated by setting the parameter `SPECPRIMARY` equal to 1 in the `spAll` file, while setting it equal to zero for all other spectroscopic observations of a given object that may be present within the survey data set.
Template Classes {#sec:templates}
================
In order to compare and select among galaxy, quasar, and stellar models objectively and with the highest statistical significance, the BOSS pipeline requires redshift and classification measurement templates that span both the full space of physical object types within the survey and the full wavelength range of the spectrograph. BOSS expands on SDSS-I/II in both regards, and hence requires a new set of pipeline templates, which we now describe.
Galaxies {#subsec:galtemp}
--------
The `idlspec2d` galaxy redshifts for SDSS-I/II were measured using templates generated from 480 galaxies observed on SDSS plate 306, MJD 51690.[^4] Redshifts for this training set were established by modeling each spectrum across a range of trial redshifts as a linear combination of (1) the leading two components of a PCA analysis of 10 velocity-standard stars in M67 observed by SDSS-I plate 321 on MJD 51612, (2) a set of common optical emission lines modeled as narrow Gaussian profiles, and (3) a low-order polynomial. The adopted redshift for each galaxy was taken from the location of the minimum-$\chi^2$ value localized to sub-grid accuracy, in the same manner described in §\[subsec:zmeasure\] above. Using these redshifts, the training-sample spectra were transformed to a common rest-frame baseline, and input to an iterative PCA procedure that accounts for measurement errors and missing data (e.g., @Tsalmantza12 and references therein). The leading four “eigenspectra” from this procedure were taken to define the galaxy redshift template basis for SDSS-I/II. For the commissioning analysis of BOSS spectra, these same templates were used for measuring galaxy redshifts, despite their lack of $z=0$ coverage redward of 9300Å and their under-representation of post-starburst galaxies (which appear with more frequency in the BOSS CMASS sample than in SDSS-I/II).
To generate a new redshift template set for use in automated analysis of BOSS spectra, we select a set of BOSS galaxies with redshifts over the interval $0.05 < z < 0.8$ that are well-measured by the original SDSS templates. To increase S/N and flatten the coverage of galaxy parameter space before performing a PCA to generate the template set, we bin together galaxies with similar 4000Åbreak strengths ($D4000$, @Balogh99) and redshifts, for the purposes of stacking their spectra. We use a $D4000$ range from 1.0 to 2.2 with a binning interval of 0.2, and a redshift binning interval of 0.05. In some $D4000$–redshift bins, we further subdivide the galaxies into several H$\delta_A$ sub-bins. The number of sub-bins depends on the number of galaxies in each $D4000$–redshift bin: if the total number is smaller than 600, we do not divide further into H$\delta_A$ sub-bins; if the number is in the range 600–1200, we divide into two sub-bins; and if there are greater than 1200, we divide into three sub-bins.
We also select a set of “post-starburst” galaxies from the BOSS galaxy sample, defined by having either $$D4000<1.3 ~~\mbox{and}~~ (H\delta_A + H\gamma_A)/2 > 7$$ or $$(H\delta_A + H\gamma_A)/2 > {\rm max}[-17.50 \times
D4000 + 29.25, ~3].$$ This criterion leads to a sample of about 2400 post-starburst galaxies, which we divide into five bins in redshift with equal numbers of galaxies per bin. We then stack the rest-frame spectra of all galaxies in each bin, to generate a set of high-S/N stacked spectra across the range of parameters indicated.
Once all these stacked spectra are in hand, we fit stellar continuum models to them [@Brinchmann04; @Tremonti04] using simple stellar population (SSP) models. Our SSP models are taken from @Maraston09 and @Maraston11, and are based on a combination of theoretical and observational stellar library data from @Rodriguez05, @Sanchez06, and @Gustafsson08. In the rest-frame wavelength range 1900–9900Å, we patch the stacked spectra with the fitted continuum models for pixels with S/N smaller than 10, pixels where the difference between models and stacks is larger than 30%, and pixels where there are no observations. We also add H$\alpha$, \[N<span style="font-variant:small-caps;">ii</span>\], and \[S<span style="font-variant:small-caps;">ii</span>\] emission lines for cases where these lines fall outside the range of observed wavelengths used to generate the stacked spectra. This is accomplished by selecting galaxies with similar $D4000$, H$\delta_A$, and dust extinction as the galaxies used to make the stacks, and computing the median values of $\sigma_{\rm H\alpha}/\sigma_{\rm H\beta}$, $f_{\rm H\alpha}/f_{\rm H\beta}$, $\sigma_{\rm [N\textsc{ii}]}/\sigma_{\rm [O\textsc{iii}]}$, $f_{\rm [N\textsc{ii}]}/f_{\rm [O\textsc{iii}]}$, $\sigma_{\rm [S\textsc{ii}]}/\sigma_{\rm [O\textsc{iii}]}$, and $f_{\rm [S\textsc{ii}]}/f_{\rm [O\textsc{iii}]}$ for these comparison samples (here, $\sigma$ is Gaussian line dispersion and $f$ is line flux). By multiplying these ratios with the appropriate line width or flux of H$\beta$ or \[O<span style="font-variant:small-caps;">iii</span>\] from the stacks, we predict the line widths and fluxes for H$\alpha$, \[N<span style="font-variant:small-caps;">ii</span>\] and \[S<span style="font-variant:small-caps;">ii</span>\] to be added to the stacked spectra, which we do using a Gaussian model for each line.
At the end of this process, we have 160 stacked and patched spectra. We augment these data with a sample of 28 type-II quasars (e.g., @Zakamska03 [@Reyes08]) identified within the BOSS spectroscopic data set (see the discussion in §\[sec:issues\]). This full set of spectra is then used as input to the rest-frame spectrum PCA algorithm to generate the four-component BOSS galaxy redshift template basis, which is shown in the top panel of Figure \[fig:templates\].
Quasars {#subsec:qsotemp}
-------
Quasar redshift templates are generated from a training sample of targets selected from the SDSS DR5 quasar catalog [@Schneider07] and targeted for re-observation with the BOSS spectrographs. The targets were chosen from the catalog at random, while enforcing as uniform a distribution as possible in redshift. As of 2011 June 10, 571 objects from this sample had been observed by BOSS. Removal of three spectra for localized cosmetic defects gives a training sample of 568 BOSS quasars. The distribution in redshift of the targeted sample and the observed sample is shown in Figure \[fig:qsohist\]. The observed sample is weighted more heavily above redshift $z = 2.2$, in accordance with overlapping BOSS quasar sample priorities. We keep this weighting in the training set, since we want our redshifting performance to be particularly well tuned for the redshift range of interest to the BOSS Ly$\alpha$ forest program.
Using the redshifts given by @Schneider07, we shift these training spectra to their rest frames and perform a PCA of the sample, with iterative replacement to fill in missing data. The top four principal components are retained and used as the linear basis set for our automated redshift and classification measurements, and are shown in the middle panel of Figure \[fig:templates\].
We do not employ the redshift estimates of @Hewett10 for the quasar-template training sample because the current BOSS pipeline is not configured to incorporate the absolute-magnitude information that would be necessary to take advantage of the increased precision afforded by these redshifts. Future BOSS pipeline versions may incorporate the @Hewett10 approach. We note that the primary criterion for BOSS spectroscopic pipeline performance on quasar targets is to minimize catastrophic redshift failures. Several detailed approaches to maximizing quasar redshift precision are being investigated within the BOSS quasar science working group, but all of these rely on having essentially correct initial quasar redshifts from the `idlspec2d` pipeline and/or visual inspection procedures [@Paris12].
Stars {#subsec:startemp}
-----
Although stellar science is not a primary goal of BOSS, the redshift pipeline must successfully flag stars from within the galaxy and quasar target samples of the survey. There is currently no comprehensive library of observed stellar spectra covering the full usable wavelength range of the BOSS spectrograph and the full H-R diagram. To assemble a set of stellar templates suitable to BOSS spectrum classification, we use a hybrid approach that extends data from the Indo-US observational stellar spectrum library [@Valdes04], selected to provide uniform coverage of the space of stellar atmosphere parameters $T_{\mathrm{eff}}$, $\log g$, and \[Fe/H\]) using theoretical atmosphere models computed using the MARCS [@Gustafsson08 for cool stars], ATLAS [@Kurucz05 for intermediate stars], and CMFGEN [@Hillier98 for hot stars] codes, obtained via the curated POLLUX spectrum database [@Palacios10].
### Template spectrum creation
We start with the full database of 1273 Indo-US stellar spectra, which have a resolution of approximately 1Å, a reduced pixel scale of 0.4Å, spectral coverage over 3400$<\lambda<$9500, and good flux calibration for most stellar types. The original radial-velocity zeropoints for the library were established either from literature or from velocity-standard cross-correlations. Since classification is the primary function of these spectra within the BOSS pipeline, we do not attempt any further refinement of these velocity zeropoints.
We initialize a “bad pixel” mask for each Indo-US spectrum based upon the zero-value Indo-US pixel mask convention. Furthermore, we define the following telluric absorption bands, and mask all pixels within them: 6850Å–6950Å, 7150Å–7350Å, 7560Å–7720Å, 8105Å–8240Å, $>$8900Å. We then select the subset of spectra that meet the conditions of (1) wavelength coverage from at least 3500Å to 8900Å, (2) good data over at least 75% of their pixels, (3) flux calibration with a non-flat (i.e., stellar) standard, and (4) no single gap within the spectrum larger than 200Å (the largest adopted telluric band width). These cuts result in a sample of 879 spectra covering spectral types from O6 to M8, but exclude carbon stars (which are fluxed with a flat SED in the Indo-US library).
We then take the 1040 model atmosphere spectra from the POLLUX database ranging in temperature from 3000K to 49000K, convolved and binned to the resolution and sampling of the Indo-US spectra. For each Indo-US spectrum in our subset, we loop over all model atmospheres and determine the multiplicative scaling of the model that minimizes the sum of squared data-minus-model residuals over non-masked pixels. We adopt the model spectrum that gives the overall minimum sum of squared residuals as being the “best fit” for a particular data spectrum.
The “best fit” model spectrum for each data spectrum is used to extend the data wavelength coverage and interpolate over the data gaps as follows. We define a running window of $\pm 400$ pixels ($\pm 160$Å) about an output pixel of interest, and determine the multiplicative scale and tilt to apply to the model over that window in order to give the best (least squares) fit to the non-masked data pixels over that same range. The scaled and tilted model is evaluated at the central pixel to define the new, locally scaled model spectrum, and the process is repeated over the entire spectrum by sliding the window. For pixels centered outside the outermost pixel of data coverage on the red and blue ends, the scale and tilt at the outermost data-covered pixel are used. The data and “sliding-scaled” model spectra are combined into a single output spectrum by assigning 100% model in pixels where the data have no coverage, defining a 100-pixel (40Å) transition region on either side of data gaps where the output spectrum is a weighted combination of the data and the sliding-scaled model, and varying the weight linearly from 0% model $+$ 100% data to 100% model $+$ 0% data over the transition region. Finally, we convolve and bin these output spectra down to the typical resolution (about 3Å FWHM) and reduced-spectrum sampling ($\Delta \log_{10} \lambda = 0.0001$ per pixel) of the BOSS data, also transforming from air to vacuum wavelengths to match the BOSS spectrum convention.
### Archetype subset selection
From these 879 patched and extended stellar spectra, our goal is to select a representative subset of “archetypes” that provide sufficient coverage of stellar parameter space to perform automated spectroscopic star–galaxy and star–quasar separation, while not attempting overly detailed stellar analysis that is beyond the scope of the BOSS science mission (cf. @Lee08).
We first visually inspect the template database and remove a single spectrum with noticeable data quality issues in an unmasked data region (Indo-US ID\#33111, 5450Å $< \lambda < $ 6000Å). We also select the 12 template spectra that have significant emission lines, and retain each of them for our final archetype set. This leaves 866 spectra from which to select the remainder of our archetype sample.
To select a subset of archetypes from the remaining set of templates, we wish to make use of a measure of the degree of similarity or difference between any two spectra. We first restrict our attention to the wavelength range 3400Å–11000Å, corresponding to $N_{\mathrm{pix}} = 5099$ pixels at the processed 69kms$^{-1}$ BOSS spectrum pixel scale. We then re-normalize all the template spectra to satisfy $$\sum_{i=1}^{N_{\mathrm{pix}}} f_i^2 = N_{\mathrm{pix}} ~,$$ where $f_i$ is the flux density (in the $f_{\lambda}$ sense) in pixel $i$. We define a statistic $s^2$ measuring the quality of spectrum $f^{\prime}$, scaled by a factor $a$, as a model for spectrum $f$: $$s^2 = \sum_{i=1}^{N_{\mathrm{pix}}} (f_i - a f^{\prime}_i)^2 ~.$$ With our normalization convention, the best-fit (minimum-$s^2$) scaling is simply given by $$a_{\mathrm{best}} = N^{-1}_{\mathrm{pix}} \sum_{i=1}^{N_{\mathrm{pix}}} f_i f^{\prime}_i ~,$$ and the value of $s^2$ at this best scaling is given by $$s^2_{\mathrm{best}} = N_{\mathrm{pix}} (1 - a_{\mathrm{best}}^2) ~.$$ Note that $a_{\mathrm{best}}$ and $s^2_{\mathrm{best}}$ are symmetric under the interchange of $f$ and $f^{\prime}$: the amplitude and fit quality of one template to another does not depend upon which one is taken as the “data” and which one as the “model”. Thus $s^2_{\mathrm{best}}$ can be regarded as a measure of how different two templates are from one another.
We compute the matrix of $s^2_{\mathrm{best}}$ between all pairs of templates in our set, and determine our archetype list in an iterative procedure. We set a threshold of 7.5 for the maximum $s^2_{\mathrm{best}}$ allowable between two spectra in order for one spectrum to be an acceptable representative for the other. This threshold was selected heuristically to tune the size and diversity of the final sample. We then identify the single template spectrum within the sample that has the most $s^2_{\mathrm{best}} < 7.5$ matches to the rest of the sample. This spectrum and all the spectra that it matches are removed from further consideration, and the process is iterated until all spectra have been accounted for in this manner. For our chosen threshold, this process identifies 105 archetypes out of 866 analyzed templates. When added to the 12 emission-line templates, this yields a set of 117 stellar templates for our automated spectrum classification algorithm. Spectra fit by these templates are tagged with the stellar subclass listed in the Indo-US database, along with the library identification number of the archetype spectrum.
### Special stellar subclasses
Several subclasses of star appear with some frequency in the BOSS target sample, but are not represented in the (flux-calibrated) Indo-US library. For these subclasses, representative training samples from within the BOSS data set are identified based upon classification using SDSS-I/II stellar templates. New templates are derived by averaging the spectra of these training sets within a PCA framework. The six subclasses and the number of training spectra for each of them are: (1) 47 carbon stars; (2) 50 “hotter” white dwarfs with $u - g < 0.3$; (3) 50 “cooler” white dwarfs with $u - g \ge 0.3$; (4) 19 calcium white dwarfs; (5) 31 magnetic white dwarfs; and (6) 50 L dwarfs. In addition, a sample of 18 cataclysmic variable stars (CVs) with prominent emission lines is used to define a CV star eigenbasis of 3 PCA modes, which is shown in the bottom panel of Figure \[fig:templates\]. Because of the use of multiple eigenvectors rather than a single average spectrum, CV stars are treated as an object class separate from other stars in the automated classification analysis.
Performance and Verification {#sec:performance}
============================
Table \[table:dr9summary\] provides a summary of the BOSS DR9 spectroscopic data set analyzed by the redshift and classification pipeline described in this work, along with a number of summary performance statistics that we now examine. Additional checks on the `idlspec2d` pipeline performance for galaxy targets in comparison with the `zcode` cross-correlation redshift software described by @Cannon06 are presented in @Dawson12, and additional discussion of pipeline quasar classification and redshift performance is found in @Paris12. The BOSS DR9 sample contains 831,000 spectra. Of these, about 0.2% are lost to unplugged fibers and spectra falling along bad CCD columns. Approximately 92% of the BOSS DR9 spectra are of unique objects (as defined by a 2$^{\prime\prime}$ positional match). The remaining 8% are repeat spectra from overlapping plates or repeat observations of the same plate.
Galaxy redshift completeness and purity {#subsec:galcomp}
---------------------------------------
Using the `Z_NOQSO` redshift measurement convention as described in §\[subsec:z\_noqso\], we achieve an automated completeness (i.e., `ZWARNING_NOQSO == 0` rate) of 98.7% for the CMASS sample and 99.9% for the LOWZ sample (from Table \[table:dr9summary\]). Restricting further to objects that are spectroscopically classified as galaxies (`CLASS_NOQSO == GALAXY`), we find combined targeting and measurement completeness percentages of 95.4% for CMASS and 99.2% for LOWZ. These percentages satisfy the BOSS science requirement of at least 94% overall galaxy redshift success. For the CMASS sample, about 70% of the (small) survey inefficiency is due to targeting stars and star–galaxy superpositions rather than galaxies, and about 30% arises from known redshift measurement failures.
To verify the completeness and quantify the purity of the automated galaxy redshifting and classification, we make use of a “truth table” generated by the first author from the visual inspection of 4864 galaxy spectra taken on eight plates observed during 2010 March.[^5] We focus primarily on the CMASS sample, as this higher-redshift (and thus lower S/N) sample poses the greatest challenge to the software. Of the inspected spectra, 3666 are CMASS targets that are above the fiber-magnitude threshold, not unplugged, and not falling on bad CCD columns. From among these 3666 galaxy-sample spectra, 3627 have confidently measured pipeline redshifts and classifications, giving an automated completeness of 98.9%, consistent with the completeness of the full DR9 CMASS sample from above. Of this subset, 3500 are classified as galaxies (as opposed to stars) by the pipeline, giving a 95.5% overall sample completeness including target-selection efficiency, which is also consistent with the sample-wide value.
To quantify the purity of the CMASS spectroscopic redshift sample, we first search for “catastrophic” impurities in the CMASS redshift sample, defined as spectra for which the pipeline reports a confident galaxy classification and redshift, but for which the visual inspection yields a confident classification (of any class) with a redshift that differs by greater than $\Delta z = 0.005$. This search yields three such spectra out of 3500: two are definite galaxy–M-star superpositions, and the other is a possible galaxy-galaxy superposition (for which the pipeline redshift is more convincing in retrospect than the inspection redshift). We next check for less clearly defined impurities, defined as spectra for which the pipeline reports a confident galaxy classification and redshift, but for which the visual inspection does not yield a confident result. This search identifies 10 such spectra, six of which are plausible pipeline redshifts with subjective visual judgments of low S/N, and the remaining four of which are due to artifacts associated with spectrum combination across the spectrograph dichroic at 6000Å(see Item \[item:crosstalk\] in §\[sec:issues\]). Taking the three superposition spectra and the four artifact spectra as genuine contaminants, we find a CMASS sample impurity rate of about 0.2%, satisfying the 1% maximum catastrophic redshift failure rate specified as the scientific requirement for BOSS.
To check for the possibility of recoverable incompleteness, we examine CMASS spectra for which the visual inspections yield a confident galaxy classification and redshift, but for which the automated pipeline yields either no confident result (i.e., `ZWARNING_NOQSO > 0`), or a classification as a star. There are 26 such spectra, which break down as follows: 11 low-S/N spectra for which the pipeline’s lack of confidence is statistically defensible; 5 definite or possible galaxy–galaxy superpositions; 4 definite or possible star–galaxy superpositions; 3 spectra with artifacts; 2 broadline AGN mistaken for stars (but with correct quasar-class redshifts that are excluded by the `Z_NOQSO` convention); and 1 narrow-line AGN for which the pipeline confuses \[O<span style="font-variant:small-caps;">iii</span>\] 5007 and H$\alpha$. Taking the 11 noisy but visually convincing redshifts and the three AGN spectra to represent the recoverable sample, we find an excess incompleteness of about 0.4% relative to the maximum attainable given the data.
To further assess the effects of star–galaxy superpositions (for which the pipeline takes no special approach), we search a set of 57910 CMASS spectra from 150 plates for instances of a best-fit non-quasar class of GALAXY and a next-best non-quasar class of STAR, and examine these spectra visually for the presence of significant stellar features. From this sample, we find 103 possible and 58 probable star–galaxy superpositions, indicating a total CMASS star–galaxy superposition rate of between 0.1% and 0.2%. These star–galaxy superpositions that are given a spectroscopic class of GALAXY are a source of sample impurity, as the galaxies are typically neither bright enough nor of the correct color to fall within the CMASS color–magnitude selection cuts on their own. Any star–galaxy superpositions classified as STAR by the pipeline are excluded from the large-scale structure analysis and contribute only to target-selection inefficiency.
Our visual inspection set also contains 568 LOWZ galaxies brighter than the fiber-magnitude threshold. All of these spectra are confidently classified and redshifted by both the pipeline and the visual inspection, with three classified as stars. This is consistent with the automated completeness and stellar contamination rate for the full LOWZ sample, with no detectable incidence of catastrophic failures.
Galaxy redshift precision
-------------------------
Redshift errors are calculated from the curvature of the $\chi^2$ function in the vicinity of the minimum value that is used to determine the best-fit redshift measurement. To assess the accuracy of these statistical error estimates, we make use of a set of 27170 repeat observations of CMASS targets and 7503 repeat observations of LOWZ targets within the DR9 data set. We reference all repeat observations to the `SPECPRIMARY` observation of a given object, and scale the redshift difference between the two observations by the quadrature sum of the error estimates from the two epochs. We then construct a histogram of these scaled velocity differences and fit it with a Gaussian function. If the estimated errors accounted for all the statistical uncertainty, these fitted Gaussians would have a dispersion parameter of unity. Figure \[fig:z\_err\_hist\] shows the results of this analysis, with fitted dispersions of $\sigma = 1.34$ for the LOWZ sample and $\sigma = 1.19$ for the CMASS sample. Thus, while slightly underestimated, the redshift errors are impressively close to being statistically accurate. The greater scatter (relative to the statistical error estimates) for the LOWZ sample suggests that systematic effects become more important at higher S/N.
This analysis of repeat spectra also yields 44 CMASS re-observations that have absolute redshift differences $|\Delta z| > 0.005$ between the two epochs. These are primarily due to galaxy–galaxy superpositions at distinct redshifts, un-masked spectrum artifacts, and a number of type II quasars for which broad \[O<span style="font-variant:small-caps;">iii</span>\] 5007 emission is confused with H$\alpha$ in one epoch but not the other (see Item \[item:type2\] in §\[sec:issues\]). The implied 0.16% CMASS redshift impurity rate is consistent with the value found from the truth-table tests of §\[subsec:galcomp\]. For the LOWZ repeat observations, two spectra yield $|\Delta z| > 0.005$, both of which are galaxy–galaxy superpositions.
For all CMASS and LOWZ targets, we also compute the distribution of estimated redshift errors as a function of redshift. These distributions are shown in Figure \[fig:v\_err\_hist\]. In all cases, typical errors are a few tens of kms$^{-1}$ even when scaled up to reflect the super-statistical scatter displayed in Figure \[fig:z\_err\_hist\]. These errors are well below the 300kms$^{-1}$ redshift precision requirement of the BOSS galaxy large-scale structure science analyses.
Galaxy redshift success dependence {#subsec:zfaildepend}
----------------------------------
As in any redshift survey, spectroscopic S/N is the primary determinant of redshift success in BOSS. Figure \[fig:sdss\_snr\_zrate\] shows the dependence of the CMASS galaxy redshift failure rate as a function of the median spectroscopic signal-to-noise ratio over the SDSS $r$, $i$, and $z$ bandpass ranges, which represent the most relevant regions of the spectrum for measuring continuum redshifts of passive galaxies over the redshift interval $z \approx 0.4$–0.8. Failure is defined in the sense of `ZWARNING_NOQSO > 0`, so that targets confidently identified as stars are counted as a success for the pipeline even though they represent a failure in the larger sense of galaxy targeting and redshift measurement. We see a decrease in the failure rate as a function of $r$-band S/N up to S/N$_r \simeq 3$, where an asymptotic minimum of $\approx 5 \times 10^{-3}$ is reached. For CMASS spectra with S/N$_r = 3$, the typical value for both S/N$_i$ and S/N$_z$ is approximately 6, consistent with the S/N values in those bands at which the asymptotic failure rate is reached in Figure \[fig:sdss\_snr\_zrate\].
Galaxy magnitude correlates strongly with spectroscopic S/N and hence with redshift success: this is the motivation for the formal CMASS sample limit of $i$-band magnitude brighter than 21.5 within a 2$\arcsec$-diameter BOSS fiber. To gauge the dependence of redshift completeness on this limit, Figure \[fig:ifibrate\] shows the CMASS sample redshift failure rate as a function of $i_{\mathrm{fiber}}$, selecting the best single observation of each target. Targets fainter than $i_{\mathrm{fiber}} = 21.5$ are available from a more permissive CMASS cut applied during commissioning observations. At the formal CMASS cutoff, the marginal failure rate is about 7%.
The characteristics of the BOSS spectrograph optics and CCD detectors produce a weak dependence of redshift success rate on fiber identification number along the linear spectrograph slit-heads. Figure \[fig:fiberid\] presents this dependence for the CMASS sample. This figure is generated only for targets brighter than the $i_{\mathrm{fiber}} < 21.5$ cut, but includes all survey spectra of each target (i.e., no `SPECPRIMARY` cut) so as to give an unbiased picture of performance versus fiber number. The upturns near fiber numbers 1, 500, and 1000 are associated with the edges of the spectrograph camera fields of view, and are described further in Item \[item:badfiber\] in §\[sec:issues\] below. The effects of isolated bad CCD columns are also evident, and are described in Item \[item:badcol\] in §\[sec:issues\]. The failure rate is slightly higher on average for fibers above 500, corresponding to a lower end-to-end survey-averaged throughput for the optics and CCDs of spectrograph 2 as compared to those of spectrograph 1.
In principle, variations in the quality of sky foreground subtraction can also affect spectroscopic redshift success. In practice, we do not see this effect in BOSS. Figure \[fig:skyres\] shows the spectrum of systematic sky-subtraction residual flux measured from the sky-subtracted blank-sky fibers of a representative BOSS plate, calculated by subtracting statistical spectrum pixel error estimates in quadrature from the root-mean-square (RMS) residual spectrum across all sky fibers on the plate. At the positions of bright OH air-glow lines, the systematic residuals are generally at or below 1% of the sky flux. To test whether the redshift failure rate is affected significantly by variations in sky-subtraction quality, we quantify the level of residual flux from the sky-subtraction process in each plate as the RMS flux in all sky-subtracted blank-sky fibers over the wavelength range 8300Å to 10400Å, where the effects of OH air-glow lines are particularly pronounced. Figure \[fig:sky\_zrate\] displays the results of this test, with RMS residual flux expressed both in units of estimated statistical significance and in units of specific flux. In both cases, there is no discernible correlation between sky-subtraction residual scale and redshift failure rate. The two conclusions we draw are that (1) the quality of BOSS sky subtraction is uniformly high, and (2) residual variations in the quality of this sky subtraction do not significantly affect redshift measurement for the passive, continuum-dominated CMASS galaxies.
Quasar redshift success
-----------------------
Unlike the BOSS galaxy samples, the BOSS quasar sample does not have a stated requirement on automated classification and redshift success. The entire quasar target sample is being manually inspected to provide a catalog of visually verified classifications and redshifts [@Paris12], for which the automated BOSS pipeline redshifts provide the initial default value. From Table \[table:dr9summary\], we find that the `idlspec2d` pipeline reports a confident classification and redshift (i.e., `ZWARNING == 0`) for about 79% of the unique spectra of the BOSS quasar target sample. The majority of the remaining 21% of the quasar sample observations are low-S/N spectra of faint targets. Approximately 51.5% of the unique observed quasar sample targets are spectroscopically confirmed as quasars; most of the confidently classified non-quasar spectra are stars (typically of spectral type F) occupying the same region of color space as quasars in the targeted redshift range. However, only 33.6% of the unique target sample are confirmed quasars at the redshifts $2.2 < z < 3.5$ which are the focus of the BOSS Ly$\alpha$ forest analysis [@Dawson12]. Figure \[fig:qso\_conf\_rate\] presents the spectroscopic confirmation rate for quasars in this redshift range $2.2 < z < 3.5$, as a function of median S/N per pixel over the $g$-band wavelength range.
The full comparison of visual redshifts and pipeline redshifts for BOSS quasar-sample targets is presented in @Paris12, and is beyond the scope of this current work. We note two particular statistics here. First, the visual inspections provide a 1.7% increase in the sample of $2.2 < z < 3.5$ quasars beyond those that are confidently identified by the automated pipeline. Second, 0.6% of the quasars identified confidently by the pipeline at $2.2 < z < 3.5$ either have redshifts in disagreement by $|\Delta z| > 0.05$ with the visual-inspection values, or do not have confident visual identification despite having been inspected. The latter are due mostly to extremely broad absorption-line quasars and to line mis-identifications. The overall conclusion, however, is that the completeness and purity of the automated quasar classification and redshift measurement is quite high.
Figure \[fig:z\_err\_hist\_qso\] shows the distribution of error-scaled redshift differences for 1464 repeat BOSS observations of confirmed quasars, as well as the redshift-dependent distributions of statistical single-epoch redshift error estimates, analogous to Figures \[fig:z\_err\_hist\] and \[fig:v\_err\_hist\] for galaxies. For quasars, the *statistical* pipeline redshift errors are underestimated by a factor of approximately two, although the true errors in the pipeline quasar redshifts are likely dominated by systematic effects. Eight of the repeat observations, or about 0.5%, give a redshift difference of $|\Delta z| > 0.05$, consistent with the rate of catastrophic errors found by the comparison with the visual inspections. The redshift range 1.0–2.0 is particularly difficult since the observed optical spectra do not have either the narrow \[O<span style="font-variant:small-caps;">iii</span>\] 5007 line or the strong Ly$\alpha$ line to guide the template fit.
Stellar radial velocity precision
---------------------------------
We now briefly examine the precision and accuracy of BOSS stellar radial velocities based on stellar repeat observations. Specifically, we identify 8174 repeat observations of objects classified as `STAR` with `ZWARNING` $== 0$ for both epochs. In Figure \[fig:starvdiff\], we plot the velocity difference between the two epochs of these repeats against the quadrature sum of their statistical error estimates. We see that the distribution becomes tighter at higher S/N as expected, with reasonably good agreement between estimated statistical error and actual velocity differences above approximately 15kms$^{-1}$ in combined statistical error (or approximately 10kms$^{-1}$ in single-epoch error). Subtracting the statistical error estimates in quadrature from the half-difference between the 84$^{\mathrm{th}}$ and 16$^{\mathrm{th}}$ percentile velocity differences, and dividing by a factor of $\sqrt{2}$ to convert to a single-epoch value, we find a systematic radial-velocity floor of approximately 4.5kms$^{-1}$ at the high S/N end, comparable to the 4kms$^{-1}$ precision attained for bright stars by the SEGUE project in SDSS-I/II [@Yanny09].
Known Issues {#sec:issues}
============
In order to freeze a set of reductions for collaboration analysis and public release, we have accepted the presence of a number of known outstanding issues in the software that either were deemed small enough in a statistical sense within the survey, or were discovered after the software freeze deadline. These issues are documented in the following list, and several are illustrated in Figure \[fig:crappo\].
1. PCA fits of the `GALAXY` and `QSO` classes can sometimes yield unphysical basis combinations at low S/N. This effect is part of the motivation for the `Z_NOQSO` redshifts described in §\[subsec:z\_noqso\], and is illustrated in panel “a” of Figure \[fig:crappo\]. In order to enable a targeting-blind spectroscopic classification of the sort used in SDSS-I/II, this effect could be remedied by priors on physical PCA coefficient combinations, or by non-negativity requirements on archetype-based models such as are used for non-CV stellar classifications in `idlspec2d`. These alternatives are the subject of ongoing development for future BOSS data releases.
2. \[item:type2\] A small number of type II quasars [e.g., @Zakamska03] at redshift $z \sim 0.5$ are selected by the CMASS cuts due to their colors, but their obscured-AGN spectra are not typical of the majority of galaxies used to train the galaxy redshift templates. The inclusion of several such systems in the galaxy-template training set has addressed this issue partially, but a number of these objects have a best-fit galaxy-template redshift that confuses broad \[O<span style="font-variant:small-caps;">iii</span>\] 5007 for H$\alpha$. Their quasar-template redshifts are generally correct, but due to the `Z_NOQSO` redshift strategy employed for the BOSS galaxy samples (§\[subsec:z\_noqso\]), their adopted redshifts are often in error (see panel “b” of Figure \[fig:crappo\].) Since these objects represent such a small percentage of the BOSS galaxy target samples, these errors were deemed acceptable for DR9 galaxy-clustering analyses.
The fundamental problem is that the spectra of type II quasars are sufficiently different from the spectra of most BOSS galaxies that we cannot span the space of both categories with the current number of PCA templates (four) in the single `GALAXY` basis set. In future BOSS data releases, we anticipate addressing this issue through either higher-dimensional basis sets with physical coefficient priors, sub-division of the `GALAXY` class into several subclasses each with its own basis set, or an archetype-based galaxy redshifting algorithm.
3. \[item:crosstalk\] A small number of spectra are affected by cross-talk from bright stars (generally spectrophotometric standards) in neighboring fibers. This is often manifested in a strong break feature at the dichroic transition around 6000Å(see panel “c” of Figure \[fig:crappo\]), due to different levels of cross-talk between the red and blue arms of the spectrograph [@Smee12]. These effects appear to occur less frequently at later survey dates, presumably because of improvements in the operating focus of the BOSS spectrographs. We intend to address these effects in future BOSS data releases through improvements in the extraction codes, and to flag any spectra that remain compromised. No masking of this effect is implemented for BOSS DR9 data, however, except to the extent that it sometimes triggers a `ZWARNING` flag.
4. \[item:badfiber\] As discussed in §\[subsec:zfaildepend\] and shown in Figure \[fig:fiberid\], the BOSS redshift success rates are somewhat dependent on fiber number in the sense that fibers near the edge of the spectrograph camera fields of view (`FIBERID` values near 1, 500, and 1000) have lower success rates. Longer-term development of new extraction codes based on the 2D PSF-modeling approach of @Bolton10 is ongoing, and may mitigate this problem to a significant extent.
5. \[item:badcol\] A few columns in the BOSS CCDs are bad only in a transient sense, and are not included in the bad-column masks applied to the CCD frames. These columns lead to occasional spectrum artifacts concentrated near particular fiber numbers (see panel “d” of Figure \[fig:crappo\]) that are not masked or flagged.
6. White-dwarf, L-dwarf, carbon-star, and cataclysmic-variable star subclasses have less accurate template radial-velocity zero-points in comparison to the stellar archetypes derived from the Indo-U.S. library. This issue may be rectified in future data releases, although the primary role of stellar templates in BOSS will remain to correctly classify and set aside non-galaxies and non-quasars.
7. Spectra showing superpositions of two objects are not systematically identified and flagged by the pipeline. While the majority of BOSS spectra are of single objects, superpositions are occasionally found to occur. In some cases, the inclusion of the polynomial terms in the redshift model fitting leads to fits of almost equal quality for the two components individually, leading to a `SMALL_DELTA_CHI2` flag in the `ZWARNING` (or `ZWARNING_NOQSO`) mask. In other cases, one component is dominant and is identified by the pipeline as the confident classification and redshift, but with the second component typically identified by one of the lower-quality fits reported in the `spZall` file. Various examples of superposition spectra are displayed in Figure \[fig:crappo\], including star–star (panel “e”), star–galaxy (panel “f”), and galaxy–galaxy (panel “g”). A systematic search for superposition spectra in the BOSS data set by the BOSS Emission-Line Lens Survey (BELLS, @Brownstein12) has discovered a large sample of strong gravitational lens galaxies.
Summary and Conclusion {#sec:summary}
======================
We have described the “1D” component of the `idlspec2d` pipeline that provides automated redshift measurement and and classification for the SDSS-III BOSS DR9 data set, which comprises 831,000 optical spectra. This software is substantially similar to the `idlspec2d` redshift analysis code used for SDSS-I/II data, but has been upgraded with new templates and several new algorithms for application to the BOSS project, and has been presented in great detail for the first time in this work. The pipeline also provides additional parameter measurements, including emission-line fits for all objects, and velocity-dispersion likelihood curves for objects classified as galaxies. The redshift success rate of the `idlspec2d` pipeline is well in excess of the scientific requirements of the BOSS project. The software provides first-principles estimates of statistical redshift errors that are Gaussian distributed and accurate to within small correction factors. The “2D” component of the `idlspec2d` pipeline that extracts spectra from raw CCD pixels is the subject of @Schlegel12. Full data-model information for both the 2D and 1D BOSS pipeline outputs can be found at the SDSS-III DR9 website (<http://www.sdss3.org/dr9/>).
Development work continues on data-reduction software for BOSS, both in the calibration and extraction of spectra, and in the classification and redshift analysis procedures. Subsequent BOSS data releases will be accompanied by similar documentation of the implemented results of this ongoing development.
Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is <http://www.sdss3.org/>.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, The University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
This research has made use of the POLLUX database (`http://pollux.graal.univ-montp2.fr`) operated at LUPM (Université Montpellier II - CNRS, France) with the support of the PNPS and INSU.
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[^1]: Throughout this paper, we will refer to the earlier SDSS phases collectively as SDSS-I/II.
[^2]: The DR9 tagged version of `idlspec2d` can be obtained at `www.sdss3.org/svn/repo/idlspec2d/tags/v5_4_45/`.
[^3]: `http://www.sdss3.org/dr9/`
[^4]: These spectra are tabulated in the file `eigeninput_gal.dat` within the `templates` subdirectory of the `idlspec2d` product.
[^5]: The plates are: 3804 of MJD 55267; 3686, 3853, and 3855 of MJD 55268; and 3687, 3805, 3856, and 3860 of MJD 55269.
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bibliography:
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title: On the number of graphs without large cliques
---
On the number of graphs without large cliques
Institute of Theoretical Computer Science\
ETH Zürich, 8092 Zürich, Switzerland\
[{[moussetf|rnenadov|steger]{}}[@inf.ethz.ch]{}]{}
<span style="font-variant:small-caps;">Abstract.</span>
In 1976 Erdős, Kleitman and Rothschild determined the number of graphs without a clique of size $\ell$. In this note we extend their result to the case of forbidden cliques of increasing size. More precisely we prove that for $\ell_n \le (\log n)^{1/4}/2$ there are $$2^{(1-1/(\ell_n-1))n^2/2+o(n^2/\ell_n)}$$ $K_{\ell_n}$-free graphs of order $n$. Our proof is based on the recent hypergraph container theorems of Saxton, Thomason and Balogh, Morris, Samotij, in combination with a theorem of Lovász and Simonovits.
Introduction
============
Let $F$ be an arbitrary graph. A graph $G$ is called $F$-free if $G$ does not contain $F$ as a (weak) subgraph. Let $f_n(F)$ denote the number of (labeled) $F$-free graphs on $n$ vertices. As every subgraph of an $F$-free graph is also $F$-free, we trivially have $f_n(F)\ge 2^{\ex(n,F)}$, where $\ex(n,F)$ denotes the maximum number of edges of an $F$-free graph on $n$ vertices. It is well known [@t41; @es46; @ES66] that $$\ex(n,F)= \left(1-\frac1{\chi(F)-1}\right)\frac{n^2}2 + o(n^2).$$ Erdős, Kleitman and Rothschild [@ekr76] showed that in the case of cliques, i.e., for $F=K_\ell$, this lower bound actually provides the correct order of magnitude. Erdős, Frankl and Rödl [@EFR] later showed that a similar result holds for all graphs $F$ of chromatic number $\chi(F)\ge 3$: $$\label{eq:fixedl}
{f_n(F)} = 2^{(1+o(1))\ex(n,F)}.$$\[eq:struc\] Note that these results just provide the asymptotics of $\log_2(f_n(F))$. Extending an earlier result from [@ekr76] for triangles, Kolaitis, Prömel and Rothschild [@KPR] determined the typical structure of $K_{\ell}$-free graphs by showing that almost all of them are $(\ell-1)$-colorable. Thus, $$f_n(K_\ell) = (1+o(1))\cdot\text{col}_n(\ell-1),$$ where $\text{col}_n(\ell)$ denotes the number of (labeled) $\ell$-colorable graphs on $n$ vertices. An asymptotic for $\text{col}_n(\ell)$ is given in [@PS95]. For additional results and further pointers to the literature see e.g. [@ABBM11; @bbs04; @BS11; @PS92]. All of the above results consider the case of a [*fixed*]{} forbidden graph $F$. Much less is known if the size of the forbidden graph $F$ increases with the size of the host graph $G$. The study of such situations was started only recently by Bollobás and Nikiforov [@BollobasNikiforov10]. They showed that for any sequence $(F_n)$ of graphs with $v(F_n)=o(\log{n})$ one has $$\label{eq:bollobas-nikiforov}
\log_2{f_n(F_n)} =\left(1-\frac{1}{\chi(F_n)-1}\right)\binom{n}{2}+o(n^2)\text.$$ It is interesting to note that the proof of completely avoids the use of the regularity lemma, a common tool for attacking this kind of questions. Indeed, because of the tower-type dependence of the size of an $\varepsilon$-regular partition on the parameter $\varepsilon$ (see [@Gowers97]), it seems hard to adapt the regularity-based proof of to the case of forbidden subgraphs of non-constant size. Furthermore, observe that is only non-trivial if the chromatic number $\chi(F_n)$ is bounded. In particular, it does not determine $\log_2 f_n(K_{\ell_n})$ for an increasing sequence $\ell_n$ of positive integers, because the term $o(n^2)$ [swallows]{} the lower-order term $\binom{n}{2}/(\ell_n-1)$.
The aim of this paper is to provide the first non-trivial result for forbidden cliques of increasing size:
\[thm:1\] Let $(\ell_n)_{n\in\mathbb{N}}$ be a sequence of positive integers such that for every $n\in \mathbb N$, we have $3\leq \ell_n\le(\log{n})^{1/4}/2$. Then $$\log_2{f_n(K_{\ell_n})}= \left(1-\frac{1}{\ell_n-1}\right)\binom{n}{2}+o(n^2/\ell_n).$$
Our proof is based on the recent powerful hypergraph container results of Balogh, Morris, Samotij [@bms12] and Saxton and Thomason [@SaxtonThomason12].
The upper bound on $\ell_n$ in our theorem is an artifact of our proof. We have no reason to believe that this bound is tight. In fact, it is not unconceivable that the statement from Theorem \[thm:1\] holds up to the size of a maximal clique in the random graph $G_{n,1/2}$ which is known to be $(2+o(1))\log_2 n$. We leave this question to future research.
Note also that, similarly to the result of Erdős, Kleitman and Rothschild [@ekr76], our theorem just provides the asymptotics of the logarithm of $f_n(K_{\ell_n})$. However, our paper has already stimulated further research, and very recently a structural result in the spirit of Kolaitis, Prömel and Rothschild has been established by Balogh et al. [@BLS].
Hypergraph Containers
=====================
In the proof of Theorem \[thm:1\], we make use of the hypergraph container theorem proved independently by Saxton and Thomason [@SaxtonThomason12] and Balogh, Morris and Samotij [@bms12]. Before we state this theorem, we introduce some notation.
Let $H$ be an $r$-uniform hypergraph with the average degree $d$. Then for every $\sigma \subseteq V(H)$, we define the *co-degree* $$d(\sigma) = |\{e \in E(H) : \sigma \subseteq e\}|.$$ Moreover, for every $j \in [r]$, we define the $j$-th *maximum co-degree* $$\Delta_j = \max{\{d(\sigma) : \sigma\subseteq V(H)\text{ and }|\sigma|=j\}}.$$ Finally, for any $p\in(0,1)$, we define the function $$\Delta(H,p) = 2^{\binom{r}{2}-1}\sum_{j=2}^{r}2^{-\binom{j-1}{2}}\frac{\Delta_j}{d p^{j-1}}.$$ We will use the following version of the general hypergraph container theorem.
\[thm:container\] There exists a positive integer $c$ such that the following holds for all positive integers $r$ and $N$. Let $H$ be an $r$-uniform hypergraph of order $N$. Let $0\leq p\leq 1/(cr^{2r})$ and $0< \varepsilon< 1$ be such that $\Delta(H,p)\leq \varepsilon/(cr^r)$. Then there exists a collection $\mathcal{C}\subseteq \mathcal{P}(V(H))$ such that
(i) every independent set in $H$ is contained in some $ C\in \mathcal{C}$,
(ii) for all $C\in\mathcal{C}$, we have $e(H[C]) \leq \varepsilon e(H)$, and
(iii) the number $|\mathcal{C}|$ of containers satisfies $$\log{|\mathcal{C}|} \leq cr^{3r}(1+\log(1/\eps))Np\log(1/p).$$
Theorem \[thm:container\] is an easy consequence of Theorem 5.3 in the paper of Saxton and Thomason [@SaxtonThomason12]; it is derived exactly as Corollary 2.7 (also therein), the only difference being that we are precise about the dependence on the edge size $r$. We also mention that our notation deviates slightly from that used in [@SaxtonThomason12]: we write $p$ instead of $\tau$ and we use $\Delta(H,p)$ as an upper bound for the function $\delta(H,\tau)$ used by Saxton and Thomason. Finally, let us just note without further explanation that Theorem \[thm:container\] is much weaker than the general container theorem, although it is sufficient for the purposes of this note (and is simpler to state and apply).
As a corollary of Theorem \[thm:container\] we prove the following version tailored for a collection of $K_{\ell}$-free graphs, with $\ell$ being a function of $n$.
\[cor:container\] For every constant $\delta>0$ and sequence $(\ell_n)_{n\in\mathbb N}$ such that $3\le \ell_n\le(\log{n})^{1/4}/2$, the following holds for all large enough $n\in\mathbb N$: there exists a collection $\mathcal{G}$ of graphs of order $n$ such that
(i) every $K_{\ell}$-free graph of order $n$ is a subgraph of some $G \in \mathcal{G}$,
(ii) every $G\in\mathcal{G}$ contains at most $\delta\binom{n}{\ell_n}/e^{\ell_n}$ copies of $K_{\ell_n}$, and
(iii) the number $|\mathcal{G}|$ of graphs in the collection satisfies $$\log{|\mathcal{G}|} \leq \delta n^2/\ell_n.$$
Let us assume that $n$ is large enough and write $\ell:=\ell_n$.
Let $H$ be a hypergraph defined as follows: the vertex set of $H$ is the edge set of $K_n$, and the edges of $H$ are the edge sets of subgraphs of $K_{n}$ isomorphic to $K_{\ell}$. Observe that the graph $H$ is an $\binom{{\ell}}{2}$-uniform hypergraph of order $\binom{n}{2}$ with $e(H)=\binom{n}{{\ell}}$. With some foresight, we would like to apply Theorem \[thm:container\] with $$\varepsilon = \delta e^{-{\ell}} \quad \text{and} \quad p = n^{- (\log{\ell})/(2\ell^2)}\label{eq:p}$$ to the hypergraph $H$. We first verify that this is indeed possible, that is, that $$\label{eq:delta_c}
\Delta(H, p) \leq \delta \left(c\cdot e^{\ell}\binom{\ell}{2}^{\binom{\ell}{2}}\right)^{-1}$$ and $$\label{eq:p_c}
p \leq \left( c\cdot \binom{\ell}{2}^{2\binom{\ell}{2}}\right)^{-1}\text,$$ for every positive integer constant $c$.
Let us start with the values $\Delta_j$ for $H$. Consider some $\sigma\subseteq V(H)$ with $|\sigma|=j$, where $1\leq j \leq \binom{\ell}{2}$. We can view $\sigma$ as a subgraph of $K_n$ with $v(\sigma) = \left| \bigcup \{e:e\in \sigma\} \right|$ vertices and $|\sigma|=j$ edges. The co-degree $d(\sigma)$ is then simply the number of ways in which we can extend this graph to a copy of $K_{\ell}$ in $K_{n}$. If $v(\sigma) > \ell$ then clearly $d(\sigma) = 0$, and otherwise $$d(\sigma) = \binom{n-v(\sigma)}{{\ell}-v(\sigma)}\leq n^{{\ell}-v(\sigma)}.$$ Note that $j\leq \binom{v(\sigma)}{2}$ implies that $$v(\sigma)\geq \frac{1+\sqrt{1+8j}}{2}> \frac12+\sqrt{2j}\text,$$ giving the bound $$\Delta_j\leq n^{\ell-1/2-\sqrt{2j}}\text.$$ On the other hand, using that that $\ell\leq (\log{n})^{1/4}/2$ and that $n$ is sufficiently large, the average degree $d$ of $H$ is $$d = \binom{n-2}{{\ell}-2} \geq \left(\frac{n}{{\ell}}\right)^{{\ell}-2}\ge n^{\ell-1.9}\text,$$ so for $2\leq j \leq \binom{\ell}{2}$, we have $$\frac{\Delta_j}{dp^{j-1}}\leq n^{1.4-\sqrt{2j}+(j-1)(\log{\ell})/(2\ell^2)}\text.$$ Using the fact that $\log(\ell)/\ell\leq 1/e$ holds for all $\ell>0$, we have, for every $2\leq j\leq
\binom{\ell}{2}$, that $$\sqrt{2j}-\frac{(j-1)(\log{\ell})}{2\ell^2} \ge
\sqrt{2j}-\frac{(\log{\ell})\sqrt{j}}{2\sqrt{2}\ell} \ge \sqrt{2j}- \frac{\sqrt{j}}{2e\sqrt{2}}
\geq 2-\frac{1}{2e}\text,$$ whence, for sufficiently large $n$, $$\frac{\Delta_j}{dp^{j-1}}\leq n^{1.4-2+1/(2e)} \leq n^{-1/4}\text.$$ Then, using $\ell \le (\log{n})^{1/4}/2$, for large enough $n$, we get $$\Delta(H, p) \leq e^{\ell^4} \cdot \sum_{j = 2}^{\binom{\ell}{2}} 2^{- \binom{j - 1}{2}} \cdot
n^{-1/4}
\leq e^{\ell^4} n^{-1/4}
\leq \delta/(ce^{\ell^4})\text,$$ which easily implies the desired bound on $\Delta(H, p)$. On the other hand, again using $\ell\le (\log{n})^{1/4}/2$, we have $$\label{eq:upper_p}
p = n^{-(\log{\ell})/(2\ell^2)}\leq 1/(c\ell^{4\ell^2})$$ for all large enough $n$, so $p$ satisfies . Therefore, we can apply Theorem \[thm:container\] with parameters $\varepsilon$ and $p$.
We now turn to the construction of the family $\mathcal{G}$. Let $\mathcal{C}$ be a collection of subsets of $V(H)$ given by Theorem \[thm:container\]. We show that the family of graphs $$\mathcal{G} = \{ ([n], C) : C \in \mathcal{C}\}$$ satisfies the claim.
Suppose that $I$ is some $K_{\ell}$-free graph on the vertex set $[n]$. Then its edge set $E(I)$ is an independent set in $H$, and thus there exists $C \in \mathcal{C}$ such that $E(I) \subseteq C$. Therefore there exists $G \in \mathcal{G}$ such that $I$ is a subgraph of $G$, and the property *(i)* holds. Furthermore, since $e(H[C]) \leq \varepsilon e(H)$ for each $C \in \mathcal{C}$, it follows that the number of copies of $K_{\ell}$ in each $G \in \mathcal{G}$ is also bounded by $\binom{n}{\ell} / e^{\ell}$, satisfying property *(ii)*. It remains to show that $\log{|\mathcal{C}|} = o(n^2 / \ell)$, which then implies property *(iii)*. Straightforward calculation yields that for large enough $n$, we have $$\begin{aligned}
\log{|\mathcal{C}|} &\leq c\binom{{\ell}}{2}^{3\binom{{\ell}}{2}}(1+\ell-\log\delta)\binom{n}{2}p\log(1/p) \\
&\stackrel{\text{\eqref{eq:upper_p}}}{\leq} \ell^{3\ell^2}(1+\ell-\log\delta)n^2 \left( \ell^{-4\ell^2}\cdot \log (c\ell^{7\ell^2}) \right) \\
&\leq \delta n^2/\ell,
\end{aligned}$$ where in the second line, we used together with the fact that $p\log (1/p)$ is monotonically decreasing. This finishes the proof of the corollary.
The requirement that $\ell_n\leq (\log{n})^{1/4}/2$ cannot be significantly improved upon with the same method. Indeed, the requirement that $\Delta(H,p)=o(1)$ implies that $2^{\binom{\binom{\ell_n}{2}}{2}}\Delta_2/d= o(1)$, which, since $\Delta_2/p=n^{-1+o(1)}$, implies $\ell_n=O((\log{n})^{1/4})$. We also note that the proof shows that, in fact, we have $\log{|\mathcal G|}= n^2e^{-\Omega(\ell_n^2\log{\ell_n})}$, which is much stronger than the bound $\log{|\mathcal G|}=o(n^2/\ell_n)$ that we need for the proof of Theorem \[thm:1\].
Proof of Theorem \[thm:1\]
==========================
Let us start with the easy part – proving the lower bound. Consider the $(\ell_n-1)$-partite *Turán graph*. That is, let $T$ be the complete $(\ell_n-1)$-partite graph of order $n$ whose partite sets have size either $\lceil n/(\ell_n-1) \rceil$ or $\lfloor n/(\ell_n-1) \rfloor$. Clearly, $T$ is a $K_{\ell_n}$-free graph, as is every subgraph of $T$. As there are at least $$2^{e(T)} \geq 2^{(\frac{n}{{\ell_n}-1}-1)^2\binom{{\ell_n}-1}{2}} \geq
2^{(1-\frac{1}{{\ell_n}-1})\binom{n}{2}+o(n^2/{\ell_n})}$$ subgraphs of $T$, the lower bound on the number of $K_{\ell_n}$-free graphs of order $n$ follows.
Now we turn to proving the upper bound. We show that for every $\delta > 0$ and large enough $n$, we have $$\log{f_n(K_{\ell_n})} \leq \left(1-\frac{1-\delta}{{\ell_n}-1}\right)\binom{n}{2}+\delta n^2/{\ell_n}.$$
We use the following Theorem of Lovász and Simonovits.
\[thm:supersat\] Let $n$ and $\ell$ be positive integers. Then every graph of order $n$ with at least $$\left(1-\frac{1}{t}\right)\frac{n^2}{2}$$ edges contains at least $\left(\frac{n}{t}\right)^{\ell}\binom{t}{\ell}$ copies of $K_{\ell}$.
Using Theorem \[thm:supersat\] together with Corollary \[cor:container\], we can now finish the proof of Theorem \[thm:1\] as follows. Fix some $\delta>0$ and assume that $n$ is large enough. Write $\ell:=\ell_n$ and apply Corollary \[cor:container\] for $\delta := \delta ^{1/\delta}$. We deduce that there exists a collection $\mathcal{G}$ of at most $2^{\delta n^2/\ell}$ graphs of order $n$ such that each contains at most $\delta^{1/\delta}\binom{n}{\ell}/e^{\ell}$ copies of $K_{\ell}$ and every $K_{\ell}$-free graph of order $n$ is a subgraph of some $G \in \mathcal{G}$.
By Theorem \[thm:supersat\], if a graph $G$ of order $n$ has at least $$\left(1-\frac{1-\delta}{{\ell}-1}\right)\frac{n^2}{2}$$ edges, then the number of copies of $K_{\ell}$ in $G$ is at least $$k(\ell) := \left(\frac{n(1-\delta)}{{\ell}-1}\right)^{\ell}\binom{({\ell}-1)/(1-\delta)}{{\ell}}.$$ If $\ell\geq 1/\delta$, then $(\ell-1)/(1-\delta)\geq \ell$ and we can use the bounds $ (\frac{a}b)^b\le \binom{a}b\le (\frac{ea}b)^b$ to obtain $$k(\ell)\geq \frac{n^{\ell}}{{\ell}^{\ell}} > \binom{n}{{\ell}}/e^{{\ell}}\text.$$ For $\ell<1/\delta$, we use the definition of the (generalized) binomial coefficient to deduce that, in this case, $$k(\ell)\geq \frac{n^{\ell}}{\ell!}\cdot \prod_{i=1}^{\ell-1}\left(1-\frac{i(1-\delta)}{\ell-1}\right)
> \delta^{1/\delta}\binom{n}{\ell}\text.$$ Since every $G\in\mathcal{G}$ has at most $\delta^{1/\delta}\binom{n}{\ell}/e^{\ell}$ copies of $K_{\ell}$, we deduce that every $G\in\mathcal G$ has fewer than $$\left(1-\frac{1-\delta}{{\ell}-1}\right)\frac{n^2}{2}$$ edges. We can now count the number of $K_{\ell}$-free graphs by counting the number of subgraphs of order $n$ of the graphs in $\mathcal{G}$, $$f_n(K_{\ell}) \leq |\mathcal{G}| \cdot 2^{(1-\frac{1-\delta}{{\ell}-1})\frac{n^2}{2}}
= 2^{(1-\frac{1-\delta}{{\ell}-1})\frac{n^2}{2}+\delta n^2/{\ell}},$$ completing the proof of the theorem.
|
---
abstract: 'We give numerical integration results for Feynman loop diagrams such as those covered by Laporta [@laporta01] and by Baikov and Chetyrkin [@baikov10], and which may give rise to loop integrals with UV singularities. We explore automatic adaptive integration using multivariate techniques from the [ParInt]{} package for multivariate integration, as well as iterated integration with programs from the [Quadpack]{} package, and a trapezoidal method based on a double exponential transformation. [ParInt]{} is layered over [MPI]{} (Message Passing Interface), and incorporates advanced parallel/distributed techniques including load balancing among processes that may be distributed over a cluster or a network/grid of nodes. Results are included for 2-loop vertex and box diagrams and for sets of 2-, 3- and 4-loop self-energy diagrams with or without UV terms. Numerical regularization of integrals with singular terms is achieved by linear and non-linear extrapolation methods.'
author:
- E de Doncker
- F Yuasa
- K Kato
- T Ishikawa
- J Kapenga
- O Olagbemi
bibliography:
- './bib2.bib'
- './wi.bib'
- './bibjk.bib'
title: |
Regularization with Numerical Extrapolation for Finite and\
UV-Divergent Multi-loop Integrals
---
Feynman loop integrals ,UV singularities ,multivariate adaptive integration ,numerical iterated integration ,asymptotic expansions ,extrapolation/convergence acceleration
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge the support from the National Science Foundation under Award Number 1126438, and the Center for High Performance Computing and Big Data at Western Michigan University. This work is further supported by Grant-in-Aid for Scientific Research (15H03668) of JSPS, and the Large Scale Simulation Program Nos. 15/16-06 and 16/17-21 of KEK.
References {#references .unnumbered}
==========
|
addtoreset[equation]{}[section]{}
\
[We show that a class of type IIA vacua recently found within the 4 effective approach corresponds to compactification on $\ads_4 \times \S^3 \times \S^3/\Z_2^3$. The results obtained using the effective method completely match the general ten-dimensional analysis for the existence of 1 warped compactifications on $\ads_4 \times \M_6$. In particular, we verify that the internal metric is nearly-Kähler and that for specific values of the parameters the Bianchi identity of the RR 2-form is fulfilled without sources. For another range of parameters, including the massless case, the Bianchi identity is satisfied when D6-branes are introduced. Solving the tadpole cancellation conditions in 4 we are able to find examples of appropriate sets of branes. In the second part of this paper we describe how an example with internal space $\C\P^3$ but with non nearly-Kähler metric fits into the general analysis of flux vacua. ]{}
Introduction {#intro}
============
Four-dimensional 1 supersymmetric vacua of type II supergravity with fluxes can be analyzed directly in 10 or by means of an effective potential formalism in 4. In this work we point out that a class of type IIA vacua, with geometric fluxes switched on, that were found using the latter method [@cfi] corresponds to compactification on $\ads_4 \times \S^3 \times \S^3/\Z_2^3$. The results obtained using the effective formalism are in complete accord with the general conditions for the existence of $\ads_4 \times \M_6$ vacua [@lt; @gmpt1; @gmpt2]. This is a particular example of the equivalence between the higher and lower dimensional approaches considered lately in greater generality [@km1; @Cassani].
In the $\ads_4 \times\S^3 \times \S^3/\Z_2^3$ compactification, that we study in depth, we show that the internal metric is nearly-Kähler. In [@bc] it was first proven that when $\M_6$ is nearly-Kähler there are consistent vacua of massive IIA supergravity with 1 supersymmetry in $\ads_4$. As also remarked in [@bc], besides $\S^3 \times \S^3$, there are other six-dimensional compact spaces that admit a nearly-Kähler metric, namely $\S^6$, $\C\P^3$ and $SU(3)/U(1)^2$ [@ssbook]. However, these spaces are not group manifolds and cannot be treated in a simple effective approach based on adding geometric fluxes to a toroidal compactification. It would be interesting to formulate all nearly-Kähler compactifications within the effective four-dimensional approach. A first step in this direction is the Kaluza-Klein reduction on nearly-Kähler spaces [@Kashani]. The case of $SU(3)/U(1)^2$ has been considered in [@hp].
A property of nearly-Kähler compactifications is that for special values of the fluxes the Bianchi identity for the RR 2-form can be satisfied without adding sources [@bc; @lt]. For other ranges of parameters it is necessary to add O6-planes, D6-branes, or both, wrapping 3-cycles in the internal space. In any case, including D6-branes is required to generate charged chiral multiplets. In the $\S^3 \times \S^3/\Z_2^3$ compactification we will present examples of supersymmetric D6-branes that can be included to fulfill the Bianchi identity or equivalently to cancel tadpoles. This problem was first addressed in [@adhl] where it was argued that a certain setup of D6-branes could cancel the tadpoles. We find similar results at the time we go further in proving tadpole cancellation because we supply the explicit background fluxes.
The second part of this paper is devoted to describing how other 1, $\ads_4$ vacua of massless IIA supergravity, discovered long time ago [@np; @vst; @stv; @pp], fit into the modern analysis of flux vacua. In these compactifications the internal space can be $\C\P^3$ or $SU(3)/U(1)^2$, but the metric is not nearly-Kähler. We will focus on the $\C\P^3$ example, but the analysis can be easily extended to $SU(3)/U(1)^2$. We give explicit expressions for the metric and the fluxes and then find the Killing spinor that allows to derive the fundamental forms that define the $SU(3)$ structure.
The organization of this paper is as follows. In section 2 we summarize the conditions for the existence of 1 $\ads_4$ vacua derived from the 10 theory. We also discuss the issue of solving the Bianchi identity for the RR 2-form with or without sources. In section 3 we study compactification on $\ads_4 \times \S^3 \times \S^3/\Z_2^3$ by describing the internal space in terms of a set of structure constants, the so-called geometric fluxes, known to give 1 vacua from the analysis of the 4 effective potential. We then explain how the Bianchi identity for $F_2$ can be satisfied in general by adding sources and present as well a concrete configuration of D6-branes in the massless case. There is an important interplay with the results in the 4 effective formalism that are collected in appendix A. Section 4 deals with the compactification on $\ads_4 \times \C\P^3$ that provides an example where the internal space is not nearly-Kähler. In appendix B we show that the proposed metric and background fluxes in $\C\P^3$ do satisfy the equations of motion and preserve 1 supersymmetry in 4.
Review of supersymmetric conditions in 10 {#d10}
=========================================
We are interested in 1 compactifications of type IIA supergravity with fluxes turned on and warped product geometry ds\^2 = e\^[2A(y)]{} ds\_4\^2 + ds\_6\^2 , \[geo\] where $ds_4^2$ and $ds_6^2$ are respectively the line elements of $\ads_4$ and the internal compact space. The general conditions that these vacua must fulfill were derived in [@lt] using Romans massive action [@romans] and also in [@gmpt1; @gmpt2] starting with the democratic formulation of IIA supergravity [@demo]. In this note we use the results and notation of [@gmpt2] that are more suited to compare with the effective potential approach.
By assumption, the internal manifold has strictly $SU(3)$ structure, i.e. it admits only one nowhere vanishing invariant spinor which in turn allows to write a fundamental 2-form $J$ and a holomorphic 3-form $\Omega$ satisfying the relations J=0 ; \^\* = -3 J J J . \[su3\] In the most general supersymmetric solution of the equations of motion, the warp factor and the dilaton are constants related by $\phi=3A$. Moreover, the characteristic forms $J$ and $\Omega$ must meet the conditions dJ= 2m e\^[-A]{} ; d=-3 m e\^[-A]{} J\^2 -i\_2 J , \[derj\] where $\cw_2$ is a real primitive 2-form. Here $\widehat \Omega = -i e^{i(\a+\b)} \Omega$, with $\a, \b$, phases that enter in the normalization of the 10 supersymmetry parameters (see [@gmpt2] for more details). The equations of motion also require $(\a - \beta)$ to be a constant.
Besides the constant $\tilde m$, the solutions depend on the IIA mass parameter $m$. These two real quantities are combined into the complex constant = e\^[-i(-)]{} (m + im) . \[mudef\] The parameter $\mu$ enters in the covariant derivative of the 4 gravitino and it turns out to be related to the cosmological constant through $\Lambda = -3 |\mu|^2$. This $\Lambda$ is defined with respect to the unwarped ${\rm AdS}_4$ metric.
In the solution the field strengths are determined to be[^1]
[lclcl]{} H = 2 m e\^[-A]{} & ; & F\_0 = - 5 m e\^[-4A]{} & ; & F\_2 = -e\^[-3A]{} \*d - 3m e\^[-4A]{} J\
F\_4 = -32 m e\^[-4A]{} J\^2 & ; & F\_6 = 12 m e\^[-4A]{} J\^3 & .
\[fluxsol\] The relation to the NSNS and RR forms is given by H = dB + ; F\_p = d C\_[p-1]{} - H C\_[p-3]{} + ( e\^B) |\_[p]{} . . \[hfdef\] The barred quantities are background fluxes and $\ov{F} = \ov{F_0} + \ov{F_2} + \ov{F_4} + \ov{F_6}$ is a formal sum.
Clearly, (\[derj\]) implies $J \wedge dJ=0$ and $d(\re \widehat{\Omega})=0$. This means that the internal space is always a half-flat manifold. If the torsion class $\cw_2$ vanishes the internal space is nearly-Kähler and the RR 2-form simplifies to F\_2 = - 3 e\^[-4A]{} J . \[f2nk\] This implies in particular that $dF_2 \not=0$ in nearly-Kähler compactifications.
The Bianchi identities for $H$ and $F_4$ are automatically satisfied. On the other hand, for the RR 2-form the generic results imply $dF_2-F_0H \not=0$. The situation is not hopeless because there might be further contributions due to D6-branes or O6-planes wrapping 3-cycles in the internal space. Actually, the Bianchi identity (BI) for $F_2$ is equivalent to tadpole cancellation conditions for the RR $C_7$ form that couples to such sources.
Following the prescription of [@gmpt2] we assume that the sources are smeared instead of localized. This means that in the BI D6-branes and O6-planes can be represented by additional 3-forms in the internal space. This is actually the only consistent possibility for the $\ads_4$ vacua in which the warp factor must be constant. Upon including smeared sources the BI becomes dF\_2 - F\_0 H + A\_3 = 0 , \[tadf2\] where $A_3$ is the Poincaré dual to internal 3-cycles wrapped by D6-branes or O6-planes. By virtue of (\[hfdef\]), this identity can be written purely in terms of background fluxes as $d\ov{F}_2 - F_0 \ov{H} + A_3 = 0$.
A property of the 1 $\ads_4$ vacua is that $H \propto dJ$. Thus, the form $A_3$ is necessarily exact. In consequence, to saturate the Bianchi identity of the RR 2-form, or equivalently to cancel $C_7$ tadpoles, the sources need not wrap non-trivial 3-cycles. This point has been known for some time [@adhl; @Cascales] and further elaborated recently [@km2]. Due to the special properties of $\ads_4$ such D6-branes can still be stable.
When $F_0 \not=0$ there could be a solution of (\[tadf2\]) without sources even if $dF_2 \not=0$. Indeed, when the internal space is nearly-Kähler from the above results it follows that dF\_2 - F\_0 H = e\^[-5A]{}(15m\^2 - m\^2) . \[tadf3\] Therefore, it is possible to avoid sources, i.e. $A_3=0$, provided that $\tilde m^2 = 15 m^2$. This interesting fact was first obtained in [@bc] and later in [@lt]. On the other hand, if $\tilde m^2 \not= 15 m^2$, sources must be added to fulfill the Bianchi identity. For instance, if $\tilde m^2 > 15 m^2$ a solution can be achieved by adding only D6-branes. This follows because supersymmetric 3-cycles are calibrated by $\re \Omega$ and in this case $\int_{\M_6}\re \Omega \wedge A_3 > 0$. Here we are taking $\widehat \Omega = -i \Omega$ according to results in appendix A.
It is also feasible to satisfy the Bianchi identity without sources and $F_0=0$ simply when $dF_2 =0$. Clearly, in this situation the internal space cannot be nearly-Kähler. Instead, the torsion class $\cw_2$ must be non-zero. Examples of this type were actually found several years ago [@np; @vst; @stv; @pp]. In section \[ccp3\] we discuss in detail the case of compactification on $\C\P^3$.
Flux compactification on $\bm{{\rm AdS}_4 \times \S^3 \times \S^3}$ {#cs3s3}
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We are interested in 1 type IIA vacua in presence of geometric fluxes $\omega^P_{MN}$ together with NSNS and RR fluxes. Such solutions can be viewed as compactifications in which the internal space has a basis of globally defined 1-forms satisfying d\^P = -\^P\_[MN]{} \^M \^N , \[gflux\] where the $\omega^P_{MN}$ are the structure constants of some Lie group $G$. If the Killing form $\K_{MN}=\omega^P_{MR} \omega^R_{NP}$ is non-degenerate, $G$ is semisimple and furthermore it is compact if $\K_{MN}$ is negative definite. If $G$ is not semisimple, but it has a discrete compact sub-group $\Gamma$, the internal space can be compactified by taking the quotient $G/\Gamma$. This is the case of the nil and solvmanifolds studied in [@gmpt2]. In this note we rather study the situation where $G$ is compact and the internal space is the $G$ group manifold. In particular, we want to show that in a class of supersymmetric $\ads_4 \times \M_6$ vacua found in [@cfi] the structure constants are actually those of $SU(2) \times SU(2)$ and the internal space is $\S^3 \times \S^3$ realized as $SU(2) \times SU(2)\times SU(2)/ SU(2)_{\rm diag}$.
The number of independent geometric fluxes $\omega^P_{MN}$ can be reduced by imposing additional conditions on the internal space. We will enforce a $\Z_2\times \Z_2$ symmetry whose generators act as \_2 & : & (\^1, \^2, \^3, \^4, \^5, \^6) (-\^1, -\^2, \^3, -\^4, -\^5, \^6)\
\_2 & : & (\^1, \^2, \^3, \^4, \^5, \^6) (\^1,-\^2, -\^3, \^4, -\^5, -\^6) . \[z2z2\] Furthermore, keeping in mind the eventual need for orientifold planes to cancel tadpoles, the geometric fluxes are required to be invariant under an orientifold involution $\sigma$ which is also a $\Z_2$ symmetry given by : \^i \^i ; \^[i+3]{} -\^[i+3]{} , i=1,2,3 . \[oaction\] In the end only twelve geometric fluxes survive and they are further constrained by the Bianchi identities following from (\[gflux\]). In the $\ads_4$ solutions found in [@cfi] there are only four independent parameters $a$ and $b_i$ which appear in the structure equations
[lcl]{} d\^1 = -a \^[56]{} - b\_1 \^[23]{} & ; & d\^4 = - b\_2\^[53]{} - b\_3 \^[26]{}\
d\^2 = -a \^[64]{} - b\_2 \^[31]{} & ; & d\^5 = - b\_1\^[34]{} - b\_3 \^[61]{}\
d\^3 = -a \^[45]{} - b\_3 \^[12]{} & ; & d\^6 = - b\_1\^[42]{} - b\_2 \^[15]{} . \[abis\]
The notation $\eta^{12}=\eta^1 \wedge \eta^2$, etc. is understood.
For future purposes we record the 2, 3 and 4-forms invariant under the $\Z_2\times \Z_2$ symmetry. These are
[lclclcl]{} \_1 = - \^[14]{} & ; & \_0 =\^[123]{} & ; & \_0 =\^[456]{} & ; & \_1 = \^[2536]{}\
\_2 = - \^[25]{} & ; & \_1 =\^[156]{} & ; & \_1 =\^[423]{} & ; & \_2 = \^[1436]{}\
\_3 = - \^[36]{} & ; & \_2 =\^[426]{} & ; & \_2 =\^[153]{} & ; & \_3 = \^[1425]{}\
& & \_3 =\^[453]{} & ; & \_3 =\^[126]{} & . & \[allforms\]
Notice that $\a_I$ and $\tilde \omega_i$ are even whereas $\b_I$ and $\omega_i$ are odd under the orientifold involution. The normalization is \_[\_6]{} \_i \_j = \_[\_6]{} \_i \_j = \_6 \_[ij]{} , \[ccdef\] where $\cv_6$ is a constant to be computed later on. When the geometric fluxes $a$ and $b_i$ are zero, the internal space can be compactified into a flat six-dimensional torus. Moreover, the $\Z_2\times \Z_2$ symmetry that is assumed implies that this torus is a product of three $\T_i^2$. Each 2-torus has a basis of 1-forms $(\eta^i, \eta^{i+1})$, a Kähler modulus (area) $t_i$ and a complex structure parameter $\tau_i$ that must be real for consistency with the orientifold involution. With this picture in mind we take the metric on $\M_6$, with $a, b_i \not=0$, to still be given by ds\_6\^2= \_[i=1]{}\^3 (\^i)\^2 + t\_i \_i (\^[i+3]{})\^2 . \[metricm6\] By construction, $t_i > 0$ and $\tau_i > 0$. Clearly, $\sqrt{g_6}= t_1 t_2 t_3$. Integrating gives the volume ${\rm Vol}(\M_6)= \cv_6 \, t_1 t_2 t_3$, where $\cv_6$ is the normalization constant defined above.
The hermitian almost complex structure corresponding to the metric is J= - t\_1 \^[14]{} - t\_2 \^[25]{} - t\_3 \^[36]{} = t\_i \_i . \[jm6\] The associated holomorphic (3,0) form can be written as = (\^1 - i\_1 \^4) (\^2 - i\_2 \^5) (\^3 - i\_3 \^6) . \[om6\] These $J$ and $\Omega$ satisfy (\[su3\]) so that they provide an $SU(3)$ structure on the internal space $\M_6$. Notice also that under the orientifold involution, $J \to - J$ and $\Omega \to \Omega^*$.
From (\[abis\]) we find that $dJ$ and $d\Omega$ are not zero but $J \wedge dJ$ and $d(\im \Omega)$ do vanish. Thus, the $\M_6$ defined by (\[abis\]) is a half-flat manifold. Additional properties must be fulfilled for $\M_6$ to serve as internal space in an 1 supersymmetric $\ads_4$ vacua of type IIA. Moreover, it is necessary to turn on particular NSNS and RR background fluxes. Now, from the discussion in [@cfi] we know that a solution is obtained with a precise set of fluxes invariant under the $\Gamma=\Z_2^3$ group of symmetries (\[z2z2\]) and (\[oaction\]). Furthermore, in this solution the variables $t_i$ and $\tau_i$ that enter in the metric satisfy specific relations. In the following our strategy is to use these results to continue analyzing the properties of the $\M_6$ at hand.
In the appendix we review the conditions of [@cfi] to obtain $\ads_4 \times \M_6$ supersymmetric minima. The fluxes allow a configuration with $t_1=t_2=t_3=t$, where $t$ is completely fixed. A crucial property is that the structure constants $a$ and $b_i$ must all have the same sign. Also, the second equation in (\[realsu\]) together with the explicit form of the moduli, c.f. (\[defrsu\]), gives the very useful relations b\_i \_j \_k = 3a \_i\^2 = , i = j = k . \[crux\] We then find dJ = 32 (\_1 ) ; d = \_1 J J ; \_1 = \[djdo\] In general the exterior derivatives of $J$ and $\Omega$ can be expressed in terms of torsion classes (see e.g. [@grana]). In our case, from (\[djdo\]) we easily see that the only non-zero class is $\cw_1$. This is precisely the condition for the internal space to be nearly-Kähler.
It is a simple exercise to compute the Killing form for the structure constants given in (\[abis\]). We find = -4 [diag]{}(b\_2b\_3, b\_1b\_3, b\_1 b\_2, a b\_1, a b\_2, ab\_3) . \[killf\] Now, recall that to obtain $\ads_4 \times \M_6$ supersymmetric minima the geometric fluxes $a$ and $b_i$ must all have the same sign. Therefore, $\K$ is non-degenerate and negative-definite. We might guess that the semisimple compact algebra being six-dimensional is that of $SU(2)\times SU(2)$. Indeed, after performing the change of basis
[lcl]{} \^1 = \^1 + \^4 & ; & \^1 = \^1 - \^4\
\^2 = \^2 + \^5 & ; & \^2 = \^2 - \^5\
\^3 = \^3 + \^6 & ; & \^3 = \^3 - \^6 , \[xibasis\]
the structure equations become d\^i = -12 \_[ijk]{} \^i \^j ; d \^i = -12 \_[ijk]{} \^i \^j . \[newstr\] This confirms that the underlying algebra is that of $SU(2)\times SU(2)$.
We can take the $\xi^i$ and $\hat \xi^i$ to be two sets of $SU(2)$ left invariant 1-forms. Concretely, \^1 & = & d+ d\
\^2 &= & -d+ d \[xihsu2\]\
\^3& = & d+ d , and similarly for the $\xi^i$. The range of angles is $0 \leq \hat \theta \leq \pi$, $0 \leq \hat \phi \leq 2\pi$ and $0 \leq \hat \psi \leq 4\pi$.
Our claim that the internal space is $\S^3 \times \S^3$ is supported by the explicit form of the metric in the new basis. Substituting (\[xibasis\]) into (\[metricm6\]) readily gives ds\_6\^2 = . \[s3s3metric\] This is an Einstein metric that belongs to a family of homogeneous metrics on $\S^3 \times \S^3$ [@gpp]. The isometry group is $SU(2)^3$ [@amv; @aw]. There are two $SU(2)$’s from the left actions that leave $\xi^i$ and $\hat \xi^i$ separately invariant, and a further $SU(2)$ from a simultaneous right action by the same element on $\xi^i$ and $\hat \xi^i$. >From the metric and the explicit realization of the $SU(2)$ 1-forms the volume of $\S^3 \times \S^3$ can be evaluated to be (§\^3 §\^3) = \_6 t\^3 , \[vols3s3\] where $\cv_6$ is precisely the normalization constant introduced in (\[ccdef\]).
In the new basis the fundamental forms $J$ and $\Omega$ are given by J & = & (\^1 \^1 + \^2 \^2 + \^3 \^3) \[joebasis\]\
& = & - (\^1 + e\^[2i/3]{} \^1) (\^2 + e\^[2i/3]{} \^2) (\^3 + e\^[2i/3]{} \^3) . Similar expressions have appeared in the literature some time ago [@adhl] and more recently [@km2].
At this point we must remember that our actual model is constrained by some specific symmetries. Indeed, the geometric fluxes (\[abis\]), as well as the NSNS and RR backgrounds (\[fluxbg\]), have been chosen to be invariant under the group $\Gamma=\Z_2^3$ of transformations given by the geometric $\Z_2 \times \Z_2$ (\[z2z2\]) and the orientifold involution $\sigma$ (\[oaction\]). The action of $\sigma$ amounts to exchange of the spheres, $\xi^i \leftrightarrow \hat \xi^i$, which is clearly a symmetry of the metric. On the other hand, the geometric $\Z_2 \times \Z_2$ corresponds to \_2 & : & (\^1, \^2, \^3, \^1, \^2, \^3) (-\^1, -\^2, \^3, -\^1, -\^2, \^3)\
\_2 & : & (\^1, \^2, \^3, \^1, \^2, \^3) (\^1, -\^2, -\^3, \^1, -\^2, -\^3) \[z2z2xi\] which also leaves the metric invariant. The effect of these latter symmetries is to restrict the range of the angles that define the 1-forms, c.f. (\[xihsu2\]). The first and second $\Z_2$’s imply respectively $\hat \theta \equiv - \hat \theta$, and $\hat \psi \equiv - \hat \psi$ simultaneously with $\hat \phi \equiv - \hat \phi$, and analogous for the unhatted angles. In the end we truly have internal space $\S^3 \times \S^3/\Gamma$, with volume given by $\cv_6 t^3/8$. We will write (§\^3 §\^3/) = t\^3 , \[vols3s3G\] where $\cc=\cv_6/8=4\pi^4/(ab_1b_2b_3)^{3/2}$.
The nearly-Kähler metric on $\S^3 \times \S^3$ is also invariant under the order three transformation : \^i - \^i ; \^i \^i - \^i . \[betadef\] This $\beta$-symmetry proves useful when studying properties of 3-cycles on $\S^3 \times \S^3$ [@adhl].
D6-branes on $\bm{\S^3 \times \S^3}$ and Bianchi identity for $\bm{F_2}$
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When $dF_2 \not=0$, the Bianchi identity for the RR 2-form can still be fulfilled by adding appropriate sources. The task is to find the 3-form $A_3$ that satisfies (\[tadf2\]) and is the Poincaré dual of the 3-cycles wrapped by the sources.
In general, $A_3$ is some combination of the 3-forms of the internal space so that it is important to characterize these forms, specially knowing that $A_3$ must be exact. For $\S^3 \times \S^3$ the third Betti number is equal to two and the third cohomology is rather simple. The two representative closed 3-forms are easier to describe in the $(\xi^i, \hat \xi^i)$ basis. In fact, they are basically the volume forms of each $\S^3$, namely h= ; h= - . \[hhhdef\] The normalization has been chosen so that h h = = \^[123456]{} . \[hnorm\] From the six remaining 3-forms that can be constructed there are three exact combinations given by $d(\xi^i \wedge \hat \xi^i)$. The corresponding forms in terms of the $\eta^M$ basis are found using the map (\[xibasis\]). In particular, it follows that a \^[456]{} = b\_1 \^[423]{} = b\_2 \^[153]{} = b\_3 \^[126]{} = ()\^[1/4]{} (h + h) , \[exf\] where each equality is modulo exact forms.
Let us now study the homology. Our discussion resembles that in [@aw] and [@adhl]. In $\S^3 \times \S^3$ we can identify three special 3-cycles as explained below.
The locus $\hat \xi^i =0$. By definition this is the first 3-sphere $\S_1^3$. From the metric (\[s3s3metric\]), ds\_6\^2 |\_[\^i=0]{} = ds\_3\^2(§\_1\^3) = (\^i)\^2 . \[s1metric\] >From the $\Omega$ form we find that $\im \Omega \big|_{\hat \xi^i=0}=0$, and moreover |\_[\^i=0]{}= - \^[123]{}= -[dvol]{}(§\_1\^3) . \[reos1\] This shows that the charge of a brane wrapping $\S_1^3$ is $-1$, it would be an anti D6-brane in our conventions. For a D6-brane the 3-sphere must be wrapped in reverse orientation. We will define the corresponding 3-cycle to be $D_1 = (-\S_1^3)$.
The locus $\xi^i =0$. By definition this is the second sphere $\S_2^3$. We now find that |\_[\^i=0]{}= -[dvol]{}(§\_2\^3) . \[reos2vol\] Thus, a brane wrapping $\S_2^3$ has charge $-1$ and it is an anti D6-brane in our conventions. Since $\im \Omega \big|_{\xi^i=0}=0$, we surmise that the supersymmetric D6-brane must wrap the 3-cycle $D_2 = (-\S_2^3)$.
The locus $\xi^i= \hat \xi^i$. By definition this is the diagonal 3-sphere $\S_D^3$. It is easy to check that $\im \Omega \big|_{\xi^i=\hat \xi^i}=0$. Besides, from the metric (\[s3s3metric\]) and the $\Omega$ form we deduce |\_[\^i=\^i]{}= [dvol]{}(§\_D\^3) . \[reosdvol\] Due to some extra factors now there is a plus sign in front so that the charge of a brane wrapping the diagonal 3-sphere is a D6-brane with charge $+1$. We will denote $D_0 = \S_D^3$.
The three 3-cycles discussed above, $D_0$, $D_1$ and $D_2$, cannot be independent since the third Betti number of $\S^3 \times \S^3$ is two. In fact there is a linear relation among these cycles that will become clear when we discuss the corresponding dual 3-forms.
In general, given a 3-form $X$ integrated over one of the 3-cycles $D_i$, the Poincaré dual form $Y_i$ to $D_i$ in $\M_6=\S^3 \times \S^3$ is such that \_[D\_i]{} X=\_[\_6]{} X Y\_i . \[poincaredef\] For example, for $D_1=(-\S_1^3)$ we find Y\_1 = - , \[duald1\] where $\hat h$ is defined in (\[hhhdef\]). To demonstrate this we can choose X= [dvol]{}(D\_1)=- \^[123]{}=- \^[123]{} , \[xf1\] so that $\int _{D_1} X= V_3$. On the other hand we can also compute \_[\_6]{} X (-) = V\_3 . \[checkduald1\] In a similar fashion we obtain the dual to $D_2=(-\S_2^3)$ to be Y\_2 = - , \[duald2\] where $h$ is defined in (\[hhhdef\]).
We can now compute the intersection number of the 3-cycles $D_1$ and $D_2$ by means of the representative dual 3-forms. This is D\_2D\_1 =\_[D\_1]{} Y\_2 = \_[\_6]{}Y\_2 Y\_1 = \_[\_6]{} h =1 . \[int12\] This agrees with the analysis of [@aw].
We still need to find the dual 3-form of the diagonal 3-sphere $D_0$. In this case it is convenient to use the $\eta^M$ basis. We notice that $\xi^i = \hat \xi^i$ amounts to going to the locus $\eta^4=\eta^5=\eta^6=0$. Either by changing variables or by evaluating directly in (\[om6\]), we obtain (D\_0)= \^[123]{} . \[dvoldeta\] It then follows that the dual 3-form is given by Y\_0 = \^[456]{} , \[duald0\] where we have used that $\tau_1\tau_2\tau_3=(27/ab_1b_2b_3)^{1/4}$ as implied by (\[crux\]).
As mentioned before, there must be a linear relation among the three supersymmetric 3-cycles that have been identified. The claim is that $$D_0+D_1+D_2= 0 \ ,
\label{linD}$$ in homology. This can be simply understood in terms of the dual 3-forms. In fact, from (\[exf\]) we have $Y_0= \frac{h+\hat h}{\cv_6}$, up to exact forms. Therefore, in cohomology, $Y_0+Y_1+Y_2= 0$, modulo exact forms. This confirms the validity of (\[linD\]).
The remaining intersection numbers are also easily calculated. We find for instance $D_0\cdot D_2 = \int_{\M_6}Y_0 \wedge Y_2 =1$. In general, D\_i D\_j = \_[\_6]{}Y\_i Y\_j = \_[j,i-1]{} -\_[j,i+1]{} , \[intij\] where the indices are defined modulo 3. These are the intersection numbers found in [@aw]. In particular they satisfy, $D_i\cdot ( D_0+D_1+D_2)=0$, consistent with (\[linD\]).
We will now carry the discussion in the quotient space $\S^3\times \S^3/\Gamma$ with $\Gamma=\Z_2^3$. To the 3-cycles, $D_i$ in the covering space we associate $D_i^\prime$ with corresponding dual forms $Y_i^\prime$ in the quotient. Closely following [@aw], let us assume that the lifting to the covering space $\M_6=\S^3\times \S^3$ is given by the map (Y\_0\^,Y\_1\^, Y\_2\^) && (Y\_0, 8 Y\_1, 8 Y\_2)\
(D\_0\^, D\_1\^,D\_2\^) && (D\_0, 8 D\_1, 8 D\_2) . \[lifti\] With this Ansatz we then obtain for instance, D\_2\^D\_1\^& = & \_ [\_6]{} Y\_2\^Y\_1\^ = \_ [\_6]{} 8Y\_2 8 Y\_1 = 18 \_ [\_6]{} 8Y\_2 8Y\_1 =8\
D\_0\^D\_2\^& = & \_ [\_6]{} Y\_0\^Y\_2\^= \_ [\_6]{} Y\_0 8Y\_2 = 18 \_ [\_6]{} Y\_0 8Y\_2 =1 , \[intsq\] where we have defined $\cs_6=\M_6/\Gamma=\S^3\times \S^3/ \Gamma$ to streamline expressions. As expected, this is consistent with the normalization \_ [\_6]{} \^ [123456]{}= = \[MGvolume\] where ${\cc}t^3$ is the volume of $\cs_6$. We will see that these intersection numbers also arise in our model 1 in 4 discussed in appendix A.
According to [@aw], the 3-cycle $D_0^\prime$ corresponds to $D_0^\prime= S_D^3/{\Gamma}$. Namely, $D_0=S_D^3$ is an 8-fold cover of $D_0^\prime$. Since cycles are not independent, this indicates that wrapping $N$ D6-branes around each of the cycles $D_i^\prime$ with $i=1,2$, requires wrapping $D_0^\prime$ $8N$ times. In other words, 8D\_0\^+D\_1\^+D\_2\^= 0 , \[linDp\] which is true by virtue of the map (\[lifti\]) and the relation (\[linD\]).
With all the information collected so far we can already establish a connection to our model 1 explained in appendix A. In this model, with mass parameter $F_0=0$, we found that tadpoles could be cancelled by a setup of supersymmetric D6-branes wrapping particular factorizable 3-cycles in the $\eta^M$ basis. The concrete configuration is summarized in table \[adsm0\] where the 3-cycles are explicitly given. It consists of a stack of $8N_B$ D6-branes wrapping a cycle $\Pi_A$, $N_B$ D6-branes wrapping a cycle $\Pi_B$, plus $N_B$ D6-branes wrapping the mirror cycle $\widetilde{\Pi}_B$. In the model, the geometric flux parameters satisfy $a=b_i=2N_B/c$, where $c$ is related to the RR 2-form background. Interestingly enough, it is possible to represent these factorizable cycles in terms of the supersymmetric 3-cycles in $\S^3\times \S^3/\Gamma$. In fact, the following identifications can be made \_A= (1,0)\^3 D\_0\^; \_B = (-1,1)\^3 D\_2\^; \_B=(-1,-1)\^3 D\_1 \^\[idenmod1\] Evidence for these matchings comes from the equivalence of the loci described in both the $\eta^M$ and the $(\xi^i,\hat \xi^i)$ basis, and from agreement of the intersection numbers. For instance, $\Pi_A \cdot \Pi_B =1 = \D_0^\prime \cdot \D_2^\prime$ and $\Pi_B \cdot \widetilde{\Pi}_B = 8 = \D_2^\prime \cdot \D_1^\prime$. Besides, below we will check that the corresponding dual 3-forms do coincide.
Based on the above results from the analysis of supersymmetric 3-cycles in $\S^3\times \S^3/\Gamma$ we conclude that a setup of D6-branes wrapping the cycles $D_0^\prime$, $D_1^\prime$ and $D_2^\prime$, will lead to tadpole cancellation. Otherwise stated, the corresponding dual 3-forms must add up to the precise 3-form $A_3$ needed to saturate the Bianchi identity. To substantiate this claim we will examine the Bianchi identity for the RR 2-form in more detail. The starting point is equation (\[c7tadpole\]). For sources wrapping space-time the RR 7-form can be written as $C_7={\rm dvol}_4 \wedge X$, where $X$ is some 3-form in the internal space which we take to be $\cs_6=\S^3\times \S^3/\Gamma$. Then, (\[c7tadpole\]) leads to \_[\_6]{} X (d\_2 - F\_0 ) + \_a N\_a Q\_a \_[\_a]{} X = 0 . \[c7tadpolevol\] The factor of $\sqrt\cc$ is necessary because we are writing $d\ov{F}_2$ and $\ov{H}$ in a basis of forms with normalization (\[MGvolume\]) or analogous in terms of the $(\xi^i, \hat \xi^i)$ 1-forms.
To continue, recall that $\int_{\Pi_a} \hspace{-2mm} X = \int_{\cs_6} \hspace{-1mm} X \wedge Y_a^\prime$, where the 3-form $Y_a^\prime$ is the Poincaré dual of the 3-cycle $\Pi_a$. Thus, from the above integral we arrive at the BI d\_2 - F\_0 + \_a N\_a Q\_a Y\_a\^= 0 . \[biovo\] In terms of the notation in section \[d10\] we have A\_3 = \_a N\_a Q\_a A\_3\^a , \[a3a\] where $A_3^a=\sqrt{\cc}\, Y_a^\prime$ is the contribution of each individual source. Recall that $N_a$ is the number of D6-branes or O6-planes wrapping the 3-cycle $\Pi_a$ and $Q_a$ is the corresponding charge.
In the following we focus on the massless case $F_0=0$ as in model 1 of appendix A. As argued in section \[d10\], when $m=0$, necessarily sources of positive charge must be included to satisfy the BI. In this case, in our $\S^3\times \S^3/\Z_2^3$ compactification, from previous results we know that $d\ov{F}_2$ is given by d\_2 = - dJ = - \_1 . \[bgf2\] In the $\eta^M$ basis this yields the rather simple expansion d\_2 = -c(3a\^[456]{}- b\_1 \^[423]{}-b\_2 \^[153]{}- b\_3 \^[126]{}) . \[bianchimzero\] Our results for tadpole cancellation in model 1 in appendix A suggest a solution to the BI, $d\ov{F}_2 + A_3=0$. Concretely we propose that in this situation $A_3$ can be written as A\_3 = N\_B(8 A\_3\^A + A\_3\^B + A\_3\^B) , \[a3sum\] because $N_A=8N_B$ and $Q_A=Q_B=1$. Indeed, it is straightforward to check that the BI is satisfied with A\_3\^A & = & \^[456]{} ,\
A\_3\^B & = & -(\^[456]{} + \^[423]{} + \^[153]{} + \^[126]{} + \^[123]{} + \^[156]{} + \^[426]{} + \^[453]{} ) , \[a3m0\]\
A\_3\^B & = & -(\^[456]{} + \^[423]{} + \^[153]{} + \^[126]{} - \^[123]{} - \^[156]{} - \^[426]{} - \^[453]{} ) , as long as $a=b_i=2N_B/c$, which precisely guarantees tadpole cancellation.
To close our argument we compare the dual 3-forms $Y_a^\prime$ with those found before for the supersymmetric 3-cycles in $\S^3\times \S^3/\Z_2^3$. We find Y\_A\^& = & 1 A\_3\^A = \^[456]{} = Y\_0 = Y\_0\^\
Y\_B\^& = & 1 A\_3\^B = 8 (-)= 8 Y\_2= Y\_2\^\[alleluja\]\
\_B\^& = & 1 \_3\^B = 8 ( -)= 8 Y\_1= Y\_1\^ . Therefore, the cycles wrapped by D6-branes correspond to the “quotient spheres” $D_0^\prime$, $D_1^\prime$ and $D_2^\prime$, as already anticipated in (\[idenmod1\]).
As we might suspect, a more generic solution to the BI can be obtained as we now explain. Again in the massless case, the problem is to solve d\_2 + \_a N\_a Q\_a Y\_a\^= 0 . \[biovo2\] In general we can attempt a solution with 3 stacks of D6-branes wrapping the supersymmetric quotient 3-spheres so that A\_3 = \_a N\_a Q\_a Y\_a\^= (N\_0 Y\_0\^+ N\_1 Y\_1\^+ N\_2 Y\_2\^) , \[a3generic\] setting the charges to 1. Now, as suggested by (\[linDp\]), we choose $N_0=8N$, $N_1=N_2=N$. Then, A\_3 = 8 N( Y\_0+ Y\_1+ Y\_2) = (3a\^[456]{}- b\_1 \^[423]{}-b\_2 \^[153]{}- b\_3 \^[126]{}) , \[a3generic2\] where we used the lifting (\[lifti\]) and the expansions of the dual forms in the $\eta^M$ basis. Comparing with (\[bianchimzero\]) we see that the BI is satisfied provided that $$c= \frac {2 N}{(ab_1b_2b_3)^{\frac14}} \ .
\label{cgeneric}$$ In the 4 formulation developed in section A.1, this generic solution can be associated to a particular configuration of supersymmetric D6-branes similar to model 1. The setup consists of $N_B$ D6-branes wrapping $\Pi_B=(-1,k) \otimes (-1,\ell) \otimes (-1,m)$, where $(k,\ell,m)$ are positive integers, $N_B$ D6-branes along the mirror 3-cycle $\widetilde{\Pi}_B$, plus $N_A=8N_B$ D6-branes wrapping $\Pi_A=(1,0)^3$. It is not difficult to check that tadpoles are cancelled, and $\Pi_B$ is supersymmetric, as long as $ac=2N_B$, $b_1c=2N_B \ell m$, $b_2c=2N_B k m$ and $b_3c=2N_B k \ell$. Combining these parameters we reproduce (\[cgeneric\]) with $N=N_B \sqrt{k\ell m}$.
To finish this section we would like to comment on the massless spectrum originating from the configuration of D6-branes. The interpretation is that in $\S^3\times \S^3/\Gamma$ a setup of $N_B$ D6-branes wrapping each of the cycles $D_1^\prime$ and $D_2^\prime$, as well as $N_A=8N_B$ D6-branes wrapping $D_0^\prime$, allows to satisfy the BI. These D6-branes produce an anomaly-free spectrum with gauge group $U(N_A) \times U(N_B) \times U(N_B)$ and massless matter content (,,[**1**]{}) + (,[**1**]{},) + 8([**1**]{},,) , \[specinters\] consistent with the intersection numbers of the 3-cycles. Notice that the spectrum is chiral and, therefore it cannot be continuously deformed away. This signals the stability of the D6-brane configuration.
The above spectrum is the same as in model 1 in appendix A. We are assuming that the curvature of the 3-spheres wrapped by the D6-branes is large. In fact, the radius is controlled by the size modulus $t$ whose vev can turn out large, for instance by adjusting the RR flux $e_0$ [@cfi]. On the other hand, the fact that the D6-branes wrap 3-spheres can have interesting consequences. For instance, since the first Betti number of $\S^3$ is zero, open string massless scalar moduli are not expected. In the lines of [@cg] these, adjoint, scalars would become massive through $\mu $ terms in the effective superpotential[^2]. This could be an appealing feature from a phenomenological perspective.
So far we have concentrated here in massless type IIA without orientifold planes. Extensions to more general cases can in principle be worked out and could lead to attractive models from the phenomenological point of view.
Flux compactification on $\bm{{\rm AdS}_4 \times \C\P^3}$ {#ccp3}
=========================================================
Compactification of massless type IIA supergravity on ${\rm AdS}_4 \times \C\P^3$ have been studied in detail in [@np; @vst; @stv]. The idea was to look for solutions similar to the Freund-Rubin compactification of eleven-dimensional supergravity. Thus, a non-trivial background for the RR 4-form, $F_4 \propto {\rm dvol}_4$ is turned on. By Hodge duality this is equivalent to $F_6 \propto {\rm dvol}_6$. The solutions are unwarped and have constant dilaton. There is no $H$ flux. The RR 2-form flux can be chosen to be $F_2 \propto J$, where $J$ is the fundamental form of $\C\P^3$. When the internal metric is given by the Fubini-Study metric the equations of motion are satisfied. Furthermore the Bianchi identity for $F_2$ is automatic because $dJ=0$. It can be shown that an extended 6 supersymmetry is preserved.
Applying the general results reviewed in section \[d10\] we can see that for $m=0$ there is a solution with 1 supersymmetry when the metric in $\C\P^3$ is chosen to be nearly-Kähler. However, in this case the Bianchi identity for $F_2 \propto J$ is not satisfied because $dJ \not=0$. Presumably the tadpoles could be cancelled by adding D6-branes. The third homology of $\C\P^3$ is trivial but there could exist supersymmetric 3-cycles.
Yet another 1 solution with $m=0$ can be found by choosing the metric on $\C\P^3$ and the RR 2-form flux to descend from the metric of the squashed seven-sphere which gives an 1 solution of 11 supergravity [@dnp]. In this case the $\C\P^3$ metric is not Einstein and therefore not nearly-Kähler. According to the general analysis it must be that the metric is such that the two torsion classes $\cw_1$ and $\cw_2$ are different from zero. In fact setting $\widehat\Omega=-i\Omega$ in (\[derj\]) tells us that dJ= 32 \_1 ; d= \_1 J\^2 + \_2 J , \[nonnk\] where $\cw_1=\frac43 \tilde m e^{-A}$ and $\cw_2$ is a real primitive 2-form. Substituting in (\[fluxsol\]) then gives F\_2= -14 \_1 J + \^\*(\_2 J) , \[f2nnk\] where we have put the warp factor to zero. In principle it is then feasible to attain $dF_2=0$ even if $dJ\not=0$. Below we try to check these claims.
The generic metric on $\C\P^3$ can be constructed as a bundle with base $\S^4$ and fiber $\S^2$. Denoting by $(\theta, \varphi)$ the coordinates of the $\S^2$ this means that ds\_6\^2 = ds\_4\^2 + \^2 (d- \^1 + \^2)\^2 + \^2 \^2 (d- (\^1 + \^2) + \^3)\^2 , \[cp3metric\]\
where $d\tilde s_4^2$ is the line element of $\S^4$ and $\ca^A$ is the self-dual $SU(2)$ instanton potential on $\S^4$. More explicitly, ds\_4\^2 = d\^2 + 14 \^2 \^A \^A ; \^A = \^2 2 \^A . \[s4mi\] The $\Sigma^A$, $A=1,2,3$, are left-invariant $SU(2)$ 1-forms for which we use coordinates \^1 & = & d+ d ,\
\^2 & = & - d+ d , \[s3forms\]\
\^3 & = & d+ d , Notice that $d\Sigma^A = -\oh \epsilon_{ABC} \Sigma^B \wedge \Sigma^C$.
In the following we will employ a flat Sechsbein defined as e\^1 & = & d; e\^j = \^[j-1]{} , j=2,3,4 ,\
e\^5 & = & (d- \^1 + \^2) , \[bein6\]\
e\^6 & = & (d- (\^1 + \^2) + \^3) . In the flat basis the Ricci tensor of the $\C\P^3$ metric is diagonal with components R\_[ab]{}=(3-\^2) \_[ab]{} ; R\_[ij]{}=(\^2 + 1[\^2]{}) \_[ij]{} , \[riccicp3\] where $a,b=1, \cdots, 4$, and $i,j=5,6$.
Taking $\lambda^2=1$ gives the standard Einstein metric that is Kähler. A second Einstein metric that is nearly-Kähler is obtained setting $\lambda^2=\frac12$. In both cases the Einstein equation of motion of type IIA supergravity can be solved with $F_2 \propto J$. Another solution can be found choosing $\lambda^2=\frac15$ and turning on an appropriate RR 2-form flux. Concretely, $F_2$ must be F\_2 = -(e\^[12]{} + e\^[34]{}) - (e\^[13]{} + e\^[42]{}) - (e\^[14]{} + e\^[23]{}) - 1 e\^[56]{} . \[f2cp3\] It can be checked that $dF_2=0$ and $\nabla_m F^{mn}=0$. Moreover, we will see that $F_2$ is of the expected form (\[f2nnk\]), with $\cw_2 \not=0$. In appendix B we will check that all equations of motion are satisfied and that 1 supersymmetry is preserved.
As already stressed in [@np; @vst; @stv], the new $\C\P^3$ compactification of massless IIA supergravity is directly related to compactification of 11 supergravity on the squashed seven-sphere [@dnp]. Indeed, the metric on the squashed $\S^7$ can be written as[^3] ds\_7\^2 = (d- A)\^2 + ds\_6\^2 , \[ss7\] where $ds_6^2$ is the above metric on $\C\P^3$ and the gauge potential $A$ is such that $dA$ gives precisely the RR 2-form background displayed in (\[f2cp3\]). When $\lambda^2=\frac15$ this seven-dimensional metric is Einstein and admits only one Killing spinor.
The fundamental forms $J$ and $\Omega$ can be obtained from the Killing spinor in six dimensions. Details are presented in appendix B. The main results are J & = & -(e\^[12]{} + e\^[34]{}) - (e\^[13]{} + e\^[42]{}) - (e\^[14]{} + e\^[23]{}) + e\^[56]{} ,\
& = & (e\^[126]{} + e\^[346]{}) + (e\^[136]{} + e\^[426]{}) + (e\^[125]{} + e\^[345]{})\
& & - (e\^[135]{} + e\^[425]{}) - (e\^[146]{} + e\^[236]{}) , \[jocp3\]\
& = & -(e\^[125]{} + e\^[345]{}) - (e\^[135]{} + e\^[425]{}) + (e\^[126]{} + e\^[346]{})\
& & - (e\^[136]{} + e\^[426]{}) + (e\^[145]{} + e\^[235]{}) . These forms satisfy (\[su3\]).
The torsion classes are found after computing the exterior derivatives that turn out to be exactly of the form (\[nonnk\]) with \_1 = ; \_2 J = J\^2 - 6e\^[1234]{} . \[w1w2cp3\] Both $\cw_1$ and $\cw_2$ are real. In fact, $d\im \Omega=0$. We can check that $\cw_2 \wedge J \wedge J = 0$ so that $\cw_2$ is primitive. It also follows that \^\*(\_2 J) = 2J - 6e\^[56]{} . \[dualw2\] With all this information it is a simple exercise to verify that the RR 2-form $F_2$ given in (\[f2cp3\]) can indeed be written as (\[f2nnk\]) when $\lambda^2=\frac15$.
Final remarks {#conclu}
=============
The original motivation behind this paper was to identify the internal space implicit in a class of 1 type IIA $\ads_4$ vacua obtained using the effective 4 formalism. As we have explained, this internal space turns out to be $\S^3 \times \S^3/\Z_2^3$ with a nearly-Kähler metric. This property, together with the structure of background fluxes, is in complete agreement with the general results derived from supersymmetry conditions and equations of motion in 10.
Unlike the Minkowski case, $\ads_4$ 1 type IIA compactifications have the peculiarity that the equations of motion can be solved in the absence of orientifold planes of negative tension. In the 4 approach this can be simply understood from the tadpole cancellation equations that include fluxes and sources [@cfi]. In 10, as reviewed in section 2, this follows from the Bianchi identity for the RR 2-form [@gmpt2]. In the $\S^3 \times \S^3/\Z_2^3$ compactification we have found explicit solutions of the tadpole cancellation conditions and used them to construct configurations of D6-branes that allow to solve the Bianchi identity in 10.
A second motivation of our work was to find a concrete example of 1 type IIA compactification to $\ads_4$ in which the internal space is not nearly-Kähler. This possibility is allowed by the general analysis of flux vacua, it corresponds to the case in which both torsion classes $\cw_1$ and $\cw_2$ are different from zero. Previous attempts to construct examples of this sort failed because the Bianchi identity for the RR 2-form could not be fulfilled [@lt]. Our contribution has been to observe that some solutions of massless type IIA supergravity discovered long time ago [@np; @vst; @stv; @pp] do fit within the general framework of $\ads_4$ flux vacua while the internal manifold does not have a nearly-Kähler metric. We considered compactification on $\C\P^3$ and showed that both torsion classes $\cw_1$ and $\cw_2$ are different from zero. Moreover, the background of the RR 2-form has the correct expression in terms of the torsion classes. Another example with both $\cw_1$ and $\cw_2$ non zero, already studied in [@hp], which has as internal space the coset $SU(3)/U(1)^2$, can be treated along the same lines as in section \[ccp3\].
In this note we have exemplified the validity and applicability of the effective 4 approach to uncover properties of 10 flux vacua. It is clearly desirable to extend the effective formalism to compactifications with more generic internal manifolds. In the future we hope to join efforts in pursuing further research on this interesting subject.
[**Acknowledgments**]{}
We thank B. Acharya, M. Graña, L. Ibáñez and J. Maldacena for useful comments. We are specially grateful to P. Cámara and S. Theisen for clarifying explanations and sharing their notes, and to Y. Oz for bringing [@np] to our attention. A.F. acknowledges the hospitality of the Max-Planck-Institut für Gravitationsphysik while preparing this paper. This work has been partially supported by the European Commission under the RTN European Program MRTN-CT-2004-503369, the Comunidad de Madrid through Proyecto HEPHACOS S-0505/ESP-0346, and the AECI (Agencia Española de Cooperación Internacional). G.A. acknowledges the hospitality of the International Centre for Theoretical Physics where part of this work was done. The work of G.A. is partially supported by ANPCyT grant PICT 11064 and CONICET PIP 5231.
Appendix A: Effective approach in 4 {#appA .unnumbered}
===================================
This appendix serves several purposes. First we give a concise account of the effective action for 4, 1 type IIA toroidal orientifolds [@gl; @dkpz; @vz]. We then describe to some extent the specific model that turns out to be related to compactification on $\ads_4 \times \S^3 \times \S^3$. We will also show that the results are in complete agreement with those derived from supersymmetry conditions and equations of motion in 10. Finally, we discuss tadpole cancellation equations and provide configurations of supersymmetric D6-branes that solve these equations.
In the 4 effective formalism the analysis of vacua is based on the superpotential generated by RR, NSNS and geometric fluxes. In type IIA orientifolds the flux induced superpotential can be written as W = \_[\_6]{} e\^[J\_c]{} + \_c ( + dJ\_c) . \[fullw\] The complexified forms defined as J\_c = B + iJ ; \_c= C\_3 + i(e\^[-]{} ) , \[cforms\] are expanded in the basis of invariant 2 and 3-forms, with coefficients given by the moduli fields. In the model we are considering these fields are the axiodilaton $S$, together with three complex structure $U_i$ and three Kähler moduli $T_i$. The relevant expansions are J\_c = iT\_i \_i ; \_c= iS \_0 - iU\_i \_i . \[cformexps\] As we saw in section \[cs3s3\], $J=t_i \omega_i$. The NSNS 2-form can also be expanded in terms of the $\omega_i$ as $B=-v_i \omega_i$. The $v_i$ are the Kähler axions and indeed the Kähler moduli are $T_i=t_i+i v_i$. For the axiodilaton and complex structure moduli we can substitute (\[om6\]) to obtain S s = e\^[-]{} ; U\_i u\_i = s \_j \_k , j = k . \[defrsu\] The corresponding axions arise from the RR 3-form given by $C_3 = -\im S \a_0 + \im U_i \a_i$.
To compute the superpotential we need expansions for the background fluxes. We follow the approach of [@cfi] where the fluxes are chosen to comply with the $\Z_2 \times \Z_2$ symmetry (\[z2z2\]). Thus, just as $J_c$ and $\Omega_c$, the fluxes are to be expanded in the basis of invariant forms displayed in (\[allforms\]). Furthermore, since we are assuming that the moduli are those of toroidal IIA orientifolds, the fluxes are required to conform to the orientifold involution (\[oaction\]). This means that $\ov{F}_0$ and $\ov{F}_4$ are even, whereas $\ov{H}$, $\ov{F}_2$ and $\ov{F}_6$ are odd under the orientifold involution [@gl]. The upshot is that background fluxes have the following expansions
[lclcl]{} = h\_0 \_0 + h\_i \_i & ; & \_0 = - M & ; & \_2 = q\_i \_i\
\_4 = e\_i \_i & ; & \_6 = e\_0 \_0 \_0 & .
\[fluxbg\] The exterior derivative of these fluxes and the Kähler form $J$ are found using (\[abis\]) that define the internal space.
The scalar potential of the moduli has the standard 1 expression V= e\^K { \_[=S,T\_i, U\_i]{} (+ \^\*)\^2 |D\_W|\^2 - 3|W|\^2 } , \[spot\] where we already assumed the classical Kähler potential $K = - \sum_{\Phi=S,T_i, U_i} \, \log(\Phi + \Phi^*)$, and $D_\Phi W = \partial_\Phi W + W \partial_\Phi K$. Supersymmetric AdS minima are determined by the condition $D_\Phi W=0$.
To obtain the model analyzed in [@cfi] one chooses RR fluxes $q_i=-c$ and $e_i=e$ so that a configuration with $T_i=T$ is allowed. The resulting superpotential is simply[^4] =e\_0 + 3ieT + 3c T\^2 + iM T\^3 + (ih\_0 - 3a T)S - \_[k=0]{}\^3 (ih\_k + b\_kT) U\_k . \[supT\]
This superpotential admits supersymmetric AdS minima provided that the fluxes satisfy the constraint 3h\_k a + h\_0 b\_k = 0 ; k=1,2,3 . \[homega\] In this case the real parts of the axiodilaton and complex structure moduli are completely determined in terms of the Kähler modulus according to as = 2t(c-Mv) ; 3 as = b\_k u\_k ; k=1,2,3 . \[realsu\] Recall that $s=\re S$, $u_k=\re U_k$, $t=\re T$ and $v=\im T$ and that the real part of the moduli are positive definite. Thus, (\[realsu\]) requires that the geometric fluxes $a$ and $b_k$ be of the same sign. For the $S$ and $U_i$ axions only one linear combination is fixed, this is 3a S +b\_k U\_k = 3e + (3h\_0 - 7a v) - v(3h\_0 - 8a v) . \[axionfix\] To have the minimum with $T_i=T$ it must also be that $b_1 \im U_1 = b_2 \im U_2 = b_3 \im U_3$.
The vacuum expectation value for the Kähler modulus depends on whether the mass parameter $M$ vanishes or not. When $M=0$ it is found that v=v\_0= ; 9c t\^2 = e\_0 - - . \[valmzero\] In this case (\[realsu\]) implies that necessarily there is a flux of the RR 2-form, i.e. $c \not= 0$, and furthermore that $ac > 0$ and $c b_k > 0$. Background fluxes $\ov{H}$ and $\ov{F}_4$ can be absent but then the Freund-Rubin flux $\ov{F}_6 \sim e_0$ must be turned on.
When $M \not= 0$ the Kähler axion satisfies the cubic equation 160(v-v\_0)\^3 + 294(v\_0-)(v-v\_0)\^2 + 135(v\_0-)\^2(v-v\_0) + v\_0\^2(v\_0-) + (e\_0 a - e h\_0) = 0 . \[cubiceq\] The real part of the Kähler modulus is now determined from t\^2= 15 (v-)(v-v\_0) . \[valtm\] The solution for $v$ must be real and such that $t^2 > 0$.
Let us now check that the above results agree with the general analysis in 10. To begin observe that we have $d \im \Omega=0$ compared to $d \re \widehat \Omega=0$. We find that in order to match the 4 and 10 results we need to make the choice = - i; + =0 [mod]{} 2 . \[wophase\] The full exterior derivatives of $J$ and $\Omega$ are given in (\[djdo\]).
The next step is to express the field strengths in terms of the background fluxes and the moduli. In the case at hand, with $q_i=-c$, $e_i=e$, $T_i=T$, it is possible to write most forms in terms of $J$ and $\Omega$. For example, $B=-v J/t$, $\ov{F}_4=e J^2/2t^2$, and so on. After substituting in (\[hfdef\]) we find H & = & (h\_0 - 3 av) ,\
F\_2 & = & J , \[hfd4\]\
F\_4 & = & ,\
F\_6 & = & . All these expressions greatly simplify upon using (\[axionfix\]) and (\[cubiceq\]). In the end we obtain dJ & = & e\^ ; d= e\^ J\^2 ; H = 25 M e\^ , \[fd4\]\
F\_0 & = & -M ; F\_2 = J ; F\_4 = - J\^2 ; F\_6 = J\^3 . These agree with (\[derj\]) and (\[fluxsol\]) provided that =3A ; m = 5 e\^[4A]{} ; m = e\^[4A]{} . \[checkft\] The relation between the dilaton and the warp factor is precisely the same found in the ten-dimensional analysis.
It is also interesting to compute the cosmological constant which follows from the value $V_0$ of the scalar potential at the minimum. For the AdS minimum, $V_0 = -3 e^K |W_0|^2$. To determine $W_0$ we can substitute the vevs of the moduli in (\[supT\]). A more general approach is to use the original form of the superpotential (\[fullw\]). Using previous results to evaluate the integrand at the minimum we arrive at e\^[J\_c]{} + \_c ( + dJ\_c) |\_0 = 3 (m +i m) e\^[-4A]{} J\^3 . \[intcero\] This shows that the superpotential at the minimum is proportional to the complex constant $\mu$. More precisely, $|W_0|^2=16 t^6 e^{-8A} |\mu|^2{\cc}^2$. For the classical Kähler potential, $e^K = (2^7 t^3 s u_1 u_2 u_3\,{\cc})^{-1}$, which can be rewritten as $e^K=e^{4\phi}/128 t^9 {\cc}^3$. Therefore, V\_0 = - ||\^2 = . \[ccd4\] where $\Lambda=-3 |\mu|^2$ is the cosmological constant and $M_P^2 = 8 e^{2A-2\phi} {\cc}t^3$. The moduli above are evaluated at the minimum and we are taking $\a^\prime=1$.
A.1 {#tadin4 .unnumbered}
-----
The fluxes induce tadpoles for the RR 7-form $C_7$ that can also couple to D6-branes and O6-planes. In general these objects span space-time and wrap a 3-cycle in $\M_6$. The RR 7-form can then be written as $C_7={\rm dvol}_4 \wedge X$, where $X$ is some 3-form in the internal space, which can be expanded in a convenient basis. We denote by $\Pi_a$ the 3-cycle wrapped by a stack of $N_a$ $\D6_a$-branes or $\O6_a$-planes. The coupling of $C_7$ in the action has contributions from fluxes and from the sources, namely \_[\_4 \_6]{} C\_7 (d\_2 - F\_0 ) + \_a N\_a Q\_a \_[\_4 \_a]{} C\_7 , \[c7tadpole\] where $Q_a=1$ for D6-branes and $Q_a=-4$ for O6-planes. Here we are considering the internal space to be $\cs_6=\S^3\times \S^3/\Z_2^3$. The factor $\sqrt{\cc}$ must be included for consistency with the normalization of the 1-form basis (see \[MGvolume\]).
As usual in the 4 effective formulation, it appears useful to consider factorizable 3-chains \_a=(n\_a\^1, m\_a\^1)(n\_a\^2, m\_a\^2) (n\_a\^3, m\_a\^3) , \[facpi\] where $n_a^i$ $(m_a^i)$ are the wrapping numbers along the $\eta^i$ $(\eta^{i+3})$. In particular, there is a basis of 3-chains $\Pi_{ijk}$ spanning the $\{i,j,k\}$ directions. For instance, $\Pi_{123}=(1,0)\otimes(1,0) \otimes(1,0)$, etc.. To each $\Pi_{ijk}$ we have a corresponding “dual” 3-form $\eta^{ijk}$ such that $$\int_{ \Pi_{i^{\prime}j^{\prime}k^{\prime}}}\eta^{ijk}= \frac{1}{\sqrt{\cc}}\, \delta_{i,i^{\prime}}
\, \delta_{j,j^{\prime}}\, \delta_{k,k^{\prime}} \ .
\label{intetaijk}$$ Integrating over the 3-chain $\Pi_a$ then gives, $\int_{ \Pi_a}\eta^{123}=\frac{1}{\sqrt{\cc}} \, n_a^1 n_a^2 n_a^3$, $\int_{ \Pi_a}\eta^{156}=\frac{1}{\sqrt{\cc}} \, n_a^1 m_a^2 m_a^3$, and so on.
It is worth noticing that the basis manifolds $\Pi_{ijk}$ are not necessarily closed cycles and, therefore, neither is $\Pi_a$, for generic wrappings. As an example, consider the exact form $d (\xi^1 \wedge {\hat \xi}^1) = 2 \sqrt{ab_1b_2b_3}
(a \eta^{456}+ b_1\eta^{423}-b_2 \eta^{153}-b_3 \eta^{126})$, then, $ \int_ {\Pi_{456}}d (\xi^1 \wedge \xi^1)=\frac{2}{\sqrt{\cc}} \,\sqrt{ab_1b_2b_3}$, indicating that the manifold $\Pi_{456}$ has a boundary (see [@marche] for a related discussion). We rely on tadpole cancellation and supersymmetry to restrict to the adequate wrappings for D6-branes. When the orientifold action (\[oaction\]) is implemented there are eight O6-planes along $\otimes_i(1,0)$ and image D6-branes wrapping $\otimes_i (n_a^i, -m_a^i)$ must be included.
To preserve the same supersymmetries as the background the D6-branes must wrap cycles $\Pi_a$ such that $$\theta_a^1+\theta_a^2+\theta_a^3=0\quad{\rm mod}\quad 2 \pi \ ,
\label{susyd6}$$ where the angles are measured in accordance with \^j\_a = . \[tans\] Recall that the $\tau_i$ are the complex structure moduli that enter in the metric as shown in (\[metricm6\]). In the vacuum we are considering they satisfy (\[crux\]).
>From the supersymmetric constraint (\[susyd6\]) it follows that $$\tau_1\tau_2\tau_3 m_a^1 m_a^2 m _a^3-\tau_1 m_a^1 n_a^2 n_a^3- \tau_2 n_a^1
m_a^2 n_a^3 - \tau_3 n_a^1n_a^2 m_a^3 = 0 \ .
\label{tg}$$ This condition amounts to $\im \Omega \big |_{\Pi_a} =0$. In fact, the supersymmetric cycles are calibrated by $\re \Omega$ and the condition on the angles is equivalent to $\re \Omega \big |_{\Pi_a} = {\rm dvol}(\Pi_a)$. Besides, the factorizable cycles satisfy $J \big |_{\Pi_a} =0$.
Substituting the fluxes and the data for the sources in (\[c7tadpole\]) we obtain the tadpole cancellation equations. The conditions receiving flux contributions are \_a N\_a Q\_a n\_a\^1 n\_a\^2 n\_a\^3 +(M h\_0 -3a c) & = & 0 ,\
\_a N\_a Q\_a n\_a\^1 m\_a\^2 m\_a\^3 + (M h\_1 + b\_1 c) & = & 0 , \[tadb\]\
\_a N\_a Q\_a m\_a\^1 n\_a\^2 m\_a\^3 + (M h\_2 + b\_2 c) & = & 0 ,\
\_a N\_a Q\_a m\_a\^1 m\_a\^2 n\_a\^3 + (M h\_3 + b\_3 c) & = & 0 . The sum in $a$ includes $\O6_a$-planes, when orientifold actions are performed, as well as and their orientifold images if necessary. Finally, there are fluxless conditions \_a N\_a Q\_a m\_a\^1 m\_a\^2 m\_a\^3 & = & 0 ,\
\_a N\_a Q\_a m\_a\^1 n\_a\^2 n\_a\^3 & = & 0 , \[tadfless\]\
\_a N\_a Q\_a n\_a\^1 m\_a\^2 n\_a\^3 & = & 0 ,\
\_a N\_a Q\_a n\_a\^1 n\_a\^2 m\_a\^3 & = & 0 . When the orientifold action (\[oaction\]) is implemented these last four constraints are automatically satisfied once images are included.
When $M \not=0$ the tadpole equations admit a solution without branes or O-planes. This happens because $h_k=-h_0 b_k /3a$ and then all flux tadpoles can cancel simultaneously when $Mh_0 = 3ac$ [@cfi]. Now, the relations (\[checkft\]) and (\[valtm\]) imply that this condition is equivalent to $\tilde m^2 = 15 m^2$. As explained in section \[d10\] this is precisely the case when the internal space is nearly-Kähler and no sources are necessary to satisfy the Bianchi identity for $F_2$. In 10 we have further seen that when $\tilde m^2 > 15 m^2$ the Bianchi identity can be satisfied by adding sources of positive charge. In the 4 approach it is indeed evident that whenever $Mh_0 < 3ac$ the tadpoles can be cancelled by adding only $\D6$-branes.
$N_a$ $(n_a^1,m_a^1)$ $(n_a^2,m_a^2)$ $(n_a^3,m_a^3)$
------- ----------------- ----------------- -----------------
$N_A$ $(1,0)$ $(1,0)$ $(1,0 )$
$N_B$ $(-1,1)$ $ (-1,1)$ $(-1,1)$
$N_B$ $(-1,-1)$ $ (-1,-1)$ $(-1,-1)$
: Wrapping numbers for D6-branes in model 1[]{data-label="adsm0"}
To present examples of tadpole cancellation with only D6-branes we will consider the case $M=0$ in the $\S^3 \times \S^3$ compactification that we have been analyzing. A first model consists of the factorizable D6-branes shown in table \[adsm0\]. The third stack is the mirror image, $\tilde m_B^i= -m_B^i$, of the second and it is included to saturate the fluxless tadpole equations. We also take $N_A=8 N_B$. The first stack has $\theta_A^i=0$, hence it is supersymmetric independently of the values of the complex structure parameters. On the other hand, substituting the wrapping numbers in (\[tadb\]) gives the relations $2N_B=ac=b_1c=b_2c=b_3c$, needed for tadpole cancellation. Next, using that $\tau_i = b_i \sqrt{3 a/b_1 b_2 b_3}$, we find $\tau_1=\tau_2=\tau_3=\sqrt3$. We can then check that the second and third stack are also supersymmetric. In fact, computing $\tan \theta_B^i$ for the second shows that $\theta_B^i=2\pi/3$. Assuming that the $\Pi_a$ cycles have large volume, in this model 1 the resulting gauge group is $U(N_A) \times U(N_B) \times U(N_B)$. According to the intersections between cycles, the matter content consists of chiral multiplets transforming as (,,[**1**]{}) + (,[**1**]{},) + 8([**1**]{},,) . \[specuno\] The multiplicity 8 of the last representation arises from the intersection number between the cycle B and its mirror. Since $N_A=8N_B$ this chiral spectrum is free of gauge anomalies.
$N_a$ $(n_a^1,m_a^1)$ $(n_a^2,m_a^2)$ $(n_a^3,m_a^3)$
------- ----------------- ----------------- -----------------
$N_0$ $(1,0)$ $(1,0)$ $(1,0 )$
$N_1$ $(1,0)$ $ (0,-1)$ $(0,1)$
$N_2$ $(0,1)$ $ (1,0)$ $(0,-1)$
$N_3$ $(0,1)$ $ (0,-1)$ $(1,0)$
: Wrapping numbers for D6-branes in model 2[]{data-label="oadsm0"}
There are other D6-brane configurations capable of canceling tadpoles. Some are equivalent to the setup in model 1 but others belong to a different class. For instance, in table \[oadsm0\] we display a model 2 with four stacks of branes that are all supersymmetric independently of the complex structure moduli. To cancel tadpoles the numbers of branes in each stack must be related to the fluxes by $N_0=3ac$ and $N_i = b_ic$. In this model the resulting spectrum is non-chiral.
Appendix B: Supersymmetric vacua of massless type IIA supergravity in 10 {#appB .unnumbered}
========================================================================
In this appendix we tersely summarize some basic aspects of compactification of massless IIA supergravity on ${\rm AdS}_4 \times \M_6$. We will review the case when the internal space is $\C\P^3$ and appropriate fluxes are turned on so that there is a vacuum with 1 supersymmetry in 4 [@np; @vst; @stv]. Our main goal is to explicitly find the six dimensional Killing spinor in order to determine the fundamental forms $J$ and $\Omega$ that define the $SU(3)$ structure. We will follow and refer to the discussion of [@np] where the equations of motion and the supersymmetry transformations are spelled out in full detail.
We consider a class of vacua with background metric of the product form (\[geo\]) but to simplify the warp factor $A$ is fixed to zero. The dilaton $\phi$ is assumed to be constant whereas the NS 2-form and its field strength are taken to vanish. On the contrary, there are non-trivial RR fluxes. For the 4-form one makes the Freund-Rubin Ansatz F\_ = 3 f \_ ; \_[0123]{}= , \[marc\] while other components are zero. For the RR 2-form there is a flux $F_{mn}$ through $\M_6$ to be specified shortly. Under these conditions the equations of motion reduce to R\_ & = & -12 e\^[/2]{} f\^2 g\_ ,\
R\_[mn]{} & = & 6 e\^[/2]{} f\^2 g\_[m n]{} + e\^[3/2]{} F\_[mp]{} F\_n\^[ p]{} , \[eommz\]\
e\^F\_[mn]{} F\^[mn]{} & = & 24 f\^2 ; \_m F\^[mn]{}=0 . The Einstein equation in 4 shows that space-time can indeed be taken to be ${\rm AdS}_4$ with cosmological constant $\Lambda= -12 e^{\phi/2} f^2 $.
To characterize the internal space we still need to specify the flux $F_2$. We will see that it is consistent to take $\M_6$ to be $\C\P^3$ with metric given in (\[cp3metric\]), while $F_2$ can be set equal to the 2-form (\[f2cp3\]). This RR 2-form satisfies the equation of motion and the properties F\_[mn]{} F\^[mn]{} = 2(2\^2 + 1[\^2]{}) ; F\_[ac]{} F\_b\^[ c]{}= \^2 \_[ab]{} ; F\_[ik]{} F\_j\^[ k]{}= 1[\^2]{} \_[ij]{} , \[f2props\] where $a,b,c=1, \cdots, 4$, and $i,j,k=5,6$, are flat indices.
Once the flux $F_2$ is given, the dilaton equation of motion implies that the vevs $e^\phi$, $f$ and the metric parameter $\lambda$ are related by e\^(2\^2 + 1[\^2]{}) = 12 f\^2 . \[flambda\] Substituting in the 6 Einstein equation we then find that in the flat basis the Ricci tensor is diagonal with components R\_[ab]{}=e\^[3/2]{} (3\^2+1[\^2]{}) \_[ab]{} ; R\_[ij]{}= e\^[3/2]{} (\^2 + 1[\^2]{}) \_[ij]{} . \[riccid6\] The Ricci tensor of the generic $\C\P^3$ metric has precisely this structure. Comparing with (\[riccicp3\]) we see that the dilaton vev has to be $e^\phi=1$. Moreover, the parameter $\lambda$ must be such that 5 \^2 - 6 + 1[\^2]{} = 0 . \[lequ\] There is a solution with $\lambda^2=1$ for which the metric is Einstein. We are more interested in the solution with $\lambda^2=1/5$. In this case, from (\[flambda\]) it transpires that the Freund-Rubin parameter is fixed to be $f^2=9/20$.
We now discuss the requirements for residual supersymmetry in 4. We will employ exactly the same conventions of [@np] for the 10 Dirac matrices. In 6 we basically adopt the matrices used in [@dnp] in 7. With flat indices these are \_1 & = & i \_0 ; \_2 = \_1 ; \_3 = \_2 ; \_4 = \_3\
\_5 & = & i\_5 \_1 ; \_6 = i\_5 \_2 ; \_0 = \_ 1 \_6 = i\_5 \_3 , \[gammad6\] where $\sigma_i$ are the Pauli matrices. The 4-dimensional matrices are \_0 =
(
[cc]{} 0 & 1\
1 & 0
)
; \_a =
(
[cc]{} 0 & \_a\
-\_a & 0
)
, \[gammad4\] and $\gamma_5= - i \gamma_0 \gamma_1 \gamma_2 \gamma_3$. We will also need the charge conjugation matrix in 6 which in our conventions is given by $C=\Gamma_2 \Gamma_ 4\Gamma_6$.
To study the conditions for the vacuum to preserve supersymmetry we first write the 10-dimensional parameter as $\epsilon \otimes \eta$, where $\epsilon$ and $\eta$ are respectively spinors in four and six dimensions. We then substitute the vevs of all fields in the supersymmetry transformations of the fermionic fields which in 10, IIA supergravity are the gravitino and the dilatino. We refer the reader to reference [@np] for the explicit equations of these transformations. From the dilatino variation we obtain (S + 2f)= 0 , \[etaev\] where the matrix $S$ depends on the RR 2-form flux as S = F\_[mn]{} \^[mn]{} \_0 . \[sdef\] For the $F_2$ background in (\[f2cp3\]), $S$ turns out to have eigenvalues ${\mbox{\small$1/\lambda$}}$, ${\mbox{\small$(2\lambda^2-1)/\lambda$}}$, and $-{\mbox{\small$(2\lambda^2+1)/\lambda$}}$, with degeneracies $4,2$ and 2 respectively. Remarkably, for the case of interest with $\lambda^2=1/5$ and $f^2=9\lambda^2/4$, $S$ can have an eigenvalue $-2f$ as long as we take $f=3\lambda/2$. The corresponding eigenvector has the simple form =
(
[c]{} s\_1\
s\_2\
s\_3\
s\_4\
0\
0\
0\
0
)
; s\_1 = - ; s\_4 = , \[etafull\] where $s_2$ and $s_3$ in principle depend on all internal coordinates.
From the gravitino variation $\delta \psi_\mu$, using (\[etaev\]), we find D\_- e\^[/4]{} f \_5 \_= 0 . \[killingads\] This is the expected equation for the supersymmetry parameter in ${\rm AdS}_4$ with cosmological constant $\Lambda= -12 e^{\phi/2} f^2 $. Finally, from the variation $\delta \psi_m$ we obtain the Killing equation D\_m - 2 \_m - 14 F\_m\^[ n]{} \_n \_0 = 0 , \[killingcp3\] where we have set $e^\phi=1$. For the covariant derivative acting on spinors we use the conventions of [@dnp].
It remains to solve the Killing equation to determine the unknown functions $s_2$ and $s_3$ in $\eta$. From the $\psi$ component we find s\_2 = i e\^[-i]{} s\_3 . \[s2s3\] It further follows that $s_3$ is completely independent of the $\S^4$ coordinates $(\psi, \alpha, \beta. \gamma)$, but depends on the $\S^2$ variables as s\_3 = e\^[i]{} e\^[-i/2]{} 2 , \[s3final\] where $\delta$ is a constant phase. The normalization guarantees that the Weyl spinors \_= 2 \[etaweyl\] satisfy $\eta_\pm^{\dagger} \eta_\pm =1$. The phase $\d$ is fixed by imposing the reality condition $\eta_+^* = C \, \eta_-$.
We are now ready to compute the fundamental forms $J$ and $\Omega$ defined by J\_[mn]{} = i \_-\^ \_[mn]{} \_- ; \_[mnp]{} = \_+\^ \_[mnp]{} \_- . \[joeta\] In the end we obtain the results reported in section \[ccp3\]. We stress that there is a unique Killing spinor $\eta$ so that the internal manifold has $SU(3)$ structure and there is 1 supersymmetry in 4.
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[^1]: The sign differences with respect to equation (7.9) in [@gmpt2] are due to our conventions ${}^* J = J^2/2$ and ${}^*1 = J^3/6$, where ${}^*$ is the Hodge dual in six dimensions.
[^2]: We thank P. Cámara for these observations.
[^3]: The metric on the squashed $\S^7$ is $ds_7^2 = d\tilde s_4^2 + \lambda^2( d\sigma^A - \ca^A)^2$, where $\sigma^A$ is a second set of $SU(2)$ left-invariant 1-forms. To recover (\[ss7\]) just set $\sigma^1 = \sin \varphi \, d\theta + \sin \theta \cos \varphi \, d\tau$, $\sigma^2 = -\cos \varphi \, d\theta + \sin \theta \sin \varphi \, d\tau$, $\sigma^3 = -d\varphi + \cos \theta \, d\tau$.
[^4]: A volume factor ${\cc}$ appears here due to normalization (\[MGvolume\]).
|
---
abstract: 'Ordering of the Heisenberg spin glass in four dimensions (4D) with the nearest-neighbor Gaussian coupling is investigated by equilibrium Monte Carlo simulations, with particular attention to its spin and chiral orderings. It is found that the spin and the chirality order simultaneously with a common correlation-length exponent $\nu_{CG}=\nu_{SG}\simeq 1.0$, [*i.e.*]{}, the absence of the spin-chirality decoupling in 4D. Yet, the spin-glass ordered state exhibits a nontrivial phase-space structure associated with a continuous one-step-like replica-symmetry breaking, different in nature from that of the Ising spin glass or of the mean-field spin glass. Comparison is made with the ordering of the Heisenberg spin glass in 3D, and with that of the 1D Heisenberg spin glass with a long-range power-law interaction. It is argued that the 4D might be close to the marginal dimension separating the spin-chirality decoupling/coupling regimes.'
author:
- Hikaru Kawamura
- Shinichirou Nishikawa
title: Ordering of the Heisenberg spin glass in four dimensions
---
I. Introduction
===============
The Heisenberg spin-glass (SG) model, or the Edwards-Anderson model with the isotropic Heisenberg exchange interaction [@EA], has been considered as the standard model of many real SG materials [@review]. In realistic three spatial dimensions (3D), earlier studies in the 80’s suggested that the isotropic Heisenberg SG did not exhibit an equilibrium SG transition at any finite temperature in apparent contrast to experiments [@Banavar82; @McMillan85; @Olive86; @Matsubara91; @Yoshino93]. Then, a proposal was made in 1992 that the model might exhibit a finite-temperature transition in the chiral sector, with the standard SG order occurring at a temperature $T_{SG}$ lower than the chiral-glass (CG) ordering temperature $T_{CG}$, [*i.e.*]{}, $T_{SG} < T_{CG}$ [@Kawamura92]. The occurrence of such separate spin and chirality transitions is now called “spin-chirality decoupling” [@Kawamura10]. Chirality is a multispin variable representing the sense or the handedness of local noncoplanar spin structures induced by spin frustration. A possible counter-view to such a picture might be that the spin and the chirality order at a common finite temperature [@Matsubara; @LeeYoung03; @Campos06; @LeeYoung07; @Fernandez09]. Although there is no complete consensus, recent simulations point to the occurrence of the spin-chirality decoupling in 3D [@Matsumoto02; @HukuKawa05; @Campbell07; @VietKawamura09a; @VietKawamura09b]. For example, Ref.[@VietKawamura09b] reported that $T_{SG}$ was lower than $T_{CG}$ by about $10\sim 15$%.
To get further insight into the issue, it might be useful to extend the space dimension from original $d=3$ to general $d$-dimensions. In $d=1$, the Heisenberg SG with a short-range (SR) interaction exhibits only a $T=0$ transition. In $d=2$, recent calculations suggested that the vector SG model, either the three-component Heisenberg SG [@KawaYone03] or the two-component [*XY*]{} SG [@Weigel08], exhibited a $T=0$ transition but with the spin-chirality decoupling, [*i.e.*]{}, the CG correlation-length exponent $\nu_{CG}$ was greater than the SG correlation-length exponent $\nu_{SG}$, meaning that this $T=0$ transition was characterized by two distinct diverging length scales, each associated with the chirality and with the spin. In the opposite limit of $d\rightarrow \infty$, the model is known to reduce to the mean-field (MF) model, [*i.e.*]{}, the Sherrington-Kirkpatrick (SK) model. The Heisenberg SK model is known to exhibit a single finite-temperature transition, with no spin-chirality decoupling. In high but finite $d$, Monte Carlo (MC) study by Imagawa and Kawamura suggested that the spin-chirality decoupling did not occur for $d=5$, whereas the situation in $d=4$ appeared somewhat more marginal [@ImaKawa03a].
Another useful way of attacking the issue might be to study the one-dimensional (1D) Heisenberg SG with a long-range (LR) power-law interaction proportional to $1/r^{\sigma}$ ($r$ the spin distance). Indeed, several studies both for the Ising and the Heisenberg SGs suggested that the 1D LR SG model with a power-law exponent $\sigma$ might show the ordering behavior analogous to the $d$-dimensional SG model with a SR interaction [@BinderYoung86; @Katzgraber03; @Katzgraber05; @Katzgraber09; @Leuzzi99; @Leuzzi08; @VietKawamura10a; @VietKawamura10b; @SharmaYoung]. Even a simple empirical formula relating $\sigma$ and $d$, $d=2/(2\sigma-1)$, was proposed [@BinderYoung86], though the relation is only approximate.
Recent MC calculation on the 1D LR Heisenberg SG by Viet and Kawamura suggested that the spin-chirality decoupling occurred for $\sigma \geq \sigma_c$, but did not occur for $\sigma \leq \sigma_c$, $\sigma_c$ being estimated numerically to be $\sigma_c\simeq 0.8$ [@VietKawamura10a; @VietKawamura10b]. If one applies the approximate $d-\sigma$ correspondence formula quoted above [@BinderYoung86], the critical dimension below which the spin-chirality decoupling is expected would be $d_c\simeq 3.3$, suggesting that $d=4$ might lie near the margin of, slightly on the side of the spin-chirality coupling regime. Of course, the above $d-\sigma$ correspondence formula is only approximate, and even whether $d=4$ is greater or smaller than $d_c$ is not clear. Previous simulation on the $4d$ Heisenberg SG, which dealt with the linear size of $L\leq 10$, was not definitive concerning the occurrence of the spin-chirality decoupling in 4D [@ImaKawa03a].
Under such circumstances, the purpose of the present paper is first to clarify whether the spin and the chirality are decoupled or not in the $4d$ Heisenberg SG by simulating larger systems than the ones studied in Ref.[@ImaKawa03a]. Since $d=4$ is expected to be close to the marginal dimension concerning the spin-chirality decoupling, we wish to see what kind of ordering behavior is realized for the spin and the chirality near the marginal dimension.
The present paper is organized as follows. In §II, we introduce our model and explain some of the details of our MC simulation. Various physical quantities are defined in §III. The results of our MC simulations, including the spin and the chiral correlation lengths, the spin and the chiral Binder ratios, are presented in §IV. The SG and CG transition temperatures are determined by carefully examining the size dependence of the finite-size data. In §V, critical properties of the spin and of the chirality are investigated by means of a finite-size scaling analysis. Finally, section §VI is devoted to summary and discussion.
II. The model and the method {#secModel}
============================
The model we consider is the isotropic classical Heisenberg model on a 4D hyper-cubic lattice, with the nearest-neighbor Gaussian coupling. The Hamiltonian is given by $${\cal H}=-\sum_{<ij>}J_{ij}\vec{S}_i\cdot \vec{S}_j\ \ ,
\label{eqn:hamil}$$ where $\vec{S}_i=(S_i^x,S_i^y,S_i^z)$ is a three-component unit vector, and $<ij>$ sum is taken over nearest-neighbor pairs on the lattice. The nearest-neighbor coupling $J_{ij}$ is assumed to obey the Gaussian distribution with zero mean and variance $J^2$, which is taken to be unity ($J=1$) in the following. The temperature $T$ is measured in units of $J$.
We perform equilibrium MC simulations on the model. The lattices are hyper-cubic lattices with $N=L^{4}$ sites with $L=6$, 8, 10, 12, 16 and 20. We impose periodic boundary conditions in all four directions. Sample average is taken over 1300, 1200, 840, 590, 430 and 256 independent bond realizations for $L=6$, 8, 10, 12, 16 and 20, respectively. Error bars of physical quantities are estimated by sample-to-sample statistical fluctuations over the bond realization.
In order to facilitate efficient thermalization, we combine the heat-bath and the over-relaxation methods with the temperature-exchange technique [@HukushimaNemoto]. For each heat-bath sweep we perform 11, 15, 19, 23, 31 and 55 over-relaxation sweeps, while the total number of temperature points in the temperature-exchange process are taken to be 35, 51, 59, 55, 55 and 60 for $L=6$, 8, 10, 12, 16 and 20, respectively. Care is taken to be sure that the system is fully equilibrated. Equilibration is checked by following the procedures of Ref.[@VietKawamura09b].
III. Physical quantities
========================
In this section, we define various physical quantities measured in our simulations.
For the Heisenberg spin, the local chirality at the $i$-th site and in the $\mu$-th direction $\chi_{i\mu}$ may be defined for three neighboring Heisenberg spins by a scalar $$\chi_{i\mu}=
\vec{S}_{i+{\hat{e}}_{\mu}}\cdot
(\vec{S}_i\times\vec{S}_{i-{\hat{e}}_{\mu}}),$$ where ${\hat{e}}_{\mu}\ (\mu=x,y,z,u)$ denotes a unit vector along the $\mu$-th axis. There are in total $4N$ local chiral variables. We define an “overlap” for the chirality. We prepare at each temperature two independent systems 1 and 2 described by the same Hamiltonian (1) with the same interaction set. We simulate these two independent systems 1 and 2 in parallel with using different spin initial conditions and different sequences of random numbers.
The $k$-dependent chiral overlap, $q_\chi(\vec k)$, is defined as an overlap variable between the two replicas 1 and 2 as a scalar $$q_\chi(\vec k) =
\frac{1}{4N}\sum_{i=1}^N\sum_{\mu=x,y,z,u}
\chi_{i\mu}^{(1)}\chi_{i\mu}^{(2)}e^{i\vec k\cdot \vec r_i},$$ where the upper suffixes (1) and (2) denote the two replicas of the system, and $\vec r_i$ is the position vector of the site $i$.
The $k$-dependent spin overlap, $q_{\alpha\beta}(\vec k)$, is defined by a [*tensor*]{} variable between the $\alpha$ and $\beta$ components of the Heisenberg spin, $$q_{\alpha\beta}(\vec k) =
\frac{1}{N}\sum_{i=1}^N S_{i\alpha}^{(1)}S_{i\beta}^{(2)}e^{i\vec k\cdot \vec r_i},
\ \ \ (\alpha,\beta=x,y,z).$$ In term of the $k$-dependent overlap, the CG and the SG order parameters are defined by the second moment of the overlap at a wavevector $k=0$, $$q_{CG}^{(2)}=\frac {[\langle | q_{\chi}(\vec 0)|^2 \rangle]} {\overline{\chi}^{4}},$$ $$q_{SG}^{(2)} = [\langle q_{\rm s}(\vec 0)^2\rangle]\ ,
\ \ \
q_{\rm s}(\vec k)^2 = \sum_{\alpha,\beta=x,y,z} \left| q_{\alpha\beta}(\vec k) \right| ^2,$$ where $\langle \cdots \rangle$ represents a thermal average and $[\cdots ]$ an average over the bond disorder. The CG order parameter $q_{CG}^{(2)}$ has been normalized here by the mean-square amplitude of the local chirality, $$\overline{\chi}^{2}=\frac{1}{4N}\sum_{i=1}^N\sum_{\mu=x,y,z,u}[\langle \chi_{i\mu}^2\rangle],$$ which remains nonzero only when the spin has a noncoplanar structure locally. The local-chirality amplitude depends on the temperature and the lattice size only weakly.
Finite-size correlation length $\xi_L$ is defined by $$\xi_L =
\frac{1}{2\sin(k_\mathrm{m}/2)}
\sqrt{ \frac{ [\langle q(\vec 0)^2 \rangle] }
{[\langle q(\vec{k}_\mathrm{m})^2 \rangle] } -1 },$$ for each case of the chirality and the spin, $\xi_{CG}$ and $\xi_{SG}$, where $\vec{k}_{\rm m}=(2\pi/L,0,0,0)$ with $k_{\textrm{m}}=|\vec k_{\textrm{m}}|$. For the CG correlation length $\xi_{CG}$, we consider two distinct definitions depending on the mutual direction between the $\hat{e}_\mu$-vector appearing in the definition of the local chirality (2) and the $\vec{k}_\mathrm{m}$-vector. When $\hat{e}_\mu \parallel \vec{k}_\mathrm{m}$, [*i.e.*]{}, $\mu=x$, we call the corresponding $\xi_{CG}$ the parallel CG correlation length $\xi_{CG}^\parallel$, whereas, when $\hat{e}_\mu \perp \vec{k}_\mathrm{m}$, [*i.e.*]{}, $\mu=y,z,u$, we call the corresponding $\xi_{CG}$ the perpendicular CG correlation length. The perpendicular CG correlation length $\xi_{CG}^\perp$ is actually defined by the mean of three equivalent ones, each defined in the $\mu=y,z,u$ directions.
The CG and the SG Binder ratios are defined by $$g_{CG}=
\frac{1}{2}
\left(3-\frac{[\langle q_{\chi}(\vec 0)^4\rangle]}
{[\langle q_{\chi}(\vec 0)^2\rangle]^2}\right),$$ $$g_{SG} = \frac{1}{2}
\left(11 - 9\frac{[\langle q_{\rm s}(\vec 0)^4\rangle]}
{[\langle q_{\rm s}(\vec 0)^2\rangle]^2}\right).
\label{eqn:gs_def}$$ These quantities are normalized so that, in the thermodynamic limit, they vanish in the high-temperature phase and gives unity in the non-degenerate ordered state. In the present Gaussian coupling model, the ground state is expected to be non-degenerate so that both $g_{CG}$ and $g_{SG}$ should be unity at $T=0$.
IV. Monte Carlo results
=======================
In this section, we present the results of our MC simulations on the 4D isotropic Heisenberg SG with the random Gaussian coupling.
We show in Fig.1 the temperature dependence of the CG and the SG correlation-length ratios, $\xi_{SG}/L$ in (a), $\xi_{CG}^\perp/L$ in (b), and $\xi_{CG}^\parallel/L$ in (c). As can be seen from the figures, both the chiral $\xi_{SG}/L$ and the spin $\xi_{CG}/L$ curves cross at temperatures which are weakly $L$-dependent. Magnified views of the crossing-temperature range are shown in Figs.2(a)-(c) for the spin, the perpendicular chirality and the parallel chirality, respectively.
  
  
As an other indicator of the transition, we show in Fig.3 the Binder ratios for the spin (a) and for the chirality (b). The chiral Binder ratio $g_{CG}$ exhibits a negative dip. The data of different $L$ cross on the [*negative*]{} side of $g_{CG}$. A magnified view of $g_{CG}$ in the crossing-temperature region is shown in Fig.4.
In contrast to $g_{CG}$, the spin Binder ratio $g_{SG}$ shown in Fig.3(a) exhibits no crossing in the investigated range of the temperature and the lattice size, monotonically decreasing with $L$. However, $g_{SG}$ develops a more and more singular shape with increasing $L$, a prominent peak appearing for larger $L$.
In the $L\rightarrow \infty$ limit, the Binder ratios $g_{SG}$ and $g_{CG}$ should satisfy here $g\rightarrow 0$ in the high-temperature phase, and $g=1$ at $T=0$. Hence, the asymptotic form of $g_{CG}$ in the $L\rightarrow \infty$ limit should be like the one as illustrated in the inset of Fig.3(b). In fact, such a form of $g$ is expected in a system with an ordered state exhibiting a continuous one-step-like replica-symmetry breaking (RSB) [@ImaKawa03b]. A similar form of $g_{CG}$ was observed in 3D [@HukuKawa05; @VietKawamura09a; @VietKawamura09b]. Note that the one-step-like RSB discussed here is of a continuous type, in contrast to the one-step RSB of a discontinuous type often discussed in conjunction with structural glasses. In the latter case, the negative dip of $g_{CG}$ should exhibit a negative divergence at the transition, while such a negatively divergent behavior is not observed here.
 
In order to estimate the bulk SG and CG transition temperatures quantitatively, we plot in Fig.5(a) the crossing temperatures $T_{cross} (L)$ of $\xi_{SG}/L$ versus the inverse system size $1/L_{av}$ for pairs of the sizes $L$ and $sL$ with $s=2, 5/3$ and 5/4, where $L_{av}=\frac{1+s}{2}L$. Likewise, the crossing temperatures $T_{cross} (L)$ of $\xi_{CG}^\perp/L$, $\xi_{CG}^\parallel/L$ and $g_{CG}$ are plotted versus $1/L_{av}$ in Fig.5(b). The crossing temperature $T_{cross} (L)$ is expected to obey the scaling form,$$T_{cross}(L;s)=T_g + c_s L^{-\theta}, \ \ \
\theta=\omega+\frac{1}{\nu} ,$$ where $\nu$ is the correlation-length exponent and $\omega$ is the leading correction-to-scaling exponent. We fit our data of $T_{cross} (L;s)$ for the spin or for the chirality to the above form (11), to extract the transition temperature ($T_g=T_{CG}$ or $T_{SG}$) and the exponent $\theta$ ($\theta=\theta_{CG}$ or $\theta_{SG}$) for the spin or the chirality.
 
For the spin, we perform a joint fit of $T_{cross}(L;s)$ of $\xi_{SG}/L$ for three different values of $s=2, \frac{5}{3}, \frac{5}{4}$, where the values of $T_{SG}$ and $\theta_{SG}$ are taken common while the values of $c_s$ be $s$-dependent. We then find an optimal fit for $T_{SG}=0.391(2)$ and $\theta_{SG}=4(2)$ with the associated $\chi^2$ value, $\chi^2$/DOF=0.73.
For the chirality, we have several kinds of crossing temperatures $T_{cross}(L;s)$, [*i.e.*]{}, $T_{cross}(L;s)$ of $\xi_{CG}^\parallel/L$, $\xi_{CG}^\perp/L$ and $g_{CG}$. Then, we perform a joint fit of the data of $T_{cross}(L;s)$ of these three kinds of $T_{cross}(L;s)$, each with $s=2, \frac{5}{3}, \frac{5}{4}$, where $T_{CG}$ and $\theta_{CG}$ are taken common while the values of $c_s$ be $s$-dependent. We then get $T_{CG}=0.390(1)$ and $\theta_{CG}=2.4(4)$ with the associated $\chi^2$ value, $\chi^2$/DOF=0.51.
One sees from these results that the spin and the chiral transition temperatures agree within the error bars, [*i,e,*]{} $T_{SG}=T_{CG}$ within the accuracy of 1%. This observation strongly suggests the absence of the spin-chirality decoupling in 4D, in contrast to the case of 3D where $T_{SG}$ lies below $T_{CG}$ by about $10\sim 15$%.
For the CG transition, we have another indicator, [*i.e.*]{}, the negative-dip temperature $T_{dip}(L)$ of the chiral Binder ratio $g_{CG}$, which is expected to obey the scaling form, $$T_{dip}(L;s)=T_g + c L^{-\frac{1}{\nu}}.$$ The data of $T_{dip}(L)$ are also shown in Fig.5(b). As can be seen from the figure, $T_{dip}(L)$ changes its behavior with increasing $L$. It tends to [*decrease*]{} with $L$ for smaller sizes, while it tends to [*increase*]{} with $L$ for larger sizes of $L\gtrsim 16$. Indeed, such a non-monotonic size-dependence of $T_{dip}(L)$ is expected due to the following reason. For large enough $L$, the negative-dip temperature $T_{dip}(L)$ should lie [*below*]{} the crossing temperature of $g_{CG}$, $T_{cross}(L)$. Since the exponents governing the asymptotic size dependence of $T_{dip}(L)$ and $T_{cross}(L)$ are $\theta$ and $1/\nu$ which satisfy the inequality $\theta > 1/\nu$ by definition, $T_{dip}(L)$ needs to approach $T_{CG}$ [*from below*]{} for large enough $L$. Hence, a bending-up behavior observed in $T_{dip}(L)$ for larger $L$ is a necessary changeover as expected from the argument above.
Anyway, this changeover in the observed size-dependence of $g_{dip}(L)$ makes a systematic extrapolation of $T_{dip}(L)$ difficult. Nevertheless, as can be seen from Fig.5(b), our data of $T_{dip}(L)$ for larger $L\geq 16$ seems fully consistent with the $T_{CG}$-value obtained above from the crossing temperatures. As mentioned above, the negative dip of $g_{CG}$ shown in Fig.3(b) is consistent with the occurrence of a one-step-like RSB [@HukuKawa05; @VietKawamura09a; @VietKawamura09b]. The corresponding spin Binder ratio $g_{SG}$ shown in Fig.3(b) also develops a more and more singular form with a peak structure appearing for larger $L$. If one recalls the fact that $g_{SG}$ takes a value unity at $T=0$ and approaches zero above $T_{SG}(=T_{CG})$ in the $L\rightarrow \infty$ limit, $g_{SG}$ is expected to develop a negative dip as in the case of $g_{CG}$. In the $L\rightarrow \infty$ limit, this negative dip temperature $T_{dip}$ should yield $T_{SG}$. Since $T_{SG}$ is likely to agree with $T_{CG}$ in 4D, a one-step-like RSB is expected to arise independently of the occurrence of the spin-chirality decoupling. In other words, in 4D, the Heisenberg SG is likely to exhibit a single SG transition without the spin-chirality decoupling. Yet, the SG (simultaneously CG) ordered state is peculiar in that the ordered state possesses a one-RSB-like nontrivial phase-space structure.
V. Critical properties
======================
In the previous section, we have demonstrated that, in 4D, the SG and the CG transitions are likely to take place simultaneously, [*i.e.*]{}, $T_{SG}=T_{CG}$. In this section, we study the critical properties of the transition on the basis of a finite-size scaling analysis of our MC data. In the absence of the spin-chirality decoupling, a natural expectation for the critical properties is that, as usual, the spin is a primary order parameter of the transition. Then one should have $\nu_{SG}=\nu_{CG}$ and $\eta_{SG} < \eta_{CG}$. The latter corresponds to the fact that the spin is the primary order parameter and the chirality is the composite of the spin.
We first study the critical properties of the spin by means of a finite-size scaling analysis of both $\xi_{SG}/L$ and $q^{(2)}_{SG}$. We employ the following finite-size scaling form with the leading correction-to-scaling term,
$$\frac {\xi_{SG}}{L}=\tilde X ((T-T_{SG})L^{1/\nu_{SG}})(1+aL^{-\omega_{SG}}),$$
$$q_{SG}^{(2)}=L^{-(2+\eta_{SG})}\tilde Y((T-T_{SG})L^{1/\nu_{SG}})(1+a'L^{-\omega_{SG}}) ,$$
where $a$ and $a'$ are numerical constants, while $\tilde X$ and $\tilde Y$ are appropriate scaling functions. The SG transition temperature $T_{SG}$ is fixed to $T_{SG}=0.39$ as determined in the previous section.
We begin with the finite-size scaling of $\xi_{SG}/L$ with $\nu_{SG}$ and $\omega_{SG}$ free fitting parameters. The best fit is obtained for $\nu_{SG}=1.0$ and $\omega_{SG}$=3.0. The resulting scaling plot is given in Fig.6(a). Inspecting the quality of the plot by eyes, we put the error bars as $\nu_{SG}=1.0(1)$ and $\omega_{SG}$=3(1). Note that these estimates of $\nu_{SG}$ and $\omega_{SG}$ are consistent with our above estimate of $\theta_{SG}=\omega_{SG}+\frac{1}{\nu_{SG}}=4(2)$. Next, with assuming $\nu_{SG}=1$ and $\omega_{SG}=3$, we perform the finite-size scaling analysis of $q_{SG}^{(2)}$ to obtain $\eta_{SG}=-0.3(1)$. The resulting scaling plot is shown in Fig.6(b).
We also try the type of the extended finite-size scaling analysis proposed by Campbell [*et al*]{} where the scaling variables are chosen to take a matching between the critical regime and the high temperature regime in order to get a wider scaling regime [@Campbell06]. The resulting exponent values turn out to be the same as those obtained above by the standard analysis.
 
Similar scaling analysis is also applied to the chiral degrees of freedom to estimate the chiral correlation-length exponent $\nu_{CG}$ and the chiral anomalous-dimension exponent $\eta_{CG}$. The transition temperature is fixed to $T_{CG}=0.39$ as determined in the previous section. The finite-size scaling of the chiral correlation-length ratio yields $\nu_{CG}=1.0(1)$ and $\omega_{CG}=1.7(3)$. We get the same estimates even when we use either the perpendicular or the transverse CG correlation-length ratio. The resulting scaling plot for the perpendicular one is given in Fig.7(a). These estimates of $\nu_{CG}$ and $\omega_{CG}$ are consistent with our above estimate of $\theta_{SG}=\omega_{SG}+\frac{1}{\nu_{SG}}=2.3(4)$. The finite-size scaling of the CG order parameter $q_{CG}^{(2)}$ with fixing $\nu_{CG}=1.0$ and $\omega_{CG}=1.7$ yields $\eta_{CG}=2.4(8)$. The resulting scaling plot is given in Fig.7(b). We also try the extended finite-size scaling analysis a la Campbell [*et al*]{} [@Campbell06]. Again, as in the case of the spin, the resulting exponent values turn out to be the same as those obtained above by the standard analysis.
 
Combining the exponent estimates obtained above, we finally quote as our best estimates of the spin exponents, $$\nu_{SG}=1.0\pm 0.1\ ,\ \ \ \eta_{SG}=-0.3\pm 0.1,$$ while for the chirality exponents quote $$\nu_{CG}=1.0\pm 0.1\ ,\ \ \ \eta_{CG}=2.4\pm 0.8.$$ If one applies the standard scaling or hyperscaling relations, one can estimate other SG exponents as $\alpha\simeq -2.0$, $\beta_{SG} \simeq 0.85$, $\gamma_{SG}\simeq 2.3$, and $\delta_{SG}\simeq 3.7$, [*etc*]{}.
One sees from these estimates that the correlation-length exponents $\nu$ for the spin and for the chirality agree within the error bars, [*i.e.*]{}, $\nu_{SG}=\nu_{CG}$, which indicates the existence of only one diverging length scale at the transition. This observation is fully consistent with the absence of the spin-chirality decoupling in the 4D Heisenberg SG. Our data are also not incompatible with the relation $\omega_{SG} = \omega_{CG}$ within the error bars. By contrast, the anomalous-dimension exponents satisfy the inequality $\eta_{SG} < \eta_{CG}$, indicating that the spin is the primary order parameter as usual. If one applies the scaling relation to the CG exponents $\gamma_{CG}=(2-\eta_{CG})\nu_{CG}$, one would get the CG susceptibility exponent as $\gamma _{CG}=-0.4\pm 1.1$. The estimated value of $\gamma_{CG}$ means that the CG susceptibility does not diverge, or diverges only weakly, at the transition. This observation is again consistent with the view that the primary order parameter in 4D is the spin and the chirality is only composite.
The obtained CG exponents values might be compared with the earlier estimates by Imagawa and Kawamura on the same model, [*i.e.*]{}, $\nu_{SG}=1.3(2)$ and $\eta_{SG}=-0.7(2)$ [@ImaKawa03a]. One sees that $\nu_{SG}$ agrees with our present estimate within the error bars, while $\eta_{SG}$ deviates somewhat. In view of the larger sizes employed in the present study as compared with those of ref.[@ImaKawa03a], [*i.e.*]{}, $L\leq 20$ vs. $L\leq 10$, and also of larger number of independent samples, [*e.g.*]{}, 840 vs. 80 for $L=10$, our present estimate would be more trustable.
VI. Summary and discussion
==========================
We studied equilibrium ordering properties of the 4D isotropic Heisenberg SG by means of an extensive MC simulation. By calculating various physical quantities including the correlation-length ratio, the Binder ratio and the glass order parameter up to the size as large as $L=20$ and down to temperatures well below $T_g$, we have found that $T_{SG}=0.391(2)$ is likely to coincide with $T_{CG}=0.390(1)$, which indicates that the spin and the chirality order simultaneously in the 4D Heisenberg SG, [*i.e.*]{}, the absence of the spin-chirality decoupling. If $T_{SG}$ and $T_{CG}$ are to differ, the distance in transition temperatures should be less than 1%. We also studied the critical properties of the transition on the basis of the finite-size scaling analysis. The exponents were estimated to be $\nu_{SG}=1.0(1)$ and $\eta_{SG}=-0.3(1)$ for the spin, and $\nu_{CG}=1.0(1)$ and $\eta_{CG}=2.4(8)$ for the chirality. Although the SG transition in 4D is usual in the sense that the spin is the primary order parameter, the standard exponent relations $\nu_{SG}=\nu_{CG}$ and $\eta_{SG} < \eta_{CG}$ being satisfied. Yet, the SG transition is somewhat unusual in the sense that the low-temperature SG (simultaneously CG) ordered state exhibits a nontrivial phase-space structure, [*i.e.*]{}, a continuous one-step-like RSB. Note that the type of RSB is quite different from the one observed in the Ising SG, or the one observed in the mean-field limit of both the Ising and the Heisenberg SGs.
As mentioned in §1, possible correspondence between the orderings of the $d$-dimensional SR Heisenberg SG and of the 1D LR Heisenberg SG with a power-law interaction has been suggested in the literature. Although this correspondence is by no means exact, recent numerical studies both on the Ising and the Heisenberg SGs supported such correspondence. Indeed, Katzgraber [*et al*]{} proposed a formula for the $d$-$\sigma$ correspondence, a refined version of the one mentioned in §1 [@Katzgraber09], $$d = \frac{2-\eta_{SG}}{2\sigma -1},$$ where $\eta_{SG}$ is the spin anomalous-dimension exponent of the $d$-dimensional SR system. Now, we have an estimate of $\eta_{SG}$ for the $d=4$ Heisenberg SG as $\eta_{SG}\simeq -0.3$. Substituting this into the r.h.s. of eq.(17) and putting $d=4$, we get $\sigma=0.79$. Together with the recent numerical estimate of the borderline value of $\sigma_c$ separating the spin-chirality coupling/decoupling regimes, $\sigma_c\simeq 0.8$ [@VietKawamura10a; @VietKawamura10b], the $d$-$\sigma$ correspondence suggests that the 4D lies very close to the borderline dimensionality of the spin-chirality coupling/decoupling, on the coupling regime only slightly. Such a view on the basis of the $d$-$\sigma$ correspondence seems fully consistent with our present MC results.
In fact, the correspondence holds also for the critical exponents. In the $d$-$\sigma$ analogy, the exponent $\nu_{SG}$ of the 1D LR model should be related to that of the $d$-dimensional SR model via the relation, $\nu_{SG}$\[1D-LR\]=$d\times \nu_{SG}$\[$d$D-SR\] [@Larson]. Then, our 4D result suggests that the corresponding 1D LR model should be characterized by the exponent $\nu_{SG}\simeq 4\times 1.0=4$. Meanwhile, ref.[@VietKawamura10b] gave $\nu_{SG}=3.6(5)$ and $\nu_{CG}=4.0(5)$ for $\sigma=0.8$ so that the expected relation is indeed satisfied. All these results suggest that $d=4$ probably lies fairly close to the borderline dimensionality of the spin-chirality decoupling/coupling, may even lie just at the border.
The authors are thankful to T. Okubo and T. Obuchi for useful discussion. This study was supported by Grant-in-Aid for Scientific Research on Priority Areas “Novel States of Matter Induced by Frustration” (19052006 & 19052007). We thank ISSP, University of Tokyo, YITP, Kyoto University, and Cyber Media Center, Osaka University for providing us with the CPU time.
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author:
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Guangming Lang$^{1,2,3}$ [^1]\
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title: 'Double-quantitative $\gamma^{\ast}-$fuzzy coverings approximation operators'
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> [**Abstract.**]{} In digital-based information boom, the fuzzy covering rough set model is an important mathematical tool for artificial intelligence, and how to build the bridge between the fuzzy covering rough set theory and Pawlak’s model is becoming a hot research topic. In this paper, we first present the $\gamma-$fuzzy covering based probabilistic and grade approximation operators and double-quantitative approximation operators. We also study the relationships among the three types of $\gamma-$fuzzy covering based approximation operators. Second, we propose the $\gamma^{\ast}-$fuzzy coverings based multi-granulation probabilistic and grade lower and upper approximation operators and multi-granulation double-quantitative lower and upper approximation operators. We also investigate the relationships among these types of $\gamma-$fuzzy coverings based approximation operators. Finally, we employ several examples to illustrate how to construct the lower and upper approximations of fuzzy sets with the absolute and relative quantitative information.
>
> [**Keywords:**]{} Double-quantitative approximation operators; Grade rough sets; Probabilistic rough sets; $\gamma-$fuzzy covering approximation space; $\gamma^{\ast}-$fuzzy coverings information system\
Introduction
============
Rough set theory, proposed by Pawlak in 1982, is an important mathematical tool for dealing with imprecise and uncertain information in data analysis. In theoretical aspects, by extending the equivalence relation, Pawlak’s model has been generalized to covering-based rough sets, fuzzy covering-based rough sets, dominance rough sets, fuzzy rough sets, rough fuzzy sets, decision-theoretic rough sets, double-quantitative rough sets, multi-granulation rough sets, and so on. In application aspects, rough set theory has been successfully applied to various fields such as machine learning, data mining, image processing, and knowledge discovery, and the applied fields are being increasing with the development of rough set theory.
Among all generalizations of Pawlak’s model, fuzzy covering rough set theory computes the lower and upper approximations of fuzzy sets in fuzzy covering approximation spaces, and provides an important mathematical tool for knowledge discovery. So far, many types of fuzzy covering-based lower and upper approximation operators have been proposed with respect to different backgrounds. Especially, the fuzzy $\gamma-$covering based lower and upper approximation operators introduced by Ma[@Ma1; @Ma2] builded the link between the fuzzy covering rough set theory and Pawlak’s model, which provides an effective approach to studying the fuzzy covering approximation spaces from the view of Pawlak’s rough sets. In practical situations, there are a lot of fuzzy covering approximation spaces. Especially, there exists a great number of fuzzy covering information systems. To perform knowledge discovery of fuzzy covering information systems, we should construct the effective lower and upper approximation operators for the fuzzy covering approximation spaces. There are many effective rough set models such as probabilistic rough sets, grade rough sets, double quantitative rough sets, and multi-granulation rough sets, and we should study how to construct the lower and upper approximation operators for fuzzy covering approximation spaces with the advantages of different rough set models.
The purpose of this work is shown as follows. First, we propose the fuzzy $\gamma$-covering based probabilistic lower and upper approximation operators, as extensions of probabilistic lower and upper approximation operators, in fuzzy $\gamma$-covering approximation spaces. We present the fuzzy $\gamma$-covering based grade lower and upper approximation operators as extensions of grade lower and upper approximation operators. We also discuss the relationship between the fuzzy $\gamma$-covering based probabilistic operators and the fuzzy $\gamma$-covering based grade operators. Second, we provide the fuzzy $\gamma$-covering based disjunctive double-quantitative lower and upper approximation operators in fuzzy $\gamma$-covering approximation spaces. We propose the fuzzy $\gamma$-covering based conjunctive double-quantitative lower and upper approximation operators. We also discuss the relationship between the fuzzy $\gamma$-covering based disjunctive and conjunctive double-quantitative approximation operators and the fuzzy $\gamma$-covering based probabilistic and grade approximation operators. Third, we present the fuzzy $\gamma^{\ast}$-coverings based multi-granulation probabilistic and grade lower and upper approximation operators in fuzzy $\gamma^{\ast}$-coverings information systems. We also discuss the relationship between the $\gamma^{\ast}$-coverings based multi-granulation probabilistic and grade approximation operators and the fuzzy $\gamma^{\ast}$-coverings based probabilistic and grade approximation operators. Fourth, we provide the fuzzy $\gamma^{\ast}$-coverings based disjunctive and conjunctive multi-granulation double-quantitative lower and upper approximation operators. We also discuss the relationship between the fuzzy $\gamma^{\ast}$-coverings based double-quantitative multi-granulation approximation operators and the fuzzy $\gamma^{\ast}$-coverings based multi-granulation probabilistic and grade approximation operators.
The reminder of this paper is organized as follows. Section 2 reviews the related works. In Section 3, we recall some basic concepts of covering rough set theory, probabilistic rough set theory, grade rough set theory, and fuzzy covering rough set theory. Section 4 proposes the fuzzy $\gamma-$covering based probabilistic and grade lower and upper approximation operators. In Section 5, we present the fuzzy $\gamma-$covering double-quantitative lower and upper approximation operators. Section 6 introduces the fuzzy $\gamma^{\ast}-$coverings multi-granulation lower and upper approximation operators. In Section 7, we construct the fuzzy $\gamma^{\ast}-$coverings multi-granulation double-quantitative lower and upper approximation operators. The paper ends with conclusions in Section 8.
Review of related works
=======================
In this section, we review some works related to double-quantitative rough set theory, multi-granulation rough set theory, and fuzzy covering-based rough set theory.
Double-quantitative rough set theory[@Fan1; @Fang1; @Li4; @Xu1; @Zhang1; @Zhang3; @Zhang4; @Zhang5], as the combination of probabilistic rough sets[@Liang1; @Liu1; @Liu2; @Ma2; @Sang1; @Sun1; @Yao1; @Yao2; @Yao3; @Yao5; @Yu1] and grade rough sets[@Liu4; @Huang1; @Yao6], considers the relative and absolute quantitative information when constructing the lower and upper approximation operators. For example, Fan et al.[@Fan1] proposed a couple of double-quantitative decision-theoretic rough fuzzy set models based on logical conjunction and logical disjunction operation and discuss rules and the inner relationship between two models. Fang et al.[@Fang1] presented the probabilistic graded rough set as an extension of Pawlak’s rough set and grade rough sets and double relative quantitative decision-theoretic rough set models. Li et al.[@Li4] provided double-quantitative decision-theoretic rough sets and studied its properties. Xu et al.[@Xu1] proposed the lower and upper approximations of generalized multi-granulation double-quantitative rough sets by introducing the lower and upper support characteristic functions. They also constructed the approximation accuracy to show the advantage of the proposed model. Zhang et al.[@Zhang2; @Zhang3] provided information architecture, granular computing and rough set model in the double-quantitative approximation space of precision and grade. They also proposed two basic double-quantitative rough set models of precision and grade and their investigation using granular computing.
Many efforts have focused on multi-granulation rough set theory[@Feng1; @Li2; @Li3; @Liang2; @Lin1; @Lin2; @Liu3; @Qian2; @Qian3; @Raghavan1; @She1; @Wu1; @Xu1; @Xu2; @Xu3; @Yang3; @Zhang1]. For example, Feng et al.[@Feng1] proposed variable precision multi-granulation decision-theoretic fuzzy rough sets. Li et al.[@Li2] presented a comparative study of multi-granulation rough sets and concept lattices via rule acquisition. Li et al.[@Li3] provided multi-granulation decision-theoretic rough sets in ordered information systems. Liang et al.[@Liang2] established an efficient feature selection algorithm with a multi-granulation view. Lin[@Lin1] introduced an approach to feature selection via neighborhood multi-granulation fusion. Lin et al.[@Lin2] presented a fuzzy multi-granulation decision-theoretic approach to multi-source fuzzy information systems. Liu et al.[@Liu3] provided the multi-granulation rough sets in covering context. Qian et al.[@Qian2; @Qian3] introduced multi-granulation rough set theory by extending Pawlak’s model. Raghavan et al.[@Raghavan1] explored the topological properties of multi-granulation rough sets. She et al.[@She1] deeply studied topological structures and properties of multi-granulation rough sets. Wu et al.[@Wu1] proposed a formal approach to granular computing with multi-scale data decision information systems. Xu et al.[@Xu1; @Xu2; @Xu3] considered variable, fuzzy and ordered multi-granulation rough set models. Yang et al.[@Yang3] updated multi-granulation rough approximations with increasing of granular structures. Zhang et al.[@Zhang1] provided constructive methods of rough approximation operators and multi-granulation rough sets.
Fuzzy covering-based rough set theory[@Chen1; @Deng1; @D'eer1; @Feng2; @Huang1; @Lang1; @Li1; @Ma1; @Restrepo1; @Seselja1; @Wang1; @Yang1; @Yang2; @Yao4; @Zhang6] has attracted more and more attention. For example, based on fuzzy covering and binary fuzzy logical operators, Deng[@Deng1] proposed an approach to fuzzy rough sets in the framework of lattice theory, and presented a link between the generalized fuzzy rough approximation operators and fundamental morphological operators. Feng et al.[@Feng2] defined a novel pair of belief and plausibility functions by employing a method of non-classical probability model and the approximation operators of a fuzzy covering. Huang et al.[@Huang1] presented an intuitionistic fuzzy graded covering rough set and studied its properties. Li et al.[@Li1] showed a general framework for the study of covering-based fuzzy approximation operators in which a fuzzy set can be approximated by some elements in a crisp or a fuzzy covering of the universe. Ma[@Ma1] presented the concepts of fuzzy $\gamma$-covering and fuzzy $\gamma$-neighborhood and two new types of fuzzy covering rough set models which links covering rough set theory and fuzzy rough set theory. $\check{S}e\check{s}elja$[@Seselja1] investigated lattice-valued covering, or fuzzy neighboring relation arising from a given lattice-valued order and showed that every L-fuzzy covering is obtained by synthesis of crisp coverings arising from the corresponding cut orderings. Wang et al.[@Wang1] proposed the concepts of consistent and compatible mappings with respect to fuzzy sets and constructed a pair of lower and upper rough fuzzy approximation operators by means of the concept of fuzzy mappings. Yang et al.[@Yang2] presented the definition of fuzzy $\gamma$-minimal description and a novel type of fuzzy covering-based rough set model and investigate its properties. Yao et al.[@Yao4] introduced the concepts of fuzzy positive region reduct, lower approximation reduct and generalized fuzzy belief reduct, and investigated the relationships among these reducts. Zhang et al.[@Zhang6] proposed the generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings and studied its properties.
Preliminaries
=============
In this section, we briefly recall some basic concepts of covering rough set theory, probabilistic and grade rough set theory, and fuzzy $\gamma-$covering approximation spaces.
Covering approximation spaces
-----------------------------
In this section, we recall the concepts of coverings, the lower and upper approximation operators.
[@Zakowski1] Let $U$ be a non-empty set $($the universe of discourse$)$. A non-empty sub-family $\mathscr{C}\subseteq
\mathscr{P} (U)$ is called a covering of $U$ if
$(1)$ every element in $\mathscr{C} $ is non-empty;
$(2)$ $\bigcup \{C \mid C \in \mathscr{C} \}=U$, where $\mathscr{P} (U)$ is the powerset of $U$.
Unless stated otherwise, $U$ is a finite universe, the covering $\mathscr{C}$ consists of finite number of sets, and the ordered pair $(U, \mathscr{C}) $ is called a covering approximation space, which is an extension of Pawlak’s model using the equivalence relation. Especially, the incomplete information system is corresponding to a covering approximation space or a covering information system in fact.
We employ an example to illustrate how to construct the covering approximation space as follows.
Let $U=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}$ be eight cars, $C=\{price\}$ the attribute set, the domain of $price$ is $\{high, middle, low\}$. The specialists $A$ and $B$ are employed to evaluate these cars and their evaluation reports are shown as follows: $$\begin{aligned}
high_{A}&=&\{x_{1}, x_{4}, x_{5}, x_{7}\}, middle_{A}=\{x_{2},
x_{8}\}, low_{A}=\{x_{3}, x_{6}\};\\
high_{B}&=&\{x_{1}, x_{2}, x_{4}, x_{7}, x_{8}\},
middle_{B}=\{x_{5}\}, low_{B}=\{x_{3}, x_{6}\},\end{aligned}$$ where $high_{A}$ denotes the cars belonging to high price by the specialist $A$, and the meanings of other symbols are similar. Since their evaluations are of equal importance, we should consider all their advice. Therefore, we drive the covering approximation space $(U,\mathscr{C}_{price})$, where $\mathscr{C}_{price}=\{high_{A\vee B},
middle_{A\vee B}, low_{A\vee B}\}$, and $$\begin{aligned}
high_{A\vee B}&=&high_{A}\cup high_{B}=\{x_{1}, x_{2}, x_{4}, x_{5},
x_{7}, x_{8}\};\\ middle_{A\vee B}&=&middle_{A}\cup
middle_{B}=\{x_{2}, x_{5}, x_{8}\};\\ low_{A\vee B}&=&low_{A}\cup
low_{B}=\{x_{3}, x_{6}\}.\end{aligned}$$
[@Zhu1] Let $(U,\mathscr{C})$ be a covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, and $N(x)=\bigcap\{C_{i}\mid x\in C_{i}\in \mathscr{C}\}$ for $x\in U$. Then the lower and upper approximations of $X\in P(U)$ are defined as follows: $$\begin{aligned}
\underline{R}(X)&=&\{x\in U|N(x)\subseteq X\};\\
\overline{R}(X)&=&\{x\in U|N(x)\cap X\neq \emptyset\}.\end{aligned}$$
The neighborhood $N(x)$ of $x\in U$ is constructed using the covering $\mathscr{C}$, and $\{N(x)|x\in U\}$ is also a covering of $U$, but the granularity of the covering $\{N(x)|x\in U\}$ is finer than the covering $\mathscr{C}$, and the lower and upper approximation operators given by Definition 3.3 is very effective for computing the lower and upper approximations of sets in the covering approximation spaces.
Probabilistic and grade lower and upper approximation operators
---------------------------------------------------------------
In this section, we recall the probabilistic and grade lower and upper approximation operators in the covering approximation spaces.
[@Yao1] Let $(U,\mathscr{C})$ be a covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $N(x)=\bigcap\{C_{i}\mid x\in C_{i}\in \mathscr{C}\}$ for $x\in U$, $P(X|N(x))=\frac{|X\cap N(x)|}{|N(x)|}$, and $0\leq \beta \leq \alpha \leq 1$. Then the probabilistic lower and upper approximations of the set $X\in P(U)$ are defined as follows: $$\begin{aligned}
\overline{R}_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|N(x))\geq\beta \};\\
\underline{R}_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|N(x))\geq\alpha \}.\end{aligned}$$
We observe that the probabilistic lower and upper approximation operators are proposed by generalizing Definition 3.3, which compute the lower and upper approximations of sets using the relative quantitative information, and the conditions of the probabilistic lower and upper approximation operators are looser than Definition 3.3.
The positive, boundary, and negative regions of the set $X\in P(U)$ using the probabilistic lower and upper approximation operators are constructed as follows: $$\begin{aligned}
POS_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|N(x))\geq\alpha \};\\
BOU_{(\alpha,\beta)}(X)&=&\{x\in U\mid \beta\leq P(X|N(x))<\alpha \};\\
NEG_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|N(x))<\beta \}.\end{aligned}$$
[@Yao6] Let $(U,\mathscr{C})$ be a covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $N(x)$ the neighborhood of $x\in U$, and $k\in R$. Then the grade lower and upper approximations of the set $X\in P(U)$ are defined as follows: $$\begin{aligned}
\overline{R}_{k}(X)&=&\{x\in U\mid \Sigma_{y\in U}|X\cap N(x)|>k\};\\
\underline{R}_{k}(X)&=&\{x\in U\mid \Sigma_{y\in U}|X^{c}\cap N(x)|\leq k\}.\end{aligned}$$
We see the grade lower and upper approximation operators are different from the probabilistic lower and upper approximation operators, which compute the lower and upper approximations of sets using the absolute quantitative information, but there are some similarities between them, and they can be transferred into each other under some conditions.
The positive, lower boundary, upper boundary, and negative regions of the set $X\in P(U)$ are computed by Definition 3.5 as follows: $$\begin{aligned}
POS_{k}(X)&=&\overline{R}_{k}(X)\cap \underline{R}_{k}(X);\\
NEG_{k}(X)&=&(\overline{R}_{k}(X)\cup \underline{R}_{k}(X))^{c};\\
LBO_{k}(X)&=&\underline{R}_{k}(X)-\overline{R}_{k}(X);\\
UBO_{k}(X)&=&\overline{R}_{k}(X)-\underline{R}_{k}(X);\\
BOU_{k}(X)&=&LBO_{R_{k}}(X)\cup UBO_{R_{k}}(X).\end{aligned}$$
Fuzzy $\gamma-$covering approximation spaces
--------------------------------------------
In this section, we recall some concepts of fuzzy covering approximation spaces.
[@Zadeh1] Let $\mu_{A}$ be a mapping from $U$ to $[0,1]$ such as $\mu_{A}:
U\longrightarrow [0,1]:$ $ x\longrightarrow \mu_{A}(x),$ where $x\in U$, and $\mu_{A}$ is the membership function. Then $A$ is referred to as a fuzzy set.
We denote the family of all fuzzy subsets of $U$ and $\mu_{A}(x)$ as $\mathscr{F}(U)$ and $A(x)$, respectively, for simplicity. For any $A,B\in \mathscr{F}(U)$, if $A(x)\leq B(x)$ for any $x\in U$, then we say $A$ is contained in $B$, denoted as $A\subseteq B$. Especially, $A=B$ if and only if $A\subseteq B$ and $B\subseteq A$. We also have $(A\cup B)(x)=A(x)\vee B(x)$, $(A\cap B)(x)=A(x)\wedge B(x)$, and $A^{c}(x)=1-A(x)$ for any $x\in U$.
We also employ an example to illustrate the union, intersection, and complement of fuzzy sets as follows.
(Continuation from Example 3.2) Let $A$ and $B$ be fuzzy subsets of the universe $U$ as follows: $$\begin{aligned}
A&=&\frac{1}{x_{1}}+\frac{0.6}{x_{2}}+\frac{0}{x_{3}}
+\frac{0.8}{x_{4}}+\frac{1}{x_{5}}
+\frac{0}{x_{6}}+\frac{0.8}{x_{7}}+\frac{1}{x_{8}}
;\\
B&=&\frac{1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.6}{x_{3}}
+\frac{1}{x_{4}}+\frac{0}{x_{5}}
+\frac{0.8}{x_{6}}+\frac{1}{x_{7}}+\frac{0.8}{x_{8}}.\end{aligned}$$
By Definition 3.6, we have that $A(x_{1})=1,A(x_{2})=0.6,A(x_{3})=0,A(x_{4})=0.8,
A(x_{5})=1,A(x_{6})=0,A(x_{7})=0.8,A(x_{8})=1,$ $B(x_{1})=1,B(x_{2})=0,B(x_{3})=0.6,B(x_{4})=1,
B(x_{5})=0,B(x_{6})=0.8,B(x_{7})=1,$ and $B(x_{8})=0.8.$ We also have $A\cap B$, $A\cup B$, and $A^{c}$ as follows: $$\begin{aligned}
A\cap B&=&\frac{1}{x_{1}}+\frac{0}{x_{2}}+\frac{0}{x_{3}}
+\frac{0.8}{x_{4}}+\frac{0}{x_{5}}
+\frac{0}{x_{6}}+\frac{0.8}{x_{7}}+\frac{0.8}{x_{8}}
;\\
A\cup B&=&\frac{1}{x_{1}}+\frac{0.6}{x_{2}}+\frac{0.6}{x_{3}}
+\frac{1}{x_{4}}+\frac{1}{x_{5}}
+\frac{0.8}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}};\\
A^{c}&=&\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{1}{x_{3}}
+\frac{0.2}{x_{4}}+\frac{0}{x_{5}}
+\frac{1}{x_{6}}+\frac{0.2}{x_{7}}+\frac{0}{x_{8}}.\end{aligned}$$
[@Ma1] A fuzzy $\gamma-$covering of $U$ is a collection of fuzzy sets $\mathscr{C}^{\ast}\subseteq \mathscr{F}(U)$ which satisfies
$(1)$ every fuzzy set $C^{\ast}\in \mathscr{C}^{\ast}$ is non-empty, i.e., $C^{\ast}\neq\emptyset$;
$(2)$ $ \forall x\in U, \bigvee_{C^{\ast}\in
\mathscr{C}^{\ast}}C^\ast(x)\geq \gamma$.
Unless stated otherwise, $U$ is a finite universe, the fuzzy covering $\mathscr{C}^{\ast}$ consists of finite number of sets, and the ordered pair $(U, \mathscr{C}^{\ast}) $ is called a $\gamma-$fuzzy covering approximation space, as an extension of the covering approximation space.
(Continuation from Example 3.2) To evaluate these cars, specialists $A$ and $B$ are employed and their evaluation reports are shown as follows: $$\begin{aligned}
high^{\ast}_{A}&=&\frac{1}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}
+\frac{0.9}{x_{4}}+\frac{0.9}{x_{5}}
+\frac{0}{x_{6}}+\frac{0.9}{x_{7}}+\frac{0.6}{x_{8}};\\
middle^{\ast}_{A}&=&\frac{0.6}{x_{1}}+\frac{0.9}{x_{2}}+\frac{0.4}{x_{3}}
+\frac{0.4}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.5}{x_{6}}+\frac{0.5}{x_{7}}+\frac{0.9}{x_{8}};\\
low^{\ast}_{A}&=&\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.9}{x_{3}}
+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.9}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}};\\
high^{\ast}_{B}&=&\frac{0.9}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}
+\frac{0.9}{x_{4}}+\frac{0.9}{x_{5}}
+\frac{0}{x_{6}}+\frac{0.9}{x_{7}}+\frac{0.8}{x_{8}};\\
middle^{\ast}_{B}&=&\frac{0.6}{x_{1}}+\frac{0.9}{x_{2}}+\frac{0.4}{x_{3}}
+\frac{0.4}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.7}{x_{6}}+\frac{0.5}{x_{7}}+\frac{1}{x_{8}};\\
low^{\ast}_{B}&=&\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.9}{x_{3}}
+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.9}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}},\end{aligned}$$ where $high^{\ast}_{A}$ is the membership degree of each car belonging to the high price by the specialist $A$. The meanings of the other symbols are similar. Then we obtain a $0.9-$fuzzy covering approximation space $(U,\mathscr{C}^{\ast}_{price})$, where $\mathscr{C}^{\ast}_{price}=\{C^{\ast}_{high}, C^{\ast}_{middle}, C^{\ast}_{low}\}$, and $$\begin{aligned}
C^{\ast}_{high}&=&high^{\ast}_{A}\cup
high^{\ast}_{B}=\frac{1}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}
+\frac{0.9}{x_{4}}+\frac{0.9}{x_{5}}
+\frac{0}{x_{6}}+\frac{0.9}{x_{7}}+\frac{0.8}{x_{8}};\\
C^{\ast}_{middle}&=&middle^{\ast}_{A}\cup
middle^{\ast}_{B}=\frac{0.6}{x_{1}}+\frac{0.9}{x_{2}}+\frac{0.4}{x_{3}}
+\frac{0.4}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.7}{x_{6}}+\frac{0.5}{x_{7}}+\frac{1}{x_{8}};\\
C^{\ast}_{low}&=&low^{\ast}_{A}\cup
low^{\ast}_{B}=\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.9}{x_{3}}
+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.9}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}}.\end{aligned}$$
It is obvious that we can construct a fuzzy $\gamma-$covering of the universe with an attribute. Since the fuzzy covering rough set theory is very effective to handle uncertain information, the investigation of this theory becomes an important task in rough set theory.
Double-quantitative approximation operators
===========================================
In this section, we provide the concepts of the fuzzy $\gamma-$covering based probabilistic approximation operators, grade approximation operators, and double-quantitative approximation operators.
Probabilistic lower and upper approximation operators
-----------------------------------------------------
In this section, we recall the concept of the fuzzy $\gamma-$neighborhood $\widetilde{N}_{x}^{\gamma}$ of $x\in U$ as follows.
[@Ma1] Let $(U,\mathscr{C}^{\ast})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, and $\mathscr{C}^{\ast}=\{C^{\ast}_{1},C^{\ast}_{2},...,C^{\ast}_{m}\}$, and $\gamma\in (0,1]$. Then the fuzzy $\gamma-$neighborhood $\widetilde{N}_{x}^{\gamma}$ of $x\in U$ is defined as follows: $$\widetilde{N}_{x}^{\gamma}=\bigcap\{C^{\ast}_{i}\in \mathscr{C}^{\ast}\mid C^{\ast}_{i}(x)\geq \gamma\}.$$
The concept of the fuzzy $\gamma-$neighborhood operator $\widetilde{N}_{x}^{\gamma}$ is an extension of the classical neighborhood $N(x)$ in the fuzzy $\gamma-$covering approximation space, which will be applied to compute the fuzzy $\gamma-$covering based probabilistic lower and upper approximations of fuzzy sets. In what follows, we denote $\mathscr{C}^{\ast}$ and $C^{\ast}_{i}$ as $\mathscr{C}$ and $C_{i}$, respectively, for simplicity.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, and $\gamma\in (0,1]$. Then the conditional probability $P(X|\widetilde{N}_{x}^{\gamma})$ of the fuzzy event $X\in \mathscr{F}(U)$ given the description $\widetilde{N}_{x}^{\gamma}$ is defined as follows: $$\begin{aligned}
P(X|\widetilde{N}_{x}^{\gamma})=\frac{\Sigma_{y\in U}(X\cap\widetilde{N}_{x}^{\gamma})(y)}{\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)}.\end{aligned}$$
The concept of the conditional probability $P(X|\widetilde{N}_{x}^{\gamma})$ of the fuzzy event $X\in \mathscr{F}(U)$ is an generalization of the conditional probability $P(X|N(x))$ of the event $X\in P(U)$, which is helpful for studying the fuzzy $\gamma-$covering approximation space.
In what follows, we propose the concept of the fuzzy $\gamma-$covering based probabilistic lower and upper approximation operators in the fuzzy $\gamma-$covering approximation space.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, and $0\leq \beta \leq \alpha\leq 1$. Then the fuzzy $\gamma-$covering based probabilistic lower and upper approximations of the fuzzy set $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{FR}_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\beta \};\\
\underline{FR}_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\alpha \}.\end{aligned}$$
The fuzzy $\gamma-$covering based probabilistic lower and upper approximation operators $\overline{FR}_{(\alpha,\beta)}(X)$ and $\underline{FR}_{(\alpha,\beta)}(X)$ for the fuzzy set $X\in \mathscr{F}(U)$ given by Definition 4.3 are extensions of the probabilistic lower and upper approximation operators $\overline{R}_{(\alpha,\beta)}(X)$ and $\underline{R}_{(\alpha,\beta)}(X)$ for the set $X\in P(U)$ given by Definition 3.4, which construct the lower and upper approximations of fuzzy sets using the relative quantitative information.
(Continuation from Example 3.9) Taking $X=\frac{0.6}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.7}{x_{3}}
+\frac{0.8}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.6}{x_{6}}+\frac{0}{x_{7}}+\frac{0.2}{x_{8}}$, $\alpha=0.75,$ and $\beta=0.25$, we have the fuzzy $\gamma-$covering based probabilistic lower and upper approximations of $X$ as follows: $$\begin{aligned}
\underline{FR}_{(\alpha,\beta)}(X)=\{x_{3},x_{6}\}\text{ and }
\overline{FR}_{(\alpha,\beta)}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, and $0\leq \beta \leq \alpha \leq 1$. Then the fuzzy $\gamma-$covering based probabilistic positive, boundary, and negative regions of the fuzzy set $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\widetilde{POS}_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\alpha \};\\
\widetilde{BOU}_{(\alpha,\beta)}(X)&=&\{x\in U\mid \beta\leq P(X|\widetilde{N}_{x}^{\gamma})<\alpha \};\\
\widetilde{NEG}_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})<\beta \}.\end{aligned}$$
The the fuzzy $\gamma-$covering based probabilistic positive, boundary, and negative regions $\widetilde{POS}_{(\alpha,\beta)}(X),$ $\widetilde{BOU}_{(\alpha,\beta)}(X), \widetilde{NEG}_{(\alpha,\beta)}(X)$ of the fuzzy set $X\in \mathscr{F}(U)$ in the fuzzy $\gamma-$covering approximation space are generalizations of the probabilistic positive, boundary, and negative regions $POS_{(\alpha,\beta)}(X), BOU_{(\alpha,\beta)}(X)$ and $NEG_{(\alpha,\beta)}(X)$ of the set $X\in P(U)$ in the covering approximation spaces.
(Continuation from Example 4.4) Taking $\alpha=0.75,$ and $\beta=0.25$, we have the fuzzy $\gamma-$covering based probabilistic positive, boundary, and negative regions of $X$ as follows: $$\begin{aligned}
\widetilde{POS}_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\alpha \}=\{x_{3},x_{6}\};\\
\widetilde{BOU}_{(\alpha,\beta)}(X)&=&\{x\in U\mid \beta\leq P(X|\widetilde{N}_{x}^{\gamma})<\alpha \}=\{x_{1},x_{2},x_{4},x_{5},x_{7},x_{8}\};\\
\widetilde{NEG}_{(\alpha,\beta)}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})<\beta \}=\emptyset.\end{aligned}$$
We study the basic properties of the fuzzy $\gamma-$covering based lower and upper approximations of sets as follows.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $0\leq \beta< \alpha\leq 1$, and $X,Y\in \mathscr{F}(U)$. Then\
$(1) \underline{FR}_{(\alpha,\beta)}(U)=U;\overline{FR}_{(\alpha,\beta)}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{FR}_{(\alpha,\beta)}(X)\subseteq \overline{FR}_{(\alpha,\beta)}(Y);\\
(3) X\subseteq Y\Rightarrow\underline{FR}_{(\alpha,\beta)}(X)\subseteq \underline{FR}_{(\alpha,\beta)}(Y);\\
(4) \overline{FR}_{(\alpha,\beta)}(X)\cup \overline{FR}_{(\alpha,\beta)}(Y)\subseteq \overline{FR}_{(\alpha,\beta)}(X\cup Y);\\
(5)\underline{FR}_{(\alpha,\beta)}(X)\cup \underline{FR}_{(\alpha,\beta)}(Y)\subseteq \underline{FR}_{(\alpha,\beta)}(X\cup Y);\\
(6) \overline{FR}_{(\alpha,\beta)}(X\cap Y)\subseteq \overline{FR}_{(\alpha,\beta)}(X)\cap \overline{FR}_{(\alpha,\beta)}(Y);\\
(7)\underline{FR}_{(\alpha,\beta)}(X\cap Y)\subseteq \underline{FR}_{(\alpha,\beta)}(X)\cap \underline{FR}_{(\alpha,\beta)}(Y);\\
(8) \alpha_{1} \leq \alpha_{2},\beta_{1} \leq \beta_{2} \Rightarrow\underline{FR}_{(\alpha_{2},\beta_{2})}(X)\subseteq \underline{FR}_{(\alpha_{1},\beta_{1})}(X);\\
(9) \alpha_{1} \leq \alpha_{2},\beta_{1} \leq \beta_{2} \Rightarrow\overline{FR}_{(\alpha_{2},\beta_{2})}(X)\subseteq \overline{FR}_{(\alpha_{1},\beta_{1})}(X).$
**Proof.** (1) For any $x\in U$, we have $P(U|\widetilde{N}_{x}^{\gamma})=\frac{\Sigma_{y\in U}(U\cap\widetilde{N}_{x}^{\gamma})(y)}{\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)}=1\geq \alpha$ and $P(\emptyset|\widetilde{N}_{x}^{\gamma})=\frac{\Sigma_{y\in U}(\emptyset\cap\widetilde{N}_{x}^{\gamma})(y)}{\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)}=0< \beta$ by Definition 4.2. So $\underline{FR}_{(\alpha,\beta)}(U)=U$ and $\overline{FR}_{(\alpha,\beta)}(\emptyset)=\emptyset$.
\(2) For $x_{0}\in \overline{R}_{(\alpha,\beta)}(X)$, we have $P(X|\widetilde{N}_{x_{0}}^{\gamma})=\frac{\Sigma_{y\in U}(X\cap\widetilde{N}_{x_{0}}^{\gamma})(y)}{\Sigma_{y\in U}\widetilde{N}_{x_{0}}^{\gamma}(y)}\geq\beta$. Since $X\subseteq Y$, we have $\Sigma_{y\in U}(X\cap\widetilde{N}_{x_{0}}^{\gamma})(y)\leq \Sigma_{y\in U}(Y\cap\widetilde{N}_{x_{0}}^{\gamma})(y)$. It follows $P(X|\widetilde{N}_{x_{0}}^{\gamma})\leq P(Y|\widetilde{N}_{x_{0}}^{\gamma})$. So $x_{0}\in \overline{R}_{(\alpha,\beta)}(Y)$. Therefore, $\overline{FR}_{(\alpha,\beta)}(X)\subseteq \overline{FR}_{(\alpha,\beta)}(Y)$.
\(3) For $x_{0}\in \underline{FR}_{(\alpha,\beta)}(X)$, we have $P(X|\widetilde{N}_{x_{0}}^{\gamma})=\frac{\Sigma_{y\in U}(X\cap\widetilde{N}_{x_{0}}^{\gamma})(y)}{\Sigma_{y\in U}\widetilde{N}_{x_{0}}^{\gamma}(y)}\geq\alpha$. Since $X\subseteq Y$, we have $\Sigma_{y\in U}(X\cap\widetilde{N}_{x_{0}}^{\gamma})(y)\leq \Sigma_{y\in U}(Y\cap\widetilde{N}_{x_{0}}^{\gamma})(y)$. It follows $P(X|\widetilde{N}_{x_{0}}^{\gamma})\leq P(Y|\widetilde{N}_{x_{0}}^{\gamma})$. We obtain $x_{0}\in \underline{FR}_{(\alpha,\beta)}(Y)$. So $\underline{FR}_{(\alpha,\beta)}(X)\subseteq \underline{FR}_{(\alpha,\beta)}(Y)$.
\(4) By Theorem 4.7(2), we have $\overline{FR}_{(\alpha,\beta)}(X)\subseteq \overline{FR}_{(\alpha,\beta)}(X\cup Y)$ and $\overline{FR}_{(\alpha,\beta)}(Y)\subseteq \overline{FR}_{(\alpha,\beta)}(X\cup Y)$ for $X,Y\in \mathscr{F}(U)$. Therefore, $\overline{FR}_{(\alpha,\beta)}(X)\cup\overline{FR}_{(\alpha,\beta)}(Y)\subseteq \overline{FR}_{(\alpha,\beta)}(X\cup Y)$
\(5) By Theorem 4.7(2), we have $\underline{FR}_{(\alpha,\beta)}(X)\subseteq \underline{FR}_{(\alpha,\beta)}(X\cup Y)$ and $\underline{FR}_{(\alpha,\beta)}(Y)\subseteq \underline{FR}_{(\alpha,\beta)}(X\cup Y)$ for $X,Y\in \mathscr{F}(U)$. So $\underline{FR}_{(\alpha,\beta)}(X)\cup\underline{FR}_{(\alpha,\beta)}(Y)\subseteq \underline{FR}_{(\alpha,\beta)}(X\cup Y)$.
\(6) and (7) The proof is similar to Theorem 4.7(3) and (4).
\(8) By Definition 4.3, we have $\underline{FR}_{(\alpha_{1},\beta_{1})}(X)=\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\alpha_{1} \}$ and $\underline{FR}_{(\alpha_{2},\beta_{2})}(X)=\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\alpha_{2} \}$. If $P(X|\widetilde{N}_{z}^{\gamma})\geq\alpha_{2}$ for $z\in U$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\alpha_{1}$ for $z\in U$ since $\alpha_{1} \leq \alpha_{2}$. Therefore, $\underline{FR}_{(\alpha_{2},\beta_{2})}(X)\subseteq \underline{FR}_{(\alpha_{1},\beta_{1})}(X)$.
\(9) By Definition 4.3, we have $\overline{FR}_{(\alpha_{1},\beta_{1})}(X)=\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\beta_{1} \}$ and $\overline{FR}_{(\alpha_{2},\beta_{2})}(X)=\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\beta_{2} \}$. If $P(X|\widetilde{N}_{z}^{\gamma})\geq\beta_{2}$ for $z\in U$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\beta_{1}$ for $z\in U$ since $\beta_{1} \leq \beta_{2}$. So $\overline{FR}_{(\alpha_{2},\beta_{2})}(X)\subseteq \overline{FR}_{(\alpha_{1},\beta_{1})}(X)$. $\Box$
Grade lower and upper approximation operators
---------------------------------------------
In this section, we propose the fuzzy $\gamma-$covering based grade lower and upper approximation operators for the fuzzy $\gamma-$covering approximation space.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $\gamma\in (0,1]$, and $k\in R$. Then the fuzzy $\gamma-$covering based grade lower and upper approximations of the fuzzy set $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{GR}_{k}(X)&=&\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)>k\};\\
\underline{GR}_{k}(X)&=&\{x\in U\mid \Sigma_{y\in U}[\widetilde{N}_{x}^{\gamma}(y)-(X\cap \widetilde{N}_{x}^{\gamma})(y)]\leq k\}.\end{aligned}$$
The fuzzy $\gamma-$covering based grade lower and upper approximation operators $\overline{GR}_{k}(X)$ and $\underline{GR}_{k}(X)$ for the fuzzy set $X\in \mathscr{F}(U)$ given by Definition 4.8 are extensions of the grade lower and upper approximation operators $\overline{R}_{k}(X)$ and $\underline{R}_{k}(X)$ for the set $X\in P(U)$ given by Definition 3.5, which construct the lower and upper approximations of fuzzy sets using the absolute quantitative information.
We employ an example to illustrate the construction of the fuzzy $\gamma-$covering based grade lower and upper approximations of sets as follows.
(Continuation from Example 4.4) Taking $k=2$, we have the fuzzy $\gamma-$covering based grade lower and upper approximations of the fuzzy set $X$ as follows: $$\begin{aligned}
\overline{GR}_{2}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\} \text{ and }
\underline{GR}_{2}(X)=\{x_{2},x_{3},x_{6},x_{8}\}.\end{aligned}$$
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $\gamma\in (0,1]$, and $k\in R$. Then the fuzzy $\gamma-$covering based grade positive, boundary, and negative regions of the fuzzy set $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\widetilde{POS}_{k}(X)&=&\overline{GR}_{k}(X)\cap \underline{GR}_{k}(X);\\
\widetilde{NEG}_{k}(X)&=&(\overline{GR}_{k}(X)\cup \underline{GR}_{k}(X))^{c};\\
\widetilde{LBO}_{k}(X)&=&\underline{GR}_{k}(X)-\overline{GR}_{k}(X);\\
\widetilde{UBO}_{k}(X)&=&\overline{GR}_{k}(X)-\underline{GR}_{k}(X);\\
\widetilde{BOU}_{k}(X)&=&\widetilde{LBO}_{k}(X)\cup \widetilde{UBO}_{k}(X).\end{aligned}$$
The fuzzy $\gamma-$covering based grade positive, lower and upper boundary, and negative regions $\widetilde{POS}_{k}(X),$ $\widetilde{NEG}_{k}(X),\widetilde{LBO}_{k}(X),$ $\widetilde{UBO}_{k}(X)$, and $\widetilde{BOU}_{k}(X)$ of the fuzzy set $X\in \mathscr{F}(U)$ in the fuzzy $\gamma-$covering approximation space are generalizations of the grade positive, lower and upper boundary, and negative regions $POS_{k}(X), NEG_{k}(X),LBO_{k}(X),$ $UBO_{k}(X),$ and $BOU_{k}(X)$ of the set $X\in P(U)$ in the covering approximation spaces.
(Continuation from Example 4.4) Taking $k=2$, we have the fuzzy $\gamma-$covering based grade positive, lower and upper boundary, and negative regions of the fuzzy set $X$ as follows: $$\begin{aligned}
\widetilde{POS}_{2}(X)&=&\overline{GR}_{2}(X)\cap \underline{GR}_{2}(X)=\{x_{2},x_{3},x_{6},x_{8}\};\\
\widetilde{NEG}_{2}(X)&=&(\overline{GR}_{2}(X)\cup \underline{GR}_{2}(X))^{c}=\emptyset;\\
\widetilde{LBO}_{2}(X)&=&\underline{GR}_{2}(X)-\overline{GR}_{2}(X)=\emptyset;\\
\widetilde{UBO}_{2}(X)&=&\overline{GR}_{2}(X)-\underline{GR}_{2}(X)=\{x_{1},x_{5},x_{6},x_{7}\};\\
\widetilde{BOU}_{2}(X)&=&\widetilde{LBO}_{2}(X)\cup \widetilde{UBO}_{2}(X)=\{x_{1},x_{5},x_{6},x_{7}\}.\end{aligned}$$
We present the basic properties of the fuzzy $\gamma-$covering based grade lower and upper approximation operators as follows.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $k,k_{1},k_{2}\in R$, and $X,Y\in \mathscr{F}(U)$. Then\
$(1) \underline{GR}_{k}(U)=U;\overline{GR}_{k}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{GR}_{k}(X)\subseteq \overline{GR}_{k}(Y);\\
(3) X\subseteq Y\Rightarrow\underline{GR}_{k}(X)\subseteq \underline{GR}_{k}(Y);\\
(4) \overline{GR}_{k}(X)\cup \overline{GR}_{k}(Y)\subseteq \overline{GR}_{k}(X\cup Y);\\
(5) \underline{GR}_{k}(X)\cup \underline{GR}_{k}(Y)\subseteq \underline{GR}_{k}(X\cup Y);\\
(6) \overline{GR}_{k}(X\cap Y)\subseteq \overline{GR}_{k}(X)\cap \overline{GR}_{k}(Y);\\
(7) \underline{GR}_{k}(X\cap Y)\subseteq \underline{GR}_{k}(X)\cap \underline{GR}_{k}(Y);\\
(8) k_{1} \leq k_{2} \Rightarrow\overline{GR}_{k_{1}}(X)\subseteq \overline{GR}_{k_{2}}(X);\\
(9) k_{1} \leq k_{2}\Rightarrow\underline{GR}_{k_{2}}(X)\subseteq \underline{GR}_{k_{1}}(X).$
**Proof.** (1) By Definition 4.8, $\Sigma_{y\in U}(U^{c}\cap \widetilde{N}_{x}^{\gamma})(y)=0\leq k$ and $\Sigma_{y\in U}(\emptyset\cap \widetilde{N}_{x}^{\gamma})(y)=0\leq k$ for any $x\in U$. It follows that $\underline{GR}_{k}(U)=U$ and $\overline{GR}_{k}(\emptyset)=\emptyset$.
\(2) For any $x_{0}\in \overline{GR}_{k}(X)$, we have $\Sigma_{y\in U}(X\cap \widetilde{N}_{x_{0}}^{\gamma})(y)\leq \Sigma_{y\in U}(Y\cap \widetilde{N}_{x_{0}}^{\gamma})(y)$. Since $\Sigma_{y\in U}(X\cap \widetilde{N}_{x_{0}}^{\gamma})(y)>k$. It follows $\Sigma_{y\in U}(Y\cap \widetilde{N}_{x_{0}}^{\gamma})(y)>k$. So $x_{0}\in \overline{R}_{k}(Y)$.
\(3) For any $x_{0}\in \underline{GR}_{k}(X)$, we have $\Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x_{0}}^{\gamma})(y)> \Sigma_{y\in U}(Y^{c}\cap \widetilde{N}_{x_{0}}^{\gamma})(y)$. Since $\Sigma_{y\in U}(X\cap \widetilde{N}_{x_{0}}^{\gamma})(y)<k$. It follows $\Sigma_{y\in U}(Y^{c}\cap \widetilde{N}_{x_{0}}^{\gamma})(y)<k$. Therefore, $x_{0}\in \underline{GR}_{k}(Y)$.
\(4) By Theorem 4.12(2), we have $\underline{GR}_{k}(X)\subseteq \underline{GR}_{k}(X\cup Y)$ and $\underline{GR}_{k}(Y)\subseteq \underline{GR}_{k}(X\cup Y)$ for $X,Y\in \mathscr{F}(U)$. It follows that $\underline{GR}_{k}(X)\cup \underline{GR}_{k}(Y)\subseteq \underline{GR}_{k}(X\cup Y)$.
\(5) By Theorem 4.12(3), we have $\overline{GR}_{k}(X)\subseteq \overline{GR}_{k}(X\cup Y)$ and $\overline{GR}_{k}(Y)\subseteq \overline{GR}_{k}(X\cup Y)$ for $X,Y\in \mathscr{F}(U)$. It follows that $\overline{GR}_{k}(X)\cup \overline{GR}_{k}(Y)\subseteq \overline{GR}_{k}(X\cup Y)$.
(6),(7) The proof is similar to Theorem 4.12(4) and (5).
\(8) By Definition 4.8, we have $\overline{GR}_{k_{1}}(X)=\{x\in U\mid \Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma})(y)>k_{1}\}$ and $\overline{GR}_{k_{2}}(X)=\{x\in U\mid \Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma})(y)>k_{2}\}$ for $X\in \mathscr{F}(U)$. For any $z\in \overline{GR}_{k_{1}}(X)$, we have $\Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma})(z)>k_{1}\geq k_{2}$. It follows that $z\in \overline{GR}_{k_{2}}(X)$. Therefore, $\overline{GR}_{k_{1}}(X)\subseteq \overline{GR}_{k_{2}}(X)$.
\(9) By Theorem 4.12, we have $\underline{GR}_{k_{1}}(X)=\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)<k_{1}\}$ and $\underline{GR}_{k_{2}}(X)=\{x\in U\mid \Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma})(y)<k_{2}\}$ for $X\in \mathscr{F}(U)$. For any $z\in \underline{GR}_{k_{1}}(X)$, we have $\Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma})(z)<k_{1}\leq k_{2}$. It follows that $z\in \underline{GR}_{k_{2}}(X)$. Therefore, $\underline{GR}_{k_{2}}(X)\subseteq \underline{GR}_{k_{1}}(X)$. $\Box$
In what follows, we discuss the relationship between the fuzzy $\gamma-$covering based probabilistic lower and upper approximation operators and the fuzzy $\gamma-$covering based grade lower and upper approximation operators.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $X\in \mathscr{F}(U)$, $\gamma\in (0,1]$, and $k\in R$. Then $$\begin{aligned}
\overline{FR}_{(\alpha,\beta)}(X)&=&\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq\beta\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)\};\\
\underline{FR}_{(\alpha,\beta)}(X)&=&\{x\in U\mid \Sigma_{y\in U}[\widetilde{N}_{x}^{\gamma})(y)-(X\cap \widetilde{N}_{x}^{\gamma})(y)]\leq \Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)\}-\alpha\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)\}\}.\end{aligned}$$
**Proof.** By Definition 4.3, we have $\overline{FR}_{(\alpha,\beta)}(X)=\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\beta \}$, which implies that $P(X|\widetilde{N}_{x}^{\gamma})\geq\beta$ for $x\in \overline{FR}_{(\alpha,\beta)}(X)$. It follows that $P(X|\widetilde{N}_{x}^{\gamma})=\frac{\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)}{\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)}\geq\beta$. So $\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq\beta\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)$. Therefore, $\overline{FR}_{(\alpha,\beta)}(X)=\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq\beta\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)\}.$
By Definition 4.3, we have $\underline{FR}_{(\alpha,\beta)}(X)=\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\alpha \}$, which implies that $P(X|\widetilde{N}_{x}^{\gamma})\geq\alpha$ for $x\in \underline{FR}_{(\alpha,\beta)}(X)$. It follows that $P(X|\widetilde{N}_{x}^{\gamma})=\frac{\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)}{\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)}\geq\alpha$. So $\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq\alpha\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)$, which implies $\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)-\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\leq\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)-\alpha\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)$. Therefore, $\underline{FR}_{(\alpha,\beta)}(X)=\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq\beta\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)\}.$ $\Box$
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $X\in \mathscr{F}(U)$, $\gamma\in (0,1]$, and $k\in R$. Then $$\begin{aligned}
\overline{GR}_{k}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\frac{k}{\Sigma_{y\in U} \widetilde{N}_{x}^{\gamma}(y)}\};\\
\underline{GR}_{k}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq 1-\frac{k}{\Sigma_{y\in U} \widetilde{N}_{x}^{\gamma}(y)}\}.\end{aligned}$$
**Proof.** By Definition 4.8, we have $\overline{GR}_{k}(X)=\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq k\},$ which implies that $\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)>k$ for $x\in \overline{GR}_{k}(X)$. It follows that $P(X|\widetilde{N}_{x}^{\gamma})=\frac{\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)}{\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)}\geq\frac{k}{\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)}$ for $x\in \overline{GR}_{k}(X)$. Therefore, $\overline{GR}_{k}(X)=\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq\frac{k}{\Sigma_{y\in U} \widetilde{N}_{x}^{\gamma}(y)}\}.$
By Definition 4.8, we have $\underline{GR}_{k}(X)=\{x\in U\mid \Sigma_{y\in U}[\widetilde{N}_{x}^{\gamma}(y)-(X\cap \widetilde{N}_{x}^{\gamma})(y)]\leq k\},$ which implies that $\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq \Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)-k$ for $x\in \underline{GR}_{k}(X)$. It follows that $P(X|\widetilde{N}_{x}^{\gamma})=\frac{\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)}{\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)}>1-\frac{k}{\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)}$ for $x\in \underline{GR}_{k}(X)$. Therefore, $\underline{GR}_{k}(X)=\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq 1-\frac{k}{\Sigma_{y\in U} \widetilde{N}_{x}^{\gamma}(y)}\}.$
Theorems 4.13 and 4.14 illustrate the relationship between the fuzzy $\gamma-$covering based probabilistic lower and upper approximations of fuzzy sets and the fuzzy $\gamma-$covering based grade lower and upper approximations of fuzzy sets, which build a bridge between two fuzzy $\gamma-$covering based approximation operators.
Double-quantitative lower and upper approximation operators
===========================================================
In this section, we present the fuzzy $\gamma-$covering based double-quantitative lower and upper approximations of fuzzy sets in the fuzzy $\gamma-$covering approximation space.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $0\leq \beta \leq \alpha \leq 1,$ and $k\in R$. Then the fuzzy $\gamma-$covering based disjunctive double-quantitative lower and upper approximations of $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\{x\in U\mid [P(X|\widetilde{N}_{x}^{\gamma})\geq\beta] \wedge [\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)>k]\};\\
\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\{x\in U\mid [P(X|\widetilde{N}_{x}^{\gamma})\geq\alpha] \wedge [\Sigma_{y\in U}[\widetilde{N}_{x}^{\gamma})(y)-(X\cap \widetilde{N}_{x}^{\gamma})(y)]\leq k]\}.\end{aligned}$$
The fuzzy $\gamma-$covering based disjunctive lower and upper approximation operators $\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)$ and $\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)$ consider the relative and absolute quantitative information, which are generalizations of disjunctive double quantitative rough set model proposed by Xu[@Xu].
We convert the fuzzy $\gamma-$covering based disjunctive double-quantitative lower and upper approximations of the fuzzy set $X\in \mathscr{F}(U)$ as follows: $$\begin{aligned}
\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq max\{\beta,\frac{k}{\Sigma_{y\in U} \widetilde{N}_{x}^{\gamma}(y)}\}\};\\
\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq max\{\alpha,1-\frac{k}{\Sigma_{y\in U} \widetilde{N}_{x}^{\gamma}(y)}\},\end{aligned}$$ and $$\begin{aligned}
\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq max\{\beta\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y), k\}\};\\
\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq max\{\alpha\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y), \Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)-k\}\}.\end{aligned}$$
We employ an example to illustrate the computing of the fuzzy $\gamma-$covering based disjunctive double-quantitative lower and upper approximations of sets as follows.
(Continuation from Example 4.4) Taking $\alpha=0.75,$ $\beta=0.25$, and $k=2$, we have the fuzzy $\gamma-$covering based disjunctive double-quantitative lower and upper approximations of the fuzzy set $X$ as follows: $$\begin{aligned}
\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)=\{x_{3},x_{6}\}\text{ and }
\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\};\end{aligned}$$
In what follows, we show the relationship between the fuzzy $\gamma-$covering based disjunctive double-quantitative lower and upper approximation operators and the fuzzy $\gamma-$covering based probabilistic and grade lower and upper approximation operators.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $0\leq \beta \leq \alpha \leq 1,$ $k\in R$, and $X\in \mathscr{F}(U)$. Then $$\begin{aligned}
\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\overline{FR}_{(\alpha,\beta)}(X)\cap \overline{GR}_{ k}(X);\\
\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\underline{FR}_{(\alpha,\beta)}(X)\cap \underline{GR}_{ k}(X).\end{aligned}$$
**Proof.** For $z\in\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\beta $ and $\Sigma_{y\in U}(X\cap \widetilde{N}_{z}^{\gamma})(y)>k$, which implies that $z\in\overline{FR}_{(\alpha,\beta)}(X)$ and $ z\in\overline{GR}_{k}(X)$. It follows that $\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)\subseteq\overline{FR}_{(\alpha,\beta)}(X)\cap \overline{GR}_{k}(X).$ For $z\in\overline{FR}_{(\alpha,\beta)}(X)\cup \overline{GR}_{k}(X)$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\beta $ and $\Sigma_{y\in U}(X\cap \widetilde{N}_{z}^{\gamma})(y)\geq k$, which implies that $z\in\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)$. Therefore, $\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)=\overline{FR}_{(\alpha,\beta)}(X)\cap \overline{GR}_{k}(X).$
For $z\in\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\alpha $ and $\Sigma_{y\in U}[\widetilde{N}_{x}^{\gamma})(y)-(X\cap \widetilde{N}_{x}^{\gamma})(y)]\leq k$, which implies that $z\in\underline{FR}_{(\alpha,\beta)}(X)$ and $ z\in\underline{GR}_{ k}(X)$. It follows that $\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)\subseteq\underline{FR}_{(\alpha,\beta)}(X)\cap \underline{GR}_{k}(X).$ For $z\in \underline{FR}_{(\alpha,\beta)}(X)\cap \underline{GR}_{k}(X)$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\alpha $ and $\Sigma_{y\in U}[\widetilde{N}_{x}^{\gamma})(y)-(X\cap \widetilde{N}_{x}^{\gamma})(y)]\leq k$, which implies that $z\in \underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X).$ Therefore, $\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)=\underline{FR}_{(\alpha,\beta)}(X)\cap \underline{GR}_{k}(X)$. $\Box$
(Continuation from Example 4.4) Taking $\alpha=0.75,$ $\beta=0.25$, and $k=2$, we have the fuzzy $\gamma-$covering based disjunctive double-quantitative lower and upper approximations of the fuzzy set $X$ as follows: $$\begin{aligned}
\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\underline{FR}_{(\alpha,\beta)}(X)\cap\underline{GR}_{2}(X)=\{x_{3},x_{6}\};\\
\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)&=&\overline{FR}_{(\alpha,\beta)}(X)\cap\overline{GR}_{2}(X)=\{x_{1},x_{2},x_{3},
x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We investigate the basic properties of the fuzzy $\gamma-$covering based disjunctive double-quantitative lower and upper approximation operators as follows.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $0\leq \beta< \alpha\leq 1$, $k\in R$, and $X,Y\in \mathscr{F}(U)$. Then\
$(1)\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(U)=U;\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)\subseteq \overline{DR}_{(\alpha,\beta)\wedge k}^{I}(Y);\\
(3) X\subseteq Y\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)\subseteq \underline{DR}_{(\alpha,\beta)\wedge k}^{I}(Y);\\
(4) \overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)\cup \overline{DR}_{(\alpha,\beta)\wedge k}^{I}(Y)\subseteq \overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X\cup Y);\\
(5)\underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)\cup \underline{DR_{(\alpha,\beta)\wedge k}}^{I}(Y)\subseteq \underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X\cup Y);\\
(6) \overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X\cap Y)\subseteq \overline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)\cap \overline{DR}_{(\alpha,\beta)\wedge k}^{I}(Y);\\
(7) \underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X\cap Y)\subseteq \underline{DR}_{(\alpha,\beta)\wedge k}^{I}(X)\cap \underline{DR}_{(\alpha,\beta)\wedge k}^{I}(Y);\\
(8) \alpha_{1} \leq \alpha_{2}, \beta_{1} \leq \beta_{2}, k_{1} \leq k_{2}\Rightarrow\underline{DR}_{(\alpha_{1},\beta_{1})\wedge k_{1}}^{I}(X)\subseteq \underline{DR}_{(\alpha_{2},\beta_{2})\wedge k_{2}}^{I}(X);\\
(9) \alpha_{1} \leq \alpha_{2}, \beta_{1} \leq \beta_{2}, k_{1} \leq k_{2}\Rightarrow\overline{DR}_{(\alpha_{1},\beta_{1})\wedge k_{1}}(X)\subseteq \overline{DR}_{(\alpha_{2},\beta_{2})\wedge k_{2}}^{I}(X).$
**Proof.** By Theorems 4.7 and 4.12, the proof is straightforward. $\Box$
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $0\leq \beta \leq \alpha \leq 1,$ $k\in R$, and $X\in \mathscr{F}(U)$. Then the fuzzy $\gamma-$covering based conjunctive double-quantitative lower and upper approximations of the fuzzy set $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\{x\in U\mid [P(X|\widetilde{N}_{x}^{\gamma})\geq\beta] \vee [\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)>k]\};\\
\underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\{x\in U\mid [P(X|\widetilde{N}_{x}^{\gamma})\geq\alpha] \vee [\Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma})(y)\leq k]\}.\end{aligned}$$
The fuzzy $\gamma-$covering based conjunctive double-quantitative lower and upper approximation operators $\overline{DR}_{(\alpha,\beta)\wedge k}^{II}(X)$ and $\underline{DR}_{(\alpha,\beta)\wedge k}^{II}(X)$ for the fuzzy set $X\in \mathscr{F}(U)$ consider the relative and absolute quantitative information, which are generalizations of conjunctive double-quantitative rough set model proposed by Xu[@Xu].
We convert the fuzzy $\gamma-$covering based disjunctive double-quantitative lower and upper approximations of the fuzzy set $X\in \mathscr{F}(U)$ as follows: $$\begin{aligned}
\overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq min\{\beta,\frac{k}{\Sigma_{y\in U} \widetilde{N}_{x}^{\gamma}(y)}\}\};\\
\underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\{x\in U\mid P(X|\widetilde{N}_{x}^{\gamma})\geq min\{\alpha,1-\frac{k}{\Sigma_{y\in U} \widetilde{N}_{x}^{\gamma}(y)}\},\end{aligned}$$ and $$\begin{aligned}
\overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq min\{\beta\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y), k\}\};\\
\underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\{x\in U\mid \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma})(y)\geq min\{\alpha\Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y), \Sigma_{y\in U}\widetilde{N}_{x}^{\gamma}(y)-k\}\}.\end{aligned}$$
We employ an example to illustrate the computing of the fuzzy $\gamma-$covering based conjunctive double-quantitative lower and upper approximations of fuzzy sets as follows.
(Continuation from Example 4.4) Taking $\alpha=0.75,$ $\beta=0.25$, and $k=2$, we have the conjunctive double-quantitative lower and upper approximations of the fuzzy set $X$ as follows: $$\begin{aligned}
\underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)=\{x_{2},x_{3},x_{6},x_{8}\}\text{ and }
\overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We show the relationship between the fuzzy $\gamma-$covering based conjunctive double-quantitative lower and upper approximation operators and the fuzzy $\gamma-$covering based probabilistic and grade lower and upper approximation operators as follows.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $0\leq \beta \leq \alpha \leq 1,$ $X\in \mathscr{F}(U)$, and $k\in R$. Then $$\begin{aligned}
\overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\overline{FR}_{(\alpha,\beta)}(X)\cup \overline{GR}_{ k}(X);\\
\underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\underline{FR}_{(\alpha,\beta)}(X)\cup \underline{GR}_{ k}(X).\end{aligned}$$
**Proof.** For $z\in\overline{DR}_{(\alpha,\beta)\wedge k}^{II}(X)$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\beta $ or $\Sigma_{y\in U}(X\cap \widetilde{N}_{z}^{\gamma})(y)\geq k$, which implies that $z\in\overline{FR}_{(\alpha,\beta)}(X)$ or $ z\in\overline{GR}_{ k}(X)$. It follows that $\overline{DR}_{(\alpha,\beta)\wedge k}^{II}(X)\subseteq\overline{FR}_{(\alpha,\beta)}(X)\cup \overline{GR}_{ k}(X).$ For $z\in\overline{FR}_{(\alpha,\beta)}(X)\cup \overline{GR}_{k}(X)$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\beta $ or $\Sigma_{y\in U}(X\cap \widetilde{N}_{z}^{\gamma})(y)\geq k$, which implies that $z\in\overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)$. Therefore, $\overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)=\overline{FR}_{(\alpha,\beta)}(X)\cup \overline{GR}_{ k}(X).$
For $z\in\underline{DR}_{(\alpha,\beta)\wedge k}^{II}(X)$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\alpha $ or $\Sigma_{y\in U}[\widetilde{N}_{x}^{\gamma})(y)-(X\cap \widetilde{N}_{x}^{\gamma})(y)]\leq k$, which implies that $z\in\underline{FR}_{(\alpha,\beta)}(X)$ or $ z\in\underline{GR}_{ k}(X)$. It follows that $\underline{DR}_{(\alpha,\beta)\wedge k}^{II}(X)\subseteq\underline{FR}_{(\alpha,\beta)}(X)\cup \underline{GR}_{ k}(X).$ For $z\in \underline{FR}_{(\alpha,\beta)}(X)\cup \underline{GR}_{ k}(X)$, we have $P(X|\widetilde{N}_{z}^{\gamma})\geq\alpha $ or $\Sigma_{y\in U}[\widetilde{N}_{x}^{\gamma})(y)-(X\cap \widetilde{N}_{x}^{\gamma})(y)]\leq k$, which implies that $z\in \underline{DR}_{(\alpha,\beta)\vee k}^{II}(X).$ Therefore, $\underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)=\underline{FR}_{(\alpha,\beta)}(X)\cup \underline{GR}_{k}(X)$. $\Box$
(Continuation from Example 4.4) Taking $\alpha=0.75,$ $\beta=0.25$, and $k=2$, we have the fuzzy $\gamma-$covering based conjunctive double-quantitative lower and upper approximations of the fuzzy set $X$ as follows: $$\begin{aligned}
\underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\underline{FR}_{(\alpha,\beta)}(X)\cup\underline{GR}_{2}(X)=\{x_{2},x_{3},x_{6},x_{8}\};\\
\overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)&=&\overline{FR}_{(\alpha,\beta)}(X)\cup\overline{GR}_{2}(X)=\{x_{1},x_{2},x_{3},x_{4},
x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We show the basic properties of the fuzzy $\gamma-$covering based conjunctive double-quantitative lower and upper approximation operators as follows.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, $0\leq \beta< \alpha\leq 1$, $k\in R$, and $X,Y\in \mathscr{F}(U)$. Then\
$(1)\underline{DR}_{(\alpha,\beta)\vee k}^{II}(U)=U;\overline{DR}_{(\alpha,\beta)\vee k}^{II}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)\subseteq \overline{DR}_{(\alpha,\beta)\vee k}^{II}(Y);\\
(3) X\subseteq Y\Rightarrow\underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)\subseteq \underline{DR}_{(\alpha,\beta)\vee k}^{II}(Y);\\
(4) \overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)\cup \overline{DR}_{(\alpha,\beta)\vee k}^{II}(Y)\subseteq \overline{DR}_{(\alpha,\beta)\vee k}^{II}(X\cup Y);\\
(5)\underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)\cup \underline{DR_{(\alpha,\beta)\vee k}}^{II}(Y)\subseteq \underline{DR}_{(\alpha,\beta)\vee k}^{II}(X\cup Y);\\
(6) \overline{DR}_{(\alpha,\beta)\vee k}^{II}(X\cap Y)\subseteq \overline{DR}_{(\alpha,\beta)\vee k}^{II}(X)\cap \overline{DR}_{(\alpha,\beta)\vee k}^{II}(Y);\\
(7) \underline{DR}_{(\alpha,\beta)\vee k}^{II}(X\cap Y)\subseteq \underline{DR}_{(\alpha,\beta)\vee k}^{II}(X)\cap \underline{DR}_{(\alpha,\beta)\vee k}^{II}(Y);\\
(8) \alpha_{1} \leq \alpha_{2}, \beta_{1} \leq \beta_{2}, k_{1} \leq k_{2}\Rightarrow\underline{DR}_{(\alpha_{1},\beta_{1})\vee k_{1}}^{II}(X)\subseteq \underline{DR}_{(\alpha_{2},\beta_{2})\vee k_{2}}^{II}(X);\\
(9) \alpha_{1} \leq \alpha_{2}, \beta_{1} \leq \beta_{2}, k_{1} \leq k_{2}\Rightarrow\overline{DR}_{(\alpha_{1},\beta_{1})\vee k_{1}}(X)\subseteq \overline{DR}_{(\alpha_{2},\beta_{2})\vee k_{2}}^{II}(X).$
**Proof.** By Theorems 4.7 and 4.12, the proof is straightforward. $\Box$
Multi-granulation double-quantitative approximation operators
=============================================================
In this section, we present the fuzzy $\gamma^{\ast}-$coverings based multi-granulation double-quantitative lower and upper approximation operators in the fuzzy $\gamma^{\ast}-$coverings information system.
Multi-granulation probabilistic approximation operators
-------------------------------------------------------
In this section, we propose the concept of the fuzzy $\gamma^{\ast}-$coverings information system.
Let $U$ be a finite universe, $\Delta^{\ast}$ a family of fuzzy $\gamma-$coverings, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\Delta^{\ast}=\{\mathscr{C}_{1},
\mathscr{C}_{2},...,\mathscr{C}_{m}\}$, and $\mathscr{C}_{i}$ a fuzzy $\gamma_{i}-$covering of $U$. Then $(U,\Delta^{\ast})$ is called a fuzzy $\gamma^{\ast}-$coverings information system, where $\gamma^{\ast}=[\gamma_{1},\gamma_{2},...,\gamma_{|\Delta^{\ast}|}]$.
The fuzzy $\gamma^{\ast}-$coverings information system is the generalization of the covering information system, and we also can view the fuzzy $\gamma^{\ast}-$coverings information system as a $\gamma-$covering approximation space, where $\gamma=min\{\gamma_{i}|1\leq i\leq m\}$.
Let $U=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}$, $\triangle^{\ast}=\{\mathscr{C}^{\ast}_{1},
\mathscr{C}^{\ast}_{2}\}$, $\mathscr{C}^{\ast}_{1}=\{C_{11},C_{12},C_{13}\}$, and $\mathscr{C}^{\ast}_{2}=\{C_{21},C_{22},C_{23}\}$, where $\gamma^{\ast}=[0.9,0.6],$ and $$\begin{aligned}
C_{11}&=&\frac{1}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}
+\frac{0.9}{x_{4}}+\frac{0.9}{x_{5}}
+\frac{0}{x_{6}}+\frac{0.9}{x_{7}}+\frac{0.8}{x_{8}};\\
C_{12}&=&\frac{0.6}{x_{1}}+\frac{0.9}{x_{2}}+\frac{0.4}{x_{3}}
+\frac{0.4}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.7}{x_{6}}+\frac{0.5}{x_{7}}+\frac{1}{x_{8}};\\
C_{13}&=&\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.9}{x_{3}}
+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.9}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}};
\\
C_{21}&=&\frac{0.6}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.2}{x_{3}}
+\frac{0.4}{x_{4}}+\frac{0.1}{x_{5}}
+\frac{0.6}{x_{6}}+\frac{0.6}{x_{7}}+\frac{0.5}{x_{8}};\\
C_{22}&=&\frac{0.5}{x_{1}}+\frac{0.3}{x_{2}}+\frac{0.6}{x_{3}}
+\frac{0.6}{x_{4}}+\frac{0.4}{x_{5}}
+\frac{0.5}{x_{6}}+\frac{0.2}{x_{7}}+\frac{0.6}{x_{8}};\\
C_{23}&=&\frac{0.2}{x_{1}}+\frac{0.6}{x_{2}}+\frac{0.2}{x_{3}}
+\frac{0.5}{x_{4}}+\frac{0.6}{x_{5}}
+\frac{0.3}{x_{6}}+\frac{0}{x_{7}}+\frac{0.3}{x_{8}}.\end{aligned}$$ Then we have a fuzzy $\gamma^{\ast}-$coverings information system $(U,\Delta^{\ast})$.
In what follows, we present the fuzzy $\gamma^{\ast}-$coverings disjunctive multi-granulation probabilistic lower and upper approximation operators in the fuzzy $\gamma^{\ast}-$coverings information system.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, and $0\leq \beta_{i} \leq \alpha_{i} \leq 1$. Then the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation probabilistic lower and upper approximations of the fuzzy set $X\in\mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\{x\in U\mid \bigwedge_{1\leq i\leq m}[P(X|\widetilde{N}_{x}^{\gamma_{i}})\geq\beta_{i}]\};\\
\underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\{x\in U\mid \bigwedge_{1\leq i\leq m}[P(X|\widetilde{N}_{x}^{\gamma_{i}})\geq\alpha_{i}]\}.\end{aligned}$$
The fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation probabilistic lower and upper approximations operators $\overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)$ and $\underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)$ of the fuzzy set $X\in\mathscr{F}(U)$ consider the relative quantitative information, which are generalizations of the disjunctive multi-granulation rough set model proposed by Qian[@Qian3].
We employ an example to illustrate the construction of the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation probabilistic lower and upper approximations as follows.
(Continuation of Example 4.4) Taking $\alpha_{1}=\alpha_{2}=0.75,$ and $\beta_{1}=\beta_{2}=0.25,$ we have $$\begin{aligned}
\underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)=\{x_{3},x_{6}\}\text{ and }
\overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We employ the following theorem to illustrate the relationship between the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation probabilistic approximations operators and the fuzzy $\gamma^{\ast}-$coverings based probabilistic approximations operators.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$covering information system, $X\in\mathscr{F}(U)$, and $0\leq \beta_{i} \leq \alpha_{i} \leq 1, 1\leq i\leq m$. Then we have $$\begin{aligned}
\overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\bigcap_{1\leq i\leq m}\overline{FR}_{(\alpha_{i},\beta_{i})}(X);\\
\underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\bigcap_{1\leq i\leq m}\underline{FR}_{(\alpha_{i},\beta_{i})}(X).\end{aligned}$$
**Proof.** The proof is straightforward by Definition 6.3. $\Box$
(Continuation of Example 4.4) Taking $\alpha_{1}=\alpha_{2}=0.75,$ and $\beta_{1}=\beta_{2}=0.25,$ we have $$\begin{aligned}
\underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\underline{FR}_{(\alpha_{1},\beta_{1})}(X)\cap \underline{FR}_{(\alpha_{2},\beta_{2})}(X)\\
&=&\{x_{3},x_{6}\}\cap \{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{3},x_{6}\};\\
\overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)
&=&\overline{FR}_{(\alpha_{1},\beta_{1})}(X)\cap \overline{FR}_{(\alpha_{2},\beta_{2})}(X)\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\cap
\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We present the following concept to illustrate the relationship between the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation probabilistic lower and upper approximation operators.
Let $\alpha^{\ast}_{1}=[\alpha_{11},\alpha_{12},...,\alpha_{1m}]$, $\alpha^{\ast}_{2}=[\alpha_{21},\alpha_{22},...,\alpha_{2m}]$, $\beta^{\ast}_{1}=[\beta_{11},\beta_{12},...,\beta_{1m}]$ and $\beta^{\ast}_{2}=[\beta_{21},\beta_{22},...,\beta_{2m}]$, where $0\leq\alpha_{ij}, \beta_{ij}\leq 1$. Then we say $(\alpha^{\ast}_{1},\beta^{\ast}_{1})\leq (\alpha^{\ast}_{2},\beta^{\ast}_{2})$ if $\alpha_{1i}\leq \alpha_{2i}$ and $\beta_{1i}\leq \beta_{2i}$ for $1\leq i\leq m$.
We provide the basic properties of the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation probabilistic lower and upper approximation operators.
Let $U$ be a finite universe, $\Delta^{\ast}$ a family of fuzzy $\gamma-$coverings, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\Delta^{\ast}=\{\mathscr{C}_{1},
\mathscr{C}_{2},...,\mathscr{C}_{m}\}$, $\mathscr{C}_{i}$ a fuzzy $\gamma_{i}-$covering of $U$, and $X,Y\in\mathscr{F}(U)$. Then\
$(1)\underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(U)=U;\overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)\subseteq \overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(Y);\\
(3) X\subseteq Y\Rightarrow\underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)\subseteq \underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(Y);\\
(4) \overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)\cup \overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)\subseteq \overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X\cup Y);\\
(5) \underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)\cup \underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)\subseteq \underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X\cup Y);\\
(6) \overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X\cap Y)\subseteq \overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)\cap \overline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X); \\
(7) \underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X\cap Y)\subseteq \underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(X)\cap \underline{MR}^{I}_{(\alpha^{\ast},\beta^{\ast})}(Y);\\
(8) \alpha^{\ast}_{1} \leq \alpha^{\ast}_{2},\beta^{\ast}_{1} \leq \beta^{\ast}_{2} \Rightarrow\underline{MR}^{I}_{(\alpha^{\ast}_{2},\beta^{\ast}_{2})}(X)\subseteq \underline{MR}^{I}_{(\alpha^{\ast}_{1},\beta^{\ast}_{1})}(Y); \\
(9) \alpha^{\ast}_{1} \leq \alpha^{\ast}_{2},\beta^{\ast}_{1} \leq \beta^{\ast}_{2} \Rightarrow\overline{MR}^{I}_{(\alpha^{\ast}_{2},\beta^{\ast}_{2})}(X)\subseteq \overline{MR}^{I}_{(\alpha^{\ast}_{1},\beta^{\ast}_{1})}(X).$
**Proof.** By Theorem 4.7 and Definition 6.9, the proof is straightforward. $\Box$
In what follows, we present the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation probabilistic lower and upper approximation operators for the fuzzy $\gamma^{\ast}-$coverings information system.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, and $0\leq \beta_{i} \leq \alpha_{i} \leq 1$. Then the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation probabilistic lower and upper approximations of the fuzzy set $X\in
\mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\{x\in U\mid \bigvee_{1\leq i\leq m}[P(X|\widetilde{N}_{x}^{\gamma_{i}})\geq\beta_{i}]\};\\
\underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\{x\in U\mid \bigvee_{1\leq i\leq m}[P(X|\widetilde{N}_{x}^{\gamma_{i}})\geq\alpha_{i}]\}.\end{aligned}$$
The fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation probabilistic lower and upper approximations operators $\overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)$ and $\underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)$ of the fuzzy set $X\in\mathscr{F}(U)$ also consider the relative quantitative information, which are generalizations of the conjunctive multi-granulation rough set model proposed by Qian[@Qian3].
We employ an example to illustrate the construction of the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation probabilistic lower and upper approximations as follows.
(Continuation of Example 5.2) Taking $\alpha_{1}=\alpha_{2}=0.75,$ and $\beta_{1}=\beta_{2}=0.25,$ we have $$\begin{aligned}
\underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\};\\
\overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We employ the following theorem to illustrate the relationship between the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation probabilistic approximations operators and the fuzzy $\gamma^{\ast}-$coverings based probabilistic approximations operators.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$covering information system, $0\leq \beta_{i} \leq \alpha_{i} \leq 1, 1\leq i\leq m$, and $X\in\mathscr{F}(U)$. Then we have $$\begin{aligned}
\overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\bigcup_{1\leq i\leq m}\overline{FR}_{(\alpha_{i},\beta_{i})}(X);\\
\underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\bigcup_{1\leq i\leq m}\underline{FR}_{(\alpha_{i},\beta_{i})}(X).\end{aligned}$$
**Proof.** The proof is straightforward by Definition 6.3. $\Box$
(Continuation of Example 5.4) Taking $\alpha_{1}=\alpha_{2}=0.75,$ and $\beta_{1}=\beta_{2}=0.25,$ we have $$\begin{aligned}
\underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)&=&\underline{FR}_{(\alpha_{1},\beta_{1})}(X)\cup \underline{FR}_{(\alpha_{2},\beta_{2})}(X)\\
&=&\{x_{3},x_{6}\}\cup \{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\};\\
\overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)
&=&\overline{FR}_{(\alpha_{1},\beta_{1})}(X)\cup \overline{FR}_{(\alpha_{2},\beta_{2})}(X)\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\cup
\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We study the basic properties of the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation probabilistic lower and upper approximations operators as follows.
Let $(U,\mathscr{C})$ be a fuzzy $\gamma^{\ast}-$covering approximation space, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$, and $X,Y\in\mathscr{F}(U)$. Then\
$(1)\underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(U)=U;\overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)\subseteq \overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(Y);\\
(3) X\subseteq Y\Rightarrow\underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)\subseteq \underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(Y);\\
(4)\overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)\cup \overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)\subseteq \overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X\cup Y);\\
(5) \underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)\cup \underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)\subseteq \underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X\cup Y);\\
(6) \overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X\cap Y)\subseteq \overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)\cap \overline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X); \\ (7)\underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X\cap Y)\subseteq \underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(X)\cap \underline{MR}^{II}_{(\alpha^{\ast},\beta^{\ast})}(Y);\\
(8) \alpha^{\ast}_{1} \leq \alpha^{\ast}_{2},\beta^{\ast}_{1} \leq \beta^{\ast}_{2} \Rightarrow\underline{MR}^{II}_{(\alpha^{\ast}_{2},\beta^{\ast}_{2})}(X)\subseteq \underline{MR}^{II}_{(\alpha^{\ast}_{1},\beta^{\ast}_{1})}(Y); \\
(9) \alpha^{\ast}_{1} \leq \alpha^{\ast}_{2},\beta^{\ast}_{1} \leq \beta^{\ast}_{2} \Rightarrow\overline{MR}^{II}_{(\alpha^{\ast}_{2},\beta^{\ast}_{2})}(X)\subseteq \overline{MR}^{II}_{(\alpha^{\ast}_{1},\beta^{\ast}_{1})}(X).$
**Proof.** By Theorem 4.7 and Definition 6.9, the proof is straightforward. $\Box$
Multi-granulation grade approximation operators
-----------------------------------------------
In this section, we present the fuzzy $\gamma^{\ast}-$coverings based multi-granulation grade lower and upper approximation operators for the fuzzy $\gamma^{\ast}-$coverings information system.
Let $K_{1}$ and $K_{2}$ be two vectors, where $K_{1}=[k_{11},k_{12},...,k_{1m}]$ and $K_{2}=[k_{21},k_{22},...,k_{2m}]$. It is said that $K_{1}\leq K_{2}$ if we have $k_{1i}\leq k_{2i}$ for any $1\leq i\leq m$.
In what follows, we first present the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation grade lower and upper approximation operators for the fuzzy $\gamma^{\ast}-$coverings information system.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, and $k_{i}\in R$. Then the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation grade lower and upper approximations of the fuzzy set $X\in\mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{MR}^{I}_{K}(X)&=&\{x\in U\mid \bigwedge_{1\leq i\leq m}[\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma_{i}})(y)>k_{i}]\};\\
\underline{MR}^{I}_{K}(X)&=&\{x\in U\mid \bigwedge_{1\leq i\leq m}[\Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma_{i}})(y)\leq k_{i}]\}.\end{aligned}$$
The fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation grade lower and upper approximations operators $\overline{MR}^{I}_{K}(X)$ and $\underline{MR}^{I}_{K}(X)$ of the fuzzy set $X\in\mathscr{F}(U)$ consider the absolute quantitative information, which are generalizations of the disjunctive multi-granulation rough set model proposed by Qian[@Qian3].
(Continuation from Example 4.4) Taking $k=2$, we have the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation grade lower and upper approximations of the fuzzy set $X$ as follows: $$\begin{aligned}
\overline{GR}_{2}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\} \text{ and }
\underline{GR}_{2}(X)=\{x_{2},x_{3},x_{6},x_{8}\}.\end{aligned}$$
We employ an example to illustrate the construction of the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation probabilistic lower and upper approximations as follows.
(Continuation of Example 6.2) Taking $K=[k_{1},k_{2}]$, where $k_{1}=k_{2}=2$, we have $$\begin{aligned}
\underline{MR}^{I}_{K}(X)=\{x_{2},x_{3},x_{6},x_{8}\}\text{ and }
\overline{MR}^{I}_{K}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We employ the following theorem to illustrate the relationship between the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation grade approximations operators and the fuzzy $\gamma^{\ast}-$coverings based grade approximations operators.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, $0\leq \beta_{i} \leq \alpha_{i} \leq 1, 1\leq i\leq m$, and $X\in\mathscr{F}(U)$. Then we have $$\begin{aligned}
\overline{MR}^{I}_{K}(X)&=&\bigcap_{1\leq i\leq m}\overline{GR}_{k_{i}}(X);\\
\underline{MR}^{I}_{K}(X)&=&\bigcap_{1\leq i\leq m}\underline{GR}_{k_{i}}(X).\end{aligned}$$
**Proof.** The proof is straightforward by Definition 6.15. $\Box$
(Continuation of Example 6.2) Taking $k_{1}=k_{2}=2,$ we have $$\begin{aligned}
\underline{MR}^{I}_{K}(X)&=&\underline{GR}_{k_{1}}(X)\cap \underline{GR}_{k_{2}}(X)\\
&=&\{x_{2},x_{3},x_{6},x_{8}\}\cap \{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{2},x_{3},x_{6},x_{8}\};\\
\overline{MR}^{I}_{K}(X)
&=&\overline{GR}_{k_{1}}(X)\cap \overline{GR}_{k_{2}}(X)\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\cap
\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We study the basic properties of the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation grade lower and upper approximations operators as follows.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, $K=[k_{1},k_{2},...,k_{m}]$, $K_{1}=[k_{11},k_{12},...,k_{1m}]$, $K_{2}=[k_{21},k_{22},...,k_{2m}]$, and $X,Y\in\mathscr{F}(U)$. Then\
$(1) \underline{MR}^{I}_{K}(U)=U;\overline{MR}^{I}_{K}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{MR}^{I}_{K}(X)\subseteq \overline{MR}^{I}_{K}(Y); \\
(3) X\subseteq Y\Rightarrow\underline{MR}^{I}_{K}(X)\subseteq \underline{MR}^{I}_{K}(Y);\\
(4) \overline{MR}^{I}_{K}(X)\cup \overline{MR}^{I}_{K}(X)\subseteq\overline{MR}^{I}_{K}(X\cup Y);\\
(5) \underline{MR}^{I}_{K}(X)\cup \underline{MR}^{I}_{K}(X)\subseteq\underline{MR}^{I}_{K}(X\cup Y);\\
(6) \overline{MR}^{I}_{K}(X\cap Y)\subseteq \overline{MR}^{I}_{K}(X)\cap \overline{MR}^{I}_{K}(Y);\\
(7) \underline{MR}^{I}_{K}(X\cap Y)\subseteq \underline{MR}^{I}_{K}(X)\cap \underline{MR}^{I}_{K}(Y);\\
(8) K_{1} \leq K_{2}\Rightarrow\underline{MR}^{I}_{K_{2}}(X)\subseteq \underline{MR}^{I}_{K_{1}}(X);\\ (9) K_{1} \leq K_{2} \Rightarrow\overline{MR}^{I}_{K_{1}}(X)\subseteq \overline{MR}^{I}_{K_{2}}(X).$
**Proof.** By Theorem 4.9 and Definition 6.15, the proof is straightforward. $\Box$
In what follows, we present the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation grade lower and upper approximation operators for the fuzzy $\gamma^{\ast}-$covering information system.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, and $k_{i}\in R$. Then the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation grade lower and upper approximations of the fuzzy set $X\in\mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{MR}^{II}_{K}(X)&=&\{x\in U\mid \bigvee_{1\leq i\leq m}[\Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma_{i}})(y)>k_{i}]\};\\
\underline{MR}^{II}_{K}(X)&=&\{x\in U\mid \bigvee_{1\leq i\leq m}[\Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma_{i}})(y)\leq k_{i}]\}.\end{aligned}$$
The fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation grade lower and upper approximations operators $\overline{MR}^{II}_{K}(X)$ and $\underline{MR}^{II}_{K}(X)$ of the fuzzy set $X\in\mathscr{F}(U)$ consider the absolute quantitative information, which are generalizations of multi-granulation rough set model proposed by Qian[@Yao].
We employ an example to illustrate the construction of the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation grade lower and upper approximations of sets as follows.
(Continuation of Example 5.2) Taking $X=\frac{0.6}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.7}{x_{3}}
+\frac{0.8}{x_{4}}+\frac{0.5}{x_{5}}
+\frac{0.6}{x_{6}}+\frac{0}{x_{7}}+\frac{0.2}{x_{8}}$, $K=[k_{1},k_{2}]$, where $k_{1}=k_{2}=2$. Then we have $$\begin{aligned}
\underline{MR}^{II}_{K}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\text{ and }
\overline{MR}^{II}_{K}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We employ the following theorem to illustrate the relationship between the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation grade approximations operators and the fuzzy $\gamma^{\ast}-$coverings based grade approximations operators.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, $0\leq \beta_{i} \leq \alpha_{i} \leq 1, 1\leq i\leq m$, and $X\in\mathscr{F}(U)$. Then we have $$\begin{aligned}
\overline{MR}^{II}_{K}(X)&=&\bigcup_{1\leq i\leq m}\overline{GR}_{k_{i}}(X);\\
\underline{MR}^{II}_{K}(X)&=&\bigcup_{1\leq i\leq m}\underline{GR}_{k_{i}}(X).\end{aligned}$$
**Proof.** The proof is straightforward by Definition 6.21. $\Box$
(Continuation of Example 6.4) Taking $k_{1}=k_{2}=2,$ we have $$\begin{aligned}
\underline{MR}^{II}_{K}(X)&=&\underline{GR}_{k_{1}}(X)\cup \underline{GR}_{k_{2}}(X)\\
&=&\{x_{2},x_{3},x_{6},x_{8}\}\cup \{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\};\\
\overline{MR}^{II}_{K}(X)
&=&\overline{GR}_{k_{1}}(X)\cup \overline{GR}_{k_{2}}(X)\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\cup
\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We show the basic properties of the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation grade lower and upper approximations operators as follows.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, $K=[k_{1},k_{2},...,k_{m}]$, $K_{1}=[k_{11},k_{12},...,k_{1m}]$, $K_{2}=[k_{21},k_{22},...,k_{2m}]$, and $X,Y\in\mathscr{F}(U)$. Then\
$(1) \underline{MR}^{II}_{K}(U)=U;\overline{MR}^{II}_{K}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{MR}^{II}_{K}(X)\subseteq \overline{MR}^{II}_{K}(Y); \\
(3) X\subseteq Y\Rightarrow\underline{MR}^{II}_{K}(X)\subseteq \underline{MR}^{II}_{K}(Y);\\
(4) \overline{MR}^{II}_{K}(X)\cup \overline{MR}^{II}_{K}(X)\subseteq\overline{MR}^{II}_{K}(X\cup Y);\\
(5) \underline{MR}^{II}_{K}(X)\cup \underline{MR}^{II}_{K}(X)\subseteq\underline{MR}^{II}_{K}(X\cup Y);\\
(6) \overline{MR}^{II}_{K}(X\cap Y)\subseteq \overline{MR}^{II}_{K}(X)\cap \overline{MR}^{II}_{K}(Y);\\
(7) \underline{MR}^{II}_{K}(X\cap Y)\subseteq \underline{MR}^{II}_{K}(X)\cap \underline{MR}^{II}_{K}(Y);\\
(8) K_{1} \leq K_{2}\Rightarrow\underline{MR}^{II}_{K_{2}}(X)\subseteq \underline{MR}^{II}_{K_{1}}(X);\\ (9) K_{1} \leq K_{2} \Rightarrow\overline{MR}^{II}_{K_{1}}(X)\subseteq \overline{MR}^{II}_{K_{2}}(X).$
**Proof.** By Theorem 4.9 and Definition 6.21, the proof is straightforward. $\Box$
Multi-granulation double-quantitative approximation operators
=============================================================
In this section, we provide the fuzzy $\gamma^{\ast}-$coverings based multi-granulation double-quantitative lower and upper approximation operators in the fuzzy $\gamma-$coverings information system.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, $0\leq \beta_{i} \leq \alpha_{i} \leq 1,$ and $ k_{i}\in R $. Then the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation double-quantitative lower and upper approximations of the fuzzy set $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)&=&\{x\in U\mid \bigwedge_{1\leq i\leq m}[P(X|\widetilde{N}_{x}^{\gamma_{i}})\geq\beta^{\ast}_{i} \wedge \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma_{i}})(y)>k_{i}]\};\\
\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)&=&\{x\in U\mid \bigwedge_{1\leq i\leq m}[P(X|\widetilde{N}_{x}^{\gamma_{i}})\geq\alpha^{\ast}_{i} \wedge \Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma_{i}})(y)\leq k_{i}]\}.\end{aligned}$$
The fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation double-quantitative lower and upper approximation operators $\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)$ and $\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)$ of the fuzzy set $X\in \mathscr{F}(U)$ consider the absolute and relative quantitative information, which are generalizations of disjunctive double quantitative rough set model proposed by Xu[@Xu1].
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, $0\leq \beta_{i} \leq \alpha_{i} \leq 1,$ and $ k_{i}\in R $. Then the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation double-quantitative lower and upper approximations of the fuzzy set $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)&=&\overline{MR}_{(\alpha^{\ast},\beta^{\ast})}^{I}(X) \cap \overline{MR}_{K}^{I}(X);\\
\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)&=&\underline{MR}_{(\alpha^{\ast},\beta^{\ast})}^{I}(X)\cap \underline{MR}_{ K}^{I}(X).\end{aligned}$$
**Proof.** The proof is straightforward by Definition 7.1. $\Box$
(Continuation of Example 6.4) Taking $\alpha_{1}=\alpha_{2}=0.75,$ and $\beta_{1}=\beta_{2}=0.25,k_{1}=k_{2}=1$ we have $$\begin{aligned}
\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)&=&\overline{MR}_{(\alpha^{\ast},\beta^{\ast})}^{I}(X) \cap \overline{MR}_{K}^{I}(X)\\
&=&\{x_{3},x_{6}\}\cap \{x_{2},x_{3},x_{6},x_{8}\}\\
&=&\{x_{3},x_{6}\};\\
\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)&=&\underline{MR}_{(\alpha^{\ast},\beta^{\ast})}^{I}(X)\cap \underline{MR}_{ K}^{I}(X)\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\cap \{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We show the basic properties of the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation double-quantitative lower and upper approximation operators as follows.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, and $X,Y\in\mathscr{F}(U)$. Then\
$(1)\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(U)=U;\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)\subseteq \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(Y);\\
(3) X\subseteq Y\Rightarrow\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge k}^{I}(X)\subseteq \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(Y);\\
(4) \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)\cup \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(Y)\subseteq \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X\cup Y);\\
(5) \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)\cup \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(Y)\subseteq \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X\cup Y);\\
(6) \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X\cap Y)\subseteq \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)\cap \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(Y);\\
(7) \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X\cap Y)\subseteq \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(X)\cap \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\wedge K}^{I}(Y);\\
(8) \alpha^{\ast}_{1} \leq \alpha^{\ast}_{2}, \beta^{\ast}_{1} \leq \beta^{\ast}_{2}, K_{1} \leq K_{2}\Rightarrow\underline{MR}_{(\alpha^{\ast}_{1},\beta^{\ast}_{1})\wedge K_{1}}^{I}(X)\subseteq \underline{MR}_{(\alpha^{\ast}_{2},\beta^{\ast}_{2})\wedge k_{2}}^{I}(X);\\
(9) \alpha^{\ast}_{1} \leq \alpha^{\ast}_{2}, \beta^{\ast}_{1} \leq \beta^{\ast}_{2}, K_{1} \leq K_{2} \Rightarrow\overline{MR}_{(\alpha^{\ast}_{1},\beta^{\ast}_{1})\wedge K_{1}}^{I}(X)\subseteq \overline{MR}_{(\alpha^{\ast}_{2},\beta^{\ast}_{2})\wedge K_{2}}^{I}(X).$
**Proof.** By Theorem 4.14, the proof is straightforward.$\Box$
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, $0\leq \beta_{i} \leq \alpha_{i} \leq 1,$ and $ k_{i}\in R $. Then the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation double-quantitative lower and upper approximations of the fuzzy set $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)&=&\{x\in U\mid \bigvee_{1\leq i\leq m}[P(X|\widetilde{N}_{x}^{\gamma_{i}})\geq\beta^{\ast}_{i} \vee \Sigma_{y\in U}(X\cap \widetilde{N}_{x}^{\gamma_{i}})(y)>k_{i}]\};\\
\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)&=&\{x\in U\mid \bigvee_{1\leq i\leq m}[P(X|\widetilde{N}_{x}^{\gamma_{i}})\geq\alpha^{\ast}_{i} \vee \Sigma_{y\in U}(X^{c}\cap \widetilde{N}_{x}^{\gamma_{i}})(y)\leq k_{i}]\}.\end{aligned}$$
The fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation double-quantitative lower and upper approximation operators $\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)$ and $\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)$ of the fuzzy set $X\in \mathscr{F}(U)$ consider the absolute and relative quantitative information, which are generalizations of conjunctive double quantitative rough set model proposed by Xu[@Xu1].
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$coverings information system, $0\leq \beta_{i} \leq \alpha_{i} \leq 1,$ and $ k_{i}\in R $. Then the fuzzy $\gamma^{\ast}-$coverings based disjunctive multi-granulation double-quantitative lower and upper approximations of the fuzzy set $X\in \mathscr{F}(U)$ are defined as follows: $$\begin{aligned}
\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)&=&\overline{MR}_{(\alpha^{\ast},\beta^{\ast})}^{II}(X) \cup \overline{MR}_{K}^{II}(X);\\
\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)&=&\underline{MR}_{(\alpha^{\ast},\beta^{\ast})}^{II}(X)\cup \underline{MR}_{ K}^{II}(X).\end{aligned}$$
**Proof.** The proof is straightforward by Definition 7.5. $\Box$
(Continuation of Example 6.4) Taking $\alpha_{1}=\alpha_{2}=0.75,$ and $\beta_{1}=\beta_{2}=0.25,k_{1}=k_{2}=1$ we have $$\begin{aligned}
\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)&=&\overline{MR}_{(\alpha^{\ast},\beta^{\ast})}^{II}(X) \cup \overline{MR}_{K}^{II}(X)\\
&=&\{x_{3},x_{6}\}\cup \{x_{2},x_{3},x_{6},x_{8}\}\\
&=&\{x_{2},x_{3},x_{6},x_{8}\};\\
\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)&=&\underline{MR}_{(\alpha^{\ast},\beta^{\ast})}^{II}(X)\cup \underline{MR}_{ K}^{II}(X)\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\cup \{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}\\
&=&\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\}.\end{aligned}$$
We present the basic properties of the fuzzy $\gamma^{\ast}-$coverings based conjunctive multi-granulation double-quantitative lower and upper approximation operators as follows.
Let $(U,\Delta)$ be a fuzzy $\gamma^{\ast}-$covering information system, and $X,Y\in\mathscr{F}(U)$. Then\
$(1)\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(U)=U;\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(\emptyset)= \emptyset;\\
(2) X\subseteq Y\Rightarrow\overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)\subseteq \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{I}(Y);\\
(3) X\subseteq Y\Rightarrow\underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee k}^{II}(X)\subseteq \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(Y);\\
(4) \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)\cup \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(Y)\subseteq \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X\cup Y);\\
(5) \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)\cup \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(Y)\subseteq \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X\cup Y);\\
(6) \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X\cap Y)\subseteq \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)\cap \overline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(Y);\\
(7) \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X\cap Y)\subseteq \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(X)\cap \underline{MR}_{(\alpha^{\ast},\beta^{\ast})\vee K}^{II}(Y);\\
(8) \alpha^{\ast}_{1} \leq \alpha^{\ast}_{2}, \beta^{\ast}_{1} \leq \beta^{\ast}_{2}, K_{1} \leq K_{2}\Rightarrow\underline{MR}_{(\alpha^{\ast}_{1},\beta^{\ast}_{1})\vee K_{1}}^{II}(X)\subseteq \underline{MR}_{(\alpha^{\ast}_{2},\beta^{\ast}_{2})\vee k_{2}}^{II}(X);\\
(9) \alpha^{\ast}_{1} \leq \alpha^{\ast}_{2}, \beta^{\ast}_{1} \leq \beta^{\ast}_{2}, K_{1} \leq K_{2} \Rightarrow\overline{MR}_{(\alpha^{\ast}_{1},\beta^{\ast}_{1})\vee K_{1}}^{II}(X)\subseteq \overline{MR}_{(\alpha^{\ast}_{2},\beta^{\ast}_{2})\vee K_{2}}^{I}(X).$
**Proof.** By Theorem 4.17, the proof is straightforward. $\Box$
Conclusions
===========
In this paper, we have presented the fuzzy $\gamma-$covering based probabilistic and grade lower and upper approximation operators. Second, we have provided the fuzzy $\gamma-$covering based double-quantitative lower and upper approximation operators. Third, we have proposed the fuzzy $\gamma^{\ast}-$coverings based multi-granulation probabilistic and grade lower and upper approximation operators. Fourth, we have presented the fuzzy $\gamma^{\ast}-$coverings based multi-granulation double-quantitative lower and upper approximation operators. Finally, we have employed several examples to illustrate how to construct the lower and upper approximations of fuzzy sets with the relative and absolute information.
There are a lot of fuzzy covering information systems in practical situations, we should further study the fuzzy covering based lower and upper approximation operators and knowledge discovery of fuzzy covering information systems, so as to build the bridge between the fuzzy covering rough set theory and other rough set models in the future.
Acknowledgments {#acknowledgments .unnumbered}
================
We would like to thank the anonymous reviewers very much for their professional comments and valuable suggestions. This work is supported by the National Natural Science Foundation of China (NO. 61673301,61603063,11526039), Doctoral Fund of Ministry of Education of China(No.201300721004), China Postdoctoral Science Foundation(NO.2013M542558,2015M580353), the Scientific Research Fund of Hunan Provincial Education Department(No.15B004).
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[^1]: Corresponding author.Tel./fax: +86 021 69585800, E-mail address: langguangming1984@tongji.edu.cn(G.M.Lang).
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---
abstract: 'Magnetic fluctuations and electrons couple in intriguing ways in the vicinity of zero temperature phase transitions – quantum critical points – in conducting materials. Quantum criticality is implicated in non-Fermi liquid behavior of diverse materials, and in the formation of unconventional superconductors. Here we uncover an entirely new type of quantum critical point describing the onset of antiferromagnetism in a nodal semimetal engendered by the combination of strong spin-orbit coupling and electron correlations, and which is predicted to occur in the iridium oxide pyrochlores. We formulate and solve a field theory for this quantum critical point by renormalization group techniques, show that electrons and antiferromagnetic fluctuations are strongly coupled, and that both these excitations are modified in an essential way. This quantum critical point has many novel features, including strong emergent spatial anisotropy, a vital role for Coulomb interactions, and highly unconventional critical exponents. Our theory motivates and informs experiments on pyrochlore iridates, and constitutes a singular realistic example of a non-trivial quantum critical point with gapless fermions in three dimensions.'
author:
- Lucile Savary
- 'Eun-Gook Moon'
- Leon Balents
bibliography:
- 'arxiv.bib'
date: 'February 4, 2014'
title: A New Type of Quantum Criticality in the Pyrochlore Iridates
---
Antiferromagnetic quantum critical points (QCPs) are controlled by the interactions between electrons and magnetic fluctuations [@sachdev2011; @lohneysen2007fermi]. In three dimensional metals with a Fermi surface, it is believed to be sufficient to consider Landau damping of the magnetic order parameter in a purely order parameter theory, which leads, following Hertz [@hertz1976; @millis1993], to mean field behavior. In two dimensions, the electronic Fermi surface and order parameter are strongly coupled, a fact which may be related to high-temperature superconductivity and associated phenomena. This problem is highly non-trivial and still an active research topic [@lee2009; @metlitski2010quantum; @mross2010controlled; @efetov2013pseudogap].
In this paper, we uncover a new antiferromagnetic QCP which is strongly coupled in [*three*]{} dimensions, engendered by spin-orbit coupled electronic structure. We consider a quadratic band-touching at the Fermi energy, as in the inverted band gap material HgTe, but having in mind the strongly correlated family of iridium oxide pyrochlores [@wan2011; @moon2012; @chen2013; @kondo2013]. The latter have chemical formula A$_2$Ir$_2$O$_7$, and an antiferromagnetic phase transition indeed occurs both as a function of temperature and at zero temperature with varying chemical pressure (ionic radius of A) [@matsuhira2011]. We show that the replacement of the Fermi surface by a point Fermi node alters the physics in an essential way, suppressing screening of the Coulomb interaction and allowing the order-parameter fluctuations to affect [*all*]{} the low-energy electrons. These two facts lead to a strongly-coupled quantum critical point.
The nodal nature of the Fermi point, happily, also enables a rather complete analysis of the problem, which we present here, using the powerful renormalization group (RG) technique. The complete theory we present is in sharp contrast to the strongly coupled Fermi surface problem in two dimensions, which remains only partially understood and controversial. Finally, the pyrochlore quantum critical point has a remarkable symmetry structure. We find that, unlike at most classical and quantum phase transitions, rotational invariance is strongly broken in the critical theory: the fixed point “remembers” the cubic anisotropy of space (and indeed takes it to an extreme limit, as explained further below). Compensating for the absence of spatial rotational invariance is, however, an [*emergent*]{} $SO(3)$ invariance of the critical field theory, which is a purely internal symmetry and unrelated to spatial rotations. The [*an*]{}isotropy in real space manifests for example in the formation of “spiky” Fermi surfaces when the system close to the QCP is doped with charge carriers, as seen in Fig. \[fig:QCP\].
To proceed with the analysis, we couple the electrons to an Ising magnetic order parameter $\phi$. This corresponds for the pyrochlore iridates to the translationally-invariant “all-in-all-out” (AIAO) antiferromagnetic state (see “inset” in Fig. \[fig:QCP\]), for which there is considerable evidence [@sagayama2013determination; @tomiyasu2012emergence; @disseler2012magnetic]. Due to the time-reversal and inversion symmetries of the paramagnetic state, electron bands are two-fold degenerate, so that band touching necessitates a minimal four-band model. Therefore the Hamiltonian is expressed in terms of four-component fermion operators $\psi$, $\psi^\dagger$, in addition to $\phi$ and the electrostatic field $\varphi$, which mediates the Coulomb interactions. The action is $$\begin{aligned}
\mathcal{S}&=&\int d^3 x d \tau \psi^{\dagger}(\alpha \partial_{\tau} + \mathcal{H}_0(-i \nabla) +i e \varphi + g M\phi)\psi \nonumber \\
&+& \int d^3 x d \tau \frac{1}{2}\left[{(\nabla\varphi)^2} + {(\nabla \phi)^2} +{(\partial_{\tau} \phi)^2} + r \phi^2 \right],
\label{eq:action}\end{aligned}$$ where the momentum cutoff ($\Lambda$) is assumed, and where the Hamiltonian density is $\mathcal{H}_0( \mathbf{k})=c_0 \mathbf{k}^2+ \sum_{a=1}^5 \hat{c}_a d_a( \mathbf{k})\Gamma_a$. Higher-order terms omitted in Eq. (\[eq:action\]) prove irrelevant at the QCP. The $d_a$’s (given in the Supp. Mat. [@suppmat]) make a complete basis of the allowed terms quadratic in $k_j$, chosen such that $d_{1,2,3}$ belong to a three-dimensional representation (often called $T_{2g}$) and $d_{4,5}$ make a two-dimensional one (commonly referred to as $E_g$), the $\Gamma_a$’s are anticommuting unit matrices, $\{\Gamma_a,\Gamma_b\}=2\delta_{ab}$, $\Gamma_{ab}=\frac{-i}{2}[\Gamma_a,\Gamma_b]$, $\hat{c}_1=\hat{c}_2=\hat{c}_3=c_1$ and $\hat{c}_4=\hat{c}_5=c_2$ (as they should since they belong to the same representation), and symmetry dictates [@suppmat] the order parameter couples via the matrix $M=\Gamma_{45}$ ($\in A_{2g}$). $e$ is the magnitude of the electron charge, and $g\in\mathbf{R}$ parametrizes the coupling strength of the fermions to the order parameter. As discussed in the Supplementary Material [@suppmat], $c_{0,1,2}$ may always be chosen positive, without loss of generality. $c_0$ parametrizes “particle-hole asymmetry”, with $c_0=0$ denoting a symmetric band structure. Also, when $c_0\leq c_1/\sqrt{6}$, in the vicinity of the Gamma point, the bands touch at and only at the Gamma point. We assume that the system parameters fall within this range, and find that this is consistent.
The model in Eq. (\[eq:action\]) has two phases. For $r>r_c \sim g^2$ (where $r_c$ is thereby defined), $\phi$ fluctuates around zero, and can be integrated out. This is a magnetically disordered state. The resulting model with Coulomb interactions alone describes a non-Fermi liquid [*phase*]{}, as first discussed by Abrikosov and Beneslavskii [@abrikosov1971; @abrikosov1974] and thoroughly revisited recently [@moon2012]. Notably, in this regime, non-trivial scaling exponents arise and the low-energy electronic dispersion renormalizes to become [*isotropic*]{}, i.e. effectively $c_1 \rightarrow c_2$ and $c_0 \ll c_1$. For $r<r_c$, the expectation value $\langle\phi\rangle \neq 0$, and replacing $\phi \rightarrow \langle \phi\rangle$ causes the two-fold degenerate bands to split, removing the quadratic touching at $\mathbf{k}=\mathbf{0}$ in favor of eight linearly-dispersing “Weyl points” along the $\langle 111\rangle$ directions: a [*Weyl semimetal*]{}.
![Quantum critical point (QCP) and quantum criticality driven by the onset of “all-in-all-out” magnetism. For $r\geq r_c$ (in this figure the star indicates $r_c$), the “Luttinger-Abrikosov-Beneslavskii” (LAB) phase occurs at $T=0$, with a quadratic Fermi node, while antiferromagnetic (AFM) “all-in-all-out” ordering occurs for $r<r_c$, with the quadratic node replaced by linear Weyl points – a Weyl semimetal. The quantum critical regime occurs at $T>0$ around $r=r_c$. Note that the quantum critical-AFM boundary (thick white line) is a true (continuous Ising) phase transition. The $E_F$ axis represents the Fermi energy and parametrizes electron or hole doping. The three-dimensional (orange) surfaces represent the shapes of the corresponding Fermi surfaces at small doping – the increased anisotropy is apparent as one moves towards the QCP. The phase transition denoted by the thick gray line is expected to exhibit critical properties appropriate to a $\mathbf{q}=\mathbf{0}$ order parameter coupled to a Fermi surface, as in the Hertz formulation [@hertz1976], though subject to the usual uncertainties regarding the theory of that problem [@lee2009; @metlitski2010quantum; @mross2010controlled]. []{data-label="fig:QCP"}](fig1){width="3.3in"}
We now turn to the critical regime. To proceed, we introduce as a formal device $N$ copies of the four fermion fields, replacing $g\rightarrow g/\sqrt{N}$ (resp. $ie\rightarrow ie/\sqrt{N}$) and $\Gamma_a \rightarrow \Gamma_a \otimes {\sf 1}_N$ (${\sf 1}_N$ is the $N\times N$ identity matrix). We organize perturbation theory in powers of $1/N$, but in the end argue that the results are asymptotically exact for the physical case $N=1$. To leading order in $1/N$, we require the two boson self-energies in Fig. \[fig:bubble\], and, using the dressed boson propagators including this correction, the fermion self-energy and vertex functions in Fig. \[fig:1oNdiags\]. These diagrams allow a full calculation of the $O(1/N)$ terms of all critical exponents. The evaluation of the diagrams is complicated by the three mass parameters of the free fermion propagators. Fortunately, a simplification is possible due to the structure of the RG. While the (inverse) mass terms $c_0$, $c_1$, and $c_2$ all have identical engineering dimensions, they, in general renormalize differently from loop corrections, and thus their ratios [*flow*]{} in the full RG treatment. We find below that, in the critical regime, $c_0/c_2, c_1/c_2\rightarrow 0$ under renormalization (arguments why this is the only reasonable choice are given in the Supplementary Material [@suppmat]). This allows technical simplifications in the loop integrals, and also has physical consequences we explore later.
In particular, in the limit $c_1/c_2 \rightarrow 0$, the interband splitting vanishes along the $\langle 111\rangle$ directions, leading to an extended singularity of the electron Green’s function. In the loop integrals determining the bosonic self-energies, this produces a divergent contribution at non-zero ${\bf k}$. Technically, with the assumptions $c_0/c_2, c_1/c_2 \ll 1$ and $c_0/c_1 < 1/\sqrt{6}$ (shown self-consistent below), the low-energy behavior (small $\omega_n,\mathbf{k}$) may be extracted as (see Supplementary Material [@suppmat]) $$\Sigma_b(\omega_n,\mathbf{k})= - r_{b}^{c}+\frac{g_b^2}{\alpha}\left(|\ln\,c_1/c_2\,||\mathbf{k}|f_b(\mathbf{\hat{k}})+\sqrt{|\omega_n|}C_b\right),
\label{eq:sigmab}$$ where $r_\phi^c = r_c \sim g^2\Lambda$, where $\Lambda$ is an upper momentum cutoff, $g_\phi=g$, $g_\varphi=ie$, and $r_\varphi^c=C_\varphi=0$ follows from charge conservation. $C_\phi\approx 1.33$ and the functions $f_b(\mathbf{\hat{k}})$ are given as integrals in the Supplementary Material [@suppmat].
Note that, at low energy, the dispersive terms in Eq. (\[eq:sigmab\]) are much larger than the bare $\mathbf{k}^2,\omega_n^2$ terms they correct, and hence dominate the renormalized Green’s functions. Thus, in the fermion self-energy and vertex correction, the renormalized boson propagator, $\mathcal{G}_b^{-1}=\mathcal{G}_{b;0}^{-1}+\Sigma_b\approx\Sigma_b+r_b$ (note $r_\varphi=0$), must be used.
![Boson self energies for the order parameter ($\Sigma_\phi$) and electrostatic ($\Sigma_\varphi$) fields.[]{data-label="fig:bubble"}](fig2){width="1.8in"}
This renormalized boson propagator corresponds to the $N=\infty$ result, and already reveals some dramatic features. First, the bosons immediately receive a large anomalous scaling dimension, equal to $1$, and their dynamics becomes damping-dominated, with dynamical critical exponents close to $2$. Second, since the damping terms which dominate $\mathcal{G}^{-1}_b$ are proportional to $g_b^2$, it implies that the fermion self-energies, which involve two interaction vertices (see Fig. \[fig:1oNdiags\]), become $g_b$ independent: this is a sign of universality at the QCP.
![$1/N$ diagrams for the fermion self-energy $\Sigma_f$ and vertex corrections $\Xi_\phi$ and $\Xi_\varphi$ (only two-loop diagrams that need be calculated, i.e. that do not vanish or cancel one another, are shown). Double lines indicate the renormalized boson propagators including the self-energies from Fig. \[fig:bubble\]. Expressions for these diagrams are given in the Supplementary Material [@suppmat].[]{data-label="fig:1oNdiags"}](fig3){width="3.3in"}
To confirm the assumed scaling of $c_1/c_2$, $c_0/c_2$, and fully determine the critical behavior, we turn to the renormalization group approach. There, as usual, we apply the following rescaling (applicable in real space) $$\begin{aligned}
&&x\rightarrow e^\ell x,\quad\tau\rightarrow e^{\int_0^\ell d\ell' z(\ell')}\tau,\quad\psi\rightarrow e^{-\int_0^\ell d\ell'\Delta_\psi(\ell')}\psi,\nonumber\\
&&\phi\rightarrow e^{-\int_0^\ell d\ell'\Delta_\phi(\ell')}\phi,\quad\varphi\rightarrow e^{-\int_0^\ell d\ell'\Delta_\varphi(\ell')}\varphi,\end{aligned}$$ where $\ell\geq0$ parametrizes the RG flow. The exponents are left allowed to be scale dependent, as is necessary [@huh2008], as we shall see below.
We evaluate the contributions to the fermion propagator and coupling constants due to a small change in the cutoff (which corresponds physically to integrating out modes to keep the rescaled cutoff unchanged). Hence, the RG flow equations are obtained by [*(i)*]{} logarithmically differentiating the fermion self energy and vertex functions with respect to the cutoff $\Lambda$ (made soft through a rapidly decaying function $|\mathbf{q}|/\Lambda\mapsto\mathcal{F}(|\mathbf{q}|/\Lambda)$) [@huh2008; @vojta2000a], and [*(ii)*]{} identifying the appropriate coefficients of the Taylor expansion (in $k_i$ and $\omega_n$) of the result [^1].
We leave most details to the Supplementary Material [@suppmat], and only give one example here. To extract the correction to the mass coefficient $c_1$, we first expand the fermion self-energy as $$\Sigma_{f}(\omega_n,\mathbf{k})=\Sigma_{f}^0 I+\sum_{a=1}^5\Sigma_{f}^a\Gamma_a,$$ and examine the $\Sigma_f^1$ component. The RG equation is then $$\frac{\partial_\ell c_1}{c_1}=z+1-2\Delta_\psi+\frac{\sqrt{2}}{c_1}\left.\left(\partial_{k_x,k_y}^2\left[\Lambda\frac{d}{d\Lambda}\Sigma_f^1\right]\right)\right|_{\omega_n=0,\mathbf{k}=\mathbf{0}}
\label{eq:floweg}$$ (we define $d_1(\mathbf{k})=k_xk_y/\sqrt{2}$ [@suppmat]). Similar expressions are obtained for the other parameters of the theory, $c_0$, $c_2$, $\alpha$, $g$ and $ie$. The latter all depend on $c_1$ and $c_2$ through $1/(N|\ln c_1/c_2\,|)$ or $1/(N|\ln c_1/c_2\,|^2)$ (expressions are expanded in small $1/|\ln c_1/c_2\,|$, see Supp. Mat. [@suppmat]). Therefore, for the six equations thereby obtained, there are four unknowns ($z$, $\Delta_\psi$, $\Delta_\phi$ and $\Delta_\varphi$) which can be chosen to keep four parameter fixed, leaving two left to flow. Here we find it is possible to keep $\alpha$, $g$, $(ie)$ and $c_2$ fixed, and thus $c_1/c_2$ and $c_0/c_2$ will flow. Note that, in doing so, we obtain a critical theory with non-zero coupling of fermions both to order-parameter and Coulomb-potential fluctuations: both effects are crucial and important in stabilizing the QCP. Finally, we obtain $$z(\ell)=2-\delta z(\ell),\; \Delta_\psi(\ell)=\frac{3+\eta_\psi(\ell)}{2},\; \Delta_b(\ell)=\frac{3+\eta_b(\ell)}{2},\label{eq:1}$$ where $\delta z=\frac{0.0634}{|\ln c_1/c_2\,|N}$, $\eta_\psi=\frac{0.287}{|\ln c_1/c_2\,|^2N}$, $\eta_\phi=1+\frac{0.510}{|\ln c_1/c_2\,|N}$ and $\eta_\varphi=1-\frac{0.127}{|\ln c_1/c_2\,|N}$.
The flow equations may be solved thanks to that of $c_1/c_2$, which is an analytically-soluble differential equation involving only $c_1/c_2$ [@suppmat]. Ultimately, we find $$(c_1/c_2)(\ell)=e^{-\frac{\upsilon_0}{\sqrt{N}}\sqrt{\ell+\ell_0}},\; \mbox{and}\; (c_0/c_1)(\ell)=\Upsilon_0 e^{-\frac{\upsilon_0'}{\sqrt{N}}\sqrt{\ell+\ell_0}},\label{eq:2}$$ with $\upsilon_0=0.202$, $\upsilon_0'=0.424$ and where $\ell_0$ and $\Upsilon_0$ are constants which depend on the system’s parameters, namely on $c_{0,1,2}(\ell=0)$. Formally, therefore both the $c_0$ and $c_1$ mass terms are irrelevant in the RG sense, but they can be “dangerously irrelevant” insofar as they control certain physical properties (see below). Note also that not only is $c_0$ irrelevant, but it also flows to zero [*faster*]{} than $c_1$, so that $c_0/c_1$ becomes small at the QCP.
Intuition for the irrelevance of $c_1$ comes from considering the fermion self-energy $\Sigma_f$, which yields the corrections to $c_{0,1,2}$ and to $\alpha$, and is given [*schematically*]{} by $\Sigma_f=\mathcal{G}_\phi MG_0M+\mathcal{G}_\varphi G_0$ (the contributions from each boson field just add up). In the first term, which represents dressing of electrons by magnetic fluctuations, the appearance of $M$, which commutes with $\Gamma_{1,2,3}$ but [*anti-*]{}commutes with $\Gamma_{4,5}$, portends “opposite” consequences for $c_1$ and $c_2$. The second term, due to Coulomb effects, tends instead to affect $c_1$ and $c_2$ identically. Our calculation shows that the former tendency prevails, and $c_1/c_2 \rightarrow 0$ under RG, as claimed above. Conversely, the fact that $c_0/c_1 \rightarrow_{\ell\rightarrow \infty} 0$ should be attributed to the effect of Coulomb forces, which suppress particle-hole asymmetry. Indeed, we have checked that if in the calculations we artificially turn off the long-range Coulomb potential, i.e. take $e=0$, the QCP is unstable and there is no direct, continuous quantum phase transition from the LAB state to the AIAO one[@suppmat].
Eqs. (\[eq:1\],\[eq:2\]) determine the properties at the QCP. We now turn to a discussion of the physical consequences. First we consider some scaling properties. For the correlation length, we need the flow equation for $\delta r=r-r_c$, the deviation from the critical point: $\partial_\ell (\delta r)=\nu^{-1} (\delta r)$, with $\nu = 1/[2-\eta_\phi(\ell)-\delta z(\ell)]$. This implies, in the usual way, that the correlation length behaves as $\xi \sim (\delta r)^{-\nu}$, up to logarithmic corrections. Also interesting is the order parameter growth in the AIAO phase. By scaling, $\langle \phi\rangle \sim \xi^{-\Delta_\phi} \sim |\delta r|^\beta$, with $\beta = \Delta_\phi \nu$. We also expect the critical temperature of the magnetic state to obey $T_{c} \sim \xi^{-z} \sim |\delta r|^{z\nu}$. In asymptopia, i.e. $\ell \rightarrow \infty$, all the $N$-dependent corrections vanish, and the exponents correspond to those of a saddle-point treatment of $\varphi,\phi$. These are still distinct from the usual order parameter mean field theory, as witnessed by the large ($\eta_\phi^\infty=\eta_\varphi^\infty=1$) anomalous dimensions in this limit, and the unconventional values $\nu^\infty=1$, $\beta^\infty= (z\nu)^\infty = 2$. The latter is noteworthy insofar as it implies an unusually wide critical fan at $T>0$ which is controlled by the QCP (see Fig. \[fig:QCP\]). The RG treatment goes beyond the saddle point in giving the corrections due to finite $c_1/c_2$, which are small only logarithmically, and thus may be significant for physically-realistic situations. For example we find $\langle \phi\rangle\sim (\delta r)^2\exp\left[\frac{13.9}{\sqrt{N}}\sqrt{\ln\frac{\delta r}{r_0}}\right]$ [@suppmat], where $r_0$ is a constant. The irrelevance of $c_0$ and $c_1$ has other, more direct, physical consequences. Because of the former, the low-energy electronic spectrum becomes approximately particle-hole symmetric. The latter has more implications. Obviously, the electronic spectrum develops pronounced cubic anisotropy, with anomalously low energy excitations along the cubic $\langle 111\rangle$ directions in momentum space. This is in stark contrast to most critical points (for example of Ginzburg-Landau type, or involving Dirac fermions), which typically have emergent spatial isotropy and even conformal symmetry and Lorentz invariance at the fixed point. These low-energy excitations manifest, for example, in the specific heat $c_v$. Since at the Gaussian level the coefficient of $T^{3/2}$ [*diverges*]{} as $c_1^{-3/2}$, we estimate, by using $\ell \sim \frac{1}{z}\ln T_0/T$ as a cut-off ($T_0$ is a microscopic energy scale), $c_v\sim \exp\left[\frac{3\upsilon_0}{2\sqrt{N}\sqrt{z}}\sqrt{\ln \frac{T_0}{T}}\right]T^{3/2}$, with $z\approx2$ [@suppmat]. The emergent anisotropy may also manifest in increasingly-“spiky” Fermi surfaces in lightly doped samples near the QCP, see Fig. \[fig:QCP\].
Although rotational symmetry is strongly broken, the vanishing of $c_1$ leads to an [*emergent internal*]{} $SO(3)$ symmetry, corresponding to rotating the $\Gamma_a$ matrices with $a=1,2,3$ amongst themselves like a vector. The generator of this symmetry is the $SU(2)$ pseudo-spin ${\bf I}$, with $$\label{eq:3}
I_a = -\frac{1}{4}\epsilon_{abc} \, \psi^{\dagger} \Gamma_{bc}\psi = \psi^{\dagger} \left(-\frac{7}{6} J_a + \frac{2}{3} J_a^3\right)\psi,$$ where $a=x,y,z=1,2,3$. Its integral has $SU(2)$ commutation relations and commutes with the fixed-point Hamiltonian.
[*Discussion.—*]{} In standard Hertz-Millis theory [@hertz1976; @millis1993], the inequality $d+z>4$ implies that the theory is above its critical dimension, and thus has mean field behavior. Although this inequality holds here, taking $z=2$, the conclusion is false. The Hertz-Millis approach assumes the fermions may be innocuously integrated out, and obtains this inequality by power-counting the $\phi^4$ term in the Landau action, which is irrelevant. Instead, here we have strong coupling of fermions with the order parameter, and the coupling term $\sim \phi \psi^\dagger \psi$ is [*marginal*]{} using $z=2$, $\Delta_\phi = 2$, $\Delta_\psi=3/2$. If one [*does*]{} integrate out the fermions, one obtains a nonanalytic $|\phi|^{5/2}$ term [@suppmat], which overwhelms the naïve $\phi^4$ one, and is again marginal by power counting. This $|\phi|^{5/2}$ dependence was obtained previously in Ref. [@kurita2013], in the context of a mean-field treatment of related transitions. Note, however, that such a mean-field analysis integrating out fermions is not justified and misses important physics.
Our critical theory has some formal similarity to the theory of a two-dimensional nodal nematic QCP in a $d$-wave superconductor [@huh2008], insofar as both theories display “infinite anisotropy”: in our case due to $c_1/c_2 \rightarrow 0$ under RG. This suggests that, as in Ref. [@huh2008], at low energy [*the perturbative expansion parameter is small for all $N$*]{}, and that therefore our results apply directly at low energy to the physical case $N=1$. This conclusion is appealing, though we have not shown it rigorously.
With the above results in hand, we comment on the connection to experiments. In the pyrochlore iridates, the QCP might be tuned by alloying the A-site atoms, e.g. Pr$_{2-2x}$Y$_{2x}$Ir$_{2}$O$_7$, or by pressurizing stoichiometric compounds nearby. The theory developed here, which relies only on cubic symmetry and strong SOC, may apply to other materials if the bands at the Fermi energy belong to the appropriate irreducible representation, and it would be interesting to search for other examples. Experimentally, the heavily-damped paramagnon could be observed in inelastic neutron or x-ray scattering. An explicit calculation of the fermion spectral function measured in angle resolved photoemission has been made neither here nor for the non-Fermi liquid paramagnetic state [@moon2012], and is an important problem for future theory. However, in general, the weak logarithmic flow of the Hamiltonian parameters signifies large self-energy corrections, and behavior somewhat similar to marginal Fermi-liquid theory may be expected.
We also mention some possible complications in the iridates. Impurity scattering is a relevant perturbation and hence important at low energy close to the band touching. Therefore, our results will apply best in the cleanest samples. Also, an accidental band crossing may occur away from the zone center, thereby shifting the Fermi level a few meV away from the nodal point. This should be addressed by [*ab initio*]{} calculations and experiments. In such a case, our results still hold for energies and/or temperatures above this shift energy. Finally, in many of the pyrochlore iridates, the A-site ion hosts rare-earth moments, which were not included here. They only weakly couple to the Ir electrons and to themselves, so are only important at low energy. On the antiferromagnetic side of the QCP, the Ir spins act as strong local effective magnetic fields, locking the A-site spins. However, when the Ir sites are not ordered, as in Pr$_2$Ir$_2$O$_7$, A-site ions will have an effect below a few Kelvins. Several authors have proposed scenarios based on RKKY interactions [@chen2012; @flint2013; @lee2013], but the quantum critical theory expounded here should be an apt starting point for a systematic analysis.
[*Acknowledgements.—*]{} We thank Cenke Xu and Yong-Baek Kim for discussions on prior work, Max Metlitski for pointing Ref. [@huh2008] to us, and acknowledge Ru Chen, Satoru Nakatsuji and Takeshi Kondo for sharing unpublished data. The integrals were performed using the Cuba library [@hahn2005], and the Feynman diagrams in Figs. \[fig:bubble\] and \[fig:1oNdiags\] drawn with JaxoDraw [@binosi2004]. L.S. and L.B. were supported by the DOE through grant DE-FG02-08ER46524, and E.-G. M. was supported by the MRSEC Program of the National Science Foundation under Award No. DMR 1121053.
SUPPLEMENTARY MATERIAL {#supplementary-material .unnumbered}
======================
In reciprocal space, the action, Eq. (1) in the main text, is $$\begin{aligned}
\mathcal{S}&=&\int_{-\infty}^{\infty}\frac{d\omega_n}{2\pi}\int_\Lambda\frac{d^3k}{(2\pi)^3}\left[\phi_{-\omega_n,-\mathbf{k}}\left(\frac{\omega_n^2}{2}+\frac{\mathbf{k}^2}{2}+\frac{r}{2}\right)\phi_{\omega_n,\mathbf{k}}+\varphi_{-\omega_n,-\mathbf{k}}\left(\frac{\mathbf{k}^2}{2}\right)\varphi_{\omega_n,\mathbf{k}}\right.\nonumber\\
&&\qquad+\; \psi^{\dagger}_{\omega_n,\mathbf{k}}\left(-\alpha\, i\omega_n+c_0\mathbf{k}^2+\sum_{a=1}^5\hat{c}_ad_a(\mathbf{k})\Gamma_a\right)\psi_{\omega_n,\mathbf{k}}\\
&&\qquad\left.+\;g\int_{-\infty}^{\infty}\frac{d\omega_n'}{2\pi}\int_\Lambda\frac{d^3k'}{(2\pi)^3}\phi_{\omega_n'-\omega_n,\mathbf{k}'-\mathbf{k}} \psi^{\dagger}_{\omega_n',\mathbf{k}'}M\psi_{\omega_n,\mathbf{k}}+ie\int_{-\infty}^{\infty}\frac{d\omega_n'}{2\pi}\int_\Lambda\frac{d^3k'}{(2\pi)^3}\varphi_{\omega_n'-\omega_n,\mathbf{k}'-\mathbf{k}} \psi^{\dagger}_{\omega_n',\mathbf{k}'}\psi_{\omega_n,\mathbf{k}}\right],\nonumber
\label{eq:actionrecipr}\end{aligned}$$
where all the notations are defined in Sec. \[sec:notations\] of the present Supplementary Material. Throughout the latter, for ease of presentation, we shift the QCP so that $r_c=0$.
Notations and symmetries {#sec:notations}
========================
In this section, we provide more information about the notations used in the main text and a more detailed discussion of the symmetries at play.
Fermion Hamiltonian {#sec:fermion}
-------------------
The fermionic Hamiltonian density in the disordered (quadratic band touching) phase reads $$\begin{aligned}
\mathcal{H}_0(\mathbf{k})&=&\alpha_1 \mathbf{k}^2 + \alpha_2 \left(\mathbf{k} \cdot \mathbf{J}\right)^2 + \alpha_3 \left(k_x^2 J_x^2 + k_y^2 J_y^2 +k_z^2 J_z^2\right) \nonumber \\
&=&c_0 \mathbf{k}^2+ \sum_{a=1}^5 \hat{c}_a d_a( \mathbf{k})\Gamma_a, \end{aligned}$$ where $\hat{c}_1=\hat{c}_2=\hat{c}_3=c_1$ and $\hat{c}_4=\hat{c}_5=c_2$. The first line uses the conventional Luttinger parameters ($\alpha_{1,2,3}$) in the $j=3/2$ matrix representation [@luttinger1956], and the second line is the form used in the main text. The Gamma matrices ($\Gamma_{a}$) form a Clifford algebra, $\{\Gamma_a,\Gamma_b\}=2\delta_{ab}$, and have been introduced as described in the literature [@murakami2004]. Note that $c_0$ quantifies the particle-hole asymmetry, while $\left|c_1-c_2\right|$ naturally characterizes the cubic anisotropy. The energy eigenvalues are $\mathsf{E}_{\pm}(\mathbf{k}) = c_0 \mathbf{k}^2 \pm
E(\mathbf{k})$, where $E(\mathbf{k})=\sqrt{\sum_{a=1}^5 \hat{c}_a^2 d_a
^2( \mathbf{k}) }$ and $$\begin{aligned}
&&d_1( \mathbf{k})=\frac{k_xk_y}{\sqrt{2}},\quad d_2( \mathbf{k})=\frac{k_xk_z}{\sqrt{2}},\quad d_3( \mathbf{k})=\frac{k_yk_z}{\sqrt{2}} \nonumber \\
&&d_4( \mathbf{k})=\frac{k_x^2-k_y^2}{2\sqrt{2}},\quad d_5( \mathbf{k})=\frac{2k_z^2 -k_x^2-k_y^2}{2\sqrt{6}}. \nonumber \end{aligned}$$ It is very important to note that, in the limit $c_{0,1} \rightarrow 0$, $E(\mathbf{k})$ and the energy spectrum $\mathsf{E}_\pm(\mathbf{k})$ become gapless along the $\langle111\rangle$ directions. When needed, a “regularization” is then possible, for example by introducing higher momentum dependence in $c_{1,2}$, e.g. $c_{1,2}\rightarrow c_{1,2}+\lambda \mathbf{k}^2$.
It is straightforward to relate the coefficients used in the main text to the Luttinger $\alpha_i$ parameters. This can be done by expressing the spin operators in terms of the Gamma matrices, using for example the equalities $$\begin{aligned}
&& J_x = \frac{\sqrt{3}}{2} \Gamma_{15} - \frac{1}{2} (\Gamma_{23} - \Gamma_{14}) \ ,\nonumber \\
&& J_y = -\frac{\sqrt{3}}{2} \Gamma_{25} + \frac{1}{2} (\Gamma_{13} + \Gamma_{24}) \ ,\nonumber \\
&& J_z = - \Gamma_{34} - \frac{1}{2} \Gamma_{12} \ ,\end{aligned}$$ where $\Gamma_{ab} = \frac{1}{2 i}
[\Gamma_a, \Gamma_b]$.
The fermion bare Green’s function is $$\begin{aligned}
G^0_{\omega_n,\mathbf{k}} = \frac{1}{-i\alpha\, \omega_n +\mathcal{H}_0(\mathbf{k})} = \frac{1 }{-i\alpha\, \omega_n + \mathsf{E}_{\epsilon}(\mathbf{k})} {\rm P}_{\epsilon}(\mathbf{k}),\nonumber \end{aligned}$$ where the sum over $\epsilon=\pm1$ is implicit and ${\rm P}_{\epsilon}(\mathbf{k}) = \frac{1}{2}\left(1+\epsilon
\frac{\mathcal{H}_0(\mathbf{k})-c_0\mathbf{k}^2}{E(\mathbf{k})}\right)$ is a projection operator, ${\rm P}_{\epsilon}^2(\mathbf{k})=1$.
Symmetries
----------
It is useful to recap the symmetries of the system in the absence of all-in-all-out order, and detail the remaining symmetries in its presence.
As defined above and in Refs. [@murakami2004] and [@moon2012], the $\Gamma_a$ matrices are even under time-reversal and inversion symmetry, while the $\Gamma_{ab}$ are even under inversion, but odd under time-reversal.
As is well-known for some semiconductors, like HgTe, the touching of four bands at the Gamma point is protected by cubic symmetries (the bands at the Gamma point belong to a four-dimensional representation of the cubic group $O_h$), and the absence of a linear term follows from time-reversal and cubic (inversion) symmetries. Moreover, thanks to inversion and time-reversal symmetries, all bands are doubly-degenerate away from the Gamma point.
The magnetic order parameter field $\phi$ transforms as follows under the symmetries of the “disordered” system. It is odd under time-reversal symmetry (since the spins $\vec{\mathsf{S}}\rightarrow-\vec{\mathsf{S}}$ under time-reversal), and so only the (time-reversal-odd) $\Gamma_{ab}$ can couple to it. It is even under inversion (since $\vec{\mathsf{S}}\rightarrow\vec{\mathsf{S}}$ under inversion), unchanged under three-fold rotations, and odd under the allowed reflections of the pyrochlore lattice. A single Gamma matrix, namely $\Gamma_{45}\propto J_xJ_yJ_z+J_zJ_yJ_x$ [@murakami2004; @moon2012] (see below), transforms identically.
The Hamiltonian at [*fixed*]{} $\mathbf{k}$, i.e.$\mathcal{H}_0(\mathbf{k})$, together with the coupling to the order parameter with $\phi\neq0$, which we call $\mathcal{H}_1(\mathbf{k})$, have the following transformation properties. For $\mathbf{k}\parallel\langle111\rangle$, $\mathcal{H}_1$ is invariant under three-fold rotations about $\mathbf{k}$, and reflections with respect to planes that contain $\mathbf{k}$. For $\mathbf{k}=\mathbf{0}$, there is additionally inversion symmetry. The symmetry group at $\mathbf{k}=\mathbf{0}$ then decomposes the four bands of interest into two two-dimensional representations. For $\mathbf{k}\neq\mathbf{0}$, symmetries do not impose bands to cross, hence making any crossings “accidental.” However, it is noteworthy that the purely quadratic Hamiltonian $\mathcal{H}_0$ we study, with $c_0\leq c_1/\sqrt{6}$ and $c_1\leq c_2/\sqrt{6}$, in the presence of the linear coupling to the order parameter $\phi\psi^\dagger\Gamma_{45}\psi$ leads inevitably to band crossings along the $\langle111\rangle$ directions.
Note that the system in the presence of an external applied magnetic field, discussed in Ref. [@moon2012], is less symmetric. The system’s Hamiltonian at fixed $\mathbf{k}$, which we call $\mathcal{H}_2(\mathbf{k})$, is only invariant under three-fold rotations about $\mathbf{k}$ if both the magnetic field and $\mathbf{k}$ point along the same $\langle111\rangle$ direction. For $\mathbf{k}=\mathbf{0}$ the system has additionally inversion symmetry, but all the representations of the symmetry group are one-dimensional anyway, and there is a priori no degeneracy at $\mathbf{k}=\mathbf{0}$. Away from $\mathbf{k}=\mathbf{0}$, any band crossing is, again, accidental.
It is important to note that, although no crossings are required by symmetry, once the crossings are found to happen, their properties are “stable” in the sense that [*(i)*]{} no symmetry-preserving perturbation will remove them, [*(ii)*]{} the dispersion along the crossings will remain linear, [*(iii)*]{} they will not move away from the $\langle111\rangle$ axes.
Couplings
---------
The long-range Coulomb interaction is described by introducing the Hubbard-Stratonovich field, $\varphi$, which couples to the density of fermions.
The all-in all-out operator is represented by the time-reversal symmetry breaking Ising field ($\phi$) corresponding to $J_x J_y J_z
+J_z J_y J_x$ in Luttinger’s notation [@luttinger1956]. In terms of the Gamma matrices, the order parameter is $\Gamma_{45} \sim J_x J_y J_z +J_z J_y J_x$. Thus, finally, the interaction part of the action is the “vertex term” given, in real space and imaginary time, by $$\begin{aligned}
\mathcal{S}_{vertex} = \int d^3 x\, d {\tau}\, {\psi^\dagger} \left[i e\, \varphi + g\, \phi\, \Gamma_{45} \right]\psi, \end{aligned}$$ where $\psi$ is the four-component spinor field. Upon extending the field space to $N$ flavors of fermions, this term becomes $$\begin{aligned}
\mathcal{S}_{vertex} \rightarrow \frac{1}{\sqrt{N}}\int d^3 x\, d {\tau}\, {\psi^\dagger} \left[i e\, \varphi + g\, \phi\, \Gamma_{45} \right]\psi.\end{aligned}$$
By appropriately transforming the Gamma matrices with transformations not belonging to the cubic group, one may show that the signs of $c_{0,1,2}$ may always be taken positive. Therefore, throughout the paper we assume $c_{0,1,2}\geq0$. We also assume $c_0\leq
c_1/\sqrt{6}$, i.e. we assume the two sets of bands have opposite curvatures in all directions at the Gamma point, or, in other words that the Fermi energy goes through the band touching point.
Green’s function and self-energy conventions
--------------------------------------------
We use the following conventions for the boson Green’s functions, $\mathcal{G}_{b;\omega_n,\mathbf{k}}$ with $b=\phi,\varphi$, fermion Green’s function, $G_{\omega_n,\mathbf{k}}$, boson self-energies, $\Sigma_b(\omega_n,\mathbf{k})$ and fermion self-energy, $\Sigma_f(\omega_n,\mathbf{k})$: $$\begin{aligned}
\mathcal{G}_{\varphi;\omega_n,\mathbf{k}}&=&\langle\varphi_{-\mathbf{k}}\varphi_{\mathbf{k}}\rangle=\frac{1}{\mathbf{k}^2+\Sigma_\varphi(\mathbf{k})},\nonumber\\
\mathcal{G}_{\phi;\omega_n,\mathbf{k}}&=&\langle\phi_{-\omega_n,-\mathbf{k}}\phi_{\omega_n,\mathbf{k}}\rangle=\frac{1}{\mathbf{k}^2+\omega_n^2+r+\Sigma_\phi(\omega_n,\mathbf{k})},\nonumber\\
G^{\mu\nu}_{\omega_n,\mathbf{k}}&=&\langle\psi_{\omega_n,\mathbf{k}}^{\mu}{\psi_{\omega_n,\mathbf{k}}^{\nu}}^\dagger\rangle\nonumber\\
&=&\left[-i\alpha\omega_n+\mathcal{H}_0(\mathbf{k})+\Sigma_f(\omega_n,\mathbf{k})\right]^{-1},\nonumber\end{aligned}$$ where $\mu,\nu=1,..,4$ (or $1,..,4N$) but are omitted throughout. The “bare propagators” are denoted with the subscript or superscript “$0$.”
Asymptotic limits of the bosonic self-energies {#sec:asymp}
==============================================
We first evaluate the boson self-energies in the large-$N$ limit. They are given by $$\begin{aligned}
&&\Sigma_b(\omega_n,\mathbf{k})=\\
&&\;\;\frac{g_b^2}{N}\int_\Lambda\frac{d^3q}{(2\pi)^3}\int_{-\infty}^{+\infty}\frac{d\Omega_n}{2\pi}\,{\rm Tr}\left[G^0_{\Omega_n,\mathbf{q}}M_bG^0_{\Omega_n+\omega_n,\mathbf{q}+\mathbf{k}}M_b\right],\nonumber\end{aligned}$$ where $g_\varphi=ie$, $g_\phi=g$, $M_\varphi=I$ and $M_\phi=\Gamma_{45}$ ($I$ is the identity matrix). Here the subscript $\Lambda$ in the $q$ integral indicates that an ultraviolet cutoff is required to keep $\Sigma_b(0,\mathbf{0})$ finite. This determines the (non-universal) location of the QCP. However, we seek the corrections to this term for non-zero frequency and momenta, which are cutoff independent, and will be therefore obtained below without further discussion of $\Lambda$. We will return later to the role of the cutoff when considering fermionic self-energy terms, and treat it in more detail. The explicit expression for $\Sigma_b(\omega_n,\mathbf{k})$ at $c_0\leq c_1/\sqrt{6}$ is
$$\Sigma_b(\omega_n,\mathbf{k}) \label{eq:sigmabsmallc0}=\frac{-g_b^2}{\alpha}\sum_{\epsilon=\pm}\int_\Lambda\frac{d^3q}{(2\pi)^3}\left[\frac{E_{\mathbf{q},\mathbf{k}}^++E_{\mathbf{q},\mathbf{k}}^-+\epsilon\,2c_0\mathbf{q}\cdot\mathbf{k}}{\alpha^2\omega_n^2+\left(E_{\mathbf{q},\mathbf{k}}^++E_{\mathbf{q},\mathbf{k}}^-+\epsilon\,2c_0\mathbf{q}\cdot\mathbf{k}\right)^2}\right]
\left(1-\frac{F_{b;\mathbf{q},\mathbf{k}}}{E_{\mathbf{q},\mathbf{k}}^+E_{\mathbf{q},\mathbf{k}}^-}\right),\nonumber$$
where $E_{\mathbf{q},\mathbf{k}}^\pm=E(\mathbf{q}\pm\mathbf{k}/2)$ and $F_{b;\mathbf{q},\mathbf{k}}=\sum_{a=1}^5(\varepsilon_a)^b\hat{c}_a^2d_a(\mathbf{q}-\mathbf{k}/2)d_a(\mathbf{q}+\mathbf{k}/2)$, with $\varepsilon=(1\;1\;1\;-1\;-1)$ and $b=0$ (resp. $b=1$) for $b=\varphi$ (resp. $b=\phi$). Note that $\Sigma_b$ is $O(1)$ (and not $O(1/N)$); mathematically this is because of the trace, which yields a factor of $N$.
As mentioned above, the boson self-energy $\Sigma_\phi(0,\mathbf{0})$ is finite but depends upon the cutoff ($\Sigma_\phi(0,\mathbf{0})$ is proportional to $\Lambda$). Again, this determines the location of the QCP at $N=\infty$, and when we focus on the critical theory, this zero-frequency zero-momentum contribution is exactly cancelled by the bare value of $r$. Hence we are left with the corrections at non-zero frequency and momenta, which we isolate by considering the self-energy difference $\Sigma_b(\omega_n,\mathbf{k})-\Sigma_b(0,\mathbf{0})$ (for $b=\varphi$ the second term is zero by charge conservation). This difference is finite and cutoff independent. In the $c_{0,1}\rightarrow0$ limit, which will be the case in the critical theory, the self-energy differences show logarithmic divergences, i.e. contain $|\ln c_1/c_2\,|$. Conveniently, as mentioned in the main text, the latter will act as a control parameter [@huh2008], in addition to $N$, in the critical theory.
In the following, we thereby obtain the one-loop bosonic self energy, $$\begin{aligned}
&&\Sigma_b(\omega_n,\mathbf{k})-\Sigma_b(0,\mathbf{0})\label{eq:sigmabexpr}\\
&&\qquad=\frac{g_b^2}{\alpha}\left(|\mathbf{k}|f_b(\mathbf{\hat{k}})|\ln
c_1/c_2\,|+\sqrt{|\omega_n|}\,C_b\right) \nonumber.\end{aligned}$$ For future convenience, we take henceforth $c_2=1$ and denote $c=c_1$. It is straightforward to obtain the coefficients of the frequency dependences, $C_b$. Because $\Sigma_b$ is [*larger*]{} than the bare term at $r=0$, which goes as $\mathbf{k}^2+\omega_n^2$, throughout this work, we take $\mathcal{G}_b\rightarrow\Sigma_b^{-1}$, where $\mathcal{G}_b$ is a full boson Green’s function. Finally, note that we used an expansion in small $1/|\ln c_1/c_2\,|$ of $\Sigma_b^{-1}$, i.e. of the inverse of Eq. (\[eq:sigmabexpr\]), in some of the calculations.
By evaluating $\Sigma_b(\omega_n,\mathbf{0})-\Sigma_b(0,\mathbf{0})$, we find $C_\varphi=0$ and $C_\phi=1.33$ taking $\alpha=1$, $c_1=0$ and $c_2=1$. Note that in the $c_1/c_2 \rightarrow 0$ limit, the frequency dependence is subdominant and the bosonic propagator becomes static.
We now extract the non-trivial logarithmic momentum dependence, $f_b(\mathbf{\hat{k}})$.
Coefficient of the logarithm
----------------------------
As mentioned above, when $c_{0,1}=0$, to which the theory flows at the QCP, the energy $E(\mathbf{k})$ and spectrum $\mathsf{E}_\pm(\mathbf{k})$ vanish for any $\mathbf{k}\parallel\langle111\rangle$, which renders the self-energy difference, $\Sigma_b(\omega_n,\mathbf{k})-\Sigma_b(0,\mathbf{0})$, [*divergent*]{}. The appearance of a divergence is subtle: for general $\mathbf{k}$, the denominator in Eq. (\[eq:sigmabsmallc0\]) appears relatively well-behaved since the singularity occurs only when [*both*]{} $\mathbf{q}+ \mathbf{k}/2$ [*and*]{} $\mathbf{q}- \mathbf{k}/2$ lie along a $\langle 111\rangle$ axis. The singularity actually arises from the regions of integration at large $|\mathbf{q}|$ along these directions, where $|\mathbf{k}| \ll |\mathbf{q}|$, so that [*both*]{} energies are small. We analyze it below. In the limit $0\leq c_0 \ll c_1 \ll c_2=1$ (i.e. with $c_1$ [*nonzero*]{} and small), which is the actual behavior in the RG [*flows*]{}, the divergence is removed, and the result is large in $|\ln c_1/c_2\,|$. In this subsection, we extract the leading result in this limit. Notably, in this limit, the result is independent of $c_0$, and can be approximated by taking simply $c_0=0$.
To extract the coefficient of the logarithm, $f_b(\mathbf{\hat{k}})$, we rotate to bases whose $x$-axes point along one of the $\langle111\rangle$ directions, and make a change of variables such that $$\left\{\begin{array}{l}
\mathbf{\hat{e}}_1=(s_1,s_2,s_3)/\sqrt{3}\\
\mathbf{\hat{e}}_2=(0,s_2,-s_3)/\sqrt{2}\\
\mathbf{\hat{e}}_3=(-2s_1,s_2,s_3)/\sqrt{6}
\end{array}\right.
\;\mbox{and}\quad \mathbf{q}=\frac{Q}{c_1}\mathbf{\hat{e}}_1+u\mathbf{\hat{e}}_2+v\mathbf{\hat{e}}_3,$$ where $s_i=\pm1$ (allows to span the eight $\langle111\rangle$ directions). This rewriting is chosen so that for $Q,u,v$ of $O(1)$, the region near the $(s_1 s_2 s_3)$ ray is singled out. The Jacobian of this coordinate transformation is $\mathcal{J}_0=|s_1s_2s_3/c_1|$. Now, we rewrite the functions involved in the integrand of the self-energies, Eq. (\[eq:sigmabsmallc0\]), in these new coordinates, and we obtain the leading asymptotic behavior of each such function at small $c_1$.
For example, we find $${E_{\mathbf{q},\mathbf{k}}^\pm}\approx\frac{1}{c_1}\epsilon^\pm_{Q,u,v;k_1,k_2,k_3}
\quad\mbox{and}\quad
{F_{b;\mathbf{q},\mathbf{k}}}\approx\frac{1}{c_1^2}\gamma^b_{Q,u,v;k_1,k_2,k_3},$$ where the $\epsilon_\pm$ and $\gamma^b$ ($b=\varphi,\phi$) are functions of $\{Q,u,v,k_1,k_2,k_3\}$ (and of course of the $s_i$’s) [*only*]{}. We are then in a position to take the logarithmic derivatives of the boson self-energies. A major simplification thereby occurs: the frequency dependence drops out of $\Sigma_b(\omega_n,\mathbf{k})-\Sigma_b(\omega_n,\mathbf{0})$. We find $$\begin{aligned}
&&\frac{\alpha}{g_b^2}c_1\,\partial_{c_1}\left[\Sigma_b(\omega_n,\mathbf{k})-\Sigma_b(\omega_n,\mathbf{0})\right]\label{eq:fint}\\
&&\qquad=\sum_{s_1,s_2,s_3=\pm1}\int_{0}^{+\infty}\frac{dQ}{2\pi}\int_{-\infty}^{+\infty}\frac{du}{2\pi}
\int_{-\infty}^{+\infty}\frac{dv}{2\pi}\; \mathcal{K}_{s_1s_2s_3}^b\nonumber\\
&&\qquad=f_b(\mathbf{k})=|\mathbf{k}|f_b(\mathbf{\hat{k}}),\end{aligned}$$ where $$\begin{aligned}
\label{eq:mathcalK}
\mathcal{K}^b_{s_1,s_2,s_3} & =& 9\sqrt{2}\,Q
\Big[ \frac{3(a_0-h^b_0)}{a_0^{5/2}}
\\
&&+\;
\frac{2\left(h^b(a_++\sqrt{a_+}\sqrt{a_-}+a_-)-3a_+a_-\right)}{a_-^2a_+^{3/2}+a_+^2a_-^{3/2}}
\Big]. \nonumber\end{aligned}$$ In the above formula, we introduced several expressions: $$\begin{aligned}
\kappa &=& s_1s_2k_xk_y + s_1s_3k_xk_z + s_2s_3k_yk_z\\
h_{0}^\phi &=& 3\left[Q^2 - 2\left(u^2 +
v^2\right)\right]\\
a_0 = h_0^\varphi &=& 3\left[Q^2 + 2\left(u^2 + v^2\right)\right]\\
h^\phi &=& h_0^\phi + \left(\mathbf{k}^2 -\kappa\right)\\
h^\varphi &=& h_0^\varphi - \left(\mathbf{k}^2 -\kappa\right)\\
a_\pm &=& a_0 + \left(\mathbf{k}^2 -\kappa \pm
3\sqrt{2}u(s_2k_y-s_3k_z)\right. \nonumber\\
&&\left. \pm \sqrt{6}v(s_2k_y + s_3k_z - 2s_1k_x)\right),\end{aligned}$$ where all the functions defined above, namely $h_0^b$, $a_0$, $h^b$, $a_\pm$, and $\mathcal{K}^b$ ($b=\phi,\varphi$), are taken at $\{Q,u,v,k_x,k_y,k_z\}$ (and are also functions of the $s_i$’s although we have written the latter explicitly for $\mathcal{K}^b$ only). Note that the integrations over $u$ and $v$ are taken all the way from $-\infty$ to $+\infty$ although the sum over the eight directions, $\sum_{s_1,s_2,s_3}$ is also taken. This is because, for non-zero $c_1$, the $u,v$ integrations have a priori upper bounds of order $Q/c_1$, which is taken to infinity. In the present order of limits, all contributions arise from regions of angular width of order $c_1$ from the $\langle111\rangle$ rays.
The integrals, Eq. (\[eq:fint\]), are evaluated thanks to the Cuba library, using the “Cuhre” routine [@hahn2005].
Approximation
-------------
![Plot of $f_\phi(\mathbf{\hat{k}})/f_\phi\!\left(001\right)$. The line represents a $\langle111\rangle$ direction. The whole surface can be obtained from the plotted points by applying cubic symmetries (note that the set of plotted points is larger than the minimal set of points). The yellow surface is a sphere of radius $f_\phi\!\left(001\right)$.[]{data-label="fig:fplot"}](fig4.png){width="2in"}
Since $f_b$ is very smooth (see Fig. \[fig:fplot\]), we approximate it by a low-order polynomial of $\mathbf{k}$ in order to be able to take accurate derivatives of $f_b$ as required to compute the flow of $c_1$ (and $c_2$) – see Sec. \[sec:RGeqs\]. Imposing cubic symmetry, the most general polynomial to order six can take the form $$\frac{1}{f_b(\mathbf{\hat{k}})}\approx m_1^b+m_2^b\left(\hat{k}_x^4+\hat{k}_y^4+\hat{k}_z^4\right)+m_3^b\,\hat{k}_x^2\hat{k}_y^2\hat{k}_z^2,$$ and fits with $m_1^\phi=2.356$, $m_2^\phi=-0.130$ and $m_3^\phi=4.136$ and $m_1^\varphi=-4.704$, $m_2^\varphi=0.264$ and $m_3^\varphi=-8.253$ provide excellent approximations: the square roots of the means of the squares are $R_\phi=0.0049$ and $R_\varphi=0.0049$, where $R_b=\frac{1}{N_{\rm pts}}\sqrt{\sum_{i=1}^{N_{\rm pts}}\frac{\left((1/f^b_i)-{\rm fit}^b_i\right)^2}{(1/f^b_i)^2}}$.
RG equations {#sec:RGeqs}
============
As discussed in the main text, twenty-four Feynman diagrams are necessary to determine the RG equations: two boson self-energies, $\Sigma_b$, given in Sec. \[sec:asymp\], two fermion self-energies, $\Sigma_{f;b}$, and twenty vertex corrections, the one-loop $\Xi_{b;(1);b'}$ (four) and the two-loop $\Xi_{b;(2);b',b'',\eta}$ (sixteen, twelve of which either vanish identically or cancel out one another), with $b,b',b''=\varphi,\phi$ and $\eta=\pm1$. The notation is expected to be transparent, and the expressions can be read off in Eqs. (\[eq:sigmafeq\]–\[eq:xi2\]). We proceed like in Refs. [@vojta2000a; @huh2008], i.e. we find the corrections to the parameters of the theory by evaluating the former when a small change in the cutoff is applied. It physically corresponds to integrating out modes to keep the rescaled cutoff unchanged. In practice, we [*(i)*]{} use soft momentum-cutoffs for the integrals, implemented by the use of a rapidly decaying function $|\mathbf{q}|/\Lambda\mapsto\mathcal{F}(|\mathbf{q}|/\Lambda)$, with e.g. $\mathcal{F}$ belonging to the function space $\mathcal{L}^2(\mathbf{R})$, [*(ii)*]{} compute the logarithmic derivatives with respect to the cutoff $\Lambda$ of the fermion self-energy and vertices, [*(iii)*]{} identify the appropriate coefficients of the Taylor expansion (in $k_i$ and $\omega_n$) of the result. The choice of a soft cutoff is fairly arbitrary, but helps to avoid spurious singularities induced by “ringing” at the spectral edge. The derivative with respect to $\Lambda$ serves to extract the [*incremental*]{} change in the band parameters due to a small change of cutoff, as in the Wilsonian view of RG. The momentum and frequency expansion allows identification of the renormalization of each term of the Hamiltonian independently.
Diagram expressions
-------------------
The fermion self-energy is $$\begin{aligned}
&&\Sigma_{f}(\omega_n,\mathbf{k})
=\sum_{b=\varphi,\phi}\frac{-g_b^2}{N} \label{eq:sigmafeq}\\
&&\quad\times\int\frac{d^3q}{(2\pi)^3}\int_{-\infty}^{+\infty}\frac{d\Omega_n}{(2\pi)}\frac{M_bG^0_{\Omega_n,\mathbf{q}}M_b
\mathcal{F}\left(\frac{|\mathbf{q}|}{\Lambda}\right) \mathcal{F}\left(\frac{|\mathbf{k}-\mathbf{q}|}{\Lambda}\right)}{\Sigma_b({\omega_n-\Omega_n,\mathbf{k}-\mathbf{q}})-\Sigma_b(0,\mathbf{0})},\nonumber\end{aligned}$$ where two cutoff functions $\mathcal{F}$ are present because both fermion lines in the self-energy should be cutoff, i.e. the momenta of all the electrons in the theory are taken within the cutoff. Similarly, the vertex corrections [*at zero external momenta and frequencies*]{} are $\Xi_b^0=\Xi_{b;(1)}^0+\Xi_{b;(2)}^0$, with $$\begin{aligned}
\label{eq:xi1}
&&\Xi_{b;(1)}^0=\sum_{b'=\varphi,\phi}\frac{g_b g_{b'}^2}{N^{3/2}}\\
&&\quad\times\int\frac{d^3q}{(2\pi)^3}\int_{-\infty}^{+\infty}\frac{d\Omega_n}{2\pi}\frac{M_{b'}G^0_{\Omega_n,\mathbf{q}}M_bG^0_{\Omega_n,\mathbf{q}}M_{b'}\mathcal{F}^2\left(\frac{|\mathbf{q}|}{\Lambda}\right)}{\Sigma_{b'}({\Omega_n,\mathbf{q}})-\Sigma_{b'}({0,\mathbf{0}})},\nonumber\end{aligned}$$ and
$$\begin{aligned}
\label{eq:xi2}
&&\Xi_{b;(2)}^0=-\sum_{b',b''=\varphi,\phi;\eta=\pm}\frac{g_b g_{b'}^2g_{b''}^2}{N^{5/2}}\int\frac{d^3q_1}{(2\pi)^3}\int\frac{d^3q_2}{(2\pi)^3}\int_{-\infty}^{+\infty}\frac{d\Omega_{n,1}}{2\pi}\int_{-\infty}^{+\infty}\frac{d\Omega_{n,2}}{2\pi}\\
&&\qquad\qquad\times\frac{M_{b'}G^0_{\Omega_{n,2},\mathbf{q}_2}M_{b''}{\rm Tr}\left\{G^0_{\Omega_{n,1},\mathbf{q}_1}M_{b'}G^0_{\Omega_{n,1}+\eta\Omega_{n,2},\mathbf{q}_1+\eta\mathbf{q}_2}M_{b''}G^0_{\Omega_{n,1},\mathbf{q}_1}M_b\right\}}{\left[\Sigma_{b'}({\Omega_{n,2},\mathbf{q}_2})-\Sigma_{b'}({0,\mathbf{0}})\right]\left[\Sigma_{b''}({\Omega_{n,2},\mathbf{q}_2})-\Sigma_{b''}({0,\mathbf{0}})\right]}\mathcal{F}\left(\frac{|\mathbf{q}_1|}{\Lambda}\right) \mathcal{F}\left(\frac{|\mathbf{q}_2|}{\Lambda}\right) \mathcal{F}\left(\frac{|\mathbf{q}_1+\eta\mathbf{q}_2|}{\Lambda}\right).\nonumber\end{aligned}$$
All other diagrams are smaller in a $1/N$ expansion. By using for example $\partial_{-i\alpha\omega_n}G^0_{\omega_n,\mathbf{k}}=-(G^0_{\omega_n,\mathbf{k}})^2$, one can show that the two-loop diagrams, $\Xi_{b;(2);b',b'',\eta}^0$, with identical internal boson propagators ($b'=b''$) cancel out one another upon performing the sum over $\eta=\pm1$ (and even vanish identically in the case $b=\phi$). The remaining two-loop diagrams correcting the Coulomb vertex ($b=\varphi$ and $b'\neq b''$) can also be shown to vanish, for example by noticing that only the $b'=b''$ diagrams can renormalize $g$. Therefore, only four two loop diagrams (those with $b=\phi$ and $b'\neq b''$), shown in Fig. 3 of the main text, need be calculated. Careful observation shows all contributions are equal, and an explicit calculation yields a finite integral, which converges to a nonzero value multiplied by $g/(N^{3/2}|\ln c_1/c_2\,|^2)$. This is actually subdominant (for $c_1/c_2 \ll 1$) to the contribution from the one loop vertex correction, although it is of the same order in $1/N$.
Flow equations
--------------
We find the following RG flow equations (“beta-functions”). The flow of $\alpha$, the coefficient of the frequency in the fermion self-energy, is $$\frac{\partial_\ell
\alpha}{\alpha}=3-2\Delta_\psi+\frac{1}{\alpha}\left.\left(\partial_{-i\omega_n}\left[D_\Lambda\Sigma_f^0\right]\right)\right|_{\omega_n=0,\mathbf{k}=\mathbf{0}}.
\label{eq:betaalpha}$$ where $D_\Lambda=\Lambda\frac{d}{d\Lambda}$. As usual, the last term of the right-hand-side corresponds in the RG procedure to the “rescaling” (or integration of momenta), while the other terms correspond to the “renormalization” [@kardar2007]. The “anisotropic” coupling of the fermions to the bosons leads to “anisotropic” corrections to the coefficients of the fermion Hamiltonian: $$\frac{\partial_\ell
c_j}{c_j}=z+1-2\Delta_\psi+\frac{\left(\delta\Sigma_f\right)_j^0}{c_j},\qquad
j=0,1,2,
\label{eq:betac0}$$ where $$\left(\delta\Sigma_f\right)_j^0=
\begin{cases}
\frac{1}{2}\left.\left(\partial_{k_x,k_x}^2\left[D_\Lambda\Sigma_f^0\right]\right)\right|_{\omega_n=0,\mathbf{k}=\mathbf{0}}
& \mbox{for }j=0\\
\sqrt{2}\left.\left(\partial_{k_x,k_y}^2\left[D_\Lambda\Sigma_f^1\right]\right)\right|_{\omega_n=0,\mathbf{k}=\mathbf{0}}&
\mbox{for }j=1\\
\sqrt{2}\left.\left(\partial_{k_x,k_x}^2\left[D_\Lambda\Sigma_f^4\right]\right)\right|_{\omega_n=0,\mathbf{k}=\mathbf{0}}& \mbox{for }j=2
\end{cases}.
\label{eq:dLambdaetc}$$ The RG equations for the coupling constants are simply: $$\frac{\partial_\ell
g_b}{g_b}=z+3-\Delta_\phi-2\Delta_\psi+M_b^{-1}\frac{\left[D_\Lambda\Xi_b^0\right]}{g_b/\sqrt{N}},
\label{eq:betag}$$ for $g_\phi=g,g_\varphi=ie$ and $M_\phi=\Gamma_{45}$, $M_\varphi=I$. The right-hand-sides of the equations eventually involve angular integrals that can be performed numerically, and which are obtained using the identities: $$\begin{cases}
\int_0^\infty dq\;\frac{1}{q}\Lambda\frac{d}{d\Lambda}\left[\mathcal{F}^2(q/\Lambda)\right]=1\\
\int_0^\infty dq\; \Lambda\frac{d}{d\Lambda}\left[\frac{\mathcal{F}(q/\Lambda)\mathcal{F}'(q/\Lambda)}{\Lambda}\right]=0\\
\int_0^\infty dq\;
q\Lambda\frac{d}{d\Lambda}\left[\frac{\mathcal{F}(q/\Lambda)\mathcal{F}''(q/\Lambda)}{\Lambda^2}\right]
=0\\
\int_0^\infty dq_1 \frac{\Lambda}{q_1}\frac{d}{d\Lambda}\left[\mathcal{F}(q_1/\Lambda) \mathcal{F}(q_1\tilde{q}_2/\Lambda) \mathcal{F}(q_1(1+\tilde{q}_2)/\Lambda)\right]=1
\end{cases}$$ (for any $\tilde{q}_2$), since $\mathcal{F}(0)=1$ and $\mathcal{F}(+\infty)=0$.
In practice, to calculate the flows of $\alpha$ and $c_0$, from Eqs. (\[eq:betaalpha\]) and (\[eq:betac0\]) with $j=0$, we shift the internal momentum in the integrands of $\Sigma_f$ (see Eq. (\[eq:sigmafeq\])), i.e. $\mathbf{q}\rightarrow\mathbf{q}+\mathbf{k}$. As a result, the derivatives with respect to the frequency $\omega_n$ or momenta $k_i$ involve the fermionic part of the integrands. Proceeding otherwise to obtain the equation for $c_0$ leads to a divergent integral. For the flow of $c_2$, where the derivatives with respect to either part of the integral converge, we have checked that both “methods” give the same result. The integrals from the vertex corrections converge, in particular, we find the double integrals in $[D_\Lambda \Xi_{b;(2)}^0]$ are subdominant (equal to a finite number times $1/|\ln c_1/c_2\,|^2$, the latter factor coming solely from the two inverse boson propagators), even upon taking $c_{0,1}=0$ directly in $G^0$.
Details of the flows of $c_1$ and $c_2$
---------------------------------------
Because the results are crucial to the physics, we give the details of the calculation of the beta functions for $c_1$ and $c_2$. Applying the derivatives in Eq. (\[eq:dLambdaetc\]) with $j=1,2$ to the “boson parts” of the integrand in the self-energies using the approximations discussed in Sec. \[sec:asymp\], and expanding $\Sigma_b^{-1}$ in small $1/|\ln c_1/c_2\,|$, we find:
$$\begin{aligned}
\frac{\left(\delta\Sigma_f\right)_1^0}{c_1}&=&-\frac{\sqrt{2}}{8\pi|\ln c_1/c_2\,|N}\int\frac{d\mathbf{\hat{q}}}{(2\pi)^2}\frac{d_1(\mathbf{\hat{q}})}{E_\mathbf{\hat{q}}}\left\{(m_1^\varphi+m_1^\phi)\mathcal{N}_{1,1}+(m_2^\varphi+m_2^\phi)\mathcal{N}_{1,2}+(m_3^\varphi+m_3^\phi)\mathcal{N}_{1,3}\right\}\\
\frac{\left(\delta\Sigma_f\right)_2^0}{c_2}&=&-\frac{\sqrt{2} }{8\pi|\ln c_1/c_2\,|N}\int\frac{d\mathbf{\hat{q}}}{(2\pi)^2}\frac{d_4(\mathbf{\hat{q}})}{E_\mathbf{\hat{q}}}\left\{(m_1^\varphi-m_1^\phi)\mathcal{N}_{2,1}+(m_2^\varphi-m_2^\phi)\mathcal{N}_{2,2}+(m_3^\varphi-m_3^\phi)\mathcal{N}_{2,3}\right\},\end{aligned}$$
where $$\begin{aligned}
\mathcal{N}_{1,1}&=&3\hat{q}_x\hat{q}_y\\
\mathcal{N}_{1,2}&=&5 \hat{q}_x \hat{q}_y \left( -8
\hat{q}_x^2\hat{q}_y^2-4 \hat{q}_x^2\hat{q}_z^2-4 \hat{q}_y^2
\hat{q}_z^2\right.\nonumber\\
&&\left.+3 \hat{q}_x^4+3 \hat{q}_y^4+7 \hat{q}_z^4\right)\\
\mathcal{N}_{1,3}&=&-\hat{q}_x \hat{q}_y \hat{q}_z^2 \left(6
\hat{q}_x^2\hat{q}_z^2-43 \hat{q}_x^2\hat{q}_y^2+6 \hat{q}_y^2
\hat{q}_z^2\right.\nonumber\\
&&\left.+10 \hat{q}_x^4+10 \hat{q}_y^4-4 \hat{q}_z^4\right)\\
\mathcal{N}_{2,1}&=&2\hat{q}_x^2-\hat{q}_y^2-\hat{q}_z^2\\
\mathcal{N}_{2,2}&=&24\hat{q}_x^2\hat{q}_y^2\hat{q}_z^2 -
21\hat{q}_x^4 \left(\hat{q}_y^2+\hat{q}_z^2\right)+\hat{q}_y^4
\left(42 \hat{q}_x^2 -5\hat{q}_z^2\right)\nonumber\\
&&+\hat{q}_z^4 \left(42 \hat{q}_x^2 -5\hat{q}_y^2\right)+2 \hat{q}_x^6-5\hat{q}_y^6-5\hat{q}_z^6\\
\mathcal{N}_{2,3}&=&\hat{q}_y^2 \hat{q}_z^2 \left(-31 \hat{q}_x^2
\hat{q}_y^2-31 \hat{q}_x^2
\hat{q}_z^2+4\hat{q}_y^2\hat{q}_z^2\right.\nonumber\\
&&\left. +30 \hat{q}_x^4+2\hat{q}_y^4+2\hat{q}_z^4\right).\end{aligned}$$ The relative signs of the terms coming from $\Sigma_\phi$ originate from the “opposite” commutation relations of $\Gamma_{1,2,3}$ and $\Gamma_{4,5}$ with $\Gamma_{45}$, i.e.$[\Gamma_a,\Gamma_{45}]=0$ for $a=1,2,3$ and $\{\Gamma_a,\Gamma_{45}\}=0$ for $a=4,5$. Note that this is true before implementing any approximation or assumption on the magnitude of $c_1/c_2$. If $e=0$, it is then obvious that the flows of $c_1$ and $c_2$ will take different directions, i.e. that the ratio $c_1/c_2$ will be either relevant or irrelevant, or in other words, will flow either to infinity or zero. Hence a calculation taking $c_1/c_2$ large or small from the beginning is for sure valid. We find that $c_1/c_2\rightarrow0$ occurs for $e=0$ (see below). When $e\neq0$, the situation is not as clear-cut, but taking $c_1/c_2$ small, as when $e=0$, proves to be self-consistent as shown below. We can also justify it [*a posteriori*]{} as follows. $c_1/c_2\rightarrow+\infty$ would lead to a situation where the coupling term $\phi\psi^\dagger\Gamma_{45}\psi$ commutes with the bare Hamiltonian at the critical point, hence [*removing*]{} all fluctuations due to the coupling to the order parameter, which is supposed to drive the transition through the fluctuations it induces. Such a choice seems therefore unreasonable. The situation where $c_1/c_2\rightarrow c^*$, a fixed constant, although perhaps seemingly more reasonable, would imply the existence of a universal ratio, when none seems to be natural. Hence, the limit $c_1/c_2\rightarrow0$ seems to be the only reasonable limit to be taken. $c_0/c_1\rightarrow0$ is also consistent.
Exponents
---------
Keeping $\alpha, c_2, g$ and $e$ constant, i.e. setting the corresponding flow equations to zero, the dynamical critical exponent and the field dimensions are $$\begin{aligned}
&&z= 2 -\frac{a_z}{N|\ln c_1/c_2\,|}, \qquad \Delta_{\psi} = \frac{3}{2}+\frac{a_\psi}{N|\ln c_1/c_2\,|^2},\nonumber\\
&&\Delta_{\phi} = \frac{3}{2} + \left[\frac{1}{2} +\frac{a_{\phi}}{N|\ln
c_1/c_2\,|}\right],\\
&&\Delta_{\varphi} = \frac{3}{2} + \left[\frac{1}{2} -\frac{a_{\varphi}}{N|\ln c_1/c_2\,|}\right], \nonumber\end{aligned}$$ where $a_{z}=0.063$, $a_{\psi} =0.143$, $a_{\phi}=0.255$, and $a_\varphi=0.063$. The anomalous dimensions are then simply $\delta z=a_z/(N|\ln
c_1/c_2\,|)$, $\eta_\psi=2a_\psi/(N|\ln c_1/c_2\,|^2)$, $\eta_\phi=1+2a_\phi/(N|\ln c_1/c_2\,|)$ and $\eta_\varphi=1-2a_\varphi/(N|\ln c_1/c_2\,|)$, as given in the main text.
Solutions to the flow equations
-------------------------------
Finally, we obtain $$\begin{aligned}
&&\partial_\ell \left(\frac{c_1}{c_2}\right)
=-\frac{c_1}{c_2}\frac{Y}{N|\ln c_1/c_2\,|},\\
&&\partial_\ell \left(\frac{c_0}{c_1}\right) =-\frac{c_0}{c_1}\frac{W}{N|\ln c_1/c_2\,|},
\label{eq:c1c2c0c1flows}\end{aligned}$$ with $Y=0.020$ and $W=0.043$. These equations are solved analytically by $$\left(c_1/c_2\right)(\ell)=e^{-\frac{\upsilon_0}{\sqrt{N}}\sqrt{\ell+\ell_0}},\quad
\left(c_0/c_1\right)(\ell)=\Upsilon_0e^{-\frac{\upsilon_0'}{\sqrt{N}}\sqrt{\ell+\ell_0}},$$ where $\upsilon_0=\sqrt{2Y}$ and $\upsilon_0'=\sqrt{2}W/\sqrt{Y}$, and where $\ell_0$ and $\Upsilon_0$ are constants which depend on $c_{0,1,2}(\ell=0)$.
Note that, as mentioned in the main text, [*in the absence of Coulomb interactions*]{}, we find $(c_1/c_2)(\ell)=e^{-\frac{0.359}{\sqrt{N}}\sqrt{\ell+\ell_1}}$ and $(c_0/c_1)(\ell)\propto e^{\frac{0.240}{\sqrt{N}}\sqrt{\ell+\ell_1}}$ ($\ell_1$ is a constant), i.e. $c_0/c_1$ is found to be a [*relevant*]{} parameter in that case. The latter means that, eventually, $c_0$ reaches $c_1/\sqrt{6}$, point at which Fermi surfaces start to develop, rendering our theory invalid and the heretofore studied critical point unstable. This would correspond to a Lifshitz transition.
Physical quantities
===================
We are now in a position to calculate the behavior of some physical quantities. We first extract the critical exponent of the correlation length. The associated RG flow is $$\frac{\partial_\ell r}{r} = z+3-2\Delta_\phi,\quad\mbox{i.e.}\quad
\partial_\ell r=\nu^{-1}(\ell)r,$$ with $$\nu^{-1}(\ell)=1-\frac{2a_\phi+a_z}{N|\ln c_1/c_2\,|}.$$ So $$\partial_\ell r=\left(1-\frac{A}{\sqrt{N}\sqrt{\ell+\ell_0}}\right)r
,\quad\mbox{with}\quad A=\frac{2a_\phi+a_z}{\upsilon_0},$$ i.e. $A=2.836$, which is solved into $$r(\ell)=r_0 e^{\ell-\frac{2A}{\sqrt{N}}\sqrt{\ell+\ell_0}},
\label{eq:rflow}$$ where $r_0$ is a constant which depends on $r(\ell=0)$. We can easily invert $r=r(\ell)$ to $\ell=\ell(r)$ by taking the log of Eq. (\[eq:rflow\]), squaring both sides and solving the quadratic equation. We get: $$\begin{aligned}
\ell
&\approx&\ln\frac{r}{r_0}+\frac{2A}{\sqrt{N}}\sqrt{\ln\frac{r}{r_0}+\ell_0},
\label{eq:lfnofr}\end{aligned}$$ where we have kept only terms to order $1/\sqrt{N}$.
Order parameter exponent
------------------------
We first extract the exponent $\beta$ and its logarithmic correction, i.e. how $\langle\phi\rangle$ behaves with $r$. We write $$\frac{\phi(\ell+d\ell)-\phi(\ell)}{\phi(\ell+d\ell)}\approx d\ell\Delta_\phi(\ell),$$ and integrate both sides from $0$ to $\ell$. Using Eq. (\[eq:lfnofr\]), we obtain $$\begin{aligned}
\frac{\phi}{\phi_0}&\sim&\left(\frac{r}{r_0}\right)^2\exp\left[2\,\frac{5a_\phi+2a_z}{\upsilon_0\sqrt{N}}\sqrt{\ln\frac{r}{r_0}+\ell_0}\right]\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\times\exp\left[\frac{-2a_\phi\sqrt{\ell_0}}{\upsilon_0\sqrt{N}}\right],\nonumber\end{aligned}$$ with $2\,\frac{5a_\phi+2a_z}{\upsilon_0}=13.867$ and $2a_\phi/\upsilon_0=2.523$ (in the main text, we absorbed $\ell_0$ in the definition of $r_0$). Contrary to more conventional problems, like the usual Ising model, where $\beta_{\rm Ising}=1/2$ in three spatial dimensions, the bosonic order parameter here grows very slowly as one moves away from the critical point on the ordered side of the transition. This can be seen to be due to the massive fluctuations of the boson field due to the strong coupling to the fermions.
Specific heat
-------------
At the critical point (or in the quantum critical region), temperature is the only relevant parameter, so thermal properties receive intriguing corrections in our critical theory. Since the fermion is well-defined ($\eta_{\psi} \rightarrow 0$), the thermal average of the energy is $$\langle \mathsf{E}\rangle = \sum_{i=\pm,\mathbf{k}}\langle
n_{i,\mathbf{k}}\rangle\mathsf{E}_i(\mathbf{k}) = \sum_{i,\mathbf{k}}\frac{2}{e^{\beta\mathsf{E}_{i}(\mathbf{k})}+1}\mathsf{E}_{i}(\mathbf{k}),$$ with, for $\langle\phi\rangle=0$, $c_0=0$ and $c_2=1$, $\mathsf{E}_{\pm}(\mathbf{k})=\pm\frac{\mathbf{k}^2}{\sqrt{6}}\sqrt{1+3(c_1^2-1)w^2_{\mathbf{\hat{k}}}}$ where $w_{\mathbf{\hat{k}}}=\hat{k}_x^2\hat{k}_y^2+\hat{k}_x^2\hat{k}_z^2+\hat{k}_y^2\hat{k}_z^2$. To lowest order, we find $$\begin{aligned}
C_V=\partial_T\langle \mathsf{E}\rangle&\approx&\frac{15 (4-\sqrt{2}) 6^{3/4}\sqrt{\pi}}{16}\zeta(5/2)
\frac{T^{3/2}}{c_1^{3/2}}\\
&\approx&22.1\exp\left[\frac{3\upsilon_0}{2\sqrt{N}\sqrt{z}}\sqrt{\ln\frac{T_0}{T}}\right]T^{3/2},\nonumber\end{aligned}$$ where $3\upsilon_0/(2\sqrt{z})\approx0.215$ (we use $z\approx2$). To obtain the last line, we used the approximation $e^\ell=\left(T(\ell)/T_0\right)^{1/z}$ and thereby solved the RG equation of $c_1$ in terms of temperature. The logarithmic correction to the $T^{3/2}$ law is a signature of the fact that $c_1$ becomes scale (temperature)-dependent in the quantum critical region.
Mean-field theory
=================
In this section we consider the behavior in the ordered phase according to naïve mean field theory, i.e. a saddle-point evaluation of the $\varphi$ and $\phi$ integrals. The former saddle point is simply $\varphi=0$, i.e. there are no effects of the long-range Coulomb interactions at the mean field level. The saddle point value of $\phi$ is non-zero in the antiferromagnetic phase. It is governed by the effective action which consists of the bare one (second line of Eq. (1) of the main text) [*plus*]{} the contribution obtained by integrating out the fermions.
The fermionic contribution to the effective action, for constant $\phi$, is simply the space-time integral of the total ground state energy density of the electrons. This is obtained by summing up the energy of occupied single-particle states.
In the saddle point approximation, the Hamiltonian density of the fermions is $$\mathcal{H}^\psi_{\rm MF}[\phi]=c_0\mathbf{k}^2+\sum_{a=1}^5\hat{c}_ad_a(\mathbf{k})\Gamma_a+\phi\Gamma_{45},$$ and we therefore have the ground state energy density $$\mathcal{E}_{\rm
MF}^\psi\left[\phi\right]=\sum_{\alpha=1}^2\int_\Lambda
\frac{d^3k}{(2\pi)^3}\, \mathsf{E}_\mathbf{k}^\alpha[\phi],
\label{eq:gsen}$$ ($\mathsf{E}_\mathbf{k}^{1,2}$ are the single-particle lowest-energy bands, with $\mathsf{E}^{1,2}_\mathbf{k}[\phi=0]=\mathsf{E}_-(\mathbf{k})$). Here, by diagonalizing $\mathcal{H}_{\rm MF}^\psi[\phi]$, we obtain $$\begin{aligned}
&&\mathsf{E}^{1,2,3,4}_{\mathbf{k}}[\phi]=c_0\mathbf{k}^2\\
&&\quad\pm\frac{1}{\sqrt{6}}\sqrt{c_2^2\mathbf{k}^4+2(c_1^2-c_2^2)w_{\mathbf{\hat{k}}}^2\mathbf{k}^4\pm6\sqrt{2}c_1\mathbf{k}^2w_{\mathbf{\hat{k}}}\phi+6\phi^2}\nonumber,\end{aligned}$$ where we define $\mathbf{k}^4=(\mathbf{k}^2)^2$, and where $1,2,3,4$ correspond to the signs $\{--,-+,+-,++\}$, respectively.
From scaling, $\mathsf{E}_\mathbf{k}^{1,2} \sim \mathbf{k}^2$, and hence, from Eq. (\[eq:gsen\]), one expects that the singular scaling contributions to the effective action behave as $\mathcal{E}_{\rm
MF}^\psi \sim |\mathbf{k}|^5 \sim |\phi|^{5/2}$, where we used $\phi \sim \mathbf{k}^2$, which follows dimensionally from $\mathcal{H}_{\rm MF}^\psi$. This describes only the singular contributions. Since $\mathcal{E}_{\rm
MF}^\psi$ is an even function of $\phi$, we expect it to contain constant and quadratic terms as well (which are cutoff-dependent). Indeed one can verify by direct expansion in $\phi$ that the integrals which arise from Eq. (\[eq:gsen\]) as coefficients of unity and $\phi^2$ are finite, but if one proceeds to the following order, the coefficient of $\phi^4$ is divergent. This is due to the presence of the $t |\phi|^{5/2}$ term.
To extract the coefficient $t$, we take three derivatives of $\mathcal{E}_{\rm MF}^\psi[\phi]$ with respect to $\phi$. We find an integral whose integrand goes as $1/|\mathbf{k}|^6$ at large $|\mathbf{k}|$, so that the result is integrable in that region. One then simply rescales $\mathbf{k} \rightarrow \mathbf{k}/\sqrt{|\phi|}$, and takes the limit of small $\phi$ (i.e. $\Lambda/\sqrt{\phi}\rightarrow+\infty$). This makes the singular behavior explicit, and in this limit we find $\partial_{\phi,\phi,\phi}^3 \mathcal{E}_{\rm
MF}^{\psi}\left[\phi\right]=1.079/\sqrt{\phi}$, i.e. $t=1.079\times\frac{8}{15}=0.575$, where the coefficient was determined by a numerical integration taking the fixed-point values $c_0=c_1=0$. Therefore, $t|\phi|^{5/2}$ is indeed the lowest-order nonanalytical term. Hence, putting everything together, and looking at the boson action with the fermions integrated out, we have: $$\begin{aligned}
\mathcal{S}_{\rm MF}\left[\phi\right]&=&V\int
d\tau\left[r\phi^2+\mathcal{E}^\psi_{\rm MF}[\phi]\right]\\
&\sim&V\int
d\tau\left[r'\phi^2+t|\phi|^{5/2}\right],\end{aligned}$$ all other terms being irrelevant. Above, $r'$ includes the $\phi^2$ terms in $\mathcal{E}^\psi_{\rm MF}[\phi]$. Most importantly, we obtained positive $t>0$, so that when $r'<0$, a stable minimum action configuration exists, describing a continuous –but unconventional– transition at the mean field level.
[^1]: Note that a number of technicalities are involved in this calculation, in particular regarding the convergence of the differentiated functions; All are discussed in the Supplementary Material [@suppmat].
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abstract: 'We introduce a new dataset of 293,008 high definition (1360 x 1360 pixels) fashion images paired with item descriptions provided by professional stylists. Each item is photographed from a variety of angles. We provide baseline results on 1) high-resolution image generation, and 2) image generation conditioned on the given text descriptions. We invite the community to improve upon these baselines. In this paper we also outline the details of a challenge that we are launching based upon this dataset.'
bibliography:
- 'example\_paper.bib'
nocite: '[@langley00]'
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Introduction
============
Machine learning has recently been employed in many applications pertaining to the fashion industry. The use cases range from style matching [@bossard2012apparel; @kalantidis2013getting; @liu2016deepfashion], recommendation systems in e-commerce sites [@chen2012describing; @xiao2015learning; @kiapour2015buy; @chen2015deep; @simo2015neuroaesthetics], trend prediction, the ability for customers to virtually try on clothes [@han2017viton], and clothing type classification [@liu2012street; @liang2016clothes; @veit2015learning; @zhu2017your].
The availability of large-scale datasets such as DeepFashion [@liu2016deepfashion] has fueled recent progress in applying deep learning to fashion tasks. However, there are still many aspects of the industry that computer vision methods have not been applied to. In this paper we explore the task of assisting fashion designers to share their ideas with others by translating verbal descriptions to images. Thus, given a description of a particular item, we generate images of clothes and accessories matching the description.
To explore these research directions we introduce here a new dataset of almost 300k high definition training images of clothes, and accessories accompanied by detailed design descriptions. Each description is provided by professional designers and contains fine-grained design details. Each product is photographed from multiple angles against a standardized background under consistent lighting conditions and annotated with matching items recommended by a stylist. See Figure \[dataset\_Sample1\] for examples.
In this paper we provide: 1) statistical details of the dataset, 2) detailed comparisons with existing datasets, 3) an introduction to the competition that we are launching on the task of text to image generation, with a brief explanation of the competition criteria and evaluation process, and 4) high-resolution image generation results using an approach based on the progressive growing of GANs [@karras2017progressive], and text-to-image translation results using StackGAN-v1 [@zhang2017stackgan], and StackGAN-v2 [@huang2017stacked].
![Pictures a, b and c present samples of the dataset. Each description is associated with all the images below it. And each item *ie. a, b* is photographed from different angles. We also provide each image’s attributes, and its relationship to other objects in the dataset[]{data-label="dataset_Sample1"}](imgs/pants.png){width="7.2cm"}
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(a)\
![Pictures a, b and c present samples of the dataset. Each description is associated with all the images below it. And each item *ie. a, b* is photographed from different angles. We also provide each image’s attributes, and its relationship to other objects in the dataset[]{data-label="dataset_Sample1"}](imgs/jacket.png){width="7.2cm"}
\
(b)\
![Pictures a, b and c present samples of the dataset. Each description is associated with all the images below it. And each item *ie. a, b* is photographed from different angles. We also provide each image’s attributes, and its relationship to other objects in the dataset[]{data-label="dataset_Sample1"}](imgs/last_mosaic.png){width="7.2cm"}
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(c)\
The paper is organized as follows: Section \[related\] discusses related work. Section \[dataset\] introduces the Fashion dataset, describes the collection procedure, and provides a statistical analysis of the dataset with details of our newly introduced challenge.[^1] In Section \[experiments\], we describe baseline approaches and the evaluation process, including human evaluation. Section \[conclusion\], concludes the paper and discusses future work.
Related Work {#related}
============
We first provide a summary of generative models used in text-to-image synthesis and then discuss related datasets.
Applications of Generative models
---------------------------------
Generative Adversarial Networks [@goodfellow2014generative] have been used in a wide range of applications, including photo-realistic image super-resolution [@ledig2016photo; @sonderby2016amortised], video generation [@denton2018stochastic; @denton2017unsupervised], inpainting [@belghazi2018hierarchical], image-to-image translation [@isola2017image; @zhu2017unpaired; @taigman2016unsupervised] and text-to-image synthesis [@zhang2017stackgan; @huang2017stacked; @reed2016generative; @reed2016learning; @zhang2018photographic].
Although state of the art generative models can already generate polished realistic images [@karras2017progressive], conditional generation and translation tasks are still far from high quality. We hypothesize that this shortcoming is due to a shortage of large, clean datasets.
Related datasets
----------------
To the best of our knowledge none of the currently used datasets for text-to-image synthesis were collected specifically for the purposes of exploring the text-to-image synthesis task. Below we discuss existing datasets that have been used for text-to-image and attributes-to-image synthesis and focus on a specific set of attributes which are important for image synthesis.
--------------------------------------- --------- -------------------- ------------- ------ ----- ---------- --------- --
CelebA 202,599 43x55 to 6732x8984 no 40 no multiple 10,177
CelebA-HQ 30,000 1024x1024 no 40 no multiple unknown
DeepFashion - Fashion Image Synthesis 78,979 300x300 multiple 1000 50 multiple unknown
MS COCO 328,000 varying sizes 5 per image no 80 single unknown
Caltech-UCSD Birds-200-2011 11,788 varying sizes no 312 200 single unknown
Flowers Oxford-102 8189 varying sizes no no 102 single unknown
**Fashion dataset (ours)** 325,536 1360x1360 yes no 48 multiple 78850
--------------------------------------- --------- -------------------- ------------- ------ ----- ---------- --------- --
![Distribution of the data per category. Note that the $x$ axis is in log scale.[]{data-label="categories"}](imgs/stats){height="8cm"}
**Caltech-UCSD Birds-200-2011** [@reed2016generative] was originally created for categorizing bird species, localizing their body-parts and classifying attributes. The dataset consists of 12k images , depicting 200 bird species with 28 attributes. More recent work employs this dataset for image synthesis tasks conditioned on text describing the attributes.\
**MS COCO** [@lin2014microsoft] was originally created as a benchmark for image captioning. While some works use this dataset for text-to-image generation tasks, the generated images miss fine-grained details and only capture high-level information. This is due to the fact that the textual descriptions are very high-level.\
**Flowers Oxford-102** [@nilsback2008automated] consists of 102 categories of flowers and was proposed for the task of fine-grained image classification. [@reed2016generative] collected 5 descriptions for each image in the dataset to augment it for the task of text to image generation.\
**CelebA** [@liu2015deep] contains pictures of 10k celebrities, with 20 images per person (200k images in total). Each image in CelebA is annotated with 40 attributes.\
**DeepFashion** [@liu2016deepfashion] contains over 200k images downloaded from a variety of sources, with varying image sizes, qualities and poses. Each image is annotated with a range of attributes. This publicly available dataset was mainly employed for the task of cloth retrieval and classification. As an extension of the dataset on the task of text-to-image generation, 79k images from the dataset were later annotated with more descriptive text [@zhu2017your].
Our Fashion Dataset {#dataset}
===================
The advantages of our new Fashion dataset over other contemporary datasets are as follows:
- The dataset consists of $293,008$ images ($260,480$ images for training, $32,528$ for validation, $32,528$ for test), which is larger than other available datasets for the task of text to image translation.
- We provide full HD images photographed under consistent studio conditions. There are no other datasets with comparable resolution and consistent photographing condition.
- All fashion items are photographed from $1$ to $6$ different angles depending on the category of the item. To our knowledge, this is the first dataset of this scale consisting of multiple angles of each item.
- Each product belongs to a main category and a more fine-grained category (*i.e: subcategory*). There are $48$ main categories, and $121$ fine-grained categories in the dataset. The name and density of each category is plotted in \[categories\]. Table \[subcat\] presents the number of images by category and subcategory.
- Each fashion item is paired with paragraph-length descriptive captions sourced from experts (*professional designers*). The distribution of the length of descriptions is presented in Figure \[desc\_length\].
- For each item, we also provide metadata such as stylist-recommended matched items, the fashion season, designer and the brand. We also provide the distribution of colors extracted from the text description presented in Figure \[color\]
![Distribution of the data based on colors. Note that the $x$ axis is in log scale.[]{data-label="color"}](imgs/color){height="8cm"}
{width="80.00000%"}
Our Challenge {#challenge}
=============
In addition to releasing a rich dataset, we are launching a challenge that uses our Fashion dataset for the task of text-to-image synthesis. To the best of our knowledge this is the first challenge on this task. Additionally, we encourage participants to take advantage of all information in the dataset, e.g. such as pose or category. We provide a framework that enables researchers to easily compare the performance of their models with an evaluation metric based on an *Inception Score* [@salimans2016improved]. The inception model we use for the experiments we present in Section \[experiments\] was trained on the training set for classifying the images into the categories presented in Figure \[categories\]. For the final challenge evaluation we will also provide inception scores from a model trained on the test set. However, there are a number of issues to consider when using inception scores for evaluating generative models [@barratt2018note]. For example, different implementations of the same model trained on the same dataset can result in significant differences in Inception scores. For these and other reasons our challenge will also provide a human evaluation as we outline below. Our automated evaluation platform for the challenge computes and displays the Inception score for each submission and compiles the best scores in a leader-board. We provide a comprehensive template and an easy to use service to submit a docker container that runs code, and evaluate the performance on an Amazon Web Services cloud instance. Our test set, which won’t be released, consists of descriptions of clothing items and is integrated at runtime in the challengers’ docker container.\
\
**Human Evaluation setup**:\
Inception scores do not consider the correlation between text and the given image. As such, the competition results will also be evaluated by humans. Since inception scores also have other issues [@barratt2018note] as discussed above, the competition winner will be determined based on this human evaluation. During the human evaluation phase, a fixed subset of the test-set will be randomly selected and the corresponding images will be given to a human evaluation system. Each human-evaluator will be given a text and $5$ images generated by each submission. The person’s task will be to rank these sets of images into the first, second, and third best set with respect to the given text. Each task of this nature will be given to $10$ different human-evaluators. The scores given to each image set will then be aggregated to compute final scores under the human evaluation.
Experiments with the Dataset {#experiments}
============================
In this section, we present two sets of experiments: 1) Generating high-resolution images by using the progressive GAN (P-GAN) growing technique of @karras2017progressive, and 2) text-to-image synthesis using StackGAN-v1 [@zhang2017stackgan] and StackGAN-v2 [@DBLP:journals/corr/abs-1710-10916].\
Generating high-resolution images using P-GANs
----------------------------------------------
The primary idea of Progressive Growing of GANs [@karras2017progressive] is to grow the generator and discriminator gradually and in a symmetric manner in order to produce high-resolution images. P-GAN starts with very low-resolution images and each new layer of the model improves quality and adds fine-grained details to the image generated in the prior stage. Experiments on the CelebA dataset [@liu2015deep] showed promising results and we similarly employ P-GANs to generate $1024\times 1024$ images using our fashion dataset as training data. To do this, we follow the same experimental setup and architectural details of the original P-GAN paper [@karras2017progressive][^2]
Figure \[pgan\_lr\] shows examples of images generated by P-GAN. The images exhibit global coherence and span a variety of poses and attributes ranging from color and category to accessory textures and characteristics of fashion designs.
In order to quantitatively evaluate the quality of our generated images, we compute the Inception score for the down-sampled version ($256\times 256$) of our generated images (See Table \[inception\]). The Inception score of the generated images using P-GANs is very close to the that of the original images, presented in Figure \[pgan\].
![Images generated by the P-GAN approach [@karras2017progressive][]{data-label="pgan"}](imgs/P-GAN.png){height="10cm"}
Text-to-Image synthesis:
------------------------
We employed two architectures: **StackGAN-v1** [@zhang2017stackgan] and **StackGAN-v2** [@DBLP:journals/corr/abs-1710-10916] to generate images conditioned on the their description.
**StackGAN-v1** decomposes conditional image generation into two stages. First, the *Stage-I* GAN sketches a low resolution image ($64\times 64$) with the overall shape and colors of the image conditioned on the text and a random noise vector. Subsequently, the *Stage-II* GAN refines this low-resolution image conditioned on the results of the first stage and the same text embeddings, and generates a $256\times 256$ image.
**StackGAN-v2** follows a similar architecture consisting of multiple chained generators and discriminators. The input of each stage of the chain is the output of the previous stage. One of the major differences between StackGAN-v2 and StackGAN-v1 is that these stages are trained jointly, whereas in StackGAN-v1, they are trained independently.
In our experiments, we found that the method by which we encode the textual descriptions can indeed have a big impact on the quality of the generated images. Here, we discuss the text embedding that we applied.
**Text embedding:**\
Both **StackGAN-v1** and **StackGAN-v2** condition the image generation process on $\varphi_t$, i.e. the text embedding of the corresponding image description generated from a pre-trained char-CNN-RNN encoder [@DBLP:journals/corr/ReedASL16]. It is important for the embedding of the description to correctly relate to the visual contents of the product image. We conducted our experiments using different encoders from a wide range of complexity, namely averaging word vectors, concatenating word vectors, a slightly modified encoder from the Transformer architecture [@VaswaniSPUJGKP17] and a bidirectional LSTM [@birnns].
We experimented with both pre-training these models [^3] and jointly training them with the GAN network. In the case of the Transformer’s encoder and bi-LSTM, the text embedding $\varphi_t$ is the output of the encoder of the Transformer, and the projected concatenation of the last hidden state of the forward and backward LSTM respectively. The final text embedding size for the Transformer is $1500$ and $1024$ for bi-LSTM.
We arrived at three conclusions based on our empirical experiments. First, we found that the bi-LSTM model achieves the highest category classification accuracy on the validation dataset in the pre-training process. As can be seen in Figure \[valid\_tsne\], the t-SNE [@vanDerMaaten2008] visualization of text embeddings shows relatively good separation of the categories. Secondly, we found that irrespective of the encoder architecture, pre-training the encoder model results in better correspondence between the descriptions and generated images. Finally, we found that overall, using the pre-trained bi-LSTM with fixed weights as the encoder leads to better results both visually and quantitatively.
The Inception scores reported in table \[inception\] were obtained with the pre-trained bi-LSTM encoder (with fixed weights during the training of GAN).
![Images generated from the **StackGAN-v1** model with pre-trained bi-LSTM text encoder.[]{data-label="StackGAN-v1"}](imgs/Stack-GAN-v1.png){height="10cm"}
![Images generated from the **StackGAN-v2** model with pre-trained bi-LSTM text encoder.[]{data-label="StackGAN-v2"}](imgs/Stack-GAN-v2.png){height="10cm"}
**Implementation details:**\
Throughout all the experiments, the descriptions were lowercased, tokenized and cleared of stop words[^4]. We used the first 15 tokens of the descriptions as the input sequence to the encoder model.
**StackGAN-v1:** We used the same overall architecture as [@zhang2017stackgan] [^5]. The first stage was trained for $80$ epochs, and the second stage was trained for $185$ epochs. The results can be seen in Figure \[StackGAN-v1\].
**StackGAN-v2:** After careful experimentation, we ended up using the same architecture and hyper-parameters as [@DBLP:journals/corr/abs-1710-10916] [^6]. The results can be seen in Figure \[StackGAN-v2\].
Inception Score
----------------------------------- -----------------
Fashion Real data $256\times 256$ $9.71 \pm 2.14$
StackGAN-v1 [@zhang2017stackgan] $6.50 \pm 0.05$
StackGAN-v2 [@zhang2017stackgan] $5.54 \pm 0.07$
P-GAN [@karras2017progressive] $7.91 \pm 0.15$
: Inception Scores on the validation set, i.e: trained on the Fashion train set.[]{data-label="inception"}
{width="\textwidth"}
{height="0.45\textheight"}
We can observe in the Table \[inception\] that first of all, the Inception Score of the StackGAN-V1 is better than StackGAN-V2, while the quality of the images in the StackGAN-V2 is better and the reason is due to a significant mode-collapse that we were faced to in StackGAN-V2. Another interesting point, is the fact that most of the faces in StackGAN-V1 and StackGan-V2 are blurry. It suggests that since the images are conditioned on the text, the model is focusing more on clothing material than face information.
Conclusion
==========
Recent progress in generative modeling techniques has great potential to give designers tools for rapidly visualizing and modifying ideas. While recent advances in generative models can be used to generate images of unprecedented realism, the quality of images generated from textual descriptions has so-far remained far from realistic. We believe that the lack of good datasets for this task has made it difficult to develop models for this task. In this paper, we have introduced a new Fashion themed text-to-image generation dataset, with high-quality images and extensive annotations provided by fashion experts. We provided results for 2 experiments: generating high-resolution images without providing textual descriptions as input, and generating realistic images conditioned on product description using the Fashion dataset as training data. We provide experiments with StackGAN-v1 and StackGAN-v2 models using various text encoders.
To help stimulate further research on conditional generative models, we release our dataset as part of a challenge. Detailed submission instructions are provided and our API computes the inception score (trained on the Fashion dataset). Submissions with the highest quality images as judged by human evaluators will be selected as winners in the challenge organized around this new dataset.
Acknowledgement
===============
We present our special thank to Alex Shee, for his help and support. We also thank Timnit Gebru and Archy de Berker, for assistance with comments that greatly improved the manuscript. We would also like to show our gratitude to Chelsea Moran, Valerie Becaert, Vincent Hoe-Tin-Noe, Misha Benjamin, Pedro Oliveira Pinheiro, David Vazquez, Francis Duplessis, Ishmael Belghazi, Caroline Bourbonniere and Xavier Snelgrove for their support and feedback during the course of this research. We would also like to thank SSENSE for open sourcing their data to the research community.
[^1]: The competition is part of the first workshop of Computer Vision for Fashion, Art and Design at ECCV. The challenge website is <https://fashion-gen.com/>
[^2]: Using code provided by the authors of the P-GAN paper [@karras2017progressive]: <https://github.com/tkarras/progressive_growing_of_gans>
[^3]: The pre-training step consisted of training the encoder to perform a classification task: given the item description predict its category.
[^4]: The python [NLTK](http://www.nltk.org) module was used to tokenize the descriptions by word and remove stop words
[^5]: We used the code provided by the authors of the StackGAN-v1 paper in github <https://github.com/hanzhanggit/StackGAN-Pytorch>
[^6]: We used the code provided by the authors of the StackGAN-v2 paper: <https://github.com/hanzhanggit/StackGAN-v2>
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abstract: 'In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.'
---
<span style="font-variant:small-caps;">Jacek Banasiak</span>
The paper is dedicated to Giséle Ruiz Goldstein on the occasion of her birthday
(Communicated by the associate editor name)
Introduction
============
Coagulation equations, introduced by Smoluchowski [@Smo16; @Smo17] in the discrete case and in [@muller1928allgemeinen] in the continuous one, and extended in [@becker; @blatz1945; @Melz57b; @McZi87; @vigi] to include the reverse fragmentation processes, have proved crucial in numerous applications, ranging from polymerization, aerosol formation, animal groupings, phytoplankton dynamics, to rock crushing and planetesimals formation, see a survey in [@BLL Vol. I] and, as such, they have been extensively studied in engineering, physical and mathematical sciences. We note that coagulation–fragmentation processes can be studied also in a probabilistic setting, see e.g. [@Bert06], but in this paper we shall focus on the deterministic approach that yields the following kinetic type continuous coagulation–fragmentation equation
\[PhPr001\] $$\partial_t f(t,x) = \mathcal{C}f(t,x) + \mathcal{F}f(t,x)\ , \qquad (t,x)\in {\mathbb{R}}_+^2, \label{PhPr001a}$$ with the initial condition $$f(0,x) = f^{in}(x)\ , \qquad x\in {\mathbb{R}}_+, \label{PhPr001b}$$
where the coagulation operator $\mathcal{C}$ and the fragmentation operator $\mathcal{F}$ operators are, respectively, given by $$\mathcal{C}f(x) = \frac{1}{2} \int_0^x k(x-y,y) f(x-y) f(y)\ \mathrm{d}y - \int_0^\infty k(x,y) f(y) f(x)\ \mathrm{d}y \label{PhPr002}$$ and $$\mathcal{F}f(x) = {\mathcal{A}} f(x)+ {\mathcal{B}} f(x) = - a(x) f(x) + \int_x^\infty a(y) b(x,y) f(y)\ \mathrm{d}y, \label{PhPr003}$$ for $x\in {\mathbb{R}}_+$. In , and , $f$ is the density of particles of mass $x$, the coagulation kernel $k$ is a nonnegative and measurable symmetric function defined on ${\mathbb{R}}_+^2$, the overall fragmentation rate $a$ is a nonnegative measurable function on ${\mathbb{R}}_+$ satisfying $$a \in L_{\infty,loc}([0,\infty)).
\label{aloc}$$ The daughter distribution function $b,$ sometimes referred to as the fragmentation kernel, is a nonnegative and measurable function such that for a.e. $y>0$, $$\int_0^y x b(x,y)\ \mathrm{d}x = y \;\;\text{ and }\;\; b(x,y) = 0 \;\text{ for a.e. }\; x>y. \label{PhPr004}$$ We recall that the first condition in ensures that there is no loss of matter during fragmentation events.
Typically, the analysis of the coagulation–fragmentation equations is done in the spaces $X_0 := L_1(\mathbb{R} _+, {\mathrm{d}}x)$ and $X_1:=L_1(\mathbb{R} _+, {\mathrm{d}}x)$ since the norm of a nonnegative $f$ in $X_0$, $$\|f\|_{[0]} = {\int\limits_{0}^{\infty}} f(x){\mathrm{d}}x,$$ gives the total number of particles in the system, while $$\|f\|_{[1]} = {\int\limits_{0}^{\infty}} f(x)x{\mathrm{d}}x$$ gives its total mass. It follows that by introducing some control on the evolution of large particles we can improve the properties of the involved equations. The easiest way to introduce such a control is to consider the problem in the spaces $X_m := L_1(\mathbb{R} _+, x^m {\mathrm{d}}x)$ [ and ]{} $X_{0,m} := L_1(\mathbb{R}_+,(1+x^m) {\mathrm{d}}x)$; the natural norms in these spaces will be denoted by $\|\cdot\|_{[m]}$ and $\|\cdot\|_{[0,m]}$. To shorten notation, we use the abbreviation $w_m(x) =1+x^m$.
In its full generality, is a nonlinear, nonlocal integro-differential equation with unbounded coefficients and hence its solvability presents a wide array of challenges. Early attempts, whose survey can be found in [@BLL], mostly focused on finding particular solutions to , often by quite ingenious methods. Systematic mathematical studies of date back to the 1980s and two main (deterministic) ways to approach have emerged. In the so called weak compactness method, used originally in e.g. [@BaCa90] for the discrete version of the problem and in [@Stew89] for the continuous one (see also [@ELMP03] for a more comprehensive approach), the equation is first truncated to yield a more tractable family of problems approximating . Then it is shown that the solutions to the truncated problems form a weakly compact family of functions from which one can select a subsequence converging in a suitable topology to a solution of an appropriate weak formulation of . The other method, which can be termed the operator one, was initiated in [@AizBak], and, roughly speaking, consists in considering the coagulation part as a perturbation of the linear fragmentation part. Then the theory of semigroups of operators is used to first obtain the (linear) fragmentation semigroup and hence solve by an appropriate fixed point technique. Each method has its advantages and disadvantages that make them suitable for different scenarios and thus they have been developed to large extent independently of each other. The weak compactness approach mostly uses the properties of the coagulation part and can deliver the existence of a solution for a large class of coagulation kernels but then the fragmentation part must somehow match the coagulation term; also other properties, such as mass conservation, regularity, or uniqueness of solutions, have to be proved independently under much more stringent assumptions. On the other hand, the operator method provides the existence of unique, mass conserving and classical solutions but, while being able to deal with even very singular fragmentation processes, its applications to the full problem for a long time were restricted to bounded coagulation kernels. This has changed in the recent few years with the realization that the fragmentation semigroup is analytic for a large class of fragmentation rates $a$ and the daughter distribution functions $b$. This, in turn, allowed for proving the classical solvability of even if the coagulation rate $k$ is unbounded as long as it is dominated in a suitable sense by the fragmentation rate $a$, see [@BaLa12a; @BLL13] and [@BLL Section 8.1.2]. The proofs use interpolation spaces between $X_{0,m}$ and the domain of the generator of the fragmentation semigroup in this space.
The main aim of this paper is twofold. First, we extend the results of [@BLL; @BaLa12a] by removing the assumption that the fragmentation rate is of polynomial growth. This requires a new proof of the analyticity of the fragmentation semigroup that this time is based on the Miyadera-Voigt perturbation theorem, see e.g. [@Voi77] or [@EN Corollary III.3.16], with the help of [@BaAr Lemma 4.15]. It turns out that the local in time solvability of remains the same as in [@BaLa12a], but the global one requires some new moment estimates: for the zeroth moment we adapt the ideas present in [@ELMP03; @Stew91] (see also [@BLL Lemma 8.2.27]), while for the estimates in the interpolation spaces we use the Henry-Gronwall inequality as in [@Banasiak2019]. We emphasize that, in contrast to e.g. [@DuSt96b; @EMP02; @ELMP03], we do not require any polynomial estimates on the coagulation kernel, or on the fragmentation rate; also we allow the expected number of daughter particles to be unbounded for large size of the parent particle. Second, we show that if the coefficients of satisfy the assumptions of the local existence theorem, then the solutions to the truncated problems, constructed in e.g. [@ELMP03] as the approximations to a weak solution to in the weak compactness method, converge strongly to the classical solution of on its maximal interval of existence, confirming thus the fact, not entirely surprising, that both methods agree with each other whenever they are both applicable.
**Acknowledgement.** The author is grateful to Prof. Mustapha Mokhtar-Kharroubi for the suggestions concerning the application of the Miyadera theorem to the problem and to Dr. S. Shindin, whose ideas for the estimates in the discrete case have helped to develop their counterparts in the continuous case.
Analytic fragmentation semigroup
================================
Let $X$, $Y$ be Banach spaces. The space of bounded linear operators from $X$ into $Y$ will be denoted by ${\mathcal{L}}(X, Y),$ shortened to ${\mathcal{L}}(X)$ if $X=Y$. If an operator $O$ generates a $C_0$-semigroup, this semigroup will be denoted by .
For $m \geq 0$ we introduce $$\begin{aligned}
n_m(y)&={\int\limits_{0}^{y}}b(x,y)x^m{\mathrm{d}}x, \label{nmy}\\
N_m(y)& = y^m-n_m(y);
\label{Nmy}\end{aligned}$$ then we have the inequalities, [@BLL Eqn. (2.3.16)], $$N_m(y) > 0, \quad m>1, \qquad N_1(y) = 0, \qquad N_m(y) <0, \quad 0\leq m<1.
\label{Nm}$$ First, let us assume $m\geq 1$. We define $A_m f := -af$ on $$D(A_m) =\{f \in X_{m}\; : \;af\in X_{m}\}$$ and, using the definition (\[PhPr003\]) of ${\mathcal{B}}$, by (\[nmy\]) we easily get $$\|{\mathcal{B}}f\|_{[m]} = {\int\limits_{0}^{\infty}}a(y)n_m(y)f(y) {\mathrm{d}}y <\infty, \quad f\in D(A_m)_+. \label{expB}$$ Hence, we can define $B_m = {\mathcal{B}}|_{D(A_m)}$.
Moving to $X_{0,m}$ we need to introduce some control on the number of particles produced in fragmentation events. Hence, besides (\[aloc\]) and (\[PhPr004\]), we assume that there is $l\geq 0$ and $b_0\in {\mathbb{R}}_+$ such that for any $x\in {\mathbb{R}}_+$ $$n_0(x)={\int\limits_{0}^{x}}b(y,x) {\mathrm{d}}y \leq b_0(1+x^l).
\label{ass2}$$ Similarly to $A_m$, for any $m\geq 1$ we define $A_{0,m} f := -af$ on $$D(A_{0,m}) =\{f \in X_{0,m}\; : \; af\in X_{0,m}\}.$$ Defining $B_{0,m}$ is, however, slightly more involved.
\[lem34\] If $0\leq f\in D(A_{0,m})$ with $m\geq l$, then $$\|{\mathcal{B}}f\|_{[0,m]} = {\int\limits_{0}^{\infty}}a(y)(n_m(y)+n_0(y))f(y) {\mathrm{d}}y <\infty. \label{expB'}$$ \[t3.4a\]
Let $f\in D(A_{0,m})_+$. By (\[expB\]), it suffices to estimate $$\begin{aligned}
&{\int\limits_{0}^{\infty}}\left({\int\limits_{x}^{\infty}}a(y)b(x,y)f(y) {\mathrm{d}}y\right)\!\! {\mathrm{d}}x
={\int\limits_{0}^{\infty}}a(y)f(y)\left({\int\limits_{0}^{y}}b(x,y) {\mathrm{d}}x\right) {\mathrm{d}}y\label{diss''}\\&={\int\limits_{0}^{\infty}}a(y)n_0(y)f(y) {\mathrm{d}}y
\leq
b_0{\int\limits_{0}^{\infty}}a(y)(1+y^l)f(y) {\mathrm{d}}y \leq 2 b_0{\int\limits_{0}^{\infty}}a(y)w_m(y) f(y) {\mathrm{d}}y<\infty,{\nonumber}\end{aligned}$$ where we used the estimate $$1+y^l\leq 2(1+ y^m), \label{4}$$ if $m\geq l$. Hence, we can define $B_{0,m} = {\mathcal{B}}|_{D(A_{0,m})}$ provided $m\geq l$.
\[thnewchar0\] Let $a,b$ satisfy (\[aloc\]), (\[PhPr004\]) and (\[ass2\]). Let further for some $m_0>1$ $$\liminf\limits_{y\to \infty}\frac{N_{m_0}(y)}{y^{m_0}} >0.
\label{goodchar}$$ Then
1. (\[goodchar\]) holds for all $m >1$;
2. $F_{0,m} := A_{0,m}+B_{0,m}$ is the generator of a positive analytic semigroup , on $X_{0,m}$ for any $m>\max\{1,l\}$.
1\. This result in the discrete case is due to [@Banasiak2019]. Let $y\geq 1$ and $m_0>1$. It is easy to see that is equivalent to the existence of a constant $\delta_{m_0}>0$ such that $\inf_{y\geq 1}N_{m_0}/y^{m_0} \geq \delta_{m_0}$. We have $$\frac{d}{dm} \frac{N_m(y)}{y^m} = - \frac{1}{y^m}{\int\limits_{0}^{y}} b(x,y)x^m\ln\left(\frac{x}{y}\right ){\mathrm{d}}x >0$$ and $$\frac{d^2}{dm^2} \frac{N_m(y)}{y^m} = -\frac{1}{y^m}{\int\limits_{0}^{y}} b(x,y)x^m\ln^2\left(\frac{x}{y}\right){\mathrm{d}}x <0,$$ where the differentiation under the sign of the integral is justified as $x^{m-1}(\ln x)^i$, $i=1,2$, is bounded due to $m>1$. Hence, if $\frac{N_{m_0}(y)}{y^{m_0}} > \delta_{m_0}$ for some $\delta_{m_0}> 0$ and some $m_0>1$, then $\frac{N_m(y)}{y^m}>\delta_{m_0}$ for any $m> m_0$. Further, the inequality for the second derivative shows that $m \mapsto \frac{N_m(y)}{y^m}$ is concave; that is, since $\frac{N_1(y)}{y} =0$, for $m\in (1,m_0]$ and $y\geq 1$ we obtain $$\frac{N_m(y)}{y^m} \geq \frac{N_{m_0}(y)}{y^{m_0}(m_0-1)}(m-1)\geq \frac{\delta_{m_0}(m-1)}{m_0-1},$$ which gives in the interval $(1, m_0]$.
2\. To prove the generation result, we use the Miyadera–Voigt theorem, see [@EN Corollary III.3.16] and [@BaAr Lemma 4.15]. For this we observe that (\[goodchar\]) and the positivity of $m-l$ imply that there is $r>0$ such that for $x\geq r$ we have $$\frac{n_m(x)}{x^m} \leq c'<1, \qquad \frac{b_0(1+x^l)}{1+x^m} \leq \frac{1-c'}{4},\label{pp}$$ see (\[ass2\]). Furthermore, by , there is an $\zeta>0$ such that $$\mathrm{ess}\!\!\!\sup\limits_{0\leq x\leq r} \frac{a(x)b_0(1+x^l)}{a(x)+\zeta} \leq \frac{1-c'}{4}.
\label{esss}$$ Consider the operator $(A_{0,m}-\zeta I, D(A_{0,m})$. Then for $f \in D(A_{0,m})_+$ we have $$\begin{aligned}
&\int_0^\delta \|B_{0,m} G_{A_{0,m}-\zeta I}(t)f\|_{[0,m]}{\mathrm{d}}t \\
&= \int_0^\delta\left( \int_0^\infty (1+x^m) \left(\int_x^\infty a(y)b(x,y) e^{-(a(y)+\zeta)t} f(y){\mathrm{d}}y\right){\mathrm{d}}x\right){\mathrm{d}}t\\
&\leq \int_0^\infty (1+x^m) \left(\int_x^\infty \frac{a(y)b(x,y) }{a(y)+\zeta} f(y){\mathrm{d}}y\right){\mathrm{d}}x \\
&= \int_0^\infty \frac{a(y)f(y)}{a(y)+\zeta}\left(\int_0^y (1+x^m)b(x,y){\mathrm{d}}x\right){\mathrm{d}}y = I_1+I_2,\end{aligned}$$ where, by , , and the monotonicity of $x\mapsto 1+x^m,$ $$\begin{aligned}
I_1 &:= \int_0^{r}\frac{a(y) f(y)}{a(y)+\zeta} \left(\int_0^y (1+x^m)b(x,y){\mathrm{d}}x\right){\mathrm{d}}y\leq \int_0^{r}\frac{a(y)(1+y^m)n_0(y)}{a(y)+\zeta} f(y) {\mathrm{d}}y\\
&\leq b_0\int_0^{r}\frac{a(y)(1+y^l)(1+y^m)}{a(y)+\zeta} f(y) {\mathrm{d}}y \leq \frac{1-c'}{4}\|f\|_{[0,m]}\end{aligned}$$ and, by and , $$\begin{aligned}
I_2&:= \int_{r}^\infty \frac{a(y)f(y)}{a(y)+\zeta} \left(\int_0^y (1+x^m)b(x,y){\mathrm{d}}x\right){\mathrm{d}}y\\
&\leq \int_{r}^\infty \frac{a(y)n_0(y)f(y)}{a(y)+\zeta} {\mathrm{d}}y + \int_{r}^\infty \frac{a(y)n_m(y) f(y)}{a(y)+\zeta} {\mathrm{d}}y\\
&\leq \|f\|_{[0,m]} \left(\mathrm {ess} \sup\limits_{y\geq r} \frac{a(y)b_0(1+y^l)}{(1+y^m)(a(y)+\zeta)} + c' \right)\leq \frac{3c'+1}{4}\|f\|_{[0,m]}.\end{aligned}$$ Hence $$\int_0^\delta \|B_{0,m} G_{A_{0,m}-\zeta I}(t)f\|_{[0,m]}{\mathrm{d}}t \leq I_1+I_2 \leq \gamma \|f\|_{[0,m]}$$ with $\gamma = (c'+1)/2 <1$. Therefore $B_{0,m}$ is a Miyadera perturbation of $A_{0,m}-\zeta I,$ and hence of $A_{0,m}$, see [@BaAr Lemma 4.15]. Using [@EN Exercise III.3.17(1)] and Arendt–Rhandi theorem, [@AR], we conclude that $F_{0,m} = A_{0,m}+B_{0,m}$ is the generator of an analytic positive semigroup.
Local solvability
=================
As mentioned in the introduction, the local in time solvability of can be proved exactly as in [@BaLa12a], see also [@BLL Theorem 8.1.2.1]. Certain notation and intermediate estimates will be, however, used in the proof of the global existence and thus are recalled below.
We assume that $a$ and $b$ satisfy , , and . Hence the fragmentation operator $(F_{0,m}, D( A_{0,m})) = (A_{0,m} + B_{0,m}, D(A_{0,m}))$ is the generator of an analytic positive semigroup on $X_{0,m}$ whenever $m>\max\{1,l\}.$ The coagulation kernel $k$ is assumed to be a measurable symmetric function such that, for some $K>0$ and $0 < \alpha<1$, $$0\leq k(x,y) \leq K(1+a(x))^\alpha(1+a(y))^\alpha, \quad (x,y) \in {\mathbb{R}}_+^2.\label{kass1}$$ This assumption is sufficient for the local-in-time solvability of (\[PhPr001\]). However, to prove that the solutions are global in time we need to strengthen (\[kass1\]) to $$0\leq k(x,y) \leq K\big((1+a(x))^\alpha+(1+a(y))^\alpha\big), \quad (x,y) \in {\mathbb{R}}_+^2,
\label{kass2}$$ again for $K > 0$ and $0 < \alpha<1$. Thus, using the linear operators $A_{0,m}$ and $B_{0,m}$, and the nonlinear operator $C_{0,m}$, defined via (\[PhPr002\]) but now only for $f$ in the maximal domain $$D(C_{0,m}) := \{f \in X_{0,m} : {\mathcal{C}} f \in X_{0,m}\},$$ the initial-value problem can be written as the following abstract semilinear Cauchy problem in $X_{0,m}$: $${\partial}_tf = A_{0,m}f + B_{0,m} f + C_{0,m}f, \qquad f(0) = f^{in}.
\label{feco1}$$ We note that, in general, $0 \notin \rho(F_{0,m})$ and therefore to enable us to define appropriate intermediate spaces we consider $$F_{0,m,\omega}:= F_{0,m} - \omega I = A_{0,m} - \omega I + B_{0,m} = A_{0,m,\omega} + B_{0,m},$$ where $$A_{0,m,\omega} : = A_{0,m} - \omega I,$$ assuming that $\omega$ is greater than the type of . We also assume that $\omega >1$ to simplify using (\[kass1\]) and (\[kass2\]). Clearly, $(F_{0,m,\omega}, D(A_{0,m}))$ is also the generator of an analytic semigroup $(G_{F_{0,m,\omega}}(t))_{t \ge 0} = (e^{-\omega t}G_{F_{0,m}}(t))_{t \ge 0},$ but now we have the desired property that $0 \in \rho(F_{0,m,\omega})$. Thus, as in [@BaLa12a], the intermediate spaces between $D(F_{0,m,\omega}) = D(A_{0,m,\omega})$, see [@Lun Section 2.2], satisfy $$D_{F_{0,m,\omega}}(\alpha, 1) = D_{A_{0,m,\omega}}(\alpha, 1) = X_{0,m}^{\alpha}, \qquad \alpha \in (0,1),$$ where $$X_{0,m}^{\alpha} := \left\{ f \in X_{0,m}:\; \ \int_0^\infty |f(x)|(\omega + a(x))^\alpha w_m(x)\,{\mathrm{d}}x < \infty \right\},
\label{frps1}$$ and equality of the spaces is interpreted in terms of equivalent norms, see also the Stein–Weiss theorem [@bergh1976 Corollary 5.5.4]. The natural norm on $X_{0,m}^{\alpha}$ will be denoted by $\|\cdot\|^{(\alpha)}_{[0,m]}$, and we note that $X_{0,m}^0 = X_{0,m}$, and $X_{0,m}^1 = D(A_{0,m,\omega})$.
Hence, in general, if is an analytic semigroup in $X_{0,m}$ satisfying $$\|G(t)\|_{{\mathcal{L}}(X_{0,m})} \leq M_{0,m}^{(0)}e^{\omega_{0,m}t},$$ then, by [@EN Theorem II.4.6(c)], it is a family of operators in ${\mathcal{L}}(X_{0,m}, X_{0,m}^{1})$ satisfying $$\|G(t)\|_{{\mathcal{L}}(X_{0,m},X^1_{0,m})} \leq M_{0,m}^{(1)}e^{\omega_{0,m}t}t^{-1},\quad t>0.$$ Then it follows that $G(t)\in {\mathcal{L}}(X_{0,m}, X_{0,m}^\alpha)$ and there is $M^{(\alpha)}_{0,m}$ such that $$\label{eq2.4a}
\|G(t)\|_{{\mathcal{L}}(X_{0,m},X^\alpha_{0,m})} \leq \frac{M_{0,m}^{(
\alpha)}e^{\omega_{0,m}t}}{t^\alpha}, \quad t>0,\, 0<\alpha<1.$$
Assume that $a$ and $b$ satisfy , , , and let $m > \max\{1,l\}$. Further, let $k$ satisfy (\[kass1\]). Then, for each $f^{in}\in X_{0,m,+}^{\alpha}$, there is $\tau(f^{in})>0$ such that the initial-value problem (\[feco1\]) has a unique nonnegative classical solution $$f \in C\left([0,\tau(f^{in})),
X_{0,m}^{\alpha}\right)\cap C^1\left((0,\tau(f^{in})), X^{\alpha}_{0,m}\right)\cap C\left((0,\tau(f^{in})), D(A_{0,m})\right).
\label{fprop}$$ \[fcth1\]
As we mentioned, the proof of this result is the same as of [@BaLa12a Theorem 2.2] which was proved under the additional assumption that $a$ is polynomially bounded. This assumption, however, was only needed to prove, by an alternative method, that is an analytic semigroup generated by $F_{0,m} = A_{0,m}+B_{0,m}$ in $X_{0,m}$ with a suitably bigger $m$ depending also on the growth rate of $a$. The only other consequence of the generation theorem of [@BaLa12a] is that is a quasi-contractive semigroup (that is, satisfying $\|G_{F_{0,m}}\|_{{\mathcal{L}}(X_{0,m})} \leq e^{\omega_m t}$ for some $\omega_m$) which in turn allowed in the proof of [@BaLa12a Theorem 2.2] to use the Trotter–Kato representation formula to prove that certain auxiliary semigroups are positive. This result is, however, available also by a direct analysis of the construction of these semigroups.
We recall some equalities and inequalities used in the proof of [@BaLa12a Theorem 2.2] that will be used in the sequel. First, let ${\widetilde {{\mathcal{C}}}}$ denote the bilinear form obtained from ${\mathcal{C}}$; that is, $${\widetilde {{\mathcal{C}}}}(f,f) = {\mathcal{C}}f$$ where ${\mathcal{C}}$ is defined in . Then direct calculations yield, for any measurable $\theta,$ $$\begin{aligned}
\int_0^\infty \,\theta(x)\,[\tilde{{\mathcal{C}}}(f,g)](x)\,{\mathrm{d}}x &=& \frac{1}{2}\int_0^\infty \int_0^\infty \theta(x+y)k(x,y)f(x)g(y)\,{\mathrm{d}}x{\mathrm{d}}y{\nonumber}\\
&&\phantom{xxx}- \int_0^\infty \int_0^\infty\theta(x)k(x,y) f(x)g(y)\,{\mathrm{d}}x{\mathrm{d}}y
\label{moments1}\end{aligned}$$ and, by symmetry, $$\int_0^\infty \,\theta(x)\,[{\mathcal{C}}f](x)\,{\mathrm{d}}x
= \frac{1}{2}\int_0^\infty\int_0^\infty \chi_{\theta}(x,y)k(x,y) f(x)f(y)\,{\mathrm{d}}x\,{\mathrm{d}}y\,,
\label{moments2}$$ where $$\chi_{\theta}(x,y) = \theta(x+y) - \theta(x) - \theta(y).$$ In particular, for $\theta(x) = 1 + x^m$ we will be using the elementary inequality $$(x+y)^m \leq 2^m(x^m + y^m), \qquad x,y\in {\mathbb{R}}_+^2,\, m\geq 0,$$ as well as $$0 \le (x+y)^m - x^m - y^m \le c_m \left( x y^{m-1} + x^{m-1} y \right)\ , \qquad x,y\in {\mathbb{R}}_+^2,\,m>1,\label{momestb}$$ for some $c_m$, see [@BLL Lemma 7.4.2].
Next, and (\[kass1\]) with $f,g\in X_{0,m}^{\alpha}$ and $\theta = w_m(x) = 1+x^m$ yield $$\|\tilde{{\mathcal{C}}}(f,g)\|_{[0,m]} \leq (1+2^m) K\|f\|_{[0,m]}^{(\alpha)}\|g\|_{[0,m]}^{(\alpha)}.\label{cest2}$$ On the other hand, assumption yields in a similar way $$\label{cest3}
\|{\mathcal{C}} f\|_{[0,m]} \leq 2^{m+1} K\|f\|_{[0,m]}^{(\alpha)} \|f\|_{[0,m]}.$$
Relation with weak solutions
============================
Weak solutions to are constructed as weak limits of solutions $f_r$ to the problem with the coefficients $a$ and $k$ modified as follows $$a_r(x) = \left\{\begin{array}{lcl} a(x)&\text{for}& x\leq r\\0&\text{for}& x>r,\end{array}\right.\qquad k_r(x,y) = \left\{\begin{array}{lcl} k(x,y)&\text{for}& x+y\leq r\\0&\text{for}& x+y>r,
\end{array}\right.
\label{arkr}$$ see e.g. [@ELMP03], or [@BLL Lemma 8.2.24].
Assume that the assumptions of Theorem \[fcth1\] are satisfied and $f$ is the solution to satisfying . If $(f_r)_{r>0}$ are approximate solutions defined above, then $$\lim\limits_{r\to \infty} f_r = f
\label{approx}$$ in $C([0,T], X^{\alpha}_{0,m})$ for any $T<\tau(f^{in})$.\[thappr\]
Let $0<r<\infty$. We observe that the solutions $f_r, r>0,$ to the truncated problem are unique and thus coincide with the solutions obtained by the semigroup method as follows. Let us denote by ${\mathcal{F}}_r$ the fragmentation expression restricted to $[0,r).$ Taking into account the fact that the spaces $X_{0,m,r} = L_1((0,r), w_m(x){\mathrm{d}}x)$ are invariant under the action of ${\mbox{$({G_{F_{0,m}}}(t))_{t \geq 0}$}}$ we find that the (uniformly continuous and analytic) semigroups generated by $F^{(r)}_{0,m} = F_{0,m}|_{X_{0,m,r}}$ coincide with ${\mbox{$({G_{F_{0,m}}|_{X_{0,m,r}}}(t))_{t \geq 0}$}}$ and thus all the estimates for are the same as for and independent of $r$. Also, is satisfied for $a_r$ and $k_r$ uniformly in $r$ and thus the solution $f_r$ satisfies all estimates of Theorem \[fcth1\] uniformly in $r$. Let $f$ be the classical solution to on the maximal interval $[0, \tau({{f}^{in}})),$ constructed in Theorem \[fcth1\], and let with $f^{(r)} = f|_{X_{0,m,r}}$. Introducing the error $e_r(t,x) = f^{(r)}(t,x)-f_r(t,x)$, we have $e_r(t,x)=0$ for $x>r$ and $e(0,x) =0$. Using the fact that both $f$ (see [@BaLa12a Theorem 2.2]) and $f_r$ satisfy pointwise, for $0<x<r$ we have $$\begin{aligned}
{\partial}_t e_r(t,x) &= -a(x)e_r(t,x) + {\int\limits_{0}^{\infty}}a(x) b(x,y) e_r(t,y){\mathrm{d}}x\label{e1}\\& -f^{(r)}(t,x){\int\limits_{0}^{\infty}}k(x,y)f(t,y){\mathrm{d}}y + \frac{1}{2}{\int\limits_{0}^{x}}k(x-y,y)f^{(r)}(t,x-y)f^{(r)}(t,y){\mathrm{d}}y{\nonumber}\\
&+f_r(t,x){\int\limits_{0}^{r-x}}k_r(x,y)f_r(t,y){\mathrm{d}}y - \frac{1}{2}{\int\limits_{0}^{x}}k_r(x-y,y)f_r(t,x-y)f_r(t,y){\mathrm{d}}y.{\nonumber}\end{aligned}$$ Then, we transform the coagulation part as follows $$\begin{aligned}
&\phantom{x} -f^{(r)}(t,x){\int\limits_{0}^{r-x}}k(x,y)f^{(r)}(t,y){\mathrm{d}}y -f^{(r)}(t,x){\int\limits_{r-x}^{\infty}}k(x,y)f(t,y){\mathrm{d}}y{\nonumber}\\& \phantom{x}+ \frac{1}{2}{\int\limits_{0}^{x}}k(x-y,y)f^{(r)}(t,x-y)f^{(r)}(t,y){\mathrm{d}}y{\nonumber}\\
&\phantom{x}+f_r(t,x){\int\limits_{0}^{r-x}}k(x,y)f_r(t,y){\mathrm{d}}y - \frac{1}{2}{\int\limits_{0}^{x}}k(x-y,y)f_r(t,x-y)f_r(t,y){\mathrm{d}}y{\nonumber}\\
&= -{\int\limits_{0}^{r-x}}k(x,y)(f^{(r)}(t,x)f^{(r)}(t,y)-f_r(t,x)f_r(t,y)){\mathrm{d}}y{\nonumber}\\&\phantom{x} + \frac{1}{2}{\int\limits_{0}^{x}}k(x-y,y)(f^{(r)}(t,x-y)f^{(r)}(t,y)-f_r(t,x-y)f_r(t,y)){\mathrm{d}}y {\nonumber}\\ &\phantom{x}-f^{(r)}(t,x){\int\limits_{r-x}^{\infty}}k(x,y)f(t,y){\mathrm{d}}y,\end{aligned}$$ where, for $0\leq x\leq r,$ $(x-y) + y = x\leq r$ so that $k_r(x-y,y) = k(x-y,y)$. Next we write $$f^{(r)}(t,x)f^{(r)}(t,y)-f_r(t,x)f_r(t,y) = f^{(r)}(t,y)e_r(t,x) + e_r(t,y)f_r(t,x),$$ and hence takes the form $$\begin{aligned}
{\partial}_t e_r(t,x) &= -a(x)e_r(t,x) + {\int\limits_{0}^{\infty}}a(x) b(x,y) e_r(t,y){\mathrm{d}}x\label{e2}\\& \phantom{x}-{\int\limits_{0}^{r-x}}k(x,y)(f^{(r)}(t,y)e_r(t,x) + e_r(t,y)f_r(t,x)){\mathrm{d}}y{\nonumber}\\& \phantom{x}+ \frac{1}{2}{\int\limits_{0}^{x}}k(x-y,y)(f^{(r)}(t,y)e_r(t,x-y) + e_r(t,y)f_r(t,x-y)){\mathrm{d}}y {\nonumber}\\ &\phantom{x}-f^{(r)}(t,x){\int\limits_{r-x}^{\infty}}k(x,y)f(t,y){\mathrm{d}}y =: F_{0,m} e_r(t,x) + E_1 e_r(t,x) + E_2(f^{(r)},f)(t,x).{\nonumber}\end{aligned}$$ Next, by , $$\begin{aligned}
\|E_1 e_r(t)\|_{0,m}&={\int\limits_{0}^{r}}w_m(x)\left|-{\int\limits_{0}^{r-x}}k(x,y)(f^{(r)}(t,y)e_r(t,x) + e_r(t,y)f_r(t,x)){\mathrm{d}}y\right. \\
&\phantom{x} + \left. \frac{1}{2}{\int\limits_{0}^{x}}k(x-y,y)(f^{(r)}(t,y)e_r(t,x-y) + e_r(t,y)f_r(t,x-y)){\mathrm{d}}y \right|{\mathrm{d}}x \\
&\leq L_0\|e_r(t)\|_{0,m}^{(\alpha)}(\|f_r(t)\|_{0,m}^{(\alpha)} + \|f^{(r)}(t)\|_{0,m}^{(\alpha)}) \leq L \|e_r(t)\|_{0,m}^{(\alpha)},\end{aligned}$$ where $L$ is a constant independent of $r$. Further, $$\begin{aligned}
\|E_2(f^{(r)},f)(t)\|_{0,m}&={\int\limits_{0}^{r}} w_m(x) f^{(r)}(t,x)\left(\,{\int\limits_{r-x}^{\infty}} k(x,y)f(t,y){\mathrm{d}}y \right){\mathrm{d}}x\\& \leq {\int\limits_{0}^{r}}\frac{ w_m(x) f^{(r)}(t,x)}{1+(r-x)^m}\left(\,{\int\limits_{r-x}^{\infty}} k(x,y)f(t,y)w_m(y){\mathrm{d}}y \right){\mathrm{d}}x\\
&\leq K\|f(t)\|_{0,m}^{(\alpha)} {\int\limits_{0}^{r}}\frac{ w_m(x)(1+a(x))^\alpha f^{(r)}(t,x)}{1+(r-x)^m}{\mathrm{d}}x\\
&\leq K\|f(t)\|_{0,m}^{(\alpha)}\left(\frac{ \|f(t)\|_{0,m}^{(\alpha)}}{1+ \left(\frac{r}{2}\right)^m} + \|f(t)-f^{(\frac{r}{2})}(t)\|_{0,m}^{(\alpha)}\right).\end{aligned}$$ Then, using the integral formulation of , $e_r(0)=0$ and identifying $G_{F^{(r)}_{0,m}}(t) = G_{F_{0,m}}(t)$ on $X_{0,m,r}$, we have $$e_r(t) = {\int\limits_{0}^{t}} G_{F_{0,m}}(t-s)E_1 e_r(s){\mathrm{d}}s + {\int\limits_{0}^{t}} G_{F_{0,m}}(t-s)E_2(f^{(r)},f)(s){\mathrm{d}}s.$$ Using the analyticity of , and the estimates above, we have, for any fixed $t<\tau({{f}^{in}})$, $$\begin{aligned}
\|e_r(t)\|_{0,m}^{(\alpha)} \leq c_1 {\int\limits_{0}^{t}} \frac{\|e_r(s)\|_{0,m}^{(\alpha)}{\mathrm{d}}s}{(t-s)^\alpha} + c_2\left(\frac{1}{1+ \left(\frac{r}{2}\right)^m} + \sup\limits_{0\leq s\leq t}\|f(s)-f^{(\frac{r}{2})}(s)\|_{0,m}^{(\alpha)}\right),\end{aligned}$$ where $c_1,c_2$ are uniform in $r$ and $t \in [0,\tau(f^{in}))$. Using now the Gronwall-Henry inequality in the form of [@Banasiak2019 Lemma 3.2], we obtain $$\|e_r(t)\|_{0,m}^{(\alpha)} \leq c_3\left(\frac{1}{1+ \left(\frac{r}{2}\right)^m} + \sup\limits_{0\leq s\leq t}\|f(s)-f^{(\frac{r}{2})}(s)\|_{0,m}^{(\alpha)}\right)
\label{finerr}$$ for some constant $c_3$ independent of $r$. We observe that for any $r_n\to \infty$, $(\|f(t)-f^{(\frac{r_n}{2})}(t)\|_{0,m}^{(\alpha)})_{n\in {\mathbb{N}}}$ is a monotone sequence of continuous functions converging to 0 (which is also a continuous function) and hence the convergence is uniform by Dini’s theorem. Thus $$\lim\limits_{r \to \infty} \|e_r(t)\|_{0,m}^{(\alpha)} = 0$$ uniformly in $t$ on any interval $[0,T]\subset [0,\tau({{f}^{in}}))$. Then also the approximate solutions $(f_r)_{r>0},$ extended by 0 to ${\mathbb{R}}_+$, converge to $f$ in $C([0,T],X_{0,m}^{(\alpha)})$.
Global solvability
==================
In this section we prove the following theorem
Assume that $a$ and $b$ satisfy , , and , and let $m > \max\{1,l\}$. If the coagulation kernel $k$ satisfies (\[kass2\]), then, for each $f^{in}\in X_{0,m,+}^{(\alpha)}$, the corresponding local nonnegative classical solution is global in time.
Let us fix some $m_0$ for which the assumptions of Theorem \[fcth1\] are satisfied. By a standard argument, we can assume that $f$ is defined on its maximal forward interval of existence $ [0, \tau(f^{in}))$. By [@Lun Proposition 7.1.8] (and the comment below it), if $\tau(f^{in})<\infty$, then $t\mapsto \|
f(t)\|_{[0,m_0]}^{(\alpha)}$ is unbounded as $t\to \tau(f^{in})$. Thus, to prove that $f$ is globally defined, we need to show that $t\mapsto \| f(t)\|_{[0,m_0]}^{(\alpha)}$ is *a priori* bounded on $[0,\tau(f^{in}))$. We use the following two observations. First, for $0 \leq m_1 \leq m_2,$ $X_{0,m_2}$ is continuously and densely imbedded in $X_{0,m_1}$ and hence the boundedness of $t\mapsto \Vert f(t) \Vert_{[0,m_2]}$ implies the same for $t\mapsto \Vert f(t)\Vert_{[0,m_1]}$ for all $ m_1 \in [0,m_2]$. Second, if Theorem \[fcth1\] holds for a specific $m_0$, then it is also valid in the scale of spaces $X_{0,m}$ with $m > \max\{1,l\},$ hence we can always choose an integer $m \geq \max\{2, m_0\}$ for which Theorem \[fcth1\] holds.
Next, we need to establish several inequalities valid in any space $X_{0,i,+}.$ By , (\[momestb\]) and assumption (\[kass2\]), we can deduce that for each $ i> 1$ and $f \in X_{0,i,+}$, $$\begin{aligned}
&&\int_{0}^{\infty}x^i{\mathcal{C}}f(x){\mathrm{d}}x =
\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\infty}((x+y)^i-x^i-y^i)k(x,y)f(x)f(y){\mathrm{d}}x{\mathrm{d}}y{\nonumber}\\
&&\leq K_i(\|f\|^{(\alpha)}_{[i-1]}\|f\|_{[1]} + \|f\|_{[i-1]}\|f\|_{[1]}^{(\alpha)}),
\label{Cmom1}\end{aligned}$$ where $K_i$ is a positive constant. For the case $i=1$ we have $$\int_{0}^{\infty}x{\mathcal{C}}f(x){\mathrm{d}}x = 0.$$ Turning now to the linear terms in , we recall from Theorem \[thnewchar0\], item 1., that, if $N_{m_0}(x)/x^{m_0} \geq \delta_{m_0}$ holds for some $m_0 > 1,$ $\delta_{m_0}$ and $x\geq 1$, then there is $\delta_i>0$ such that $N_i(x)/x^i \geq \delta_i>0$ for any $i > 1$ and $x\geq 1$. Hence, for $f \in D(A_{0,i})_+$, $$\begin{aligned}
&{\int\limits_{0}^{\infty}}({\mathcal{A}}f(x) +{\mathcal{B}}f(x))x^i{\mathrm{d}}x
=
-{\int\limits_{0}^{\infty}}N_i(x)a(x)f(x){\mathrm{d}}x {\nonumber}\\&= -{\int\limits_{0}^{1}}a(x) N_i(x)f(x){\mathrm{d}}x - {\int\limits_{1}^{\infty}} (a(x)+\omega)f(x) x^i \frac{N_i(x)}{x^i}{\mathrm{d}}x + \omega {\int\limits_{1}^{\infty}} f(x) N_i(x) {\mathrm{d}}x{\nonumber}\\
&\leq -{\int\limits_{0}^{1}}a(x) N_i(x)f(x){\mathrm{d}}x - \delta_i{\int\limits_{1}^{\infty}} (a(x)+\omega)f(x) x^i {\mathrm{d}}x + \omega {\int\limits_{1}^{\infty}} f(x) N_i(x) {\mathrm{d}}x{\nonumber}\\
&= - \delta_i \|f\|_{[i]}^{(1)} -{\int\limits_{0}^{1}}a(x) N_i(x)f(x){\mathrm{d}}x + \delta_i {\int\limits_{0}^{1}}(a(x)+\omega) x^if(x){\mathrm{d}}x + \omega{\int\limits_{1}^{\infty}} f(x) N_i(x) {\mathrm{d}}x{\nonumber}\\
&\leq - \delta_i \|f\|_{[i]}^{(1)} + \omega_1 \|f\|_{[i]},\label{fragest}\end{aligned}$$ where $\omega_1 = \delta_i\text{ess}\sup_{0\leq x\leq 1} a(x) + \omega(1+\delta_i)$. As for the coagulation term, for $i=1$ we have $$\int_{0}^{\infty}x({\mathcal{A}}f(x) +{\mathcal{B}}f(x)){\mathrm{d}}x = 0.$$ If we take ${{f}^{in}}$ with bounded support in $[0,\infty)$, then ${{f}^{in}} \in X^{(\alpha)}_{0,i,+}$ and, if additionally $i>\max\{1,l\}$, then the corresponding solution $(0,\tau(f^{in}))\ni t\mapsto f(t)$ is differentiable in any such $X^{\alpha}_{0,i}$ and thus in any $X_{i}, i\geq 0,$ or, in other words, any moment of the solution is differentiable. First, let us consider an integer $i \geq 2$. Then, from and , we have $$\begin{aligned}
\frac{d}{dt} \|f(t)\|_{[i]} &\leq& \omega_1 \|f(t)\|_{[i]} - \delta_i \|f(t)\|_{[i]}^{(1)} {\nonumber}\\
&& \quad + K_i(\|f(t)\|^{(\alpha)}_{[i-1]}\|f(t)\|_{[1]} + \|f(t)\|_{[i-1]}\|f(t)\|_{[1]}^{(\alpha)}).\label{firstmom}\end{aligned}$$ To simplify , we use the following auxiliary inequalities. For $i \geq 2$ and $1\leq r \leq i-1,$ we apply the Hölder’s inequality with $p=1/\alpha$ and $q =1/(1-\alpha)$ to obtain $$\begin{aligned}
\|f\|_{[r]}^{(\alpha)} &= \int_0^\infty x^r a_\omega^\alpha (x) f(x) {\mathrm{d}}x = \int_0^1 x^r a_\omega^\alpha (x) f(x) {\mathrm{d}}x + \int_1^\infty x^r a_\omega^\alpha (x) f(x) {\mathrm{d}}x{\nonumber}\\
&\leq c_a\int_0^1 x f(x) {\mathrm{d}}x + \int_1^\infty x^{(i-1)/q}f^{1/q}(x)x^{(qr-i+1)/q} a_\omega^\alpha (x) f^{1/p}(x) {\mathrm{d}}x{\nonumber}\\
&\leq c_a\| f\|_{[1]} + \left(\int_0^\infty x^{i-1}f(x){\mathrm{d}}x\right)^{1-\alpha}\left(\int_1^\infty x^{(r-(i-1)(1-\alpha))/\alpha} a_\omega(x) f(x) {\mathrm{d}}x\right)^{\alpha}{\nonumber}\\
&\leq c_a\| f\|_{[1]} + \|f\|_{[i-1]}^{1-\alpha}\left(\|f\|^{(1)}_{[i]}\right)^{\alpha}. \label{wl866}\end{aligned}$$ Note that the above derivation of uses the fact that $(r-(i-1)(1-\alpha))/\alpha \leq i-1 < i$ for $\alpha\in (0,1)$ and $r\leq i-1,$ and hence $$x^{(r-(i-1)(1-\alpha))/\alpha} \leq x^i, \qquad x \in [1,\infty).$$ Young’s inequality, with $p = 1/\alpha$ and $q = 1/(1-\alpha)$, then leads to $$\begin{aligned}
\|f\|^{(\alpha)}_{[i-1]}\|f\|_{[1]} &\leq c_a\|f\|^2_{[1]} + \|f\|_{[1]}\|f\|_{[i-1]}^{1-\alpha}\left(\|f\|^{(1)}_{[i]}\right)^{\alpha}{\nonumber}\\
&\leq c_a\|f\|^2_{[1]} + \|f\|_{[1]}\left((1-\alpha){\epsilon}^{1/(\alpha -1)}\|f\|_{[i-1]} + \alpha{\epsilon}^{1/\alpha}\|f\|^{(1)}_{[i]}\right)\end{aligned}$$ and $$\begin{aligned}
\|f\|_{[i-1]}\|f\|_{[1]}^{(\alpha)} &\leq c_a\|f\|_{[1]} \|f\|_{[i-1]}+ \|f\|_{[i-1]}^{2-\alpha}\left(\|f\|^{(1)}_{[i]}\right)^{\alpha}{\nonumber}\\
&\leq c_a\|f\|_{[1]} \|f\|_{[i-1]} + \left((1-\alpha){\epsilon}^{1/(\alpha-1)}\|f\|_{[i-1]}^{(2-\alpha)/(1-\alpha)} + \alpha{\epsilon}^{1/\alpha}\|f\|^{(1)}_{[i]}\right).\end{aligned}$$ We now apply these inequalities to the solution $t\mapsto f(t)$. Since $\|f(t)\|_{[1]} = \|f^{in}\|_{[1]}$ is constant on $[0, \tau(f^{in}))$, by choosing ${\epsilon}$ so that $\alpha{\epsilon}^{1/\alpha} K_i(\|f\|_{[1]} +1)\leq \delta_i$, we see that there are positive constants $D_{0,i}, D_{1,i}, D_{2,i}, D_{3,i}$ such that (\[firstmom\]) can be written as $$\frac{d}{dt} \|f(t)\|_{[i]} \leq D_{0,i}+ D_{1,i} \|f(t)\|_{[i]} + D_{2,i} \|f(t)\|_{[i-1]} + D_{3,i} \|f(t)\|^{(2-\alpha)/(1-\alpha)}_{[i-1]}. \label{firstmom1}$$ In particular, for $i = 2$ we obtain $$\frac{d}{dt} \|f(t)\|_{[2]} \leq D_{0,2}+ D_{1,2} \|f(t)\|_{[2]} + D_{2,2} \|f^{in}\|_{[1]} + D_{3,2} \|f^{in}\|^{(2-\alpha)/(1-\alpha)}_{[1]}, \label{firstmom2}$$ and thus $t \mapsto \|f(t)\|_{[2]}$ is bounded on $[0, \tau(f^{in}))$. Then we can use to proceed inductively to establish the boundedness of $t \mapsto \|f(t)\|_{[i]}$ for all integer $i$. Further, since for any $i>1$ we have $x^i \leq x$ for $x \in [0,1]$ and $x^i \leq x^{\lfloor i\rfloor +1}$ $$\|f\|_{[i]} \leq \|f\|_{[1]} + \|f\|_{[\lfloor i\rfloor +1]},$$ we find that all moments of the solution of order $i\geq 1$ are bounded on the maximal interval of its existence.
In the next step we show that also the moment of order 0 is bounded. In the proof we use the ideas of [@BLL Theorem 8.2.23] but while there the estimates were carried out for compactly supported approximating solutions, as in Theorem \[thappr\], here we will work with classical solutions defined for $x\in {\mathbb{R}}_+$ but only for $t\in [0, \tau(f^{in}))$.
Let us fix an integer $i>\max\{1,l\}.$ For the fragmentation term we have, as in , $${\int\limits_{0}^{\infty}} [{\mathcal{F}} f](t,x)x^i{\mathrm{d}}x \leq -\delta_i{\int\limits_{1}^{\infty}} a(x)f(t,x) x^i{\mathrm{d}}x.
\label{Fest}$$ Let us define $$\Phi(t) : = \|f(t)\|_{[i]} + \delta_i{\int\limits_{0}^{t}}{\int\limits_{1}^{\infty}} a(x) f(s,x)x^i{\mathrm{d}}x{\mathrm{d}}s.$$ We observe that, by induction, can be written as $$\frac{d \|f(t)\|_{[i]}}{dt} \leq D_{0,i}+ D_{1,i} \|f(t)\|_{[i]} + \Theta(t), \label{firstmom1a}$$ where $\Theta(t)$ is bounded on $[0,\tau({{f}^{in}}))$. Then $$\begin{aligned}
\frac{d\Phi(t)}{dt} &= \frac{d \|f(t)\|_{[i]}}{dt}+ \delta_i{\int\limits_{1}^{\infty}} a(x) f(t,x)x^i{\mathrm{d}}x \leq D_{0,i} + D_{1,i}\Phi(t) + \Theta(t)
\end{aligned}$$ and, integrating, $$\begin{aligned}
\Phi(t) &\leq e^{D_{1,i} t}\left(\Phi(0) + \frac{D_{0,i}}{D_{1,i}}(1-e^{-D_{1,i}t}) + {\int\limits_{0}^{t}}\Theta(s)e^{-D_{1,i} s}{\mathrm{d}}s\right)\end{aligned}$$ and we see that neither $\Phi,$ nor $$t\mapsto {\int\limits_{0}^{t}}{\int\limits_{1}^{\infty}} a(x) f(s,x)x^i{\mathrm{d}}x{\mathrm{d}}s
\label{Pt}$$ can blow up at $t=\tau(f^{in})$. Let us define $$P(t):= {\int\limits_{1}^{\infty}} a(x) f(s,x)w_i(x){\mathrm{d}}x{\mathrm{d}}s.$$ Using the fact that $${\int\limits_{0}^{\infty}} {\mathcal{C}} f(t,x){\mathrm{d}}x \leq 0$$ and, by , $$\begin{aligned}
{\int\limits_{0}^{\infty}} {\mathcal{F}} f(t,x){\mathrm{d}}x &\leq {\int\limits_{0}^{\infty}} (n_0(y)-1)a(y) f(t,y){\mathrm{d}}y \leq 2b_0{\int\limits_{0}^{\infty}} a(y) f(t,y) w_i(y){\mathrm{d}}y {\nonumber}\\
&\leq a_1{\int\limits_{0}^{1}} f(t,y) {\mathrm{d}}y + 2b_0 P(t),
\label{eqP1}\end{aligned}$$ on $[0,\tau({{f}^{in}})),$ where $a_1 = 2b_0 \text{ess}\sup_{y\in [0,1]}a(y)w_i(y) $, for the zeroth moment we have $$\frac{d}{dt}\|f(t)\|_{[0]} \leq a_1\|f(t)\|_{[0]} + 2b_0 P(t)$$ and hence $$\|f(t)\|_{[0]} \leq e^{a_1 t}\left(\|{{f}^{in}}\|_{[0]} + 2b_0{\int\limits_{0}^{t}} P(s){\mathrm{d}}s\right).$$ Now, using , $$\begin{aligned}
P(t)&={\int\limits_{0}^{t}}{\int\limits_{1}^{\infty}} a(x) f(s,x)w_i(x){\mathrm{d}}x{\mathrm{d}}s \leq 2{\int\limits_{0}^{t}}{\int\limits_{1}^{\infty}} a(x) f(s,x)x^i{\mathrm{d}}x{\mathrm{d}}s.
\label{Pt1}\end{aligned}$$ is bounded on $[0,\tau({{f}^{in}}))$ and hence $\|f(t)\|_{[0]}$ is bounded there as well.
To complete the proof, let $m_0>\max\{1,l\}$ be arbitrary. From the previous part of the proof, for ${{f}^{in}}$ with bounded support, the norm of the corresponding solution, $t \mapsto \|f(t)\|_{[0,m_0]},$ remains bounded on $[0,\tau({{f}^{in}}))$. We again use the properties of the analytic semigroup . The local solution $f$ satisfies the integral equation $$f(t) = G_{F_{0,m_0}}(t)f^{in} + {\int\limits_{0}^{t}}G_{F_{0,m_0}}(t-s)[{\mathcal{C}}f](s){\mathrm{d}}s$$ and thus, using and , $$\begin{aligned}
\|f(t)\|_{[0,m_0]}^{(\alpha)} &\leq C_1 \|f^{in}\|_{[0,m_0]}^{(\alpha)} + C_2{\int\limits_{0}^{t}} \frac{\|{\mathcal{C}} f(s)\|_{[0,m_0]}}{(t-s)^\alpha} {\mathrm{d}}s {\nonumber}\\
&\leq C_1 \|f^{in}\|_{[0,m_0]}^{(\alpha)} + 2^{m_0+1}KC_2{\int\limits_{0}^{t}} \frac{\|f(s)\|_{[0,m_0]}^{(\alpha)}\|f(s)\|_{[0,m_0]}}{(t-s)^\alpha} {\mathrm{d}}s{\nonumber}\\&\leq C_3 + C_4{\int\limits_{0}^{t}} \frac{\|f(s)\|_{[0,m_0]}^{(\alpha)}}{(t-s)^\alpha} {\mathrm{d}}s,\end{aligned}$$ where $C_3$ and $C_4$ are independent of time on $[0,\tau(f^{in}))$ on account of the boundedness of $t\mapsto \|f(t)\|_{[0,m_0]}$. Thus, using Gronwall–Henry inequality, for some constant $C_5$ independent of $t$, $$\|f(t)\|_{[0,m_0]}^{(\alpha)} \leq C_5, \quad t\in [0,\tau(f^{in}))$$ and hence $t \mapsto f(t)$ is a global classical solution in any $X_{0,m_0}^{\alpha}$ for which the assumptions of the theorem hold.
To prove the global existence of solutions emanating from any initial condition $f^{in}\in X_{0,m_0,+}^{\alpha}$ (and also to mild solutions) we observe that since the space of functions with bounded support is dense in $X_{0, m_0}^{\alpha}$ (respectively, $X_{0,m_0}$), a finite-time blow-up of such a solution would contradict the theorem on the continuous dependence of solutions on the initial data (which, in this case, follows from the Gronwall–Henry inequality, see [@Lun Theorem 7.1.2]) along the lines of the proof of [@BLL Theorem 8.1.1].
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abstract: 'By employing a nonlinear quantum kicked rotor model, we investigate the transport of energy in multidimensional quantum chaos. Parallel numerical simulations and analytic theory demonstrate that the interplay between nonlinearity and Anderson localization establishes a perfectly classical correspondence in the system, neglecting any quantum time reversal. The resulting dynamics exhibits a nonlinearly-induced, enhanced transport of energy through soliton wave particles.'
author:
- 'D. S. Brambila$^{1,2}$ and A. Fratalocchi$^1$'
title: 'Nonlinearly-enhanced energy transport in many dimensional quantum chaos'
---
Anderson localization is a fundamental concept that, originally introduced in solid-state physics to describe conduction-insulator transitions in disordered crystals, has permeated several research areas and has become the subject of great research interest [@sheng; @scheffold99:_local_or_class_diffus_of_light; @schwartz07:_trans_and_ander_local_in; @PhysRevLett.96.043902; @conti08:_dynam_light_diffus_three_dimen; @RevModPhys.57.287; @PhysRevLett.100.013906; @billy08:_direc_obser_of_ander_local; @roati08:_ander_local_of_non_inter; @efetov97:_super_in_disor_and_chaos; @PhysRevLett.49.509]. Theories and subsequent experiments demonstrated that disorder favors the formation of spatially localized states, which sustain diffusion breakdown and exponentially attenuated transmission in random media [@sheng]. Although many properties of wave localization are now well understood, several fundamental questions remains. Perhaps one of the most intriguing problem is related to the transport of energy. Intuitively, one can expect that disorder —by favoring exponentially localized stated— arrests in general any propagation inside a noncrystalline medium. However, the interplay between localization and disorder is nontrivial [@conti08:_dynam_light_diffus_three_dimen; @Molinari:12] and under specific conditions randomness can significantly enhance energy transport. In particular, it has been observed that quasi-crystals with multifractal eigenstates and/or material systems with temporal fluctuations of the potential (or refractive index), lead to anomalous diffusion in the phase space [@PhysRevLett.70.3915; @levi11:_disor_enhan_trans_in_photon_quasic; @PhysRevLett.39.1424; @PhysRevLett.82.4062]. This originates counterintuitive dynamics including ultralow conductivities [@PhysRevLett.70.3915], as well as the formation of mobility edges even in one dimensional systems [@PhysRevLett.82.4062]. All these studies focused on specific geometries and linear materials, while nothing is practically known about the role of nonlinearity in enhancing (or depleting) the transport of energy in disordered media. This problem acquires a strong fundamental character when refereed to the field of quantum localization. In this area, quantum-classical correspondences mediated by Anderson localization possess many implications in the irreversible behavior of time reversible systems, which are at the basis of a long standing physical dispute —i.e., the Loschmidt paradox [@Loschmidt]— as well as many fascinating quantum phenomena such as the time reversal of classical irreversible systems and the quantum echo effect [@PhysRevLett.101.074102; @PhysRevLett.61.659]. It has been argued, in particular, that microscopic chaos is at the basis of the irreversible entropy growth observed in classical systems [@gaspard98:_exper]. Time reversal, according to this interpretation, is only possible at the quantum level [@PhysRevLett.101.074102; @PhysRevLett.61.659] and sustained by Anderson localization, which breaks diffusive transport and suppresses the mixing ability of chaos [@PhysRevA.44.R3423]. However, when more dimensions are considered, numerical simulations predict that ergodicity is fully restored and diffusive transport settles is again, thus re-establishing the classical features of chaos and preventing quantum time reversal [@PhysRevLett.61.659]. Nevertheless, theoretical work reported to date considered only noninteracting systems, characterized by linear equations of motion. The Loschmidt paradox, conversely, involved the use of interacting atoms, whose interplay in the mean field regime is accounted by short and/or long ranged nonlinear responses [@PhysRevLett.94.160401; @PhysRevLett.100.240403; @griffin95:_bose_einst_conden]. Besides that, as pointed out in the literature [@PhysRevLett.101.074102], atoms interactions are of crucial importance in quantum localization and diffusion. A key question therefore lies in understanding the role of nonlinearity in transporting energy in multidimensional quantum chaos.\
In this Letter, we theoretically investigate this problem by employing both numerical simulations and analytic techniques. To pursue a general theory, we here consider the following two dimensional model: $$\begin{aligned}
\label{mod}
&i\frac{\partial\psi}{\partial t}+\nabla^2\psi+\int d\mathbf{r} R(\mathbf{r}'-\mathbf{r})\psi(\mathbf{r}')+U\psi\delta_T(t)=0,\end{aligned}$$ with $\mathbf{r}=(x,y)$, $\nabla^2=\partial^2/\partial x^2+\partial^2/\partial y^2$, $\delta_T=\sum_n\delta(t-nT)$ a periodic delta-function of period $T$, $R$ a general nonlinear response and $U(x,y)=\gamma(\cos x+\cos y)+\epsilon\cos(x+y)$ a two dimensional periodic potential with strength defined by $\epsilon$ and $\gamma$. Equation (\[mod\]) defines a two dimensional, nonlinear quantum kicked rotor: for $R=0$ it reduces to the linear quantum kicked rotator [@PhysRevLett.61.659] while for $U=0$ it corresponds to the 2D nonlinear Schrödinger equation (NLS), which represents a universal model of nonlinear waves in dispersive media [@griffin95:_bose_einst_conden]. In one dimension, conversely, Eq. (\[mod\]) generalizes the nonlinear model investigated in [@PhysRevA.44.R3423] with classical chaos parameter $K=2\gamma T$. Despite its deterministic nature, Eq. (\[mod\]) can be precisely mapped to the Anderson model with a random potential [@haake01:_quant_signat_chaos], and therefore furnishes a fundamental model for studying energy transport in random systems. The nonlinear response $n=\int d\mathbf{r} R(\mathbf{r}'-\mathbf{r})\psi(\mathbf{r}')$ is modeled as a nonlocal term following a general diffusive nonlinearity $(1-\sigma^2\nabla^2)n=|\psi|^2$, with nonlocality controlled by $\sigma$. When $\sigma=0$, the system response is local with $n=|\psi|^2$. For $\sigma\neq 0$, conversely, the system nonlinearity becomes long ranged with kernel given by $R(\mathbf{r})=\frac{1}{2\pi}K_0(\frac{\mathbf{r}}{\sigma})$, being $K_0$ the modified Bessel function of second kind. Diffusive nonlinearities are particularly interesting in the context of nonlinear optics, as they can be easily accessed in liquids [@PhysRevLett.102.083902; @fratalocchi:044101], as well as in Bose-Einstein Condensates (BEC), where they generalize previously investigated models [@PhysRevLett.95.200404; @Bang02].\
![ \[sim\] (Color Online). (a)-(b) spatial density $|\psi|^2$ distribution at (a) $t=0$ and at (b) $t=100 T$; (c) momentum diffusion $\langle P\rangle$ versus time in linear (dashed lines) and nonlinear (solid lines) conditions and for increasing coupling $\epsilon$. In the simulations we set $\sigma=0.2$, $\omega_0=0.3$, $A=4$ and $K=1.8$.](g1-01.eps){width="8.5cm"}
We begin our theoretical analysis by calculating the momentum diffusion $\langle P\rangle=\langle\psi|\frac{\hat{p}^2}{2}|\psi\rangle$ versus time, with $\hat{p}=\nabla/i$ the momentum operator and $\langle\psi| f|\psi\rangle=\int d\mathbf{r}f|\psi|^2$ the quantum average. Parallel numerical simulations are performed by a direct solution of (\[mod\]) with an unconditionally stable algorithm. In order for the field $\psi$ to explore the periodic potential $U$, we here consider wave packets whose spatial extension $\Delta r\ll 2\pi$. Figure \[sim\] summarizes our results obtained for $\sigma=0.2$, by launching at the input a gaussian beam $\psi=Ae^{-x^2/\omega_0^2}$ with waist $\omega_0=0.3$ and amplitude $A=4$ (Fig. \[sim\]a). The stochastic parameter $K$ has been set to $K=1.8>K^*$, above the stochastization threshold $K^*\approx 0.97$ where the linear classical uncoupled rotor exhibits diffusive transport in momentum space [@PhysRevLett.61.659]. For comparison, we also calculated the linear dynamics resulting from $R=0$ (Fig. \[sim\]b dotted line). As seen from Fig. \[sim\]b, the 2D nonlinear rotor behaves dramatically different with respect to its linear counterpart, demonstrating the strong role played by nonlinearity in the process. In particular, the linear system exhibits Anderson localization and diffusion suppression for $\epsilon=0$ (uncoupled condition), while for growing $\epsilon$ it shows a monotonically increasing sub-diffusion (Fig. \[sim\]b). In the nonlinear regime, conversely, Anderson localization is suppressed even for $\epsilon=0$, and the dynamics shows an erratic, random-like behavior that does not manifest any simple monotonic increase for growing values of $\epsilon$. These results are also significantly different from the nonlinear kicked rotor in one dimension [@PhysRevA.44.R3423], where nonlinearity was observed to induce localization effects.
![ \[lyap\] (Color Online). Positive Lyapunov exponent $\lambda$ versus coupling $\epsilon$, calculated for (a) Eqs. (\[sm0\]) and (b) Eq. (\[wa1\]). In the simulation we set $K=5$.](g1-02.eps){width="8.5cm"}
To theoretically understand this dynamics, we reduce the system to a nonlinear map modeling the nonlinear evolution of the ground state of Eq. (\[mod\]). This analysis is justified by the observation that the spatial field profile, despite the chaotic motion, is not significantly altered in time (Fig. \[sim\]a,c). Due to the nonintegrability of the 2D NLS equation, we found analytical expressions by a variational analysis [@PhysRevLett.95.200404; @Bang02]. In particular, we begin from the Lagrangian density $\mathcal{L}$ of Eq. (\[mod\]): $$\begin{aligned}
\label{lag}
\mathcal{L}=&\frac{i}{2}\bigg( \psi^*\frac{\partial\psi}{\partial t}-\psi\frac{\partial\psi^*}{\partial t} \bigg)-|\psi|^2\nonumber\\
&\times\bigg[U\delta_T+\frac{1}{2}\int d\mathbf{r} 'K(\mathbf{r}-\mathbf{r}')|\psi(\mathbf{r}')|^2\bigg]+|\nabla\psi|^2,\end{aligned}$$ and study the ground state for $U=0$ by the following Gaussian ansatz: $\psi=\sqrt{\frac{2P}{\pi}}\frac{e^{-r^2/a^2}}{a}$, defined by the power $P=\langle\psi|\psi\rangle$ and waist $a(t)$. By substituting the ansatz in (\[lag\]), after performing a variational derivative over $a$, we obtain a classical dynamics described by the following Hamiltonian $\mathcal{H}$: $$\begin{aligned}
\label{gs}
&\mathcal{H}=\frac{1}{2}\bigg(\frac{\partial a}{\partial t}\bigg)^2+\mathcal{V}, &\mathcal{V}=\frac{8}{a^2}-\frac{P}{2\pi\sigma^2}Z(a),\end{aligned}$$ with $Z(x)=e^x\Gamma(0,-x)$ and $\mathcal{V}$ acting as a potential for the one dimensional motion of $a$. The potential $\mathcal{V}$ has a bell shape profile that possesses a unique absolute minimum $V(a^*)$ for every combination of $P$ and $\sigma$. The fixed point $a(0)=a^*$ corresponds to a soliton wave of the system, which propagates in a translational fashion with fixed waist $a(t)=a^*$, while different initial values lead to a breather [@Snyder97] characterized by a periodic oscillation of $a$ in time. When the kicks are turned on, for $U\neq 0$, the dynamics of the ground state is perturbed by an addition of momentum $\mathbf{p}=(p_x,p_y)$, with a consequent translation of its center of mass. In order to model this dynamics, we considered the following general ansatz for the ground state evolution: $\psi=\sqrt{\frac{2P}{\pi}}\frac{e^{-(\mathbf{r}-\mathbf{r_0})^2/a^2+i\mathbf{p}(\mathbf{r}-\mathbf{r_0})/2T}}{a}$, with $\mathbf{p}(t)$, $a(t)$ and $\mathbf{r}_0(t)=[x_0(t),y_0(t)]$ Lagrangian variables whose equations of motion, after an integration from $nT$ to $(n+1)T$, are found to be: $$\begin{aligned}
\label{sm0}
&\mathbf{p}_{n+1}=\mathbf{p}_n-[\gamma_n\mathbf{g}+\epsilon_n\mathbf{u}\sin(x_0+y_0)],\nonumber\\
&\mathbf{r}_{n+1}=\mathbf{r}_n+\mathbf{p}_{n+1},\end{aligned}$$ with $\gamma_n=K e^{-\frac{a_n^2}{8}}$, $\epsilon_n=2\epsilon T e^{-\frac{a_n^2}{4}}$, $\mathbf{g}=[\sin x_0,\sin y_0]$, $\mathbf{u}=[1,1]$, $f_n\equiv f(nT)$ and $a_{n+1}$ calculated from the integration of the following equation: $$\begin{aligned}
\label{wa1}
\frac{\partial^2 a}{\partial t^2}=&-\frac{\partial\mathcal{V}}{\partial a}+\delta_Te^{-\frac{a^2}{8}}\nonumber\\
&\times[\gamma(\cos x_0+\cos y_0)+2e^{-\frac{a^2}{8}}\epsilon\cos(x_0+y_0)].\end{aligned}$$ Equations (\[sm0\]) can be regarded as a variant of the four dimensional standard map, which is randomized by time dependent coupling parameters $\gamma_n$ and $\epsilon_n$. The latter depend on Eq. (\[wa1\]), which represents the motion of a two dimensional nonlinear kicked rotor. The system possesses an $a$ dependent chaos parameter, given by $K_a=e^{-a^2/8}\gamma T$. For $K>K^*$, Eq. (\[wa1\]) is fully chaotic and can be regarded as an external noise source to Eqs. (\[sm0\]), increasing the mixing of the overall system [@PhysRevLett.61.655]. To highlight such a dynamics, we plot in Fig. \[lyap\]a and Fig. \[lyap\]b the positive Lyapunov exponent $\lambda$ [@OTT] calculated for Eqs. (\[sm0\]) and Eq. (\[wa1\]), respectively. As seen in Fig. \[lyap\]a, Eqs. (\[sm0\]) show a strong hyperchaotic behavior, with two positive Lyapunov exponents whose largest value grows linearly with $\epsilon$. Fig. \[lyap\]b, conversely, displays the chaotic nature of wave packet extension $a$, whose Lyapunov coefficient $\lambda$ increases significantly fast (quadratically) with coupling.
![ \[diff\] (Color Online). (a) Momentum diffusion $\langle\bar P\rangle$ versus time calculated from Eq. (\[mod\]) (solid lines), Eqs. (\[sm0\])-(\[wa1\]) (diamond markers) and Eq. (\[mod\]) in linear regime (dashed lines); (b) diffusion coefficient $D$ versus coupling $\epsilon$. In the simulations we set $\sigma=0.2$ and $K=5$.](g1-03.eps){width="8.5cm"}
We investigate the diffusion in momentum $\mathbf{p}$ by observing that above the stochastization threshold $K>K^*$, the change in momentum $\Delta\mathbf{ p}=\mathbf{p}_{n+1}-\mathbf{p}_n\propto K$ becomes large compared to $\pi$. The classic position $\mathbf{r}_n$, which is taken modulo $2\pi$, can be treated a random process, statistically uncorrelated in time and uniformly distributed in $[-\pi,\pi]$. The diffusion coefficient $D$ is therefore evaluated as follows: $$\begin{aligned}
\label{deq}
&D=\bigg\langle\frac{\Delta\mathbf{p}_n^2}{2}\bigg\rangle=\frac{K^2}{2}\langle \sin x_0^2\rangle\langle e^{-a_n^2/8}\rangle+2\epsilon^2 T^2\langle e^{-a_n^2/4}\rangle\nonumber\\
&\times \langle \sin (x_0+y_0)^2\rangle=\frac{K^2}{4}\langle e^{-a_n^2/8}\rangle+\epsilon^2 T^2\langle e^{-a_n^2/4}\rangle,\end{aligned}$$ To evaluate the averages $\langle e^{-a_n^2/8}\rangle$ and $\langle e^{-a_n^2/4}\rangle$, we can consider $a$ as a random variable (due to its chaotic motion in the phase-space), uniformly distributed between its oscillation extrema $a_{min}$ and $a_{max}$: $$\begin{aligned}
\label{expa}
\langle &e^{-a_n^2/\tau^2}\rangle=\frac{\sqrt{\pi}\tau}{2\Delta}\nonumber\\
&\times\bigg[\mathrm{erf}\bigg(\frac{a_{max}}{\tau}\bigg)-\mathrm{erf}\bigg(\frac{a_{min}}{\tau}\bigg)\bigg]=1+O(a_{max}^3/\tau^3)\end{aligned}$$ being $\Delta=a_{max}-a_{min}$ and having expanded the error functions up to second order, due to the smallness of their arguments $a/\tau < 1$. By substituting (\[expa\]) into (\[deq\]), we obtain the diffusion coefficient, which reads as follows: $$\label{ddd}
D=\frac{K^2}{4}+\epsilon^2 T^2$$ Equation (\[ddd\]) allows to derive interesting properties for the nonlinear dynamics of Eq. (\[mod\]). In particular, the quantum average $\langle P\rangle$ results from an hyperchaotic system described by a four dimensional standard map with random coefficients, and each realization manifests itself as a random walk in Fig. \[sim\]b. The map diffusion rate is identical to the momentum diffusion of the classical linear rotor [@PhysRevLett.61.659], hence, an additional average (in time or over an ensemble of input conditions) re-establishes a perfect classical correspondence for every coupling $\epsilon\ge 0$. It is worthwhile observing that the classical correspondence in the multidimensional linear quantum rotor is manifested only for very high coupling $\epsilon$, and in general the quantum diffusion $\langle P\rangle$ follows a fractional behavior with $\langle P\rangle\propto t^{\beta<1}$ (see e.g., [@PhysRevLett.61.659] or Fig. \[sim\]b dashed lines). As a result, the linear quantum rotor sub-diffuses at a slower rate than its classical counterpart. Conversely, Eq. (\[ddd\]) predicts a perfect classical correspondence for every coupling $\epsilon$, which is re-established thanks to nonlinear effects. In order to demonstrate this dynamics, we performed extensive numerical simulations from Eq. (\[mod\]) and calculated the average diffusion through a quantum average followed by an average over different input conditions $\langle\bar P\rangle=\int D\psi\big\langle\psi\big |\frac{\hat{p}^2}{2}\big|\psi\big\rangle$. Figure \[diff\]a summarizes our results obtained for $K=5$, $\sigma=0.2$ and by considering an initial wave packet composed by a Gaussian beam with waist $\omega_0=0.3$ and amplitude $A=4$. In complete agreement with Eqs. (\[sm0\])-(\[ddd\]), we observe a diffusive behavior $\langle\bar P \rangle\propto t$ for every $\epsilon\ge 0$ (Fig. \[diff\]a solid lines and diamond markers), whose rate is significantly faster than the linear subdiffusive dynamics (Fig. \[diff\]a dashed lines). We can therefore conclude that nonlinearity favors the energy transport in the system, increasing diffusion through nonlinear wave-particles that are faster than their linear counterparts. This result also highlights the intimate connection between the wave-particle aspects of nonlinear waves, whose quantum-classical characters cannot be singularly broken, but conversely emerge naturally after averaging over the corresponding degree of freedom. To further verify the scaling dependence predicted by Eq. (\[ddd\]), we calculated the diffusion coefficient $D$ of the nonlinear system for increasing $\epsilon$ (Fig. \[diff\]b). In perfect agreement with our theory, we observe a quadratic behavior versus the coupling parameter $\epsilon$.\
In conclusion, motivated by the large interest in the study of energy transport in complex media, we investigated the quantum-classical correspondences in many-dimensional quantum chaos. In particular, we employed a two-dimensional, nonlinear quantum kicked rotor (NQKR) and study the role of nonlinearity in enhancing or depleting energy diffusion and quantum-classical correspondences. We analytically tackled the problem by a variational analysis, reducing the dynamics to a four-dimensional standard map with random coefficients. In such an hyperchaotic system, a perfect classical correspondence is established by nonlinearity and an enhanced diffusion is observed due to solitons wave-particles, which are able to diffuse energy at a faster rate with respect to linear waves. These results show that quantum time reversal of classical irreversible systems is completely prevented in many dimensions, and demonstrate that nonlinearity can be effectively employed to increase the transport of energy in complex media. This work is expected to stimulate further theory and experiments in the broad area dealing with quantum chaos and energy transport phenomena.\
A. Fratalocchi thanks S. Trillo for fruitful discussions. We acknowledge funding from KAUST (Award No. CRG-1-2012- FRA-005).
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|
**The quantum anharmonic oscillator in the Heisenberg picture and multiple scale techniques**
\
[Université Montpellier II, F–34095 Montpellier Cedex 5, France ]{}
Multiple scale techniques are well-known in classical mechanics to give perturbation series free from resonant terms. When applied to the quantum anharmonic oscillator, these techniques lead to interesting features concerning the solution of the Heisenberg equations of motion and the Hamiltonian spectrum.
PACS: 02,03.65,04.25.\
Keywords: Multiple Scale Techniques, Quantum Mechanics, anharmonic oscillator.
0. Introduction {#introduction .unnumbered}
===============
Multiple scale techniques (MST) originated in Poincaré works have been developed by many authors, mainly in solving (partial) differential equations related to physical problems in celestial mechanics or in fluid dynamics. All these methods have a common mathematical purpose: to avoid resonances or secularities appearing in the usual or conventional perturbative theory. From a more physical point of view, one can see the MST as adaptable methods that feel the underlying physical phenomena in order to fit them. In other words, the usual perturbative theory tends to impose its choices while MST are flexible and compose with the real medium.
In this work, we apply one of the various MST to the quantum anharmonic oscillator. Such studies have been initiated by Bender and Bettencourt (B&B) in two recent papers [@BB1; @BB2]. They have found that the non resonance condition leads to a ‘"mass renormalisation‘" of the oscillator and - as a by-product - to the energy level differences of the quantum oscillator. This pioneering work was limited to the first non trivial order in MST perturbation of the coupling constant of the anharmonicity. The aim of the present paper is to extend this early study in several directions. First, we introduce an alternative framework, which turns out to be more convenient than the B&B one for performing higher order calculations. Secondly, it turns out that we are able to obtain the energy levels themselves at these perturbative orders. In the third point, we show that the diagonalization of the Hamiltonian is rather easy once the free Hamiltonian has been recast in an appropriate form. Finally, this approach leads to a natural and elegant method to find perturbatively the eigenvalues of the full Hamiltonian, far away from the original MST concept.
The paper is organized as follows. In the first section, although the classical anharmonic oscillator is studied in details in many textbooks [@JK; @N], we sketch some relevant points in order to further clarify the differences and the analogies between the classical and the quantum cases. In the second section we explain our framework and we work out the two first orders in MST perturbation, that includes the full solution of the Heisenberg equations and the energy levels. The third section is devoted to general arguments showing that the method is compelled to work at any order, due to its connection with a certain unitary transformation which diagonalizes the Hamiltonian of the anharmonic oscillator. We postpone to an appendix some explicit results: the solutions of the Heisenberg equations of motion and the energy levels of the full Hamiltonian, up to the order 6 included. .
1. The Classical case {#the-classical-case .unnumbered}
=====================
The classical anharmonic oscillator (CAO) is probably one of the most popular examples where the conventional and MST perturbative theories lead to obvious differences. Often, one speaks of Duffing equation instead, although this equation is nothing but the equation of motion of the CAO. To be precise, the Duffing equation is a second order non linear equation in the time variable, the solution of which being the position of the CAO. Starting from the CAO Lagrangian ( in units where the mass parameter is 1 ) $$L(q,\dot{q})=\dot{q}^{2}/2-\omega ^{2}q^{2}/2-gq^{4}$$ one readily gets from the Euler-Lagrange equation:
$$(e):\ddot{q}+\omega ^{2}q+4gq^{3}=0.$$
The usual formal perturbation expansion reads
$$q(t)=\sum_{n=0}^{\infty} g^{n}q_{n}(t),$$ ( with some initial conditions, say $ q(0)=Q $ and $ \dot{q}(0)=0 $ ), and the first equations one obtains from (e) are :
$$(e_{0}):\ddot{q}_{0}+\omega ^{2}q_{0}=0 ,$$
$$(e_{1}):\ddot{q}_{1}+\omega ^{2}q_{1}=-4q^{3}_{0} .$$
Then the frequency of the solution of the homogeneous part of ($e_1$) coincides with the frequency of $q_0(t)$ = $Q cos\omega t$, which generates a resonance in the solution $q_1(t)$ of the full equation ($e_1$):
$$q_{1}(t)= \frac{Q^{3}}{8\omega^2} (cos3\omega t - cos\omega t- 12 \omega t sin\omega t).$$
Hence $q_1(t)$ is unbounded and the truncated expansion $ q_{0}(t)+gq_{1}(t) $ cannot be an acceptable approximation of $q(t)$ for times $t$ larger than $\omega/Q^2 g$, however small $g$ may be. The flaw is even worse at the higher orders. It is obvious on this simple example that the perturbative solution develops spurious behaviour which is absent in the exact solution. Indeed, it is well known that the exact solution is bounded and periodic.
The main idea of MST for dealing with this problem is the introduction of new variables, independent and appropriate, and we refer to textbooks for an extensive review of the various possibilities. Here we concentrate on the anharmonic oscillator. Some methods take into account ab initio that the circular functions play a major role. For instance, in the Poincaré method, one looks for sine and cosine solutions whose argument is still $ \omega t $ but where $ \omega $ is now an arbitrary function of the coupling constant, actually $ \omega =\Sigma g^{n}\omega _{n} $. Then one has to find the $ \omega _{n} $’s , order by order, to discard the resonance. We do not insist on the application of these methods to the CAO because we believe they are not suitable for the quantum case. Another class of MST seems to be of a larger use, since there is no ‘"prerequisite‘" in these methods. The MST we will use, also called Derivative Expansion Method, belongs to this class: it promotes the time variable to be a function of the coupling constant, namely $ t_{n}=g^{n}t. $ Actually, the method is not so rough and one first extends the function depending on $ t $ to an ‘"extended‘" function depending on all the variables $ t_{n} $ ($n=0,1,2,...$) assumed to be independent [@S]. So, one introduces a new position function $ Q(T,g) $ depending on the collection $ T=\{t_{0},t_{1},t_{2},...\} $ of independent variables $ t_{n} $. This function is considered as an extension of the true position in the Lagrange formalism, which is recovered by restricting $ Q $ to the section $ t_{n}=g^{n}t $ of the $ T- $space: $q(t,g)=Q(T,g)|_{t_{n}=g^{n}t}$.
Then, forgetting temporarily any reference to the coupling constant in these $ t_{n} $ variables, one expands in power of $ g $ the new position function $ Q $ : $$Q(T,g)=\sum_{n=0}^{\infty} g^{n}Q_{n}(T).$$
One obtains from ($e$) the following set of equations, limited here at the three first orders :
$$D_{0}^{2}Q_{0}(T)+\omega ^{2}Q_{0}(T)=0\, ,$$
$$D_{0}^{2}Q_{1}(T)+\omega ^{2}Q_{1}(T)=-2D_{0}D_{1}Q_{0}(T)-4Q^{3}_{0}(T)\, ,$$
$$D_{0}^{2}Q_{2}(T)+\omega ^{2}Q_{2}(T)=-(D_{1}^{2}+2D_{0}D_{2})Q_{0}(T)-2D_{0}D_{1}Q_{1}(T)
-12Q^{2}_{0}(T)Q_{1}(T),$$
using $ d/dt=\Sigma _{n}g^{n}D_{n} $ , $ D_{n}=\partial /\partial t_{n} $ .
The basic principle of the method now consists in adjusting the $ t_{1} $ dependence of $Q(T)$ so as to eliminate the secularity in the second equation, next the $ t_{2} $ dependence of $Q(T)$ so as to eliminate the secularity in the third equation, and so on. We shall not work out the derivation here (it can be found for example in ref 3 or 4) and we merely give the solution up to the second order in $ g $ in its final form, for further classical versus quantum discussions :
$$q(t,g)=\frac{a}{2} [\exp(-i(\Omega t+b))+
\frac{\lambda}{8} (1- \frac{21 \lambda}{8}) \exp(-3i(\Omega t+b)) +
\frac{\lambda^2}{64}
\exp(-5i(\Omega t+b))] + C.C.,$$ where :
$$\label{1}
\Omega =\omega (1+\frac{3\lambda}{2} -\frac{15 \lambda^2}{16} ),
\lambda = \frac{g a^2}{\omega^2}$$
and a and b are two real integration constants fixed by the initial conditions (here unspecified).
Since the (perturbative) energy is conserved, it can be computed most easily by choosing $t=-b/\Omega $ or $ t=(\frac{\pi}{2}-b)/\Omega $ in $q(t,g)$ :
$$\label{2}
E_{c}=\frac{a^{2}\omega ^{2}}{2} (1 +\frac{9 \lambda}{4} +
\frac{25 \lambda^{2}}{64}) +O(\lambda^{3}).$$
We conclude this first section by a few comments. As far as we know, all the multiple scale techniques dealing with the secularities of the classical anharmonic oscillator are successful. However this is not a general feature, and some methods are not suitable for certain problems. Moreover it is absolutely not our purpose to discuss on a rigorous basis the mathematical aspects of the secular or non secular perturbative expansions.
2. The Quantum case : Derivation {#the-quantum-case-derivation .unnumbered}
================================
The quantum anharmonic oscillator (QAO) has been studied in the paper of B&B through the Heisenberg equation of motions for the relevant operators and we will follow this method. The main difference between the work of B&B and ours is that we will use the creation and annihilation operators to manage the problem of removing the secularities. At first sight the gain in doing this choice is not obvious and perhaps not essential. Moreover one can detect in the B&B paper an indication pointing to this direction. Let us look at the couple of equations (21) in their work [@BB1], which can be written as :
$$D_{1}Y=-CX-XC\, \, \textrm{and}\, \, D_{1}X=CY+YC$$ where $ X $ and $ Y $ are $ t_{1} $ dependent, self-adjoint operators while $ C $ is a constant, self-adjoint operator. The authors proceed with some arguments ‘"suggesting‘" the form of the solution, with the help of Weyl ordered products and Euler polynomials to deal with these equations. Of course, it seems difficult, or at least hazardous, to generalize at high orders a ‘"suggestive‘" method, which could be seen as a reminiscence of the Poincaré method, but we have more convincing arguments to leave this path. First, using $Z = X + i Y$, the previous couple of equations reduces to the single equation : $$D_{1}Z(t_{1})=-i(Z(t_{1})C+CZ(t_{1}))$$ whose solution is $ Z(t_{1})=\exp(-iCt_{1})Z(0)\exp(-iCt_{1}) $, as it is easy to check. The operator $ Z(t_{1}) $ is closely related to the creation/annihilation operators. Once derived the expression of the creation/annihilation operators, it is not necessary, in order to proceed further, to write down the position operator. Indeed, almost all the informations, the ‘"mass renormalization‘" effect and the difference of energy levels, are already contained in the argument of the exponentials. Secondly, the Heisenberg equations in terms of creation/annihilation operators are first order differential equations in place of the second order one for the position operator, which simplifies noticeabily the whole procedure. To be honest, one has the disadvantage to carry both creator and annihilator, but this is not a serious complication. Lastly, there appears also a large variation between the B&B works and ours in the status of the initial conditions: we do not use these conditions as in the classical case, which is the way taken by B&B. This point will become obvious throughout our study.
We start with the QAO Hamiltonian $ H $ written in terms of the momentum $ p $ and position $ q $ operators in convenient units $ (\hbar =\omega =1) $ : $ H=p^{2}/2+q^{2}/2+gq^{4} $, where $ g $ is assumed to be a ‘"small‘" (positive) coupling constant. Whithin the Heisenberg picture, the dynamics is governed by the equations :
$$\dot{q}= i[H,q]\, ,\, \dot{p}= i[H,p],$$
supplemented by the canonical commutation relation $ [q,p]=i $, valid at all times. The Heisenberg equations give : $ \dot{q}=p $ and $ \dot{p}=-q-4gq^{3} $. Writing as usual $ q=(a+a^{\dag })/\sqrt{2} $ and $ p=-i(a-a^{\dag })/\sqrt{2} $, the Hamiltonian becomes:
$$\label{3}
H(a,a^{\dag },g)=1/2+a^{\dag }a+g(a+a^{\dag })^{4}/4$$
together with : $$\label{4}
[a(t,g),\, a(t,g)]=1\, ,\, \, \forall t,$$ where, to avoid possible confusion later on, we have kept track of the variables $t$ and $g$.
The Heisenberg equation for the annihilator :
$$\dot{a}(t,g)=i[H(a(t,g),a^{\dag }(t,g),g),a(t,g)],$$ reads, in our case :
$$\label{5}
\dot{a}(t,g)=-i(a(t,g)+g(a(t,g)+a^{\dag }(t,g))^{3}).$$
Since the Hamiltonian is conserved, its formal solution is:
$$a(t,g)=\exp(iH(a(0),a^{\dag }(0),g)t)a(0)\exp(-iH(a(0),a^{\dag }(0),g)t),$$ with $ a(0)\equiv a(0,g) $.
We now turn on the formal series of the multitime perturbative expansion, similar to that used in the classical case. First one introduces an operator valued function $ A(T,g) $ depending on the collection $ T $ of independent variables $ t_{j} $. This function is considered as an extension of the true annihilation operator in the Heisenberg picture, which is recovered through the restriction: $$\label{6}
a(t,g)=A(T,g)|_{t_{j}=g^{j}t}.$$
Then the time derivative becomes :
$$\dot{a}(t,g)=\sum _{n\geq 0}g^{n}D_{n}A(T,g)|_{t_{j}=g^{j}t} .$$
Secondly, $ A(T,g) $ is expanded as :
$$\label{7}
A(T,g)=\sum _{n\geq 0}g^{n}A_{n}(T).$$
As for the initial conditions to be associated with the equation of motion (5), one notices that $$\label{8}
a(0,g)=\sum _{n\geq 0}g^{n}A_{n}(0).$$
This forces us to choose between two possible starting viewpoints :\
either a) : $ a(0,g) $ is taken as independent of $ g $ , which implies $$\label{9}
A_{n}(0)=0,\forall n\geq 1,$$ or b) : the previous condition is not imposed, in which case the initial values of $ a(t,g) $ must be considered as a function of $ g $.
It turns out that both approaches lead to consistent multitime expansions. In fact, the choice a) was (implicitely) adopted by B&B. However, these authors did not extend their analysis beyond the first order. In this paper, we rather follow the procedure b), which we found much more convenient, and in a sense, more natural.
The equation of motion for $ a(t,g) $ gives us the following infinite system for the $ A_{n}(T) $’s :
$$\label{10}
D_{0}A_{n}+iA_{n}=-\sum ^{n-1}_{m=0}D_{n-m}A_{m}-i\sum _{m,r,s,>,0\atop m+r+s=n-1}Q_{m}Q_{r}Q_{s}\, \, \, \, (n=0,1,2..)$$
where $ Q_{n} $ =$ A_{n}+A^{\dag }_{n} $, or explicitely :\
$ D_{0}A_{0}+iA_{0}=0, $ (10.a)
$ D_{0}A_{1}+iA_{1}=-D_{1}A_{0}-iQ_{0}^{3}, $ (10.b)
$ D_{0}A_{2}+iA_{2}=-(D_{2}A_{0}+D_{1}A_{1})-i(Q_{0}^{2}Q_{1}+Q_{0}Q_{1}Q_{0}+Q_{1}Q_{0}^{2}), $ (10.c)\
etc...
A simple check shows us that [**any**]{} formal solution of (10) generates via (6) and (7) a formal solution $ a(t,g) $ of (5). In particular, this implies that, for such a solution, $ [A(T,g),A^{\dag }(T,g)]|_{t_{j}=g^{j}t} $ is independent of $ t $. Of course, this does no mean yet that $ [A(T,g),A^{\dag }(T,g)] $ is independent of $ T $, allowing us to impose : $$\label{11}
[A(T,g),A^{\dag }(T,g)]=1\, ,\, \forall T\, ,$$ in order to insure the canonical commutation relation (4) . However, one can look for [**those**]{} solutions of (10) which are subjected to the stronger condition (11), if such solutions do exist indeed, i.e. if no inconsistencies or obstructions arise in their iterative construction. Together with (7), this entails :
$$\label{12}
\left\{ \begin{array}{ll}
\displaystyle [A_{0}(T),A^{\dag }_{0}(T)]=1 & \\
& \forall T\\
\sum _{m=0}^{n}[A_{m}(T),A^{\dag }_{n-m}(T)]=0,\, \, n\geq 1 &
\end{array}\right.$$
We are now ready to construct step by step the resonance-free solution of the problem. To zeroth order, the equation (10.a) and the first equation (12) yield :
$$\label{13}
A_{0}(T)=A_{01}(T_{1})\exp(-it_{0})$$
with $$\label{14}
[A_{01}(T_{1}),A^{\dag }_{01}(T_{1})]=1,\, \forall T_{1},$$ and the notation : $ T_{k}=\{t_{k},t_{k+1},...\}, (k=1,2,...) $.
Then, one can proceed to the first order step by inserting eq (13) into eq (10.b) :
$$D_{0}A_{1}+iA_{1}=-(D_{1}A_{01}+i(A_{01}^{2}A^{\dag }_{01}+A_{01}A^{\dag }_{01}A_{01}+A^{\dag }_{01}A_{01}^{2}))\exp(-it_{0})-i(A_{01}^{3}exp(-3it_{0})$$
$$\label{15}
+A^{\dag ^{3}}_{01}\exp(+3it_{0})+(A^{\dag 2}_{01}A_{01}+A^{\dag }_{01}A_{01}A^{\dag }_{01}+A_{01}A^{\dag 2}_{01})exp(+it_{0})).$$
Before integrating this equation, one has to get rid of the first resonant term on the right hand side, which would produce a contribution growing linearly with $ t_{0}(=t) $. This leads to the condition :
$$\label{16}
D_{1}A_{01}=-i(A_{01}^{2}A^{\dag }_{01}+A_{01}A^{\dag }_{01}A_{01}+A^{\dag }_{01}A_{01}^{2})$$
which will fix the $ t_{1} $ dependence of $ A_{01} $.
To do that, let us first introduce the self-adjoint operator $ N(T)=A^{\dag }_{0}(T)A_{0}(T) $. Thanks to (13) and its creator version, $ N(T) $ is only $ T_{1} $ dependent: $ N(T)=A^{\dag }_{01}(T_{1})A_{01}(T_{1}) $. Moreover as a consequence of (14), $ A_{01}(T_{1})N(T_{1})=(N(T_{1})+1)A_{01}(T_{1}) $. Lastly, from (16) , one observes that $ D_{1}N(T_{1})=0 $. Thus $ N $ is also independent of $ t_{1} $ and (16) can be now written in the tractable form :
$$D_{1}A_{01}=-3iA_{01}(T_{1})N(T_{2}),$$ which produces : $$A_{01}(T_{1})=A_{02}(T_{2})\exp(-3iN(T_{2})t_{1}).$$
This allows us to write down the first order annihilation operator :
$$\label{17}
A_{0}(T)=A_{02}(T_{2})exp(-i(t_{0}+3N(T_{2})t_{1})).$$
At the same time, (14) becomes : $$\label{18}
[A_{02}(T_{2}),A^{\dag }_{02}(T_{2})]=1,\forall T_{2}.$$
One can now come back to the form of (15) exempted of secularity to obtain its general solution :
$$A_{1}(T)=A^{3}_{01}(T_{1})\exp(-3it_{0})/2-A_{01}^{\dag 3}(T_{1})exp(+3it_{0})/4-3N(T_{2})A^{\dag }_{01}(T_{1})\exp(+it_{0})/2$$
$$\label{19}
+C_{1}(T_{1})exp(-it_{0}).$$
where the operator $ C_{1}(T_{1}) $ is an integration “constant”. The latter must be so adjusted, if possible, as to insure that the second equation (12) ,
$$\label{20}
[A_{0}(T),A^{\dag }_{1}(T)]+[A_{1}(T),A^{\dag }_{0}(T)]=0,$$
be fulfilled at all times $ T $. Here, it turns out that (20) is satisfied by taking simply $ C_{1}(T_{1})=0 $. One ends up with :
$$\label{21}
A_{1}(T)=A^{3}_{0}(T)/2-A^{\dag 3}_{0}(T)/4-3N(T_{2})A^{\dag }_{0}(T)/2$$
and the first order step is complete.
Before going further, some comments are in order. First, writing the position operator $ q_{0}+gq_{1} $, one notes that the coefficients of $ \exp(\pm it_{0}) $ in $ q_{0} $ get corrections coming from $ q_{1} $. It means, in the position formalism, the scheme used by B&B, that one would have to take into account the solutions of the homogeneous second order differential equation. Secondly, it appears in (21) that any power of $ exp(+it_{0}) $ (resp. $ \exp(-it_{0}) $) is multiplied by the same power of $ A^{\dag }_{01}(T_{1}) $ (resp. $A_{01}(T_{1}) $). Such a correspondance, which is specific to our way of managing the initial conditions, will be a guide throughout our study. Lastly the solution of the homogeneous equation in the classical case is different. This variation with the quantum case is due to the different status of the initial conditions.
Clearly, one can go iteratively through the higher order steps by similar (although rapidly tedious) calculations as long as the integration ‘"constants‘" analogous to $ C_{1}(T_{1}) $ can be properly adjusted. As in the first order step, we gather in eq (10.c) the terms containing $ exp(-it_{0}) $, since $ \exp(-it_{0}) $ is again (and always) solution of the homogeneous equation. Because $ D_{1}A_{1}(T) $ does not provide such a term, we just have to take into account the non derivative part of the right hand side of eq (10.c). Through an intensive use of the relation $A_{01}(T_1) N(T_2) = (N(T_2) +1) A_{01}(T_1)$, this expression can be reduced to: $ -3A_{02}(T_{2})(17N^{2}(T_{2})+7)\exp(-i t_0 -3iN(T_2)t_1)/4 $, and the non resonance condition coming from the second order reads : $$D_{2}A_{02}(T_{2})= 3iA_{02}(T_{2})(17N^{2}(T_{2})+7)/4.$$ This equation shows that $ N(T_{2}) $ is in fact independent of $ t_{2} $, too,(i.e. $N(T_2)=A^{\dag }_{03}(T_{3})A_{03}(T_{3})$ ) and we find through integration:
$$\label{22}
A_{02}(T_{2})=A_{03}(T_{3})\exp(+3i(17N^{2}(T_{3})+7)t_{2}/4),$$
whereas (18) becomes: $$\label{23}
[A_{03}(T_{3}),A^{\dag }_{03}(T_{3})]=1,\, \forall T_{3}.$$
Collecting equations (7), (17) and (22), we see that the non resonance conditions, up to the second order, imply that the first order term of the expansion of the annihilation operator is, in the variable $ t $:
$$\label{24}
a_{0}(t,g)=a_{0}(0,g)(\exp(-it(1+3gN-3g^{2}(17N^{2}+7)/4))+O(g^{3})),$$
which exhibits a large difference with the classical case : 17 is a prime number, difficult to link with the other prime number 5 coming in the CAO frequency (1) . We will discuss later on this CAO/QAO (apparent) discrepancy. Nevertheless, the result, equation (24), is in perfect agreement with the perturbative expression of the energy levels of the QAO, as calculated by standard methods :
$$\label{25}
E_{n}(g)=1/2+n+3g(1+2n+2n^{2})/4-g^{2}(1+2n)(21+17n+17n^{2})/8+O(g^{3}).$$
Indeed, a straightforward argument based on the formal expression of $a(t,g)$ in the Heisenberg picture shows us that the frequency appearing in (24) for $N=n$ should coincide with $ E_{n}(g)-E_{n-1}(g) $. This is readily checked.
Turning back on the second order equation (10.c) cleared from its resonant terms, we obtain its general solution :
$$A_{2}(T)=-15A_{0}^{3}(N-1)/4+3A_{0}^{5}/16+3(23N^{2}+7)A^{\dag }_{0}/8$$
$$\label{26}
+21(N-1)A_{0}^{\dag 3}/16-A_{0}^{\dag 5}/8+C_{2}(T_{1})\exp(-it_{0}),$$
where $ A_{0} $ and $ N $ stand for $ A_{0}(T) $ and $ N(T_{3}) $. In contrast with $C_1(T_1)$ in (19), the operator $ C_{2}(T_{1}) $ cannot be taken as vanishing, because the second condition (12),
$$[A_{0}(T),A^{\dag }_{2}(T)]+[A_{1}(T),A^{\dag }_{1}(T)]+[A_{2}(T),A^{\dag }_{0}(T)]=0,$$ would not be fulfilled. Imposing this and using eqs (12),(17),(18) and (26), one finds instead an appropriate expression for the solution of the homogeneous version of (10.c), namely :
$$\label{27}
C_{2}(T_1)\exp(-it_0)=-9A_{0}(T)(1-3N^{2})/32.$$
Let us notice that the (operator) coefficients of $ \exp(\pm it_{0}) $ which appear in the zeroth order solution get corrections from the first and second orders, and the coefficients of $ \exp(\pm 3it_{0}) $ which appear at the first order get also corrections coming from the second order. Such a behaviour still holds at the third order, as we have checked.
So far, the perturbative expression of the energy levels of the QAO (which was not our main goal) did not show up in full within our MST procedure. Yet, it can be found (without appealing to other perturbative methods) by inserting $a(t,g)$ as given by equations (6), (21), (26) and (27) in the Hamiltonian (3). Obviously, we are waiting for an expansion in powers of g polynomially dependent on $ A_{0} $ and $ A^{\dag }_{0} $, up to the second order in $ g $ :
$$H=H_{0}+gH_{1}+g^{2}H_{2}+O(g^{3}),$$
The result is that the $ H_{j} $’s are function of $ N=A^{\dag }_{0}A_{0} $, not of $ A_{0} $ and $ A^{\dag }_{0} $ separately : $$\label{28}
H=1/2+N+3g(1+2N+2N^{2})/4-g^{2}(1+2N)(21+17N+17N^{2})/8+O(g^{3}).$$
This feature, which technically appears as an accident due to many cancellations, is in fact easy to understand. One observes, at each step, the $t_0$, $t_1$, $t_2$... dependences of $A_0(T)$ arise from the exponentials only. Since the $t$-dependence must eventually disappear from the conserved quantity $H$, a proper balance between $A_0(T)$ and $A^{\dag}_0(T)$ is expected in each of the monomials $H_j$, namely as many creators as annihilators. Then, whatever are the number and the order of the $A_0$’s and $A^{\dag}_0$’s in those polynomials, the commutatiom relation (12) allows us to cast the $H_j$’s in the form of polynomials in $N= A^{\dag}_0 A_0$. Furthermore, anticipating a result to be proved in the next section, $N(T)$ is not only independent of $t_0$, $t_1$ and $t_2$ but in fact of $T$ altogether: $N=A^{\dag}_0(0) A_0(0)$. On account of the Heisenberg algebra (12) (taken at $T=0$) this implies that the spectrum of $N$ is the set of non negative integers, and the expression (28) for $H$ is in complete agreement with (25) indeed.
Finally, we have to explain the apparent discrepancy noticed earlier between the classical and quantum results. Let us write the classical energy (2) for $ \omega =1 $: $ E_{c}=a^{2}/2+3ga^{4}/8+O(g^2) $. Now for large $ n $ the quantum energy (25) reduces to : $E_q = n+3gn^2/2+O(g^2)$. Then the natural correspondance is $ a^{2}/2\rightarrow n $ plus a quantum correction so adjusted as to insure $ E_{c} = E_{q}+O(g^2) $. One finds $ a^{2}/2\rightarrow n - 3gn^2$ which, inserted in the classical frequency (1) gives $ \Omega =1+3gn-51g^{2}n^2/4 $, in agreement with the large $n$ quantum frequency from (24), derived within the MST scheme.
3. The Quantum case : General discussion {#the-quantum-case-general-discussion .unnumbered}
========================================
In the previous section, in particular on eqs (21),(26) and (27), one observes that the construction is made with two elementary bricks $ A_{0}(T) $ and $ A^{\dag }_{0}(T) $, where $ A_{0}(T) $ is the first term in the MST expansion of the annihilation operator. We have also pointed out the simple connection between the operator $ N $ and the Hamiltonian. More precisely, the first perturbative results exhibit the following features :
1\) $ N=A^{\dag }_{0}(T)A_{0}(T) $ is independent of $ t_{0} $, $ t_{1} $, $ t_{2} $..., i.e. of $T$.
2)The ‘"nonhomogeneous‘" parts of $ A_{n}(T) $ depend on $ T $ through the basic operators $ A_{0}(T) $ and $ A^{\dag }_{0}(T) $. The same is true for the ‘"homogeneous‘" parts $ C_{n}(T_{1})\exp(-it_{0}) $ which, after determination of $ C_{n}(T_{1}) $, can be recast in the form of functions of $ A_{0}(T) $ and $ A^{\dag }_{0}(T) $ only.
3\) The operators $ H $ and $ N $ commute.
If these features persist at all orders, then (putting aside any consideration of convergence) one should obtain in the limit :
$$\label{29}
A(T,g)=A_{0}(T)+\sum _{n=1}^{\infty }g^{n}F_{n}(A_{0}(T),A^{\dag }_{0}(T))$$
where : $$\label{30}
A_{0}(T)=A_{0}\exp(-i(t_{0}+\sum _{n=1}^{\infty }f_{n}(N)t_{n}))$$ together with $ [A_{0},A^{\dag }_{0}]=1 $, and where the $ F_{n} $’s are some polynomial functions of $ A_{0} $ and $ A^{\dag }_{0} $ while the $ f_{n} $’s are some polynomial functions of $ N $. We will show below that the resonance -free solutions of the perturbative multitime equations of motion do exist indeed, and have the general form (29)-(30). This means, in particular, that no obstructions are encountered in determining the integration ‘"constants‘" $ C_{n}(T_{1}) $ and giving them the appropriate form. Eqs (29) and (30) then yield :\
$$\label{31}
a(t,g)=a_{0}(t,g)+\sum _{n=1}^{\infty }g^{n}F_{n}(a_{0}(t,g),a^{\dag }_{0}(t,g))$$ where :
$$\label{32}
a_{0}(t,g)=A_{0}\exp(-i(1+\sum _{n=1}^{\infty }g^{n}f_{n}(N))t)$$
and $ N=A^{\dag }_{0}A_{0} $ is a constant operator.
Actually, these facts result from the full equivalence between the iterative process described in the previous section and the perturbative determination of an unitary transformation which brings the Hamiltonian to a diagonal form.\
In order to prove this equivalence, let us consider the spectral decomposition of the Hamiltonian : $$H(a_0,a^{\dag }_0,g) \equiv 1/2+a^{\dag }_0 a_0+g(a_0+a^{\dag }_0)^{4}/4=\sum _{n=0}^{\infty }E_{n}(g)|n,g><n,g|,$$ where $ \left\{ \mid n,g>\right\} $ is the orthonormal basis made of the “perturbed” eigenvalues of $H$ (for future convenience, $a$ is written here as $a_0$). We also introduce the ‘"unperturbed‘", orthonormal Fock basis $ \left\{ \mid n>\right\} $ induced by the operators $a_0$ and $a^{\dag}_0$, together with the unitary transformation which maps the former onto the latter : $$|n>=U(g)|n,g>(n=0,1,2...).$$
The unitary operator $ U(g) $ is determined up to a $ N $ dependent, arbitrary, right phase factor, where $ N=a^{\dag }_{0}a_{0} $.
Then if we define $ H_{d} $ as $ H_{d}(a_0,a^{\dag }_0,g)=U^{\dag}(g)H(a_0,a^{\dag }_0,g)U(g) $, we have :
$$\label{33}
H_{d}(a_0,a^{\dag }_0,g)=H(a(g),a^{\dag }(g),g),$$
where we denote by $ a(g) $ the annihilation operator transformed by $ U(g) $ :
$$\label{34}
a(g)=U^{\dag}(g)a_{0}U(g).$$
We also have :
$$H_{d}(a_0,a^{\dag }_0,g)=\sum ^{\infty }_{n=0}E_{n}(g)|n><n|.$$
In other words, the unitary transformations (34) of the dynamical variables is that one which diagonalizes the Hamiltonian in the $ \left\{ |n>\right\} $ basis. The perturbative form of eqs (33) and (34) are\
$$\label{35}
a(g)=a_{0}+\sum _{n=1}^{\infty }g^{n}a_{n}$$ and $$\label{36}
H_{d}(a_0,a^{\dag }_0,g)=\sum _{n=0}^{\infty }g^{k}H_{k}=\frac{1}{2}+a^{\dag }_{0}a_{0}+\sum _{n=1}^{\infty }g^{k}H_{k}\, \, ,$$ where the explicit expressions of the $ H_{k} $’s in terms of the $ a_{n} $’s are obtained by substituing the perturbative form (35) in $ H(a(g),a^{\dag }(g),g) $ , and expanding. For the QAO Hamiltonian we are interested in, those$H_k$’s are:\
$$H_{k}=\sum _{n=0}^{k}a^{\dag }_{m}a_{k-m}+\sum _{m,r,s,\ell \geq 0\atop m+r+s+\ell =k-1}q_{m}q_{r}q_{s}q_{l}/4\,, (k=1,2,....) \, ,$$ where $ q_{m}=a_{m}+a^{\dag }_{m} $.
The operators $ a_{n} $ ($ n=1,2,3\ldots $) in (35) are determined recursively as polynomial functions of $ a_{0} $ and $ a^{\dag }_{0} $ by requiring that :\
i) $ U(g $) be unitary indeed, or equivalently (due to Von Neumann theorem [@RS]) that the commutator of the ‘"new‘" variables $ a(g) $ and $ a^{\dag }(g) $ be canonical: $ [a(g),a^{\dag }(g)]=1,\forall g $,i.e.:\
$$\label{37}
\sum _{m=0}^{n}[a_{m},a^{\dag }_{n-m}]=\delta _{n,0},(n=0,1,2...).$$
ii\) $ H_{d} $ be diagonal indeed in the $ \left\{ \mid n>\right\} $ basis or, equivalently, that $[H_d,N]=0,\forall g $, i.e. : $$\label{38}
[H_k,N]=0,(k=1,2,3..)$$
Eqs (38) implies that all the $ H_{k} $’s are functions of the operator $ N=a^{\dag }_{0}a_{0} $ only. In particular these operators $ H_{k} $ commute between themselves.
Evidently, the equations (37) and (38) [**must**]{} admit solution for $ \left\{ a_{n}\right\} $, due to the mere existence of the unitary operator $ U(g) $ and its formal perturbative expansion. However there is no uniqueness property because of the phase freedom in the mapping $ U(g $). In our case of QAO with standard quartic interaction, the ‘"minimal‘" solution $ \left\{ a_{n}\right\} $ is such that each $ a_{n} $ is a polynomial of degree $ 2n+1 $ in $ a_{0} $ and $ a^{\dag }_{0} $ with rational coefficients, and monomials of odd degrees only : $$a_{n}=\sum _{2 k+\ell=2 n +1\atop 1\leq l_{odd}\leq 2n+1 }
(x_{n,k,l}a_{0}^{l}N^{k}+y_{n,k,l}N^{k}a_{0}^{\dag l}).$$
Let us now define the $ T $ dependent operators $ a_{n}(T) $ by:
$$\label{39}
a_{n}(T)=\exp(i\sum _{k=0}^{\infty }H_{k}t_{k})a_{n}\exp(-i\sum _{k=0}^{\infty }H_{k}t_{k}),
(n=0,1,2,...)$$
where $ t_{k}=g^{k}t $. We claim that these operators obey exactly the canonical commutation relations (12) and the differential equations (10) which serve previously to determine the $ A_{n}(T) $’s.
Because of (37) , this is immediate for the relations (12). As for the equations (10) , one first derives from (39) :
$$D_{n-m}a_{m}(T)=i\, \exp(i\sum _{k=0}^{\infty }H_{k}t_{k})[H_{n-m}(\{a_{r}\})
,a_{m}]\exp(-i\sum _{k=0}^{\infty }H_{k}t_{k})$$ which, summing up, yields
$$\label{40}
\sum _{m=0}^{n}D_{n-m}a_{m}(T)=i\sum _{m=0}^{n}[H_{n-m}(\{a_{r}(T)\}),a_{m}(T)],$$
where the commutativity of the $ H_{k} $’s has been used twice. On the other hand, using (7) together with (33) and (36) to express $ H(A(T,g),A^{\dag }(T,g),g) $ in terms of the function $ H_{k} $, one readily finds that the equations (10) read as well:
$$\label{41}
\sum _{m=0}^{n}D_{n-m}A_{m}(T)=i\sum _{m=0}^{n}[H_{n-m}(\{A_{r}(T)\}),A_{m}(T)],$$
identical to (40). This is actually true not only for the QAO but for a general interaction.
Therefore, $ \{A_{n}(T)\} $ can be identified as one of the solutions $ \{a_{n}(T)\} $, which establishes the equivalence of the two schemes, and hence the consistence of the multitime method we used, together with the general validity of the assertions 1) to 3) put forward at the begining of this section.
It is possible now to comment on the ‘"mass renormalisation‘" introduced by B&B. Since $ H_{d} $ is a pure function of $N$, $ H_{d}=H(N) $, eq(39) for $a_0(T)|_{t_{j}=g^{j}t} = A_0(T)|_{t_{j}=g^{j}t} = a_0(t,g)$ reads:
$$a_{0}(t,g)=\exp(+iH(N)t)a_{0}\exp(-iH(N)t),$$ or, by using $ a_{0}N=(N+1)a_{0} $ :
$$a_{0}(t,g)=a_{0}\exp(-i(H(N)-H(N-1))t).$$
Together with (28), this gives: $$a_{0}(t,g)=a_{0}\exp(-it(1+3gN-3g^{2}(7+17N^{2})/4) +O(g^3)).$$
The ‘"renormalisation‘" phenomenon can be pinned down to the fact that $ H(N)-H(N-1) $ is the trivial identity operator at the zeroth order, and becomes a true operator for higher orders.
As mentioned at the begining, and apparent on eqs (29) and (30), the solution $ A(T,g) $ constructed there corresponds to initial conditions depending on $ A_{0} $ [**and**]{} $ g $ . If one insists in having the perturbative solution with prescribed $ g- $independent initial condition : $ A(0,g)=a $, with $ [a,a^{\dag }]=1 $ , this is easily achieved by a few additional manipulations. Indeed, it is sufficient to invert order by order the relation :
$$a=A_{0}+\sum _{n=1}^{\infty }g^{n}F_{n}(A_{0},A_{0}^{\dag }),$$ (which is straightforward in spite of the non commutative algebra) to get :\
$$A_{0}=a+\sum _{n=1}^{\infty }g^{n}G_{n}(a,a^{\dag }),$$ and to reinsert this expression for $ A_{0} $ in eqs (29) and (30), as well as in $ N=A^{\dag }_{0}A_{0} $, truncated at the relevant order. Then, of course, the expression of $ A(T,g) $ in terms of $ a $ and $ a^{\dag } $ has no longer the ‘"simple‘" structure that it exhibits in terms of $ A_{0} $ and $ A_{0}^{\dag } $.
To conclude this section, we wish to stress again that the arguments presented there are quite general, not specific of the QAO. If one considers an Hamiltonian which is the sum of an harmonic oscillator one and a “potential” represented by a self-adjoint operator function of the position and the momentum, such an analysis can be repeated. Actually, the equivalence between MST and unitary transformation diagonalizing the Hamiltonian is likely to be a rather general feature. In particular, the previous discussion can be extended in a rather straigthforward way to systems with more than one degree of freedom.
Furthermore, the equivalence between the multitime approach and the perturbative construction of the relevant unitary transformation must have a classical counterpart. In the classical framework, multitime expansions should appear as essentially equivalent to the construction of appropriate canonical transformations, following the Poincaré - Von Zeipel method [@Gold], or some of its disguises. As a matter of fact, one can find indication of such a connection in the literature [@N; @M]. This aspect of the question, which we have not touched upon in the present paper, might deserve a further study.
4. Conclusion {#conclusion .unnumbered}
=============
In this paper, we have used the anharmonic oscillator in the Heisenberg picture as a model for investigating the practicability of the Derivative Expansion Method, of common use in classical physics, within the quantum framework. This method turns out to be successful in providing us with the perturbative expansion of the time dependent dynamical variables together with the energy levels, which we have derived explicitely up to the second order. We also have proved that this MST is equivalent to the perturbative construction of an unitary transformation diagonalizing the full Hamiltonian, leading to a step-by-step algorithm for the calculation of the previous quantities at any order, and thereby strengthening the status of the Multiple Scale Techniques in quantum mechanics.
[Acknowledgements]{} {#acknowledgements .unnumbered}
=====================
We are very grateful to Roberto Kraenkel who attracted our attention to this present topic and Miguel Manna for enlightening discussions on the perturbative multiscale theory.
Bender, C.M. and Bettencourt, L.M.A. Phys.Rev.Lett.77:4114-4117(1996) Bender, C.M. and Bettencourt, L.M.A. Phys.Rev.D54:7710-7723(1996) Jeffrey, A. and Kawahara, T. [**Asymptotic methods in nonlinear wave theory**]{}, Pitman Books, London, 1982 Nayfeh, A. [**Perturbation methods**]{}, J. Wiley and Sons, New York, 1973 Sandri, G. Nuovo Cimento, B36, 67-93, 1965 Reed, M. and Simon, B, [**Methods of Modern Mathematical Physics**]{} (Vol.1), Academic Press, New York, 1980, Chap.VIII. Goldstein, H. [**Classical Mechanics**]{}, Addison-Wesley, Reading, 1980 Morrison, J.A. [**Methods in Astrodynamics and Celestial Mechanics**]{}, Ed. by R.L. Duncombe and V,G Szebehely, Academic Press, New York, 1966 Bender, C.M. and Wu, T.T. Phys. Rev. 184, 1231 (1969)\
Appendix {#appendix .unnumbered}
========
We give below, up to order 6:
i\) the coefficients $a_n$ of the expansion (35) of the annihilator $a(g)$ in terms of $N=a^{\dag}_0 a_0$,
ii\) the coefficients $E_k(n)$ of the expansion of the energy levels: $$E_{n}(g)=\frac{1}{2}+n+\sum _{n=1}^{\infty }g^{k}E_{k}(n).$$
Both have been computed by applying the algorithm described in section 3 (eqs (37) and (38)).
i\) To be simpler, $ a $ and $ a^{\dag } $ stand for $ a_{0} $ and $ a^{\dag }_{0} $.
$ a_{1}=(2a^{3}-a^{\dag 3}-6Na^{\dag })/4 $ ;
$ a_{2}=(-9a+120a^{3}+6a^{5}-120a^{3}N+27aN^{2}+84a^{\dag }-42a^{\dag 3}-4a^{\dag 5}+42Na^{\dag 3}+276N^{2}a^{\dag })/32\, ; $
$ a_{3}=(6092a^{3}+756a^{5}+8a^{7}-8844a^{3}N+4422a^{3}N^{2}-378a^{5}N+1278aN-1062aN^{3} $
$ -1708a^{\dag 3}-464a^{\dag 5}-6a^{\dag 7}-9282Na^{\dag}+2406Na^{\dag 3}+232Na^{\dag 5}-1203N^{2}a^{\dag 3} $
$ -9042N^{3}a^{\dag })/128 $ ;
$ a_{4}=(-200645a+1546416a^{3}+380868a^{5}+9264a^{7}+40a^{9}-2975280a^{3}N+2143296a^{3}N^{2} $
$ -714432a^{3}N^{3}-322896a^{5}N+80724a^{5}N^{2}-3088a^{7}N-500298aN^{2}+162755aN^{4} $
$ +506760a^{\dag }-358200a^{\dag 3}-221832a^{\dag 5}-6696a^{\dag 7}-32a^{\dag 9}+673392Na^{\dag 3}+186464Na^{\dag 5} $
$ +2232Na^{\dag 7}+3040992N^{2}a^{\dag }-472788N^{2}a^{\dag 3}-46616N^{2}a^{\dag 5}+157596N^{3}a^{\dag 3}+1365240N^{4}a^{\dag })/2048\, ; $
$ a_{5}=(116798776a^{3}+51228696a^{5}+2189520a^{7}+20832a^{9}+48a^{11}-266946576a^{3}N $
$ +255315936a^{3}N^{2}-121842648a^{3}N^{3}+30460662a^{3}N^{4}-58614828a^{5}N+24750360a^{5}N^{2} $
$ -4125060a^{5}N^{3}-1300128a^{7}N+216688a^{7}N^{2}-5208a^{9}N+52602092aN+41073824aN^{3} $
$ -6417388aN^{5}-21539684a^{\dag 3}-28787584a^{\dag 5}-1542096a^{\dag 7}-16320a^{\dag 9} $
$ -40a^{\dag {11}}-121625250Na^{\dag }+50282976Na^{\dag 3}+32551232Na^{\dag 5}+912600Na^{\dag 7} $
$ +4080Na^{\dag 9}-47389884N^{2}a^{\dag 3}-13618080N^{2}a^{\dag 5}-152100N^{2}a^{\dag 7}-219914676N^{3}a^{\dag } $
$ +22248396N^{3}a^{\dag3 }+2269680N^{3}a^{\dag 5}-5562099N^{4}a^{\dag 3}-55675938N^{5}a^{\dag })/8192\, ; $
$ a_{6}=(-2649077789a+19854323040a^{3}+14799326898a^{5}+1000498176a^{7}+15874840a^{9} $
$ +78720a^{11}+112a^{13}-15744a^{11}N-52410470592a^{3}N+59605775856a^{3}N^{2}-37821182832a^{3}N^{3} $
$ +13464443160a^{3}N^{4}-2692888632a^{3}N^{5}-20808622800a^{5}N+11852140500a^{5}N^{2} $
$ -3324992400a^{5}N^{3}+415624050a^{5}N^{4}-817924896a^{7}N+242212752a^{7}N^{2}-26912528a^{7}N^{3} $
$ -7253184a^{9}N +906648a^{9}N^{2}-16271788323aN^{2}-6097875991aN^{4}+521267535aN^{6} $
$ +4255953324a^{\dag }-2581523304a^{\dag 3}-8101045372a^{\dag 5}-691648560a^{\dag 7}-12242240a^{\dag 9} $
$ -64720a^{\dag {11}}-96a^{\dag {13}}+12944Na^{\dag {11}}+7571823000Na^{\dag 3}+11245609120Na^{\dag 5} $
$ +562441728Na^{\dag 7}+5584128Na^{\dag 9}+35458238196N^{2}a^{\dag }-9129805056N^{2}a^{\dag 3} $
$ -6326409960N^{2}a^{\dag 5}-165946104N^{2}a^{\dag 7}-698016N^{2}a^{\dag 9}+5783860872N^{3}a^{\dag 3} $
$ +1757503840N^{3}a^{\dag 5}+18438456N^{3}a^{\dag 7}+29695249188N^{4}a^{\dag }-2055444390N^{4}a^{\dag 3} $
$ -219687980N^{4}a^{\dag 5}+411088878N^{5}a^{\dag 3}+4768483548N^{6}a^{\dag })/65536. $
ii)
$ E_{1}(n)=3(1+2n+2n^{2})/4 $ ;
$ E_{2}(n)=-(1+2n)(21+17n+17n^{2})/8 $ ;
$ E_{3}(n)=3(111+347n+472n^{2}+250n^{3}+125n^{4})/16 $ ;
$ E_{4}(n)=-(1+2n)(30885+49927n+60616n^{2}+21378n^{3}+10689n^{4})/128 $ ;
$ E_{5}(n)=3(305577+1189893n+2060462n^{2}+1857870n^{3}+1220765n^{4}+350196n^{5} $
$ +116732n^{6})/256 $ ;
$ E_{6}(n)=-(1+2n)(65518401+146338895n+213172430n^{2}+139931868n^{3}+85627929n^{4} $
$ +18794394n^{5}+6264798n^{6})/1024. $
Several number theoretic properties of the $ E_{k}(n) $’s are worth pointing out. First, all the coefficients $c_{kp}$ of $n^p$ in $E_k(n)$ are rational and positive, and the signs of the $E_k(n)$’s alternate, as it should be. Perhaps new are the following observations: whereas the denominator in the expression of $E_k(n)$ is a power of 2, the numerator is always a multiple of 3 (for integer $n$). This peculiarity was already noticed by Bender and Wu [@BW] for the ground state ($n=0$). It thus turns out to hold for the excited levels too. Also, the sum of the numerators of the coefficients $c_{kp}$ in each $E_k(n)$ is a multiple of 5. Finally, if one expresses the $E_k(n)$’s in terms of the variable $m=n+\frac{1}{2}$, one observes that they are even polynomials with positive coefficients (multiplied by $-m$ if $k$ is even). More than that, all the zeroes of these polynomials are pure imaginary. This means that all the zeroes of $ E_{k}(n) $ lie on the line $n=-\frac{1}{2} +iy$ !
|
---
abstract: 'This paper considers an architecture referred to as Cascade Region Proposal Network (Cascade RPN) for improving the region-proposal quality and detection performance by *systematically* addressing the limitation of the conventional RPN that *heuristically defines* the anchors and *aligns* the features to the anchors. First, instead of using multiple anchors with predefined scales and aspect ratios, Cascade RPN relies on a *single anchor* per location and performs multi-stage refinement. Each stage is progressively more stringent in defining positive samples by starting out with an anchor-free metric followed by anchor-based metrics in the ensuing stages. Second, to attain alignment between the features and the anchors throughout the stages, *adaptive convolution* is proposed that takes the anchors in addition to the image features as its input and learns the sampled features guided by the anchors. A simple implementation of a two-stage Cascade RPN achieves AR 13.4 points higher than that of the conventional RPN, surpassing any existing region proposal methods. When adopting to Fast R-CNN and Faster R-CNN, Cascade RPN can improve the detection mAP by 3.1 and 3.5 points, respectively. The code is made publicly available at <https://github.com/thangvubk/Cascade-RPN>.'
author:
- |
Thang Vu, Hyunjun Jang, Trung X. Pham, Chang D. Yoo\
Department of Electrical Engineering\
Korea Advanced Institute of Science and Technology\
`{thangvubk,wiseholi,trungpx,cd_yoo}@kaist.ac.kr`\
bibliography:
- 'ref.bib'
title: 'Cascade RPN: Delving into High-Quality Region Proposal Network with Adaptive Convolution'
---
Introduction
============
Object detection has received considerable attention in recent years for its applications in autonomous driving [@furgale2013toward; @hane2015obstacle], robotics [@astua2014object; @danielczuk2018segmenting] and surveillance [@conte2005meeting; @liao2017security]. Given an image, object detectors aim to detect known object instances, each of which is assigned to a bounding box and a class label. Recent high-performing object detectors, such as Faster R-CNN [@NIPS2015_5638], formulate the detection problem as a two-stage pipeline. At the first stage, a region proposal network (RPN) produces a sparse set of proposal boxes by refining and pruning a set of anchors, and at the second stage, a region-wise CNN detector (R-CNN) refines and classifies the proposals produced by RPN. Compared to R-CNN, RPN has received relatively less attention for improving its performance. This paper will focus on improving RPN by addressing its limitations that arise from heuristically defining the anchors and heuristically aligning the features to the anchors.
An anchor is defined by its scale and aspect ratio, and a set of anchors with different scales and aspect ratios are required to obtain a sufficient number of positive samples that have high overlap with the target objects. Setting appropriate scales and aspect ratios is important in achieving high detection performance, and it requires a fair amount of tuning [@Lin_2017_ICCV; @NIPS2015_5638].
An *alignment* rule is “implicitly” defined to set up a correspondence between the image features and the reference boxes. The input features of RPN and R-CNN should be well-aligned with the bounding boxes that are to be regressed. The alignment is guaranteed in R-CNN by the RoIPool [@NIPS2015_5638] or RoIAlign [@He_2017_ICCV] layer . The alignment in RPN is heuristically guaranteed: the anchor boxes are *uniformly* initialized, leveraging the observation that the convolutional kernel of the RPN *uniformly* strides over the feature maps. Such a heuristic introduces limitations for further improving detection performance as described below.
A number of studies have attempted to improve RPN by iterative refinement [@gidaris2016attend; @zhong2017cascade]. Henceforth, this paper will refer to it as Iterative RPN. The motivation behind this idea is illustrated in Figure \[fig:intro\_a\]. Anchor boxes which are references for regression are uniformly initialized, and the target ground truth boxes are arbitrarily located. Thus, RPN needs to learn a regression distribution of high variance, as shown in Figure \[fig:intro\_a\]. If this regression distribution is perfectly learned, the regression distribution at stage 2 should be close to a Dirac Delta distribution. However, such a high-variance distribution at stage 1 is difficult to learn, requiring stage 2 regression. Stage 2 distribution has a lower variance compared to that of stage 1, and thus should be easier to learn but fails with Iterative RPN. The failure is implied by the observation in which the performance improvement of Iterative RPN is negligible compared to that of RPN, as shown in Figure \[fig:intro\_b\]. It is explained intuitively in Figure \[fig:intro\_c\]. Here, after stage 1, the anchor is regressed to be closer to the ground truth box; however, this breaks the alignment rule in detection.
This paper proposes an architecture referred to as Cascade RPN to systematically address the aforementioned problem arising from heuristically defining the anchors and aligning the features to the anchors. First, instead of using multiple anchors with different scales and aspect ratios, Cascade RPN relies on a single anchor and incorporates both anchor-based and anchor-free criteria in defining positive boxes to achieve high performance. Second, to benefit from multi-stage refinement while maintaining the alignment between anchor boxes and features, Cascade RPN relies on the proposed adaptive convolution that adapts to the refined anchors after each stage. Adaptive convolution serves as an extremely light-weight RoIAlign layer [@He_2017_ICCV] to learn the features sampled within the anchors.
Cascade RPN is conceptually simple and easy to implement. Without bells and whistles, a simple two-stage Cascade RPN achieves AR 13.4 points improvement compared to RPN baseline on the COCO dataset [@COCO], surpassing any existing region proposal methods by a large margin. Cascade RPN can also be integrated into two-stage detectors to improve detection performance. In particular, integrating Cascade RPN into Fast R-CNN and Faster R-CNN achieves 3.1 and 3.5 points mAP improvement, respectively.
Related Work
============
[0.34]{}
[0.48]{} ![Iterative RPN shows limitations in improving RPN performance. (a) The target regression distribution to be learned at stage 1 and 2. The stage 2 distribution represents the error after the stage 1 distribution is learned. (b) Iterative RPN fails in learning stage-2 distribution as the average recall (AR) improvement is marginal compared to the of RPN. (c) In Iterative RPN, the anchor at stage 2, which is regressed in stage 1, breaks the alignment rule in detection.[]{data-label="fig:introduction"}](figs/stats_xy_1.png "fig:"){width="\textwidth"}
[0.48]{} ![Iterative RPN shows limitations in improving RPN performance. (a) The target regression distribution to be learned at stage 1 and 2. The stage 2 distribution represents the error after the stage 1 distribution is learned. (b) Iterative RPN fails in learning stage-2 distribution as the average recall (AR) improvement is marginal compared to the of RPN. (c) In Iterative RPN, the anchor at stage 2, which is regressed in stage 1, breaks the alignment rule in detection.[]{data-label="fig:introduction"}](figs/stats_xy_2.png "fig:"){width="\textwidth"}
[0.48]{} ![Iterative RPN shows limitations in improving RPN performance. (a) The target regression distribution to be learned at stage 1 and 2. The stage 2 distribution represents the error after the stage 1 distribution is learned. (b) Iterative RPN fails in learning stage-2 distribution as the average recall (AR) improvement is marginal compared to the of RPN. (c) In Iterative RPN, the anchor at stage 2, which is regressed in stage 1, breaks the alignment rule in detection.[]{data-label="fig:introduction"}](figs/stats_hw_1.png "fig:"){width="\textwidth"}
[0.48]{} ![Iterative RPN shows limitations in improving RPN performance. (a) The target regression distribution to be learned at stage 1 and 2. The stage 2 distribution represents the error after the stage 1 distribution is learned. (b) Iterative RPN fails in learning stage-2 distribution as the average recall (AR) improvement is marginal compared to the of RPN. (c) In Iterative RPN, the anchor at stage 2, which is regressed in stage 1, breaks the alignment rule in detection.[]{data-label="fig:introduction"}](figs/stats_hw_2.png "fig:"){width="\textwidth"}
[0.17]{} ![Iterative RPN shows limitations in improving RPN performance. (a) The target regression distribution to be learned at stage 1 and 2. The stage 2 distribution represents the error after the stage 1 distribution is learned. (b) Iterative RPN fails in learning stage-2 distribution as the average recall (AR) improvement is marginal compared to the of RPN. (c) In Iterative RPN, the anchor at stage 2, which is regressed in stage 1, breaks the alignment rule in detection.[]{data-label="fig:introduction"}](figs/intro_compare.pdf "fig:"){width="0.92\linewidth"} \[fig:my\_label\]
[0.42]{} ![Iterative RPN shows limitations in improving RPN performance. (a) The target regression distribution to be learned at stage 1 and 2. The stage 2 distribution represents the error after the stage 1 distribution is learned. (b) Iterative RPN fails in learning stage-2 distribution as the average recall (AR) improvement is marginal compared to the of RPN. (c) In Iterative RPN, the anchor at stage 2, which is regressed in stage 1, breaks the alignment rule in detection.[]{data-label="fig:introduction"}](figs/introduction.pdf "fig:"){width="0.96\linewidth"} \[fig:my\_label\]
#### Object Detection.
Object detection can be roughly categorized into two main streams: one-stage and two-stage detection. Here, one-stage detectors are proposed to enhance computational efficiency. Examples falling in this stream are SSD [@SSD], YOLO [@Redmon_2016_CVPR; @Redmon_2017_CVPR; @YOLOV3], RetinaNet [@Lin_2017_ICCV], and CornerNet [@Law_2018_ECCV]. Meanwhile, two-stage detectors aim to produce accurate bounding boxes, where the first stage generates region proposals followed by region-wise refinement and classification at the second stage, [*e*.*g*., ]{}R-CNN [@Girshick_2014_CVPR], Fast R-CNN [@Fast_RCNN], Faster R-CNN [@NIPS2015_5638], Cascade R-CNN [@Cai_2018_CVPR], and HTC [@chen2019hybrid].
#### Region Proposals.
Region proposals have become the *de-facto* paradigm for high-quality object detectors [@chavali2016object; @7182356; @hosang2014good]. Region proposals serve as the attention mechanism that enables the detector to produce accurate bounding boxes while maintaining computation tractability. Early methods are based on grouping super-pixel ([*e*.*g*., ]{}Selective Search [@uijlings2013selective], CPMC [@carreira2011cpmc], MCG [@arbelaez2014multiscale]) and window scoring ([*e*.*g*., ]{}objectness in windows [@alexe2012measuring], EdgeBoxes [@zitnick2014edge]). Although these methods dominate the field of object detection in classical computer vision, they exhibit limitations as they are external modules independent of the detector and not computationally friendly. To overcome these limitations, Shaoqing [*et al*. ]{}[@NIPS2015_5638] propose Region Proposal Network (RPN) that shares full-image convolutional features with the detection network, enabling nearly cost-free region proposals.
#### Multi-Stage RPN.
There have been a number of studies attempting to improve the performance of RPN [@gidaris2016attend; @wang2019region; @yang2016craft; @zhong2017cascade]. The general trend is to perform multi-stage refinement that takes the output of a stage as the input of the next stage and repeats until accurate localization is obtained, as presented in [@gidaris2016attend]. However, this approach ignores the problem that the regressed boxes are misaligned to the image features, breaking the alignment rule required for object detection. To alleviate this problem, recent advanced methods [@fan2019seamese; @wang2019region] rely on deformable convolution [@Dai_2017_ICCV] to perform feature spatial transformations and expect the learned transformations to align to the changes of anchor geometry. However, as there is no explicit supervision to learn the feature transformation, it is difficult to determine whether the improvement originates from conforming to the alignment rule or from the benefits of deformable convolution, thus making it less *interpretable*.
#### Anchor-based vs. Anchor-free Criterion for Sample Discrimination.
As a bounding box usually includes an object with some amount of background, it is difficult to determine if the box is a positive or a negative sample. This problem is usually addressed by comparing the Intersection over Union (IoU) between an anchor and a ground truth box to a predefined threshold; thus, it is referred to as the anchor-based criterion. However, as the anchor is uniformly initialized, multiple anchors with different scales and aspect ratios are required at each location to ensure that there are enough positive samples [@NIPS2015_5638]. The hyperparameters, such as scales and aspect ratios, are usually heuristically tuned and have a large impact on the final accuracy [@Lin_2017_ICCV; @NIPS2015_5638]. Rather than relying on anchors, there have been studies that define positive samples by the distance between the prediction points and the center region of objects, referred to as anchor-free [@FCOS; @unitbox; @FSAF]. This method is simple and requires fewer hyperparameters but usually exhibits limitations in dealing with complex scenes.
Region Proposal Network and Variants
====================================
Region Proposal Network
-----------------------
Given an image $I$ of size $W \times H$, a set of anchor boxes $\mathbb{A} = \{\boldsymbol{a}_{ij}\ | 0 < (i + \frac{1}{2})s \leq W, 0 < (j+\frac{1}{2})s \leq H\}$ is *uniformly* initialized over the image, with stride $s$. Unless otherwise specified, $i$ and $j$ are omitted to simplify the notation. Each anchor box $\boldsymbol{a}$ is represented by a 4-tuple in the form of $\boldsymbol{a} = (a_x, a_y, a_w, a_h)$, where $(a_x, a_y)$ is the center location of the anchor with the dimension of $(a_w, a_h)$. The regression branch aims to predict the transformation $\boldsymbol{\delta}$ from the anchor $\boldsymbol{a}$ to the target ground truth box $\boldsymbol{t}$ represented as follows: $$\begin{aligned}
\delta _ { x } &= \left( t _ { x } - a _ { x } \right) / a _ { w } , \quad &&\delta _ { y } = \left( t _ { y } - a _ { y } \right) / a _ { h }, \\
\delta _ { w } &= \log \left( t _ { w } / a _ { w } \right) , \quad &&\delta _ { h } = \log \left( t _ { h } / a _ { h } \right).
\end{aligned}
\label{eq:transform}$$ Here, the regressor $f$ takes as input the image feature $\boldsymbol{x}$ to output a prediction $\hat{\boldsymbol{\delta}} = f(\boldsymbol{x})$ that minimizes the bounding box loss: $$\mathcal{L}(\hat{\boldsymbol{\delta}}, \boldsymbol{\delta}) = \sum_{k \in \{x, y, w, h\}} \text{smooth}_{L_1} \left(\hat{\delta}_k - \delta_k \right),
\label{eq:l1_loss}$$ where $\text{smooth}_{L_1}(\cdot)$ is the robust $L_1$ loss defined in [@Fast_RCNN]. The regressed anchor is simply inferred based on the inverse transformation of as follows: $$\begin{aligned}
a'_x = \hat{\delta}_x a_w + a_x, \quad a'_y = \hat{\delta}_y a_h + a_y, \\
a'_w = a_w\exp(\hat{\delta}_w), \quad a'_h = a_h\exp(\hat{\delta}_h).
\label{eq:inverse_transform}
\end{aligned}$$ Then the set of regressed anchor $\mathbb{A}' = \{\boldsymbol{a}'\}$ is filtered by non-maximum suppression (NMS) to produce a sparse set of proposal boxes $\mathbb{P}$: $$\mathbb{P} = \text{NMS}(\mathbb{A}', \mathbb{S}),
\label{eq:nms}$$ where $\mathbb{S}$ is the set of objectness scores learned by the classification branch.
Iterative RPN and Variants
--------------------------
Some previous studies [@gidaris2016attend; @zhong2017cascade] have proposed iterative refinement which is referred to as Iterative RPN, as shown in Figure \[fig:arch\_irpn\]. Iterative RPN iteratively refines the anchors by treating $\mathbb{A}'$ as the new initial anchor set for the next stage and repeats Eqs. to until obtaining accurate localization. However, this approach exhibits mismatch between anchors and their represented features as the anchor positions and shapes change after each iteration.
To alleviate this problem, recent advanced methods [@fan2019seamese; @wang2019region] use deformable convolution [@Dai_2017_ICCV] to perform spatial transformations on the features as shown in Figure \[fig:arch\_irpn+\] and \[fig:arch\_garpn\] and expect transformed features to align to the change in anchor geometry. However, this idea ignores the problem that there is no constraint to enforce the features to align with the changes in anchors: it is difficult to determine whether the deformable convolution produces feature transformation leading to alignment. Instead, the proposed Cascade RPN systematically ensures the alignment rule by using the proposed adaptive convolution.
Cascade RPN
===========
[0.24]{} ![The architectures of different networks. “I”, “H”, “C”, and “A” denote input image, network head, classifier, and anchor regressor, respectively. “Conv”, “DefConv”, “DilConv” and “AdaConv” indicate conventional convolution, deformable convolution [@Dai_2017_ICCV], dilated convolution [@yu2015multi] and the proposed adaptive convolution layers, respectively.[]{data-label="fig:arch"}](figs/networks/RPN.pdf "fig:"){height="0.1\textheight"}
[0.36]{} ![The architectures of different networks. “I”, “H”, “C”, and “A” denote input image, network head, classifier, and anchor regressor, respectively. “Conv”, “DefConv”, “DilConv” and “AdaConv” indicate conventional convolution, deformable convolution [@Dai_2017_ICCV], dilated convolution [@yu2015multi] and the proposed adaptive convolution layers, respectively.[]{data-label="fig:arch"}](figs/networks/IterativeRPN.pdf "fig:"){height="0.1\textheight"}
[0.36]{} ![The architectures of different networks. “I”, “H”, “C”, and “A” denote input image, network head, classifier, and anchor regressor, respectively. “Conv”, “DefConv”, “DilConv” and “AdaConv” indicate conventional convolution, deformable convolution [@Dai_2017_ICCV], dilated convolution [@yu2015multi] and the proposed adaptive convolution layers, respectively.[]{data-label="fig:arch"}](figs/networks/IterativeRPN+.pdf "fig:"){height="0.1\textheight"}
[0.36]{} ![The architectures of different networks. “I”, “H”, “C”, and “A” denote input image, network head, classifier, and anchor regressor, respectively. “Conv”, “DefConv”, “DilConv” and “AdaConv” indicate conventional convolution, deformable convolution [@Dai_2017_ICCV], dilated convolution [@yu2015multi] and the proposed adaptive convolution layers, respectively.[]{data-label="fig:arch"}](figs/networks/GA-RPN.pdf "fig:"){height="0.1\textheight"}
[0.63]{} ![The architectures of different networks. “I”, “H”, “C”, and “A” denote input image, network head, classifier, and anchor regressor, respectively. “Conv”, “DefConv”, “DilConv” and “AdaConv” indicate conventional convolution, deformable convolution [@Dai_2017_ICCV], dilated convolution [@yu2015multi] and the proposed adaptive convolution layers, respectively.[]{data-label="fig:arch"}](figs/networks/CascadeRPN_detail.pdf "fig:"){height="0.1\textheight"}
Adaptive Convolution
--------------------
Given a feature map $\boldsymbol{x}$, in the standard 2D convolution, the feature map is first sampled using a regular grid $\mathbb{R} = \{(r_x, r_y)\}$, and the samples are summed up with the weight $\boldsymbol{w}$. Here, the grid $\mathbb{R}$ is defined by the kernel size and dilation. For example, $\mathbb{ R } = \{ ( - 1 , - 1 ) , ( - 1,0 ) , \ldots , ( 0,1 ) , ( 1,1 ) \}$ corresponds to kernel size $3\times3$ and dilation 1. For each location $\boldsymbol{p}$ on the output feature $\boldsymbol{y}$, we have: $$\boldsymbol { y } [ \boldsymbol { p }] = \sum _ { \boldsymbol { r } \in \mathbb { R } } \boldsymbol { w } [ \boldsymbol { r }] \cdot \boldsymbol { x } [ \boldsymbol { p } + \boldsymbol { r }].$$
[0.245]{} ![Illustrations of the sampling locations in different convolutional layers with $3\times3$ kernel.[]{data-label="fig:convs"}](figs/conv/conv.pdf "fig:"){width="0.95\columnwidth"}
[0.245]{} ![Illustrations of the sampling locations in different convolutional layers with $3\times3$ kernel.[]{data-label="fig:convs"}](figs/conv/dilated_conv.pdf "fig:"){width="0.95\columnwidth"}
[0.245]{} ![Illustrations of the sampling locations in different convolutional layers with $3\times3$ kernel.[]{data-label="fig:convs"}](figs/conv/deformable_conv.pdf "fig:"){width="0.95\columnwidth"}
[0.245]{} ![Illustrations of the sampling locations in different convolutional layers with $3\times3$ kernel.[]{data-label="fig:convs"}](figs/conv/adaptive_conv.pdf "fig:"){width="0.95\columnwidth"}
In adaptive convolution, the regular grid $\mathbb{R}$ is replaced by the offset field $\mathbb{O}$ that is directly inferred from the input anchor. $$\boldsymbol { y } [ \boldsymbol { p }] = \sum _ { \boldsymbol { o } \in \mathbb { O } } \boldsymbol { w } [ \boldsymbol { o }] \cdot \boldsymbol { x } [ \boldsymbol { p } + \boldsymbol { o }].$$ Let $\bar{\boldsymbol{a}}$ denote the projection of anchor $\boldsymbol{a}$ onto the feature map. The offset $\boldsymbol{o}$ can be decoupled into center offset and shape offset (shown in Figure \[fig:arch\_crpn\]): $$\boldsymbol{ o } = \boldsymbol{ o }_{\text{ctr}} + \boldsymbol{ o }_{\text{shp}},
\label{eq:offset}$$ where $\boldsymbol{ o }_{\text{ctr}} = (\bar{a}_x - p_x, \bar{a}_y - p_y)$ and $\boldsymbol{ o }_{\text{shp}}$ is defined by the anchor shape and kernel size. For example, if kernel size is $3\times 3$, then $\boldsymbol{ o }_{\text{shp}} \in \left\{ (-\frac{\bar{a}_w}{2}, -\frac{\bar{a}_h}{2}), (-\frac{\bar{a}_w}{2}, 0), \ldots, (0, \frac{\bar{a}_h}{2}), (\frac{\bar{a}_w}{2}, \frac{\bar{a}_h}{2}) \right\}$. As the offsets are typically fractional, sampling is performed with bilinear interpolation analogous to [@Dai_2017_ICCV].
#### Relation to other Convolutions.
The illustrations of sampling locations in adaptive and other related convolutions are shown in Figure \[fig:convs\]. Conventional convolution samples the features at contiguous locations with a dilation factor of 1. The dilated convolution [@yu2015multi] increases the dilation factor, aiming to enhance the semantic scope with unchanged computational cost. The deformable convolution [@Dai_2017_ICCV] augments the spatial sampling locations by learning the offsets. Meanwhile, the proposed adaptive convolution performs sampling within the anchors to ensure alignment between the anchors and features. Adaptive convolution is closely related to the others. Adaptive convolution becomes dilated convolution if the center offsets are zeros. Deformable convolution becomes adaptive convolution if the offsets are deterministically derived from the anchors.
Sample Discrimination Metrics
-----------------------------
Instead of using multiple anchors with predefined scales and aspect ratios, Cascade RPN relies on a single anchor per location and performs multi-stage refinement. However, this reliance creates a new challenge in determining whether a training sample is positive or negative as the use of anchor-free or anchor-based metric is highly adversarial. The anchor-free metric establishes a loose requirement for positive samples in the second stage and the anchor-based metric results in an insufficient number of positive training examples at the first stage. To overcome this challenge, Cascade RPN progressively strengthens the requirements through the stages by starting out with an anchor-free metric followed by anchor-based metrics in the ensuing stages. In particular, at the first stage, an anchor is a positive sample if its center is inside the center region of an object. In the following stages, an anchor is a positive sample if its IoU with an object is greater than the IoU threshold.
Cascade RPN
-----------
The architecture of a two-stage Cascade RPN is illustrated in Figure \[fig:arch\_crpn\]. Here, Cascade RPN relies on adaptive convolution to systematically align the features to the anchors. In the first stage, the adaptive convolution is set to perform dilated convolution since anchor center offsets are zeros. The features of the first stage are “bridged” to the next stages since the spatial order of the features is maintained by the dilated convolution. The pipeline of the proposed Cascade RPN can be described mathematically in Algorithm \[alg:cascade\_rpn\]. The anchor set at the first stage $\mathbb{A}^{1}$ is uniformly initialized over the image. At stage ${\tau}$, the anchor offset $\boldsymbol{o}^{\tau}$ is computed and fed into the regressor $f^{\tau}$ to produce the regression prediction $\hat{\boldsymbol{\delta}}^{\tau}$. The prediction $\hat{\boldsymbol{\delta}}^{\tau}$ is used to produce regressed anchors $\boldsymbol{a}^{\tau+1}$. At the final stage, the objectness scores are derived from the classifier, followed by NMS to produce the region proposals.
**Input**: sequence of regressors $f^{\tau}$, classifier $g$, feature $\boldsymbol{x}$ of image $I$.\
**Output**: proposal set $\mathbb{P}$.\
Uniformly initialize anchor set $\mathbb{A}^1 = \{\boldsymbol{a}^1\}$ over image $I$.\
**for** $\tau \leftarrow 1 $ **to** $T$ **do**\
Compute offset $\boldsymbol{o}^{\tau}$ of input anchor $\boldsymbol{a}^{\tau}$ on feature map using .\
Compute regression prediction $\hat{\boldsymbol{\delta}}^{\tau} = f^{\tau}(\boldsymbol{x}, \boldsymbol{o}^{\tau})$.\
Compute regressed anchor $\boldsymbol{a}^{\tau+1}$ from $\hat{\boldsymbol{\delta}}^{\tau}$ using .\
**end**\
Compute objectness score $\boldsymbol{s} = g(\boldsymbol{x}, \boldsymbol{o}^{T})$.\
Derive proposals $\mathbb{P}$ from $\mathbb{A}^{\tau+1} = \{\boldsymbol{a}^{\tau+1}\}$ and $\mathbb{S} = \{\boldsymbol{s}\}$ using NMS .
Learning
--------
Cascade RPN can be trained in an end-to-end manner using multi-task loss as follows: $$\mathcal{L} = \lambda \sum_{\tau=1}^{T} \alpha^{\tau} \mathcal{L}^{\tau}_{reg} + \mathcal{L}_{cls}.$$ Here, $\mathcal{L}^{\tau}_{reg}$ is the regression loss at stage $\tau$ with the weight of $\alpha^{\tau}$, and $\mathcal{L}_{cls}$ is the classification loss. The two loss terms are balanced by $\lambda$. In the implementation, binary cross entropy loss and IoU loss [@unitbox] are used as the classification loss and regression loss, respectively.
Experiments
===========
Experimental Setting
--------------------
The experiments are performed on the COCO 2017 detection dataset [@COCO]. All the models are trained on the `train` split (115k images). The region proposal performance and ablation analysis are reported on `val` split (5k images), and the benchmarking detection performance is reported on `test-dev` split (20k images).
Unless otherwise specified, the default model of the experiment is as follows. The model consists of two stages, with ResNet50-FPN [@Lin_2017_CVPR] being its backbone. The use of two stages is to balance accuracy and computational efficiency. A single anchor per location is used with size of $32^2$, $64^2$, $128^2$, $256^2$, and $512^2$ corresponding to the feature levels $C_2$, $C_3$, $C_4$, $C_5$, and $C_6$, respectively [@Lin_2017_CVPR]. The first stage uses the anchor-free metric for sample discrimination with the thresholds of the center-region $\sigma_{\text{ctr}}$ and ignore-region $\sigma_{\text{ign}}$, which are adopted from [@unitbox; @wang2019region], being 0.2 and 0.5. The second stage uses the anchor-based metric with the IoU threshold of 0.7. The multi-task loss is set with the stage-wise weight $\alpha^1 = \alpha^2 = 1$ and the balance term $\lambda = 10$. The NMS threshold is set to 0.8. In all experiments, the long edge and the short edge of the images are resized to 1333 and 800 respectively without changing the aspect ratio. No data augmentation is used except for standard horizontal image flipping. The models are implemented with PyTorch [@paszke2017automatic] and mmdetection [@mmdetection2018]. The models are trained with 8 GPUs with a batch size of 16 (two images per GPU) for 12 epochs using SGD optimizer. The learning rate is initialized to 0.02 and divided by 10 after 8 and 11 epochs. It takes about 12 hours for the models to converge on 8 Tesla V100 GPUs.
The quality of region proposals is measured with Average Recall (AR), which is the average of recalls across IoU thresholds from 0.5 to 0.95 with a step of 0.05. The AR for 100, 300, and 1000 proposals per image are denoted as AR$_{100}$, AR$_{300}$, and AR$_{1000}$. The AR for small, medium, and large objects computed at 100 proposals are denoted as AR$_S$, AR$_M$, and AR$_L$, respectively. Detection results are evaluated with the standard COCO-style Average Precision (AP) measured at IoUs from 0.5 to 0.95. The runtime is measured on a single Tesla V100 GPU.
Benchmarking Results
--------------------
Method Backbone AR$_{100}$ AR$_{300}$ AR$_{1000}$ AR$_{S}$ AR$_{M}$ AR$_{L}$ Time (s)
----------------------------------- ------------------ ------------ ------------ ------------- ---------- ---------- ---------- ----------
SharpMask [@pinheiro2016learning] ResNet-50 36.4 - 48.2 - - - 0.76
GCN-NS [@lu2018toward] VGG-16 (Sync BN) 31.6 - 60.7 - - - 0.10
AttractioNet [@gidaris2016attend] VGG-16 53.3 - 66.2 31.5 62.2 77.7 4.00
ZIP [@li2019zoom] BN-inception 53.9 - 67.0 31.9 63.0 78.5 1.13
RPN [@NIPS2015_5638] 44.6 52.9 58.3 29.5 51.7 61.4 **0.04**
Iterative RPN 48.5 55.4 58.8 32.1 56.9 65.4 0.05
Iterative RPN+ 54.0 60.4 63.0 35.6 62.7 73.9 0.06
GA-RPN [@wang2019region] 59.1 65.1 68.5 40.7 68.2 78.4 0.06
Cascade RPN **61.1** **67.6** **71.7** **42.1** **69.3** **82.8** 0.06
: Region proposal results on COCO 2017 `val`.
\[tab:region\_proposal\_benchmark\]
Method Proposal method \# proposals AP AP$_{50}$ AP$_{75}$ AP$_S$ AP$_M$ AP$_L$
-------- ----------------- -------------- ---------- ----------- ----------- ---------- ---------- ----------
RPN 37.0 **59.5** 39.9 21.1 39.4 47.0
Cascade RPN **40.1** **59.5** **43.7** **22.8** **42.4** **50.9**
RPN 36.6 58.6 39.5 20.3 39.1 47.0
Iterative RPN+ 38.6 58.8 42.2 21.1 41.5 50.0
GA-RPN 39.5 59.3 43.2 21.8 42.0 50.7
Cascade RPN **40.1** **59.4** **43.8** **22.1** **42.4** **51.6**
RPN 37.1 **59.3** 40.1 21.4 39.8 46.5
Cascade RPN **40.5** **59.3** **44.2** **22.6** **42.9** **51.5**
**** RPN 36.9 58.9 39.9 21.1 39.6 46.5
Iterative RPN+ 39.2 58.2 43.0 21.5 42.0 50.4
GA-RPN 39.9 **59.4** 43.6 **22.0** 42.6 50.9
Cascade RPN **40.6** 58.9 **44.5** **22.0** **42.8** **52.6**
: Detection results on COCO 2017 `test-dev`
\[tab:detection\_benchmark\]
#### Region Proposal Performance.
The performance of Cascade RPN is compared to those of recent state-of-the-art region proposal methods, including RPN [@NIPS2015_5638], SharpMask [@pinheiro2016learning], GCN-NS [@lu2018toward], AttractioNet [@gidaris2016attend], ZIP [@li2019zoom], and GA-RPN [@wang2019region]. In addition, Iterative RPN and Iterative RPN+, which are referred to in Figure \[fig:arch\], are also benchmarked. The results of Sharp Mask, GCN-NS, AttractioNet, ZIP are cited from the papers. The results of the remaining methods are reproduced using mmdetection [@mmdetection2018]. Table \[tab:region\_proposal\_benchmark\] summarizes the benchmarking results. In particular, Cascade RPN achieves AR 13.4 points higher than that of the conventional RPN. Cascade RPN consistently outperforms the other methods in terms of AR under different settings of proposal thresholds and object scales. The alignment rule is typically missing or loosely conformed to in the other methods; thus, their performance improvements are limited. The alignment rule in Cascade RPN is systematically ensured such that the performance gain is greater and more reliable.
#### Detection Performance.
To investigate the benefit of high-quality proposals, Cascade RPN and the baselines are integrated into common two-stage object detectors, including Fast R-CNN and Faster R-CNN. Here, Fast R-CNN is trained on precomputed region proposals while Faster R-CNN is trained in an end-to-end manner. As studied in [@wang2019region], despite high-quality region proposals, training a good detector is still a non-trivial problem, and simply replacing RPN by Cascade RPN without changes in the settings only brings limited gain. Following [@wang2019region], the IoU threshold in R-CNN is increased and the number of proposals is decreased. In particular, the IoU threshold and the number of proposals are set to 0.65 and 300, respectively. The experimental results are reported in Table \[tab:detection\_benchmark\]. Here, integrating RPN into Fast R-CNN and Faster R-CNN yields 37.0 and 37.1 mAP, respectively. From the results, the recall improvement is correlated with improvements in detection performance. As it has the highest recall, Cascade RPN boosts the performance for Fast R-CNN and Faster R-CNN to 40.1 and 40.6 mAP, respectively.
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Ablation Study
--------------
Baseline 1 anchor Cascade Align. AFAB Stats. IoU loss AR$_{100}$ AR$_{300}$ AR$_{1000}$
---------- ---------- ----------- ----------- ----------- -------- ---------- ------------ ------------ -------------
44.6 52.9 58.3
44.7 51.2 55.8
48.2 54.4 58.0
57.4 63.7 67.8
57.3 64.2 68.6
60.8 67.3 71.5
**61.1** **67.6** **71.7**
**+16.5** **+14.7** **+13.4**
: Ablation analysis of Cascade RPN. Here, Align., AFAB, and Stats. denote the use of alignments, anchor-free and anchor-based metrics, and regression statistics, respectively.
\[tab:ablation\]
Center Shape AR$_{100}$ AR$_{300}$ AR$_{1000}$
-------- ------- ------------ ------------ -------------
48.2 54.4 58.0
52.5 59.4 64.1
**57.4** **63.7** **67.8**
: The effects of sample metrics
\[tab:alignment\]
AF AB AR$_{100}$ AR$_{300}$ AR$_{1000}$
---- ---- ------------ ------------ -------------
55.2 61.8 66.4
**57.4** 63.7 67.8
57.3 **64.2** **68.6**
: The effects of sample metrics
\[tab:AFAB\]
#### Component-wise Analysis.
To demonstrate the effectiveness of Cascade RPN, a comprehensive component-wise analysis is performed in which different components are omitted. The results are reported in Table \[tab:ablation\]. Here, the baseline is RPN with 3 anchors per location yielding AR$_{1000}$ of 58.3. When the number of anchors per location is reduced to 1, the AR$_{1000}$ drops to 55.8, implying that the number of positive samples dramatically decreases. Even when the multi-stage cascade is added, the performance is 58.0, which is still lower than that of the baseline. However, when adaptive convolution is applied to ensure alignment, the performance surges to 67.8, showing the importance of alignment in multi-stage refinement. The incorporation of anchor-free and anchor-based metrics for sample discrimination incrementally improves AR$_{1000}$ to 68.6. The use of regression statistics (shown in Figure \[fig:intro\_a\]) increases the performance to 71.5. Finally, applying IoU loss yields a slight improvement of 0.2 points. Overall, Cascade RPN achieves 16.5, 14.7, and 13.4 points improvement in terms of AR$_{100}$, AR$_{300}$, and AR$_{1000}$ respectively, compared to the conventional RPN.
#### Acquisition of Alignment.
To demonstrate the effectiveness of the proposed adaptive convolution, the center and shape alignments, represented by the offsets in Eq. , are progressively applied. Here, the center and shape offsets maintain the alignments in position and semantic scope, respectively. Table \[tab:alignment\] shows that the AR$_{1000}$ improves from 58.0 to 64.1 using only the center alignment. When both the center and shape alignments are ensured, the performance increases to 67.8.
#### Sample Discrimination Metrics.
The experimental results with different combinations of sample discrimination metrics are shown in Table \[tab:AFAB\]. Here, AF and AB denote that the anchor-free and anchor-based metrics are applied for all stages, respectively. Meanwhile, AFAB indicates that the anchor-free metric is applied at stage 1 followed by anchor-based metric at stage 2. Here, AF and AB yield the AR$_{1000}$ of 66.4 and 67.8 respectively, both of which are significantly less than that of AFAB. It is noted that the thresholds for each metric are already adapted through stages. The results imply that applying only one of either anchor-free or anchor-based metric is highly adversarial. The both metrics should be incorporated to achieve the best results.
#### Qualitative Evaluation.
The examples of region proposal results at the first and second stages are illustrated in the first and second row of Figure \[fig:examples\], respectively. The results show that the output proposals at the second stage are more accurate and cover a larger number of objects.
#### Number of Stages.
Table \[tab:different\_stages\] shows the proposal performance on different number of stages. In the 3-stage Cascade RPN, an IoU threshold of 0.75 is used for the third stage. The 2-stage Cascade RPN achieves the best trade-off between AR$_{1000}$ and inference time.
#### Extension with Cascade R-CNN.
Table \[tab:crcnn\] reports the detection results of the Cascade R-CNN [@Cai_2018_CVPR] with different proposal methods. The Cascade RPN improves AP by 0.8 points compared to RPN. The improvement is mainly from AP$_{75}$, where the objects have high IoU with the ground truth.
Conclusion
==========
This paper introduces Cascade RPN, a simple yet effective network architecture for improving region proposal quality and object detection performance. Cascade RPN systematically addresses the limitations that conventional RPN heuristically defines the anchors and aligns the features to the anchors. A simple implementation of a two-stage Cascade RPN achieves AR 13.4 points higher than the baseline, surpassing any existing region proposal methods. When adopting to Fast R-CNN and Faster R-CNN, Cascade RPN can improve the detection mAP by 3.1 and 3.5 points, respectively.
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\# stages AR$_{100}$ AR$_{300}$ AR$_{1000}$ Time (s)
----------- ------------ ------------ ------------- ----------
1 56.0 62.2 66.3 **0.04**
2 **61.1** 67.6 71.7 0.06
3 60.9 **67.9** **72.2** 0.08
: Detection results of Cascade R-CNN with RPN and Cascade RPN (denoted by CRPN).
\[tab:different\_stages\]
Method AP AP$_{50}$ AP$_{75}$ AP$_{75}$ AP$_S$ AP$_M$
-------- ---------- ----------- ----------- ----------- ---------- ----------
RPN 40.8 **59.3** 44.3 22.0 44.2 54.2
CRPN **41.6** 59.0 **45.5** **23.0** **45.0** **55.2**
: Detection results of Cascade R-CNN with RPN and Cascade RPN (denoted by CRPN).
\[tab:crcnn\]
#### Acknowledgment.
This work was supported by Institute for Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government (MSIT) (2017-0-01780, The technology development for event recognition/relational reasoning and learning knowledge-based system for video understanding) and (No. 2019-0-01396, Development of framework for analyzing, detecting, mitigating of bias in AI model and training data)
|
---
abstract: 'Gas-rich minor mergers contribute significantly to the gas reservoir of early-type galaxies (ETGs) at low redshift, yet the star formation efficiency (SFE; the star formation rate divided by the molecular gas mass) appears to be strongly suppressed following some of these events, in contrast to the more well-known merger-driven starbursts. We present observations with the Atacama Large Millimeter/submillimeter Array (ALMA) of six ETGs, which have each recently undergone a gas-rich minor merger, as evidenced by their disturbed stellar morphologies. These galaxies were selected because they exhibit extremely low SFEs. We use the resolving power of ALMA to study the morphology and kinematics of the molecular gas. The majority of our galaxies exhibit spatial and kinematical irregularities, such as detached gas clouds, warps, and other asymmetries. These asymmetries support the interpretation that the suppression of the SFE is caused by dynamical effects stabilizing the gas against gravitational collapse. Through kinematic modelling we derive high velocity dispersions and Toomre $Q$ stability parameters for the gas, but caution that such measurements in edge-on galaxies suffer from degeneracies. We estimate merger ages to be about 100 Myr based on the observed disturbances in the gas distribution. Furthermore, we determine that these galaxies lie, on average, two orders of magnitude below the Kennicutt-Schmidt relation for star-forming galaxies as well as below the relation for relaxed ETGs. We discuss potential dynamical processes responsible for this strong suppression of star formation surface density at fixed molecular gas surface density.'
author:
- |
Freeke van de Voort,$^{1,2,3,4}$[^1] Timothy A. Davis,$^{5}$ Satoki Matsushita,$^{2}$ Kate Rowlands,$^{6}$ Stanislav S. Shabala,$^{7}$ James R. Allison,$^{8,9}$ Yuan-Sen Ting,$^{10,11,12}$ Anne E. Sansom$^{13}$ and Paul P. van der Werf$^{14}$\
$^{1}$Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118, Heidelberg, Germany\
$^{2}$Academia Sinica Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan\
$^{3}$Department of Astronomy and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720-3411, USA\
$^{4}$Astronomy Department, Yale University, P.O. Box 208101, New Haven, CT 06520-8101, USA\
$^{5}$School of Physics & Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK\
$^{6}$Department of Physics & Astronomy, Johns Hopkins University, Bloomberg Center, 3400 N. Charles St., Baltimore, MD 21218,\
USA\
$^{7}$School of Natural Sciences, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia\
$^{8}$Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, NSW 2006, Australia\
$^{9}$ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)\
$^{10}$Institute for Advanced Study, Princeton, NJ 08540, USA\
$^{11}$Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA\
$^{12}$Observatories of the Carnegie Institution of Washington, 813 Santa Barbara Street, Pasadena, CA 91101, USA\
$^{13}$Jeremiah Horrocks Institute, University of Central Lancashire, Preston, Lancashire, PR1 2HE, UK\
$^{14}$Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, the Netherlands
bibliography:
- 'sfsuppression.bib'
date: 'Accepted 2018 January 24. Received 2018 January 23; in original form 2017 December 21'
title: 'An ALMA view of star formation efficiency suppression in early-type galaxies after gas-rich minor mergers'
---
\[firstpage\]
galaxies: formation – galaxies: evolution – galaxies: star formation – galaxies: ISM – galaxies: elliptical and lenticular, cD – galaxies: kinematics and dynamics
Introduction
============
In a hierarchical vacuum-dominated cold dark matter ($\Lambda$CDM) universe, mergers play a major role in the assembly of galaxies. Galaxy mergers are capable of inducing strong star formation, fuelling black hole growth, and precipitating morphological transformations, such as the formation of bulges [e.g. @Sanders1988; @Hernquist1993; @Springel2005]. Merger events are often divided into two categories: major mergers with mass ratios larger than $1:4$ and minor mergers with mass ratios below $1:4$. Each of these can be either gas-rich (‘wet’) or gas-poor (‘dry’). Gas-poor mergers are thought to be responsible for both morphological transformation and the build-up of the mass and size of early-type galaxies (ETGs) since $z\approx2$ [e.g. @Naab2009; @Dokkum2010]. Gas-rich major mergers are usually associated with strong starbursts with extreme star formation rates [SFRs; e.g. @Sanders1988; @Barnes1991].
The star formation efficiency (SFE) is here defined as the amount of star formation per unit *molecular* gas and defined as $$\mathrm{SFE}=\mathrm{SFR}/M_\mathrm{H_2},$$ where $M_\mathrm{H_2}$ is the molecular gas mass. The SFE is the inverse of the gas depletion time, which can be as low as $10^7$ yr in extreme starburst galaxies [e.g. @Gao2004]. This unusually efficient star formation is usually explained by the dissipative collapse of the gas to the galaxy centre following a gas-rich major merger. The presence of a pre-existing interstellar medium (ISM) in spiral galaxies causes strong shocks when the gas in the two galaxies collide. These shocks and angular momentum loss drive the gas densities up and result in a nuclear starburst [e.g. @Mihos1996].
The effect of gas-rich *minor* mergers on the star formation in galaxies has not been quantified as thoroughly. Minor mergers are less violent events, but these too might be expected to increase the SFR and/or SFE [e.g. @Mihos1994; @Saintonge2012]. @Kaviraj2014 found that minor mergers play an important role in the low-redshift universe, where about 40 per cent of star formation in local spiral galaxies is induced by minor mergers. Such events also dominate the star formation budget in massive early-type systems, where they likely provide the vast majority of the star-forming gas [e.g. @Kaviraj2009; @Davis2011].
This work is an extension of that by @Kaviraj2012 and @Kaviraj2013, who selected a sample of recent minor merger remnants, based on their disturbed optical morphologies. In these recently merged galaxies, the major partner was a gas-poor ETG and the minor partner was a gas-rich dwarf. The median merger mass ratio was estimated to be 1:40, based on gas-to-dust ratios [@Davis2015]. The resulting systems are very gas- and dust-rich compared to undisturbed ETGs and have a sizeable molecular gas component ($M_\mathrm{H_2} > 10^9 $ M$_{\astrosun}$). Contrary to expectations, these coalesced systems do not host strong starbursts. The absence of strong merger-induced shocks, due to the gas-poor nature of the original host galaxy, may be responsible for this. More surprising is that the SFE was found to be suppressed by orders of magnitude, with gas depletion times that exceed the Hubble time.
The physical process that is suppressing star formation in these minor merger systems is not yet understood. Clearly such objects do not fit within a framework where the SFE is constant [e.g. @Bigiel2011] or one where mergers boost the efficiency of star formation [e.g. @Genzel2010; @Daddi2010; @Saintonge2012]. ETGs have previously been found to have somewhat lower SFE than late-type systems [e.g. @Saintonge2012; @Davis2014]. @Martig2009 found that this may be caused by gas stability in a spheroidal potential well. This could also be the reason why the star formation rate is less enhanced after minor mergers in galaxies with more prominent bulges [@Kaviraj2014]. The recent minor merger systems studied in @Davis2015 and in this work, however, have an order of magnitude greater suppression of the SFE and larger gas fractions than seen in relaxed ETGs. Gas stabilization due to the shape of the potential is therefore unlikely to be the reason for the low SFEs, at least in these extreme objects.
Star formation is a dynamical process. The cold, dense gas in galaxies follows a turbulent cascade down to the smallest scales, where it fragments and eventually forms stars. Various dynamical and environmental processes can work to suppress this cascade and affect the resulting SFE. @Davis2015 suggested that these objects may have been caught at a very specific phase in their evolution, when gas is still free-streaming towards the galaxy centre, but has not yet settled. Such streaming motions have been shown to suppress the efficiency of star formation [e.g. @Meidt2013]. In this scenario, the gas-free nature of the early-type progenitor explains why star formation suppression after minor mergers has not been observed in other systems.
This explanation is not unique, however. Shear induced by rotation can prevent the gravitational collapse of gas clouds or pull them apart, suppressing star formation [@Toomre1964; @Seigar2005]. Additionally, gravitational heating, i.e. conversion of potential energy to thermal energy during a merger, can deposit energy into the gas via weak shocks [e.g. @Johansson2009]. Such energy could stall gravitational collapse and suppress star formation. Alternatively, although no evidence for strong nuclear activity has been found in these objects [@Shabala2012], active galactic nuclei (AGN) are also expected to be triggered by mergers [@Kaviraj2015]. Therefore, nascent feedback from a central black hole may be acting to suppress star formation. Determining which, if any, of these processes are acting to suppress star formation in ETGs after a minor merger will allow us to understand its importance for merger-driven star formation in all types of galaxies.
In this paper we present results based on new observations with the Atacama Large Millimeter/submillimeter Array (ALMA) of six minor merger remnants from @Davis2015. We selected those objects with the most suppressed SFEs (below 10$^{-10}$ yr$^{-1}$). Their depletion times are above 10 Gyr, longer than those of ‘normal’ ETGs, such as those in the ATLAS$^\mathrm{3D}$ sample [@Davis2014], by a factor of three. They have typical stellar masses of about 10$^{11}$ M$_{\odot}$, with molecular gas fractions between 1 and 25 per cent.
In Section \[sec:obs\] we describe the targeted galaxies and the ALMA observations. In Section \[sec:results\] we show the resolved ISM morphologies and velocity structures, which often exhibit clear disturbances, and present results from our kinematic modelling. We discuss which process is likely responsible for the low SFEs and conclude in Section \[sec:concl\]. In this work, we assume a cosmology with $H_0=71$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_m=0.27$ and $\Omega_\Lambda=0.73$, consistent with previous work.
Observations {#sec:obs}
============
Sample
------
The galaxies targeted in this study were originally selected from the Sloan Digital Sky Survey (SDSS) photometry by @Kaviraj2012 as part of the Galaxy Zoo project [@Lintott2008]. The objects are massive, bulge-dominated galaxies with large dust lanes obscuring part of the optical light. The parent sample selected the most dust-rich objects using 250 $\mu$m luminosities observed with the *Herschel Space Telescope* [@Kaviraj2013]. From the galaxies observable from ALMA’s location, we selected the six galaxies with the lowest SFEs based on previous Institut de Radioastronomie Millimétrique (IRAM) 30 m data [@Davis2015]. Three-colour ($66\times66$ kpc$^2$) images of our six targets are shown in Figure \[fig:3colour\], using the Dark Energy Camera Legacy Survey (DECaLS; @DECaLS2016; in bands $grz$), when available, and from SDSS (@SDSSDR72009; in bands $gri$) for galaxies NGC4797 and 2MASXJ1333. Besides the clear dust lanes, indicating the likely presence of edge-on gas discs, some of the images also reveal clear disturbances of the stellar light in the outskirts of the galaxies, which point to recent minor merger events [see @Kaviraj2012].
[lllllccr]{}\
source identifier & $z$ & $D_L$ & $M_\mathrm{star}$ & $M_\mathrm{H_2}$ & SFR & SFE & conversion\
& & (Mpc) & (M$_{\astrosun}$) & (M$_{\astrosun}$) & (M$_{\astrosun}$ yr$^{-1}$) & (yr$^{-1}$) & (kpc/)\
\
2MASXJ09033081–0106127 & 0.040 & 161 & $10^{10.9\pm0.1}$ & $10^{9.64\pm0.07}$ & $0.09^{+0.05}_{-0.01}$ & $(2.06^{+1.19}_{-0.40})\times10^{-11}$ & 0.78\
2MASXJ14161186+0152048 & 0.082 & 315 & $10^{11.2\pm0.1}$ & $10^{9.89\pm0.11}$ & $0.02^{+0.06}_{-0.01}$ & $(2.58^{+7.77}_{-1.45})\times10^{-12}$ & 1.53\
NGC4797 & 0.026 & 106 & $10^{11.2\pm0.1}$ & $10^{9.17\pm0.08}$ & $0.03^{+0.03}_{-0.01}$ & $(2.03^{+2.06}_{-0.77})\times10^{-11}$ & 0.52\
2MASXJ13010083+2701312 & 0.078 & 301 & $10^{10.9\pm0.1}$ & $10^{10.31\pm0.05}$ & $0.44^{+0.44}_{-0.37}$ & $(2.16^{+2.17}_{-1.83})\times10^{-11}$ & 1.46\
2MASXJ13333299+2616190 & 0.037 & 150 & $10^{10.8\pm0.1}$ & $10^{9.71\pm0.05}$ & $0.01^{+0.06}_{-0.01}$ & $(1.95^{+11.70}_{-1.95})\times10^{-12}$ & 0.73\
CGCG013–092 & 0.035 & 142 & $10^{10.9\pm0.1}$ & $10^{9.84\pm0.04}$ & $0.28^{+0.06}_{-0.04}$ & $(4.05^{+0.94}_{-0.69})\times10^{-11}$ & 0.69\
The properties of our target galaxies are taken from @Davis2015 and presented in Table \[tab:gal\]. The source name, redshift, luminosity distance, stellar mass, SFR, SFE, and conversion factor from arcsec to kpc are listed for each source. The SFRs are derived by modelling the spectral energy distribution from the ultraviolet to the far-infrared with energy-balance code <span style="font-variant:small-caps;">magphys</span> [@Cunha2008] and therefore typically sensitive to timescales of approximately 100 Myr [@Kennicutt2012]. The errors on the SFE are unfortunately fairly large and the SFEs of the six galaxies are consistent with each other within $2\sigma$. We therefore do not expect to find strong trends with SFE within our sample.
ALMA data
---------
We observed the $^{12}$CO(1–0) line in our six dust lane ETGs with ALMA, as part of programme 2015.1.00320.S. ALMA’s 12 m antennas were used in a compact configuration, resulting in an approximately 1 beam and sensitivity to emission on scales up to $\approx30-40\arcsec$. A 1850 MHz correlator window was placed over the CO(1–0) line, yielding a continuous velocity coverage of about $1600$ km s$^{-1}$ with a raw velocity resolution of $\approx1.0$ km s$^{-1}$. This is sufficient to properly cover and sample the line. Three additional 2 GHz wide low-resolution correlator windows were simultaneously used to potentially detect continuum emission (see Section \[sec:cont\]).
The raw ALMA data were calibrated using the standard ALMA pipeline, provided by the ALMA regional centre staff. We then used the Common Astronomy Software Applications (<span style="font-variant:small-caps;">casa</span>) package to image the resulting visibility files for each track, producing a three-dimensional RA-Dec-velocity data cube (with velocities determined with respect to the rest frequency of the $^{12}$CO(1–0) line). In this work we bin the raw data to a channel width of 10 km s$^{-1}$ and use pixels of 0.3 (giving us approximately 4 pixels across the synthesized beam).
The data presented here were produced using Briggs weighting with a robust parameter of 0.5. We attempted to detect continuum emission from these sources over the full line-free bandwidth. If present, we subtracted the continuum from the line data in the $uv$ plane using the <span style="font-variant:small-caps;">casa</span> task <span style="font-variant:small-caps;">uvcontsub</span>. The continuum-subtracted dirty cubes were cleaned in regions of source emission (identified interactively) to a threshold equal to the root-mean-square (RMS) noise of the dirty channels. The clean components were then re-convolved using a Gaussian beam of full-width at half-maximum (FWHM) equal to that of the dirty beam. Finally, the residuals were added back into the clean components. This produced the final, reduced, and fully calibrated $^{12}$CO(1–0) data cubes.
The molecular masses used here assume a Galactic $X_\mathrm{CO}$ factor, which quantifies the conversion between CO integrated line flux and $\mathrm{H}_2$ mass. However, $X_\mathrm{CO}$ has been shown to vary with metallicity (see @Bolatto2013 for a review). We unfortunately do not have an independent measure of the gas-phase metallicity in our galaxies, which may be sub-solar. We could therefore be underestimating the amount of molecular gas present in these systems, possibly by an order of magnitude [@Leroy2007]. This would only reduce the SFEs even further and make our targets even more extreme objects.
Line detections
---------------
[llllllcc]{}\
source identifier & obs. date & $t_\mathrm{obs}$ & amplitude & bandpass & phase & $\theta_\mathrm{beam}$ & RMS\
& & (mins) & calibrator & calibrator & calibrator & () & (mJy beam$^{-1}$)\
\
2MASXJ09033081–0106127 & 2016-03-19 & 29.36 & J0750+1231 & J0854+2006 & J0909+0121 & $1.91\times1.32$ & 0.9\
2MASXJ14161186+0152048 & 2016-04-09 & 47.35 & Ganymede & J1337-1257 & J1410+0203 & $1.78\times1.46$ & 0.8\
NGC4797 & 2016-03-21 & 62.73 & Callisto & J1229+0203 & J1303+2433 & $1.80\times1.35$ & 1.0\
2MASXJ13010083+2701312 & 2016-05-04 & 29.86 & Callisto & J1229+0203 & J1303+2433 & $2.05\times1.29$ & 1.1\
2MASXJ13333299+2616190 & 2016-05-04 & 27.15 & J1256-0547 & J1229+0203 & J1333+2725 & $2.35\times1.03$ & 1.5\
CGCG013–092 & 2016-03-20 & 34.34 & Callisto & J1220+0203 & J1229+0203 & $1.62\times1.31$ & 1.0\
Table \[tab:obs\] summarizes our observational parameters. For each source, we list its full name (taken from the NASA/IPAC Extragalactic Database[^2]), the dates the galaxies were observed, the total integration time, the calibrator sources used (for the amplitude, bandpass, and phase), the beam size, and the RMS noise reached in 10 km s$^{-1}$ channels. The data quality and precision are sufficient to resolve the spatial structure of the molecular ISM and its line-of-sight velocity in all of our target galaxies using the CO(1-0) line. No other lines were detected. We use these data to detect and quantify morphological and kinematic disturbances in the gas in Section \[sec:results\].
Continuum detections and upper limits {#sec:cont}
-------------------------------------
[lll]{}\
source identifier & observed $\nu$ & continuum\
& (GHz) & ($\mu$Jy)\
\
2MASXJ09033081–0106127 & 97.9 (LSB) & $332\pm39$\
& 110.0 (USB) & $370\pm50$\
2MASXJ14161186+0152048 & 99.7 & $<22$\
NGC4797 & 99.3 (LSB) & $218\pm17$\
& 111.5 (USB) &$295\pm29$\
2MASXJ13010083+2701312 & 100.1 & $<23$\
2MASXJ13333299+2616190 & 104.2 & $<34$\
CGCG013–092 & 104.4 & $<80$\
Although the focus of this work is on the molecular gas, we attempted to detect continuum emission as well. Table \[tab:dust\] summarizes our findings and lists, for each source, our measurement or upper limit of the 3 mm continuum flux (right column) and the frequency at which this was observed (middle column). In case of a detection, we list the flux from the upper and lower side band (USB and LSB) separately, but combined them for our upper limits. We detected the continuum for galaxies 2MASXJ0903 and CGCG013–092 ($\approx300$ $\mu$Jy), which are both point sources at 3 mm. By comparing to the far-infrared fluxes from @Rowlands2012, we interpret this emission as arising from the Rayleigh-Jeans tail of the dust emission. We do not discuss this further here, because this is beyond the scope of this work. The lack of synchrotron-dominated spectra at 3 mm (non-detections or a spectral slope inconsistent with synchrotron emission) argues against the presence of strong AGN, but we cannot exclude low-luminosity AGN.
Results {#sec:results}
=======
\
The aim of this work is to resolve the spatial and kinematic structure of the molecular gas in the selected dust lane ETGs in order to narrow down the cause of the suppression in their SFEs. Figure \[fig:img\] therefore shows the new ALMA observations of the six selected galaxies. Left and middle panels show CO(1-0) flux contours in orange, in the left panels combined with the SDSS $r$-band image in greyscale. The contours displayed range from 3 times the RMS level to the maximum value in 3 (left) or 19 (middle) linearly spaced increments. The right panels show the CO(1-0) line-of-sight velocity contours centred on the kinematic centre as determined from their kinematic modelling (see Section \[sec:kin\]).
As expected, the gas is coincident with the dust lane seen in absorption against the stellar light, which means that both likely have the same origin. The images show that the majority of the galaxies’ molecular gas discs are asymmetrical. 2MASXJ1416 has detached gas clouds, the low-surface brightness emission extends further from the centre on one side of the disc in 2MASXJ0903 and CGCG013–092, and NGC4797 and 2MASXJ1301 exhibit a small warp in their centres (best seen by the tilt in the zero velocity contour in the velocity maps). Only galaxy 2MASXJ1333 shows no sign of asymmetries. The disturbed kinematics can also be identified in the position-velocity diagrams (see Figure \[fig:PVD\]). This provides visual evidence that the molecular gas is not completely relaxed. It also supports our conclusion that the gas was brought in by a gas-rich minor merger in the recent past.
It is interesting to compare the extent of the molecular gas in our sample to that of relaxed ETGs in the ATLAS$^\mathrm{3D}$ sample. We measure the radius of the relaxed gas disc by fitting a kinematic model to the data cube, which only reproduces the symmetric, undisturbed part (see Section \[sec:kin\]). The radius of the undisturbed gas is defined as the radius at which the model surface brightness falls below 10 M$_\odot$pc$^{-2}$, which is the surface brightness limit used by @Davis2013. The total extent, including the morphologically and kinematically disturbed part, is measured directly from the data, also using the surface brightness limit of 10 M$_\odot$pc$^{-2}$. For the relaxed galaxies, the radius of the molecular gas disc normalized by the radius of the 25 mag arcsec$^{−2}$ isophote in the $B$ band is $R_\mathrm{CO}/R_{25}=0.16$ [@Davis2013]. Unsurprisingly, the gas discs in our selected dust lane ETGs are more extended. For the undisturbed part of our gas discs, we find an average $R_\mathrm{CO}/R_{25}=0.28$ and if we include the disturbed gas this increases to $R_\mathrm{CO}/R_{25}=0.53$. This shows that our sample galaxies are extreme, not just in their total molecular gas mass, but also in the extent of their molecular gas.
\
In Figure \[fig:spec\] we compare our ALMA CO(1-0) flux measurements with those previously obtained with the IRAM 30 m telescope [@Davis2015], showing the CO(1-0) flux as a function of velocity, where the centre is obtained from our kinematic modelling (see Section \[sec:kin\]). The ALMA spectra were made by extracting a rectangular region based on the images shown in Figure \[fig:img\], then summing the unclipped data cube spatially, and dividing by the beam area. Both observations show the typical double-horned spectrum of rotating galaxies. Our interferometric observations are similar to the single dish measurements. The total flux is consistent within uncertainties, which indicates that we are likely not resolving out any structure on large scales. The spectra of galaxies 2MASXJ0903, 2MASXJ1416, 2MASXJ1301, and CGCG013–092 have an asymmetric profile, reflecting their asymmetric morphologies.
\
To study the rotation curve of the molecular gas, Figure \[fig:PVD\] shows the major-axis position-velocity diagram of each of our target galaxies. This was created by extracting a slice from the cube at the estimated position angle, with a width of 5 pixels, which were then summed spatially. Contour levels are evenly spaced between 3 times the RMS level and the peak flux in 9 increments. The disturbed structures, previously identified along the major axis in the images in Figure \[fig:img\], can also be seen here. According to our best-fit kinematic model (see Section \[sec:kin\]), they do not follow the general galaxy rotation curve, although accurate stellar photometry is needed to model the stellar mass and be absolutely certain about this. 4 out of our 6 targets clearly show asymmetrical features at large radii – galaxies 2MASXJ0903, 2MASXJ1416, NGC4797, and CGCG013–092. The asymmetry of galaxy 2MASXJ1301 is more subtle, but can be identified by noticing that the emission extends from $-8$ kpc to $+10$ kpc. Galaxy 2MASXJ1333 is the only object for which, at the current resolution, all the detected molecular gas seems to be part of a regular, relaxed gas disc.
A large fraction of the molecular gas is located in the rising part of the rotation curve. This is consistent with samples of relaxed ETGs (from ATLAS$^\mathrm{3D}$ and EDGE-CALIFA), which show that the SFE decreases as this fraction increases [@Davis2014; @Colombo2017]. This could suggest that a dynamical effect, such as the higher shear rate in this region, acts to stabilize the gas against collapse. However, the dust lane ETGs have SFEs that are much lower than those in ATLAS$^\mathrm{3D}$, so this is likely not the dominant mechanism for our extreme SFE suppressed galaxies.
The molecular gas which exists outside the central, undisturbed part of the discs appears not to be in dynamical equilibrium. Instead, it is likely flowing towards or away from the centre. Because the disturbed structures are oriented along the major axis, we consider the most likely scenario that this gas is accreting.
Kinematic modelling {#sec:kin}
-------------------
To learn more about the dynamical state of the gas and its physical parameters, we aim to construct kinematic models that reproduce our CO observations. In order to model the kinematics of the gas, we use a forward modelling approach. We assume that the gas is in a circularly symmetric disc and therefore do not try to model the asymmetrical features seen in Figure \[fig:img\] and \[fig:PVD\]. Because the disturbances are low surface density features, they do not strongly affect the model, which aims to reproduce the majority of the molecular gas (i.e. the central disc).
We used the KINematic Molecular Simulation (KinMS[^3]) tool [@Davis2013], with which we produce mock observations of a theoretical gas distribution with the same beam, pixel size, and velocity resolution as our ALMA observations. We then compare the mock observations to the real data and explore the parameter range with the Markov Chain Monte Carlo (MCMC) code KinMS\_MCMC that couples to the KinMS routines in order to get the full Bayesian posterior probability distribution. This code fits the entire CO data cube produced by ALMA, rather than simply the position-velocity diagram shown in Figure \[fig:PVD\].
The mass budget in these ETGs is dominated by their stars. Due to the lack of sufficiently high resolution optical imaging, we model a galaxy’s rotation curve by assuming it follows an arctangent [e.g. @Swinbank2012]. We force the curve to reach its maximum faster by imposing $$\begin{split}
v &= v_\mathrm{circ, flat} \mathrm{tan}^{-1}\Big(\frac{R}{r_\mathrm{norm}}\Big) \ \ \ \mathrm{for} \ \mathrm{tan}^{-1}\Big(\frac{R}{r_\mathrm{norm}}\Big) < 1 \\
&= v_\mathrm{circ, flat} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{for} \ \mathrm{tan}^{-1}\Big(\frac{R}{r_\mathrm{norm}}\Big) \ge 1,
\end{split}$$ where $v_\mathrm{circ, flat}$ is the asymptotic (or maximum) circular velocity in the flat part of the rotation curve, $R$ is the radius, and $r_\mathrm{norm}$ is the normalization radius. We fit $v_\mathrm{circ, flat}$ and $r_\mathrm{norm}$ to best match the gas velocities. We repeated our kinematic modelling based on available long-wavelength optical or near-infrared image (from SDSS or *Spitzer*) and found best fit values similar to those using the arctangent model and presented in Table \[tab:Q\].
We furthermore fit the total flux, kinematic centre, systemic velocity, position angle, and inclination of the gas disc. We additionally fit a model of the gas surface brightness profile and gas velocity dispersion (assumed to be constant throughout the disc), in order to reproduce the bulk properties of the disc. For the galaxies 2MASXJ0903, 2MASXJ1416, NGC4797, and 2MASXJ1301, we find a reasonable fit by using an exponential gas surface brightness profile. This simple form has previously been shown to be appropriate in most ETGs [@Davis2013] and provides a good match to the observed morphology and velocity of the gas in these objects. For galaxy 2MASXJ1333, however, we find that we additionally need an outer surface density cutoff in order to fit the data. Object CGCG013–092 clearly shows an X-shape in its position-velocity diagram, which is typical for molecular gas within a bar. We model this with a central ring and an outer exponential profile.
We use the derived centre, systemic velocity, and position angle to centre the previous figures. With the derived velocity dispersion and gas profile, we try to estimate whether or not the observed gas discs are stable against gravitational collapse. In order to do this, we calculate the Toomre $Q$ parameter [@Toomre1964] by evaluating $$Q=\dfrac{\kappa \sigma_\mathrm{gas}}{\pi G \Sigma_\mathrm{gas}}.$$ Here, $G$ is the gravitational constant, $\Sigma_\mathrm{gas}$ is the surface density of the molecular gas, and $\kappa$ is the epicyclic frequency (i.e. the frequency at which a gas parcel will oscillate when radially displaced). The latter is calculated by $$\kappa=\sqrt{4\Omega^2 + R\dfrac{d\Omega^2}{dR}},$$ where $R$ is the radius and $\Omega$ is the angular frequency $\Omega = v_\mathrm{circ}/R$, where $v_\mathrm{circ}$ is the circular velocity. A higher velocity dispersion means higher gas pressure stabilizing the disc against collapse. Here, we calculate $\kappa$ from our best-fitting arctangent models. $\kappa$ thus depends on the potential, but does not include the effects of any streaming or non-circular motions.
Theoretically, the disc is expected to be unstable to perturbations if $Q\lesssim1$. However, star-forming galaxies have been found to have $Q$ values above unity [e.g. @Leroy2008; @Romeo2017]. Higher threshold values have therefore also been used in order to account for asymmetric perturbations not captured in the simple model. Such perturbations are likely present in our disturbed galaxies as well and we therefore caution the reader that $Q$ by itself can probably not determine the stability of the disturbed gas discs.
Another problem is that the observed gas discs are nearly edge-on and our kinematic modelling therefore suffers from degeneracies due to non-circular rotation and beam smearing effects, although our modelling procedure attempts to take this into account. The derived velocity dispersion found is thus potentially a combination of the true velocity dispersion and the radial variation in velocities and may be spuriously high [@Barth2016]. We therefore consider our values upper limits to the true velocity dispersion of the gas.
[lclllccr]{}\
source identifier & inclination & $\sigma_\mathrm{gas}$ & $\langle Q\rangle_\mathrm{R}^{\raisebox{1.5pt}{$\scriptstyle{\rm max}$}}$ & $\langle Q\rangle_{\Sigma_\mathrm{gas}}^{\raisebox{1.5pt}{$\scriptstyle{\rm max}$}}$ & $\langle Q\rangle_\mathrm{R}^\mathrm{8~km/s}$ & $\langle Q\rangle_{\Sigma_\mathrm{gas}}^\mathrm{8~km/s}$ & $t_\mathrm{merge}$\
& () & (km s$^{-1}$) & & & & & (Myr)\
\
2MASXJ09033081–0106127 & 85 & $<40$ & $<16.3$ & $<12.6$ & 3.3 & 2.5 & 109\
2MASXJ14161186+0152048 & 76 & $<26$ & $<9.1$ & $<7.3$ & 2.8 & 2.3 & 149\
NGC4797 & 69 & $<16$ & $<19.6$ & $<22.7$ & 9.8 & 11.4 & 39\
2MASXJ13010083+2701312 & 78 & $<24$ & $<3.3$ & $<2.1$ & 1.1 & 0.7 & 171\
2MASXJ13333299+2616190 & 83 & $<6$ & $<3.1$ & $<3.8$ & 4.1 & 5.0 & $>99$\
CGCG013–092 & 78 & $<16$ & $<5.6$ & $<4.3$ & 2.8 & 2.2 & 128\
The results from our kinematic modelling are listed in Table \[tab:Q\]. The second and third column give the best fit value for the galaxies’ inclinations (with errors $<3$) and velocity dispersions. The next four columns show the derived values for $Q$ based either on the best-fit velocity dispersions (fourth and fifth columns) or on a canonical $\sigma_\mathrm{gas}=8$ km s$^{-1}$ (sixth and seventh columns; @Caldu2016). The values given include both the radial averages (fourth and sixth columns) and the surface density-weighted averages (fifth and seventh columns). The derived upper limits for the velocity dispersions are generally high, except for 2MASXJ1333. Even when assuming a canonical value of $\sigma_\mathrm{gas}=8$ km s$^{-1}$, we find that $Q>1$ and therefore theoretically stable against gravitational collapse, with the exception of 2MASXJ1301. However, these values are similar to those found in local spiral galaxies and therefore do not provide much power to discriminate between galaxies with normal and suppressed SFEs.
The last column in Table \[tab:Q\] lists the estimated merger age, $t_\mathrm{merge}$, assumed to be equal to five times the dynamical time at the transition between the undisturbed disc (based on our modelling) and disturbed gas structures (measured at 3 times the RMS level). Because the gas inside this radius appears to be relaxed, whereas the gas outside this radius is clearly not settled yet, this time can serve as a proxy for when the merger happened. This rough estimate is based on theoretical studies of the relaxation of misaligned gas discs in the potential of elliptical galaxies that found that the relaxation process typically takes a few dynamical times ($t_\mathrm{dyn}$; @Tohline1982 [@Lake1983]). Specifically, @Lake1983 found that the relaxation time, i.e. the time it takes a misaligned disc to settle into the plane, was approximately $t_\mathrm{dyn}/\epsilon$, where $\epsilon$ is the eccentricity of the potential. For a typical lenticular galaxy $\epsilon\approx0.2$ [@Mendez2008], so we therefore assume $t_\mathrm{merge}\approx5t_\mathrm{dyn}$. The true value of $t_\mathrm{merge}$ could also be higher if it takes more than five dynamical times for the gas to settle [@Voort2015b]. Our merger ages are lower than the ages of the last starburst derived by @Kaviraj2012 from broadband optical colours. Galaxy 2MASXJ1333 has a lower limit for $t_\mathrm{merge}$, because we found no disturbed gas structures in this object.
Kennicutt-Schmidt relation {#sec:KS}
--------------------------
![\[fig:KS\] Comparison between the molecular gas surface density ($\Sigma_\mathrm{gas}$) and SFR surface density ($\Sigma_\mathrm{SFR}$). The black, solid line and grey shaded region show the average relation for star-forming galaxies and its $1\sigma$ scatter [@Kennicutt1998]. Local spiral galaxies (black crosses) and starburst galaxies (black plusses) fall on this relation, whereas relaxed ETGs (open black diamonds) are slightly offset, on average [@Davis2014]. Our six dust lane ETGs are shown as red symbols, with $1\sigma$ error bars, for which we take the area of the undisturbed gas that can be modeled as a disc. The cyan symbols (connected to the corresponding red symbols by a dotted line) instead take into account the maximum extent of the disturbed features by assuming this gas is also located in a symmetric disc (note that galaxy 2MASXJ1333 shows no disturbed features). This results in the maximum possible surface area of the gas (both disturbed and undisturbed). This overestimate of the surface area moves the galaxies slightly closer to the $\Sigma_\mathrm{SFR}-\Sigma_\mathrm{H_2}$ relation, but not enough to affect our conclusions. Our sample of ETGs that have undergone a recent minor merger have molecular gas surface densities similar to spiral galaxies and relaxed ETGs, but much lower SFR surface densities, falling on overage 2 orders of magnitude below the Kennicutt-Schmidt relation. ](figures/KS.eps)
Because we spatially resolve the gas disc in all our dust lane galaxies that recently experienced a gas-rich merger, we can compare them to the observed Kennicutt-Schmidt relation [@Kennicutt1998]. Figure \[fig:KS\] shows the relation between SFR surface density, $\Sigma_\mathrm{SFR}$, and molecular gas surface density, $\Sigma_\mathrm{gas}$ for our six SFE suppressed ETGs (red symbols with $1\sigma$ error bars) compared to galaxies in the literature. This is calculated by dividing the SFR and $M_\mathrm{H_2}$ by the area of the molecular gas disc. The black, solid line and grey shaded region show the average relation for star-forming galaxies and its $1\sigma$ uncertainty, based on local spiral galaxies (black crosses) and starburst galaxies (black plusses) from @Kennicutt1998. Relaxed ETGs (black diamonds) from the ATLAS$^\mathrm{3D}$ sample fall below the Kennicutt-Schmidt relation, on average [@Davis2014].
To estimate the surface area of the molecular gas in our targets, we use our kinematic models to estimate the extent of the relaxed gas disc (measured at 3 times the RMS level, which is approximately 7 M$_\odot$pc$^{-2}$) and assume axisymmetry. The resulting surface densities are shown as the red, filled symbols. The $1\sigma$ errors are based on the errors in the SFR and $M_\mathrm{H_2}$ measurements. Since there are disturbed gas features outside the disc, the surface area may in fact be somewhat larger. We estimate the maximal extent by assuming all the detected gas is located in an axisymmetric disc and show the resulting surface densities as cyan, filled symbols (except for galaxy 2MASXJ1333, because its gas disc is not disturbed). Error bars have been omitted, but are the same as for the corresponding undisturbed measurements (i.e. the red symbols). The true values will lie in between our two estimates (connected by dotted lines), but will likely be much closer to the one which excludes the disturbed gas (i.e. the red symbols).
Note that @Kennicutt1998 included the atomic gas as well as the molecular gas within the star-forming region, whereas we (and @Davis2014) used only the total molecular gas mass within the molecular gas region (which is likely similar to the star-forming region). H<span style="font-variant:small-caps;">i</span> has been detected in the majority of our dust lane ETGs [@Davis2015], but this emission was unresolved. Adding H<span style="font-variant:small-caps;">i</span> to our $\Sigma_\mathrm{gas}$ determination, would move our points to the right in Figure \[fig:KS\] (by $0.2-0.5$ dex), only exacerbating the disagreement with the star-forming galaxies.
The $\Sigma_\mathrm{gas}$ and $\Sigma_\mathrm{SFR}$ estimate based on the undisturbed area as well as the one based on the disturbed area show that these dust lane ETGs have much lower star formation surface densities than spiral galaxies and relaxed ETGs at fixed $\Sigma_\mathrm{gas}$. Because our galaxies have molecular gas densities similar to those in spirals and relaxed ETGs, the suppression is not caused by low gas densities at least in an average sense. It remains possible that the dense molecular gas, traced by molecules such as HCN and closely related to the SFR [@Gao2004], has lower surface densities in our objects than in relaxed systems. Also if this is the case, the question remains which physical process is responsible for this.
Discussion and conclusions {#sec:concl}
==========================
We studied the morphology and kinematics of cold, molecular gas in six dust lane ETGs. These objects were selected to have very low SFEs. These systems were observed previously and are known to exhibit mild disturbances in their stellar distribution, suggesting that they underwent minor mergers in the recent past. It is possible that star formation is suppressed due to the fact that the gas brought in by the minor merger is not dynamically relaxed. If the gas is still streaming in to settle into a disc, the high velocity may have a stabilizing effect on the gas, preventing it from collapsing. We obtained spatially resolved molecular gas measurements in these galaxies with ALMA to test this hypothesis.
We detect edge-on gas discs in all our six targets, as expected. The majority of these show disturbed features, such as detached gas clouds, warps, and other asymmetries, both morphologically and kinematically. Warps are expected when the gas, which was brought in via a minor merger with its angular momentum misaligned from the stars in the host galaxy, is still settling into the stellar potential [e.g. @Voort2015b]. Even though 5 out of 6 galaxies show some assymetries indicating the gas is not fully relaxed, the majority of the molecular gas is located in the central disc, which exhibits regular rotation.
To better understand the kinematic behaviour of the gas, we fit the data cube with a kinematic model using the KinMS modelling code coupled to an MCMC code. The resulting velocity dispersions of the molecular gas are relatively high ($16-40$ km s$^{-1}$), except in galaxy 2MASXJ1333 (6 km s$^{-1}$; which also shows no disturbed features). This, in principle, supports our claim that the gas discs are not dynamically relaxed. However, due to the fact that we are viewing the discs edge-on, we are unsure how much these velocity dispersions are biased because of projection effects (caused by non-circular motions). Future observations of more face-on galaxies with low SFEs (with inclinations below 60) will enable improved modelling, especially when paired with high-resolution high resolution optical or near-infrared imaging to better constrain the galaxy rotation curve.
We believe our objects have only recently acquired their molecular gas and are therefore in a special phase of their evolution in which the gas is not yet in dynamical equilibrium with the other galaxy components. Based on the observed disturbances in the gas distribution, we derive (minor) merger ages between 39 and 171 Myr. The gas is likely still flowing towards the centre, which potentially affects their stability against fragmentation and star formation. This enhanced stability could be provided by inflowing motions [e.g. @Meidt2013] or by shear [@Seigar2005] or by excess (weak) shocks or (magnetohydrodynamic) turbulence in the molecular gas [e.g. @Cluver2010; @Padoan2011]. Other observations of star formation in tidal tails or ram pressure stripped tails have also revealed low SFEs [e.g. @Knierman2013; @Jachym2014]. Certain interacting galaxies are observed to have a large fraction of warm molecular gas, likely due to shocks and turbulence induced by gas accretion, and to show suppressed SFEs [@Alatalo2014; @Appleton2014]. These are consistent with our interpretation that the suppression of the SFE is due to dynamical effects.
The SFRs measured are sensitive to star formation in the past $\approx100$ Myr. This timescales is similar to our estimated merger ages. However, the sensitivity increases towards more recent star formation and is most sensitive to stars with ages $\lesssim5-10$ Myr. It is possible that the star formation rate in our objects is increasing with time, which would mean that the instantaneous SFR is higher than the 100 Myr-averaged SFR, in which case we would be underestimating the SFE in our recently merged galaxies. However, for this to negate the $1-2$ orders of magnitude suppression of the SFE, the SFR would have to rise by a similarly large factor over the past $5-10$ Myr for all of our objects, which we consider unlikely.
Although the lack of synchrotron emission at 3 mm argues against any active supermassive black holes in our galaxy sample, AGN feedback could eject cold molecular gas and suppress star formation. However, @Rosario2017 find normal gas fractions and SFEs in the centres of nearby Seyfert galaxies. Additionally, outflows are generally detected away from the disc and emanating from the centre, whereas the kinematically disturbed features we detected are located in the plane of the disc. Furthermore, since the molecular gas is very extended, nuclear feedback is unlikely to efficiently couple to it. We therefore disfavour the explanation that AGN feedback (or supernova feedback) could be responsible for the abnormally low SFEs.
ETGs that have recently undergone a minor merger are not the only objects that feature severely suppressed SFEs. Similarly strong suppression of the SFE has been found in post-starburst galaxies [@Kohno2002; @French2015; @Suess2017]. These are very different objects, because their star formation rates are lower than in the recent past, since they just experienced a starburst, whereas star formation rates in our ETGs are higher than before, since the star-forming gas was recently brought in by a gas-rich merger. The physical reason behind the SFE suppression may be different in these different types of galaxies, but spatially resolved observations of the molecular gas content in post-starbursts may help answer this question.
In summary, we obtained resolved observations of CO emission in six dust lane ETGs with known low SFEs. We find clear morphological and kinematic disturbances, which indicate that the suppression of star formation could indeed be due to the gas motion stabilizing the molecular gas against collapse. Gas-rich mergers with gas-rich hosts are known to result in starburst events with elevated SFEs, due to the compression of the galaxies’ ISM. The opposite effect is found here, due gas-poor nature of the hosts studied in this work and therefore the absence of strong shocks. Future studies of these effects in hydrodynamical simulations will be useful to understand the evolution of the SFE during major and minor, gas-rich and gas-poor mergers.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the referee for their comments that helped clarify the manuscript. We would also like to thank Sugata Kaviraj for helpful suggestions. FvdV is supported by the Klaus Tschira Foundation. TAD acknowledges support from a Science and Technology Facilities Council Ernest Rutherford Fellowship. SSS thanks the Australian Research Council for an Early Career Fellowship, DE130101399. Parts of this research were conducted by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. YST is supported by the Carnegie-Princeton Fellowship and the Martin A. and Helen Chooljian Membership from the Institute for Advanced Study in Princeton. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2015.1.00320.S and we thank all those involved in the proposal. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA.
\[lastpage\]
[^1]: E-mail: freeke.vandevoort@h-its.org
[^2]: https://ned.ipac.caltech.edu
[^3]: https://github.com/TimothyADavis/KinMS
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[**PRODUCTION OF A HIGGS BOSON PLUS TWO JETS IN HADRONIC COLLISIONS**]{}\
\
1.cm [*Department of Physics and Astronomy*]{}\
[*Franklin and Marshall College, Lancaster, PA 17604*]{}\
ABSTRACT
0.5cm We consider the production of a Standard Model Higgs boson accompanied by two jets in hadronic collisions. We work in the limit that the top quark is much heavier than the Higgs boson and use an effective Lagrangian for the interactions of gluons with the Higgs boson. In addition to the previously computed four-gluon process, we compute the the amplitudes involving two quarks, two gluons and the Higgs boson and those involving four quarks and the Higgs boson. We exhibit the form of our results in the small-$\ph$ and factorization limits. We present numerical results for $\sqrt{S}= 14~\TeV$ and $\sqrt{S}= 2~\TeV$. We find that the dominant processes are $gg \to ggH$ and $qg \to qg H$ with the former (latter) contributing about 60% (40%) of the cross section at $\sqrt{S}= 14~\TeV$ and the two processes each contributing about half the cross section at $\sqrt{S}= 2~\TeV$. All other processes are negligible at both energies. 0.25cm September 1996
Introduction
============
The Higgs boson is the last remaining undiscovered element of the Standard Model. Discovery of a Higgs boson (or more than one) would confirm that the Higgs mechanism is the source of electro-weak symmetry breaking while convincing evidence that Higgs bosons do not exist would necessitate another explanation for electro-weak symmetry breaking. Thus, the search for the Higgs boson is one of the fundamental quests of modern high energy physics. Current published experimental results set a lower limit on the Higgs boson mass of about 60 GeV [@mhlimit] while the $e^+e^-$ collider LEPII can be expected to extend this limit to somewhere near $80~\GeV$.
In this paper, we are concerned with the production of Higgs bosons in hadronic collisions. We are particularly interested in the so-called “intermediate mass” Higgs boson, [*i.e.*]{}, one in the mass region $80~\GeV \leq\mh \leq 200~\GeV$, although, as we will argue, our results have a somewhat larger range of applicability. Experimentally this is an extremely difficult region in which to see the Higgs boson, due to the large backgrounds to the common Higgs boson decay channels. Hence it is vital to have precise predictions for the production cross section as well as for the distribution of the produced Higgs bosons in transverse momentum and rapidity. The probability of extra particles, be they jets, W’s or top quarks, being produced along with the Higgs boson, also impacts its detection. Accompanying particles may act as tags or be confused with the Higgs decay products.
Here we discuss the production of the Higgs boson accompanied by two jets. The cross section for $g g \to ggH$ was calculated previously [@hgggg]. We compute the contributions needed for the total Higgs boson plus two jet cross section: $gg\to qqH$, $qg \to qgH$, $qq \to ggH$, $qq \to qqH$, where ‘$q$’ stands generically for a quark or anti-quark of undetermined flavor. We consider only the QCD generated processes, that is, extra quark or gluons lines attached to the basic gluon-gluon–Higgs-boson interaction. (The electroweak process, $q q \to qq H$ through $WW$ or $ZZ$ fusion does not interfere with the all quark process considered here. The interference term would be proportional to the trace of a single $SU(3)$ generator and so vanishes.)
We will work in the limit in which the top quark is much heavier than the Higgs boson and all of the energy scales in the problem. Since experiments [@cdf] place $m_{\rm top}\simeq 175~\GeV$, this limit is relevant for the intermediate mass Higgs boson. This limit is also relevant for consideration of soft and collinear radiation surrounding the production of the Higgs boson.
The organization of the paper is as follows. The effective Lagrangian is discussed in Section 2. Section 3 contains the spinor product formalism in which the amplitudes will be computed. The amplitudes for a Higgs boson plus two or three massless particles are computed in Section 4. For completeness the Higgs boson plus four gluon amplitude is presented in Section 5. Sections 6 and 7 contain the calculations of the amplitude involving a Higgs boson plus a quark anti-quark pair and two gluons and the amplitude for a Higgs boson plus two quark anti-quark pairs, respectively. The limit of our results when the momentum of the Higgs is small is presented in Section 8 and their behaviour in the factorization limits is presented in Section 9. Section 10 contains our numerical results and the Appendix contains the squares of the various amplitudes.
The Effective Lagrangian
========================
The production mechanism in which we are interested is $gg \rightarrow H$ which occurs through a quark loop where the only numerically important contribution is that of the top quark. In the limit in which the top quark is heavy, $m_{\rm top} \gg \mh$, the cross section can be computed via the following effective Lagrangian [@rusk] $${\cal L}_{\rm eff}=-{1\over 4} A H G^A_{\mu \nu} G^{A~\mu \nu},
\label{eq:leff}$$ where $G^A_{\mu \nu}$ is the field strength of the SU(3) color gluon field and $H$ is the Higgs-boson field. The effective coupling $A$ is given by $A = \a_s /(3 \pi v)$, where $v$ is the vacuum expectation value parameter, $v^2=(G_F\sqrt{2})^{-1}=(246~\GeV)^2$. The effective Lagrangian generates vertices involving the Higgs boson and two, three or four gluons. The associated Feynman rules are displayed in Fig 1. The two-gluon–Higgs-boson vertex is proportional to the tensor $$H^{\mu\nu}(p_1,p_2) = g^{\mu\nu}p_1 \cdot p_2 - p_1^\nu p_2^\mu.
\label{eq:bigh}$$ The vertices involving three and four gluons and the Higgs boson are proportional to their counterparts from pure QCD: $$V^{\mu\nu\ro}(p_1,p_2,p_3) = (p_1-p_2)^\ro g^{\mu\nu}
+ (p_2-p_3)^\mu g^{\nu\ro}
+ (p_3-p_1)^\nu g^{\ro\mu},
\label{eq:bigv}$$ and $$\begin{aligned}
X^{\mu\nu\ro\si}_{abcd} &=&
f_{abe}f_{cde}( g^{\mu\ro}g^{\nu\si} - g^{\mu\si}g^{\nu\ro} )
+f_{ace}f_{bde}( g^{\mu\nu}g^{\ro\si} - g^{\mu\si}g^{\nu\ro} )
\nonumber \\
&+&f_{ade}f_{bce}( g^{\mu\nu}g^{\ro\si} - g^{\mu\ro}g^{\nu\si} ).
\label{eq:bigx}\end{aligned}$$ It is straightforward to use this Lagrangian to obtain the ${\cal O}
(\alpha_s^3)$ contributions to the process $gg \rightarrow H$ [@sally; @zerwas]. These radiative corrections increase the lowest order rate by a factor of 1.5 to 2. As a by-product of the calculation of the ${\cal O}(\alpha_s^3)$ radiative corrections to $gg\rightarrow H$, one also obtains the cross section for $gg\rightarrow Hg $.
If the Higgs boson mass is of the same order as the top quark mass or larger the approximation entailed in the effective Lagrangian breaks down. However, even if $\mh$ is not much smaller than $m_t$, the results of the effective Lagrangian can be applied, after some modification, in the the soft and/or collinear regime. Factorization requires that as a gluon becomes soft or two particles become collinear an amplitude must factor into a divergent piece and a non-divergent piece, the divergent piece being independent of the hard process. Applied to the case of the Higgs-jet-jet amplitudes, when both outgoing jets have small $\pt$ (compared to the lowest scale in the problem, $\mh$ or $m_t$) factorization requires that the dependence on $\mh/m_t$ must be the same as the lowest order $Hgg$ amplitude. This was shown explicitly in Ref. [@hpt] for the processes $gg\to gH$, $qg \to qH$ and $q\bar q \to gH$. Since in this limit the only dependence on $m_t$ is in the overall factor, the result derived from the effective Lagrangian may simply be rescaled in order to be applied to the case when $m_t$ is not much larger than $\mh$.
Spinor Product Formalism
========================
We are interested in processes in which all the particles except the Higgs boson are massless. Each amplitude can be expressed in terms of spinors in a Weyl basis. For light-like momentum $p$ and helicity $\lam=\pm1$ we introduce spinors [@helic; @ber] $$\begin{aligned}
&&|p{\pm}\ra ={1\over 2} (1\pm \gamma_5)u(p) =
{1\over 2} (1\mp \gamma_5)v(p)\nonumber \\
%
&&\la p{\pm}| = \overline{u}(p){1\over 2} (1\mp \gamma_5)
= \overline{v}(p){1\over 2} (1\pm \gamma_5).
\label{eq:spindef}\end{aligned}$$ Polarization vectors for massless vector bosons can be written in terms of these spinors. For a gluon of momentum $k$ and positive or negative helicity $$\epsilon^{\mu}_{\pm} = { \la q{\pm}|\gamma^{\mu}|k{\pm}\ra
\over \sqrt{2}\la q{\mp}| k{\pm} \ra } \quad,
\label{eq:epsilons}$$ where the reference momentum $q$ satisfies $q^2=0$ and $q\cdot k\neq 0$ but is otherwise arbitrary; the freedom of the choice of $q$ is a reflection of gauge invariance. Each helicity amplitude can be expressed in terms of products of these spinors: $$\begin{aligned}
\la p{-}|q{+} \ra &=& {-}\la q{-}|p{+} \ra \equiv \la pq \ra, \nonumber \\
\la p{+}|q{-} \ra &=& {-}\la q{+}|p{-} \ra \equiv [ pq ], \nonumber \\
\la p{-}|q{-} \ra &=& \la q{+}|p{+} \ra = \la pp \ra = [pp] = 0.
\label{eq:spinprod}\end{aligned}$$ The spinor product $\la pq \ra$ and $[pq]$ are complex square roots of $2\, p\cdot q$, $$\begin{aligned}
&&\la pq \ra [qp] =2\, p\cdot q, \nonumber \\
%
&&\la pq \ra^* = {\rm sign}(p\cdot q) [qp].
\label{eq:spconj}\end{aligned}$$ Using the identities $$\slash p = |p{+}\ra \la p{+} | + |p{-}\ra \la p{-} |
\label{eq:helproj}$$ and $$\la p {\pm} | \g^\mu | q {\pm} \ra \g_\mu
= 2 (|q{\pm}\ra \la p{\pm} | + |p{\mp}\ra \la q{\mp} |,
\label{eq:fierz}$$ each amplitude can be written solely in terms of spinor products. The following identity and its complex conjugate are useful for simplifying the results: $$\la pq \ra \la rs \ra = \la ps \ra \la rq \ra + \la pr \ra \la qs \ra.
\label{eq:rearrange}$$
For the remainder of the paper we will use the convention that all the particles are outgoing. The amplitudes for the various processes involving two incoming massless particles and two outgoing massless particles plus a Higgs boson can then be obtained by crossing symmetry. The momenta of the massless particles are labelled $p_1,~p_2,~p_3,~p_4$ with the Higgs boson momentum being $p_\H$. Our convention is then $p_1+p_2+p_3+p_4+\ph=0$. We will use the shorthand notations $\la p_i p_j \ra = \la ij \ra$, $ [p_i p_j ] = [ij]$, $(p_i + p_j)^2 = S_{ij}$, and $(p_i + p_j + p_k)^2 = S_{ijk}$.
Two and Three Particle Plus Higgs Boson Processes
=================================================
As a preliminary to calculating the Higgs plus four-parton amplitudes, we present here the two and three parton amplitudes as examples. These amplitudes also provide the limiting forms of the four-parton amplitudes when two partons become collinear or a gluon becomes soft. The lowest order process by which Higgs bosons are created is $g g \to H$. The Feynman diagram is given in Fig. 1a. The non-zero helicity amplitudes are $++$ and $--$. We find $$\M^{++} = {1\o2} i A [12]^2 \d^{ab}.
\label{eq:hggamp}$$ The amplitude for the $--$ helicity combination can be obtained by exchanging square brackets for triangle brackets in this expression.
The lowest order processes which produce Higgs bosons with non-zero transverse momentum are $gg \to gH$, $qg \to qH$ and $q \bar q \to g H$. The third process is a crossing of the second. The relevant Feynmam diagrams are shown in Fig.’s 2 and 3. For $gggH$ the independent helicity amplitudes are (labelled according to helicities of gluons 1, 2 and 3 in order) $$\begin{aligned}
\M^{+++} &=& { g A f_{abc} \mh^4
\o \sqrt{2} \la 12 \ra \la 23 \ra \la 31 \ra } \\
\M^{-++} &=& { g A f_{abc} [23]^3
\o \sqrt{2} [12] [13] }.
\label{eq:hgggamp}\end{aligned}$$ The parity conjugate amplitudes can be obtained by exchanging square brackets for triangle brackets and multiplying by -1. Squaring these amplitudes and summing over helicities and colors leads to the known result [@hinch; @sally] $$\sum |\M(H\to ggg)|^2 = { g^2 A^2 N(N^2-1) \o S_{12} S_{13} S_{23} }
\l( S_{12}^4 + S_{13}^4 + S_{23}^4 + \mh^8 \r),
\label{eq:hgggsum}$$ where $N=3$ is the number of colors.
The helicity amplitudes for the process $H\to q\bar q g$ can be obtained similarly. Since QCD is helicity conserving in the massless limit, the quark and anti-quark must have opposite helicities. Using the momentum and color assignments in Fig. 3 we have (labeling the amplitudes by the helicity of the quark, antiquark and gluon, in that order) \^[+-+]{} = -[ig T\^a\_[ij]{} A ø ]{} [\[13\]\^2 ø\[12\]]{}, \[eq:hqqgamp\] where the $SU(3)$ generators are normalized such that $Tr(T^a T^b)={1\over 2}\delta_{ab}$. To get the parity conjugate amplitude $\M^{-+-}$ exchange square brackets with triangle brackets. To get the charge conjugates of these two amplitudes, $\M^{-++}$ and $\M^{+--}$, respectively, exchange $p_1 \leftrightarrow p_2$. Again, when squared and summed over colors and helicities, these results agree with the known expression [@hinch; @sally].
The Higgs Boson Plus Four Gluon Amplitude
=========================================
The $Hgggg$ amplitude [@hgggg] is obtained by summing the 26 Feynamn diagrams detailed in Fig. 4. Unlike the case of $Hggg$ not all the diagrams have the same color structure. To facilitate the cancellations that simplify the amplitude we introduce the [*dual color decomposition*]{}. The scattering amplitude for a Higgs boson and $n$ gluons with external momenta $p_1$, ...$p_n$, colors $a_1$,...$a_n$, and helicities $\lambda_1$,...$\lambda_n$ is written as [@parki; @parkii; @parkiii] = 2Ag\^[n-2]{} \_[perms]{} [tr]{} (T\^[a\_1]{}...T\^[a\_n]{})m(p\_1,\_1; ...;p\_n,\_n), \[eq:dual\] where the sum is over the non-cyclic permutions of the momenta. This form of the amplitude emerges when the identities f\_[abc]{} &=& -2i [Tr]{}(T\^a T\^b T\^c - T\^c T\^b T\^a),\
f\_[abe]{} f\_[cde]{} &=& -2 [Tr]{}(\[T\^a,T\^b\]\[T\^c,T\^d\]) \[eq:fabc\] are used to replace the $f_{abc}$’s with traces of combinations of $T^a$’s. The utility of the dual decomposition, Eq. \[eq:dual\], comes from the properties of the ordered sub-amplitudes $m(p_1,\epsilon_1;...;p_n,\epsilon_n)$, which we abbreviate $m(1,...,n)$: 1) they are invariant under cyclic permutations of the momenta; 2) they are independently gauge invariant ; 3) $m(1,...,n) = (-1)^n m(n,...,1)$ ; 4) they satisfy the “dual Ward identity,” which for $n=4$ is m(1,2,3,4)+m(2,1,3,4)+m(2,3,1,4)=0 ; \[eq:ward\] 5) they factorize in the soft gluon limit and in the limit in which two of the gluons are collinear; 6) they are incoherent (to leading order in the number of colors, in general, and completely for $n=4$); for $n=4$ one finds \_[colors]{} ||\^2 = [g\^2 A\^2 ø4]{} N\^2 (N\^2-1) \_[perms]{} |m(1,2,3,4)|\^2. \[eq:incoh\] (See Appendix A for the proof.)
The complete set of sub-amplitudes can be obtained from the following three [@hgggg]: &&m(1\^+,2\^+,3\^+,4\^+)=[\^41 22 3 3 44 1]{} \[eq:pppp\]\
&&m(1\^-,2\^+,3\^+,4\^+) = - [1[-]{}|/|3[-]{} \^2 \[24\]\^2 S\_[124]{} S\_[12]{} S\_[14]{}]{} -[1[-]{}|/| 4[-]{} \^2 \[2 3\]\^2 S\_[123]{} S\_[12]{} S\_[23]{}]{}\
&&-[1[-]{}|/| 2[-]{} \^2 \[3 4\]\^2 S\_[134]{} S\_[14]{} S\_[34]{}]{} +[\[2 4\] \[ 1 2 \] 2 33 4 ]{} { S\_[23]{} [1[-]{} |/| 2[-]{} 4 1]{}\
&&+ S\_[34]{} [1[-]{} |/| 4[-]{} 1 2]{} -\[2 4 \] S\_[234]{}} \[eq:mppp\]\
&&m(1\^-,2\^-,3\^+,4\^+)=-[1 2\^412 23 3441]{} -[\[34\]\^4 ]{} . \[eq:mmpp\] The structures containing $\ph$ can be expanded in terms of spinor products using Eq. \[eq:helproj\] and momentum conservation. For example $\la 1{-}|\slash \ph |3{-} \ra = - ( \la 12 \ra [23] + \la 14 \ra [43] )$. Permutations of $m(1^+,2^+,3^+,4^+)$ are obtained by permuting the momenta in the right side of Eq. \[eq:pppp\] identically. Permutations of $m(1^-,2^+,3^+,4^+)$ are obtained by permuting $p_2$, $p_3$ and $p_4$ in the right side of Eq. \[eq:mppp\] then using the cyclic and reversal properties of the sub-amplitudes. Permutations of $m(1^-,2^-,3^+,4^+)$ are obtained by permuting the momenta in the [*denominators*]{} of the right side of Eq. \[eq:mmpp\] only. It is straightforward to check that the sub-amplitudes obtained from Eqs. \[eq:pppp\]-\[eq:mmpp\] in this way obey the requisite relations. The amplitudes for the other helicity combinations can be obtained (modulo phases) by parity transformations.
The Higgs Boson Plus Quark Anti-quark and Two Gluon Amplitude
=============================================================
The $H q \bar q g g$ amplitude can be obtained from the Feynman diagrams of Fig. 5. As was the case for the $Hgggg$ amplitude, the calculation can be simplified by judicious choice of color decomposition [@kunszt; @parkii]. The amplitude for a Higgs boson, a quark–anti-quark pair with color indices $i,j$ and $n$ gluons with color indices $a_1,...,a_n$ can be written: = -i g\^n A \_[perms]{} (T\^[a\_1]{} T\^[a\_2]{} ... T\^[a\_n]{})\_[ij]{} m(p1,\_1;...;p\_n,\_n), \[eq:qdual\] where the sum runs over all $n!$ permutations of the gluons and the sub-amplitudes $m(p1,\e_1;...;p_n,\e_n)$ have an implicit dependence on the momenta and helicities of the quark and anti-quark. For the case we are interested in there are only two subamplitudes which we will label as $m(3,4)$ and $m(4,3)$ since the gluon momenta are $p_3$ and $p_4$. Like the subamplitudes for the pure gluon case these subamplitudes are separately gauge independent and factorize in the soft gluon and collinear particle limits.
As in the case of the $Hq\bar q g$ amplitude the quark and anti-quark must have opposite helicities. Labelling the helicity amplitudes by the helicity of the quark, anti-quark and the two gluons (in that order) we find m\^[+-++]{}(3,4) &=& [ 2[-]{}|/| 3[-]{} \^2 øS\_[124]{} ]{} [ \[14\] ø24 ]{} ł([1øS\_[12]{}]{} + [1øS\_[14]{}]{} ) -[ 2[-]{}|/| 4[-]{} \^2 øS\_[123]{} S\_[12]{}]{} [ \[13\] ø23 ]{}\
&+&[ 2[-]{}|/| 1[-]{} \^2 ø\[12\] 23 24 34 ]{} \[eq:qpmpp\] To get the subamplitude with the other ordering, $m^{+-++}(4,3)$, exchange $p_3 \leftrightarrow p_4$ in this expression. The other independent subamplitudes are m\^[+-+-]{}(3,4) &=& -[24\^3 ø122334]{} + [ \[13\]\^3 ø\[12\]\[14\]\[34\] ]{} \[eq:qpmpmi\]\
m\^[+-+-]{}(4,3) &=& -[ \[13\]\^2 \[23\] ø\[12\]\[24\]\[34\] ]{} + [1424\^2 ø121334]{} \[eq:qpmpmii\] The other helicity amplitudes (up to phases) can be obtained by parity $(P)$, Bose symmetry $(B)$ and charge conjugation $(C)$ transformations: |\^[-+–]{}|\^2 |\^[+-++]{}|\^2, |\^[-+++]{}|\^2 |\^[+-++]{}|\^2\_[12]{},\
|\^[+—]{}|\^2 |\^[+-++]{}|\^2\_[12]{}, |\^[-+-+]{}|\^2 |\^[+-+-]{}|\^2,\
|\^[+–+]{}|\^2 |\^[+-+-]{}|\^2\_[34]{}, |\^[-++-]{}|\^2 |\^[+-+-]{}|\^2\_[34]{}. \[eq:qsym\]
The Higgs Boson Plus Two Quark AntiQuark Pair Amplitude
=======================================================
The remaining processes producing a Higgs boson plus two jets are those involving a combination of four quarks and anti-quarks. In the case where the two pairs are of different flavors the amplitude can be obtained from the Feynman diagram in Fig. 6. In the case when the two pairs are identical there is an additional diagram which can be obtained by switching the $2 \ch 4$ in the diagram of Fig. 6. We present the amplitude for the case of two different quark pairs, since the identical case can be obtained from it. The sole independent helicity amplitude can be labelled in terms of the helicities of the 1st quark, the 1st antiquark, the 2nd quark and the 2nd antiquark (in that order): \^[+-+-]{} = iA g\^2 T\^a\_[ij]{}T\^a\_[kl]{} ł( [24\^2 ø1234]{} + [ \[13\]\^2 ø\[12\]\[34\] ]{} ). \[eq:hqqqq\] The other helicity amplitudes can be obtained by parity and charge conjugation transformations: |\^[-+-+]{}|\^2 && |\^[+-+-]{}|\^2, |\^[-++-]{}|\^2 |\^[+-+-]{}|\_[12]{}\^2,\
|\^[+–+]{}|\^2 && |\^[+-+-]{}|\_[34]{}\^2. \[eq:qqqqsym\]
The Soft Higgs Limit
====================
The effective Lagrangian, Eq. \[eq:leff\], implies that for the case of constant Higgs field $H$, [*i.e.*]{}, a Higgs boson with no momentum, the amplitude for a process containing a Higgs boson reduces to the amplitude for the process without the Higgs boson times an overall factor of $A$. For the $Hgggg$ amplitude only the helicity conserving ${+}{+}{-}{-}$ amplitude survives in the $\ph \to 0$ limit. In this limit one can show that the two terms in Eq. \[eq:mmpp\] are equal, regardless of the ordering of the momenta. As an example, consider the subamplitude m(1\^-,3\^+,2\^-,4+)=-[1 2\^413 32 2441]{} -[\[34\]\^4 ]{}. \[eq:mmppl\] In the limit $\ph\to 0$ the second term in this expression becomes &=& [3[+]{}| /p\_4 | 2[+]{} 3[+]{}| /p\_4 | 1[+]{} 4[+]{}| /p\_3 | 1[+]{} 4[+]{}| /p\_3 | 2[+]{} øS\_[13]{}S\_[32]{}S\_[24]{}S\_[41]{} ]{}\
&=&[3[+]{}| /p\_1 | 2[+]{} 3[+]{}| /p\_2 | 1[+]{} 4[+]{}| /p\_2 | 1[+]{} 4[+]{}| /p\_1 | 2[+]{} øS\_[13]{}S\_[32]{}S\_[24]{}S\_[41]{} ]{}\
&=&[ 12\^4 \[13\]\[32\]\[24\]\[41\] øS\_[13]{}S\_[32]{}S\_[24]{}S\_[41]{} ]{}, \[eq:secondt\] where momentum conservation as used in the second line. When common factors are cancelled Eq. \[eq:secondt\] is identical to the first term of Eq. \[eq:mmppl\]. The final result agrees with the well-known form of the pure gluon subamplitudes[@parki; @parkii; @parkiii].
The $Hq\bar q gg$ subamplitudes reduce to the $q\bar q gg$ subamplitudes: m\^[+-+-]{}(3,4) && - [2 \[13\]\^3 ø\[12\]\[14\]\[34\] ]{} \[eq:qqlimi\]\
m\^[+-+-]{}(4,3) && - [21424\^2 ø121334]{} \[eq:qqlimii\] Finally, the soft-Higgs limit of the $Hq \bar q q^\prime \bar q^\prime$ amplitude is \^[+-+-]{} -2iAg\^2 T\^a\_[ij]{}T\^a\_[kl]{} [24\^2 ø1234]{}, \[eq:hqqqqlim\] which has the proper relation to the amplitude for $q\bar q q^\prime \bar q^\prime$.
Factorization of the Amplitudes
===============================
The helicity amplitudes we have calculated factorize in the limit that a gluon becomes soft or two particles become collinear. This property has been established for the pure QCD processes involving quarks and gluons. We will present some representative examples of the factorization limits of our amplitudes.
The simplest cases involve the reduction of the Higgs plus 3 particle amplitudes to the Higgs plus two gluon amplitude in the appropriate limit. Discussion of these limits is facilitated by expressing the $Hgg$, $Hggg$, and $Hq\bar qg$ amplitudes in the same dual color decompositions as we used for the $Hgggg$ and $Hq\bar qgg$ amplitudes. For the $Hgg$ case there is only one subamplitude, $m_{gg}(1,2)$, which can be obtained from Eq. \[eq:hggamp\] by replacing $\delta_{ab}$ by 1. For the $Hggg$ case there are two subamplitudes but the Ward identity ensures that they are equal and opposite. The subamplitude $m_{ggg}(1,2,3)$ can be obtained from Eq. \[eq:hgggamp\] by replacing $f_{abc}$ by $-i$. The lone subamplitude for $Hq\bar qg$ is Eq. \[eq:hqqgamp\] with the factor of $T^a_{ij}$ removed.
Taking one of the gluons to be soft in $m_{ggg}(1^+,2^+,3^+)$ yields m\_[ggg]{}(1\^+,2\^+,3\^+) -ł{ [g32ø1231]{} }m\_[gg]{}(2\^+,3\^+), \[eq:mgggsoft\] where the factor in brackets is the square root of the “eikonal factor”. This is in keeping (up to phase conventions) with the general result of Mangano [*et al.*]{} [@parki]. A similar limit applies to $m_{ggg}(1^-,2^+,3^+)$. Taking two of the gluons to be collinear is accomplished by letting $p_1 \to zP$ and $p_2 \to (1-z)P$: m\_[ggg]{}(1\^+,2\^+,3\^+) && ł( [ig\[12\] ø ]{} ) ł([-iøS\_[12]{}]{} ) m\_[gg]{}(P\^+,3\^+),\
m\_[ggg]{}(1\^-,2\^+,3\^+) && ł( [ig12(1-z)\^2 ø ]{} ) ł([-iøS\_[12]{}]{} ) m\_[gg]{}(P\^+,3\^+), \[eq:mgggcoll\] which is again consistent with the general result[@parki].
The $Hgggg$ subamplitudes exhibit the same factorization properties. Notice that only the momenta which are adjacent in the argument of the subamplitude appear paired in the denominators. For example, taking $p_1$ to be soft in $m(1^+,2^+,3^+,4^+)$ gives m(1\^+,2\^+,3\^+,4\^+) = [ 42ø1241]{} m\_[ggg]{}(2\^+,3\^+,4\^+), \[eq:mggggsoft\] in agreement with the general result. The ${\cal O}(1/S_{ij})$ singularities in Eq. \[eq:mppp\] are in reality only ${\cal O}(1/\sqrt{S_{ij}})$ singularites. For example, taking $p_1 \to zP$ and $p_2 \to (1-z)P$ in $m(1^-,2^+,3^+,4^+)$ gives &&m(1\^-,2\^+,3\^+,4\^+) ł{ [(1-z)\^2 ø\[12\] ]{} [ S\_[P34]{}\^2 øP3344P]{} - [ z\^2 ø12]{} [\[34\]\^3 ø\[P3\]\[P4\] ]{} }\
&& = [ø]{} ł{ [(1-z)\^2 ø\[12\] ]{} m\_[ggg]{}(P\^+,3\^+,4\^+) - [ z\^2 ø12]{} m\_[ggg]{}(P\^-,3\^+,4\^+) }.\
\[eq:mpppcoll\]
The $Hq\bar qgg$ and $H q\bar q q^\prime \bar q^\prime$ amplitudes also factor. Taking the limit of $p_1 \parallel p_2$ as before gives: m\^[+-++]{}\_[q|q gg]{} && [(1-z)ø\[12\]]{} [ S\_[P34]{} øP3344P]{} +[zø12]{} [ \[34\]\^3 ø\[P3\] \[P4\] ]{}\
&=& [(1-z)ø\[12\]]{} m\_[ggg]{}(P\^+,3\^+,4\^+) + [zø12]{} m\_[ggg]{}(P\^-,3\^+,4\^+) \[eq:mqqggcoll\] and M\^[+-+-]{}\_[q|q q\^|q\^]{} -i g\^2 T\^a\_[ab]{}T\^a\_[dc]{} ł{ [(1-z)ø\[12\]]{} m\_[q|qg]{}\^[+-+]{} + [z ø12]{} m\_[q|qg]{}\^[+–]{} }. \[eq:mqqqqcoll\]
The $Hgggg$ subamplitudes also factor in the three-gluon channel. Letting $P = p_1 + p_2 + p_3$ and taking $P^2 \to 0$ in the ${-}{+}{+}{+}$ subamplitude gives m(1\^-,2\^+,3\^+,4\^+) \^2 [1øP\^2]{} [1P\^2 \[23\]\^2 øS\_[12]{} S\_[23]{} ]{} \~m\_[gg]{}(4+,P+) [1øP\^2]{} m(1\^-,2\^+,3\^+,P\^-), \[eq:threepole\] where $\tilde m$ is the four gluon subamplitude (without a Higgs boson) and the $\sim$ indicates equality modulo phases. Identical relations hold for the $1/S_{124}$ and $1/S_{134}$ poles. Since the four-gluon amplitude is helicity conserving the other helicity amplitudes have no three-gluon poles. Likewise there is no $1/S_{234}$ pole in $m(1^-,2^+,3^+,4^+)$.
Numerical Results and Conclusions
=================================
We will present numerical results for the CERN Large Hadron Collider (LHC) at a center-of-mass energy of $\sqrt{S}=14~\TeV$ and the Fermilab Tevatron at $\sqrt{S}=2~\TeV$. Since all the parton level cross sections are singular in the small $\pt$ limit of one of the jets we will place a $\pt$ cut on the outgoing jets. Since there are also collinear singularities we will require that the outgoing jets be separated by $\Delta R_{ij}\equiv \sqrt{\Delta \phi_{ij}^2+\Delta \eta_{ij}} \ge 0.7$. We will also require the outgoing jets have rapidity $\mid y\mid <2.5$. Since there are no singularities depending on the momentum of the Higgs boson we will allow it to be unconstrained, except for a $\pt$ cut.
The results separated according to parton processes are presented in Fig.’s 7 and 8. We see that at the LHC the all gluon process dominates as expected with the $qg \to qgH$ process and its charge conjugate contributing an additional 15%. The other processes are negligible. At the Tevatron the pure gluon process and the $qg \to qgH$ process give roughly equal contributions.
Fig.’s 9 and 10 show the result of varying the transverse momentum cut (on the jets and the Higgs boson simultaneously). We see that in both cases the cross section drops sharply with the increasing $\pt$ cut. The dependence of the cross section on the other cuts is weak. Increasing the minimum $\Delta R$ to 1.0 decreases the cross section by about 15%. Requiring that the jets be separated from the Higgs boson the same $\Delta R$ also reduces the cross section be about 15%.
In summary, we have presented the amplitudes for the production of a Higgs accompanied by two jets. We find that the cross section is around a few picobarns at the LHC and a few hundredths of a pb at the Tevatron. Our results provide the “real” corrections to Higgs production at non-zero transverse momentum. They can be combined with the virtual corrections to complete the next-to-leading order calculation.
Appendix. Squaring the Amplitudes
=================================
We first proceed to verify the incoherence of the subamplitudes for $ggggH$. Using the fact that subamplitudes are invariant (for $n=4$) under reversal of the order of the arguments we write &=& { ł\[ (T\^a T\^b T\^c T\^d) + (T\^d T\^c T\^b T\^a) \] m(1,2,3,4)\
&+& ł\[ (T\^a T\^b T\^d T\^d) + (T\^d T\^d T\^b T\^a) \] m(1,2,4,3)\
&+& ł\[ (T\^a T\^c T\^b T\^d) + (T\^d T\^b T\^c T\^a) \] m(1,3,2,4) }. \[eq:mexpl\] It is straightforward to show that the squared color factors are c\_1 &=& \_[colors]{} ł\[ (T\^a T\^b T\^c T\^d) + (T\^d T\^c T\^b T\^a) \]\^2\
&=& [ (N\^2-1)\^2 ø4N\^2]{} + [1ø16]{}ł\[ ł([5ø3]{})\^2 (N\^2-1) + N\^2(N\^2-1) \] \[eq:csquare\] and that the cross terms are c\_2 &=& \_[colors]{} ł\[ (T\^a T\^b T\^c T\^d) + (T\^d T\^c T\^b T\^a)\^\* \] ł\[ (T\^a T\^b T\^d T\^c) + (T\^c T\^d T\^b T\^a)\^\* \]\
&=& [ (N\^2-1)\^2 ø4N\^2]{} + [1ø16]{}ł\[ ł([5ø3]{})\^2 (N\^2-1) - N\^2(N\^2-1) \] \[eq:ccross\] The amplitude squared is then \_[colors]{} ||\^2&=& 4g\^4A\^2 { c\_1\
&+& c\_2 }. \[eq:tampsq\] Use of the Ward identity, Eq. (\[eq:ward\]), allows all the cross terms to be written as subamplitudes squared. When this is done and the above results for the color factor are used, Eq. (\[eq:incoh\]) follows.
The following identities were used to relate the spinor products to traces over gamma matrices[@parki]: i\_2 i\_3 ...i\_[2n]{} i\_1 &=& { i\_1 i\_2... i\_[2n]{} P\_+ }\
i\_1 i\_2 ...\[ i\_[2n]{} i\_1\] &=& { i\_1 i\_2... i\_[2n]{} P\_- }, \[eq:trace\] where $P_\pm={1\o2}(1\pm\gamma_5)$, and $\{i_1 i_2... i_n\}$ denotes the trace of $\slash p_1 \slash p_2... \slash p_n$. In order to reduce the traces which contained a factor of $\g_5$, we use the identity[@parki]: &&{i\_1 i\_2... i\_[2n]{}\_5} {j\_1 j\_2... j\_[2m]{}\_5} = {i\_1 i\_2... i\_[2n]{}} {j\_1 j\_2... j\_[2m]{}}\
&&- 2. \[eq:trproduct\]
We first consider the square of the $ggggH$ subamplitudes. For the case of $m(1^+,2^+,3^+,4^+)$, we have | m(1\^+,2\^+,3\^+,4\^+) |\^2 = [\^8 øS\_[12]{} S\_[23]{} S\_[34]{} S\_[41]{}]{}. \[eq:g4ppppsq\] In the case of $m(1^-,2^+,3^+,4^+)$, we rewrite Eq. \[eq:mppp\] in a more compact form: &&m(1\^-,2\^+,3\^+,4\^+) = - [1[-]{}|/|3[-]{} \^2 \[24\]\^2 øS\_[124]{} S\_[12]{} S\_[14]{}]{} -[1[-]{}|/| 4[-]{} \^2 \[2 3\]\^2 øS\_[123]{} S\_[12]{} S\_[23]{}]{}\
&& -[1[-]{}|/| 2[-]{} \^2 \[3 4\]\^2 øS\_[134]{} S\_[14]{} S\_[34]{}]{} +[\[2 4\] ø\[ 1 2 \] \[14\] 1 3]{} { [1[-]{} |/| 2[-]{} \^2 ø1434]{} + [1[-]{} |/| 4[-]{} \^2 1223]{} }. \[eq:mpppalt\] We then write | m(1\^-,2\^+,3\^+,4\^+) |\^2 = \_[i=1]{}\^[5]{} \_[j=1]{}\^[i]{} [[n\_[ij]{}ød\_i d\_j]{} ]{}, \[eq:g4mpppsq\] where the independent terms are: n\_[11]{} &=& [1 ø4]{} S\_[24]{}\^2{1(2[+]{}4)3(2[+]{}4)}\^2, n\_[44]{} = [1ø4]{} S\_[12]{}S\_[13]{}S\_[24]{}S\_[34]{} {1(3[+]{}4)2(3[+]{}4)}\^2\
n\_[12]{} &=& {1(2[+]{}4)324(2[+]{}3)}\^2 - 2 S\_[23]{} S\_[24]{}(S\_[12]{} S\_[24]{} [+]{} S\_[13]{} S\_[34]{} + {1243})\
&& (S\_[12]{}S\_[23]{} [+]{} S\_[14]{}S\_[34]{} [+]{} {1234})\
n\_[14]{} &=& -S\_[24]{}\
-4n\_[24]{} &=& {1(2[+]{}3)432(3[+]{}4)}{1(2[+]{}3)42}{132(3[+]{}4)}\
&& -{1(2[+]{}3)432(3[+]{}4)}({1234}\^2[-]{}4S\_[12]{}S\_[23]{}S\_[34]{}S\_[41]{})\
&& -\
&& ({1(2[+]{}3)42} [-]{} {132(3[+]{}4)})\
n\_[25]{} &=& - [1 ø4]{} S\_[23]{} {1(2[+]{}3)4(2[+]{}3)}\^2{1324}\
n\_[45]{} &=& -S\_[13]{}S\_[24]{}\
d\_1 &=& S\_[12]{} S\_[14]{} S\_[124]{}, d\_4 = S\_[12]{} S\_[13]{} S\_[14]{} S\_[34]{}. \[eq:paradigm\] The remaining terms can be obtained by switching the momenta: n\_[22]{} &=& n\_[11]{}(3 4), n\_[33]{} = n\_[11]{}(2 3), n\_[55]{} = n\_[44]{}(2 4),\
n\_[13]{} &=& n\_[12]{}(2 4), n\_[15]{} = n\_[14]{}(2 4), n\_[23]{} = n\_[13]{}(3 4),\
n\_[34]{} &=& n\_[25]{}(2 4), n\_[35]{} = n\_[24]{}(2 4), d\_5 = d\_4(2 4),\
d\_2 &=& d\_1(1 2, 3 4), d\_3 = d\_1(1 4, 2 3).\
\[eq:switches\] The two independent permutations of $m(1^-,2^-,3^+,4^+)$ squared are: |m(1\^-,2\^-,3\^+,4\^+)|\^2 &=& [S\_[12]{}\^3 øS\_[14]{} S\_[23]{} S\_[34]{}]{} +[S\_[34]{}\^3 øS\_[12]{} S\_[14]{} S\_[23]{}]{} +[{1234}\^2 - 2S\_[12]{}S\_[23]{}S\_[34]{}S\_[41]{} øS\_[14]{}\^2 S\_[23]{}\^2]{}\
|m(1\^-,3\^+,2\^-,4\^+)|\^2 &=& [S\_[12]{}\^4 + S\_[34]{}\^4 øS\_[13]{}S\_[14]{}S\_[23]{}S\_[34]{}]{}\
&&+\
&& / \[2(S\_[13]{}S\_[14]{}S\_[23]{}S\_[34]{})\^2\]. \[eq:g4mmppsq\]
Turning our attention to the $q \bar q gg H$ amplitudes, we break the subamplitude $m^{{+}{-}{+}{+}}(3,4)$ into a symmetric piece, $m_s$, and anti-symmetric piece, $m_a$, under interchange of the two gluons. The squared amplitude is then given by |\^[[+]{}[-]{}[+]{}[+]{}]{}|\^2 = 2 A\^2 g\^4 ł\[ C\_1 (|m\_[s]{}|\^2 + |m\_[a]{}|\^2) + C\_2 (|m\_[s]{}|\^2 - |m\_[a]{}|\^2) \], \[eq:qqggpmpp\] where the color factors are $C_1={(N^2-1)^2\over4N}$ and $C_2=-{(N^2-1)\over4N}$. We write: |m\_s|\^2 &=&[14]{} ł([n\_[11]{}d\_1\^2]{} + [n\_[22]{}d\_2\^2]{} + [n\_[12]{}d\_1 d\_2]{})\
|m\_a|\^2 &=&[n\_[11]{}d\_3\^2]{} + [n\_[22]{}d\_4\^2]{} + [n\_[33]{}d\_5\^2]{} - [n\_[12]{}d\_3 d\_4]{} + [n\_[13]{}d\_3 d\_5]{} + [n\_[23]{}d\_4 d\_5]{} \[eq:symmasymm\] where the numerator and denominator terms are &&n\_[11]{} = [14]{}[S\_[14]{} S\_[24]{}]{}{2(1[+]{}4)3(1[+]{}4)}\^2,n\_[22]{} = n\_[11]{}[(34)]{},\
&&n\_[33]{} = [14]{}[S\_[12]{} S\_[23]{} S\_[24]{} S\_[34]{}]{}{1(3[+]{}4)2(3[+]{}4)}\^2,\
&&n\_[12]{} = {2(1[+]{}4)314(1[+]{}3)}{2(1[+]{}4)324(1[+]{}3)}\
&&- {1324}(S\_[12]{} S\_[13]{} + S\_[24]{} S\_[34]{} + {1243}) (S\_[12]{} S\_[14]{} + S\_[23]{} S\_[34]{} + {1234})\
&&n\_[13]{} = -S\_[24]{} n\_[12]{}[(24)]{},n\_[23]{} = -S\_[23]{} n\_[13]{}[(34)]{},\
&&d\_1 = S\_[14]{} S\_[24]{} S\_[124]{},d\_2 = S\_[13]{} S\_[23]{} S\_[123]{},d\_5 = S\_[12]{} S\_[23]{} S\_[24]{} S\_[34]{},\
&&d\_4 = S\_[23]{} S\_[123]{}ł([1S\_[12]{}]{} +[12S\_[13]{}]{})\^[-1]{},d\_3 = S\_[24]{} S\_[124]{}ł([1S\_[12]{}]{}+[12S\_[14]{}]{})\^[-1]{}.\
\[eq:qqgg1sub\] For the other independent helicity amplitude, squaring yields &&|\^[[+]{}[-]{}[+]{}[-]{}]{}|\^2 = g\^4 A\^2 {C\_1 (|m\^[[+]{}[-]{}[+]{}[-]{}]{}(3,4)|\^2 + |m\^[[+]{}[-]{}[+]{}[-]{}]{}(4,3)|\^2)\
&& + 2C\_2 }, \[eq:qqggpmpm\] where\
&&},\
\
\[eq:qqgg2sub\]
For the $q \bar q q^\prime\bar q^\prime$ amplitude, the square of Eq. \[eq:hqqqq\] yields \^2 = [A\^2 g\^4 (N\^2-1)4 S\_[12]{} S\_[34]{}]{} ł\[(S\_[13]{} - S\_[24]{})\^2 + [{1243}\^2S\_[12]{} S\_[34]{}]{}\]. \[eq:hqqqqsq\] In the case of identical quark pairs, there is a second diagram whose square can be obtained by switching $1\ch 3$ in Eq. \[eq:hqqqqsq\]. The interference term which arises is -2= [-A\^2g\^4 (N\^2[-]{}1) ø4N]{} ł\[[(S\_[13]{} - S\_[24]{})\^2 {1234} - 2{1324} {1243}S\_[12]{} S\_[23]{} S\_[14]{} S\_[34]{}]{}\]. \[eq:hqqqqint\]
[**ACKNOWLEDGMENTS**]{}
The author’s would like to thank S. Dawson for her work on the initial stages of this project and for helpful discussions. S.D. and D.R. would like to thank the Hackman Scholar program at Franklin and Marshall College for financial support.
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**FIGURE CAPTIONS**
. The vertices and Feynman rules of the effective theory. The curly lines indicate gluons and the dashed lines indicate the Higgs boson.
. The Feynman diagrams for the $gggH$ amplitude. There are two more diagrams of the same form as b), where the Higgs boson attaches to gluon 1 and gluon 3.
. The Feynman diagram for the $q\bar qgH$ amplitude.
. The Feynman diagrams for the $ggggH$ amplitude. There are 12 diagrams of type a), 3 of type b), 1 of type c), 4 of type d), and 6 of type e), for a total of 26.
. The Feynman diagrams for the $q\bar q ggH$ amplitude. There is one diagram of type a), two of type b), 4 of type c) and one of type d), for a total of 8.
. The Feynman diagram for the $q\bar q q^\prime \bar q^\prime H$ amplitude. In the case when the quark pairs are identical there is a second diagram with the quark lines switched.
. The various contributions to the cross section for production of a Higgs boson plus two jets at the LHC as a function of the mass of the Higgs boson. A $\pt$ cut of 50 GeV has been placed on the jets and the Higgs boson. The labels are as follows: $gggg$ represents $gg\to ggH$; $qgqg$ represents $qg\to qgH$ and its complex conjugate; $ggqq$ represents $gg\to q\bar q H$; $qqqq$ represents all the processes involving two incoming quarks or antiquarks and two outgoing quarks or antiquarks; $ggqq$ represents $gg\to q\bar qH$.
. The various contributions to the cross section for production of a Higgs boson plus two jets at the Tevatron as a function of the mass of the Higgs boson. A $\pt$ cut of 25 GeV has been placed on the jets and the Higgs boson. The labels are the same as in Fig. 7.
. The cross section for production of a Higgs boson plus two jets at the LHC for three values of the $\pt$ cut.
. The cross section for production of a Higgs boson plus two jets at the Tevatron for three values of the $\pt$ cut.
|
---
abstract: 'Misinformation such as fake news has drawn a lot of attention in recent years. It has serious consequences on society, politics and economy. This has lead to a rise of manually fact-checking websites such as Snopes and Politifact. However, the scale of misinformation limits their ability for verification. In this demonstration, we propose a browser extension which can be used to automate the entire process of credibility assessments of false claims. Behind the scenes uses a tested deep neural network architecture to automatically identify fact check worthy claims and classifies as well as presents the result along with evidence to the user. Since is a browser extension, it facilities fast automated fact checking for the end user without having to leave the Webpage.'
author:
- Bjarte Botnevik
- Eirik Sakariassen
- Vinay Setty
bibliography:
- 'references.bib'
title: ': Browser Extension for Fake News Detection'
---
<ccs2012> <concept> <concept\_id>10002951.10003317.10003318</concept\_id> <concept\_desc>Information systems Document representation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Conclusion
==========
In this demonstration we proposed which is a browser extension to tackle the challenge of misinformation. The user can use to first identify fact check worthy claims in any news article online. Subsequently the user gets the credibility classification using a sophisticated deep neural network model. The users are also presented with the evidence from the model, and can achieve all this without leaving the Web page of the news article they are reading.
|
---
abstract: 'We examine gravitational waves in an isolated axi–symmetric reflexion symmetric NGT system. The structure of the vacuum field equations is analyzed and the exact solutions for the field variables in the metric tensor are found in the form of expansions in powers of a radial coordinate. We find that in the NGT axially symmetric case the mass of the system remains constant only if the system is static (as it necessarily is in the case of [*spherical*]{} symmetry). If the system radiates, then the mass decreases monotonically and the energy flux associated with waves is positive.'
address: |
Department of Physics, University of Toronto\
Toronto, Ontario M5S 1A7, Canada
author:
- 'N. J. Cornish, J. W. Moffat and D. C. Tatarski'
title: |
Gravitational Waves from an Axi–symmetric Source\
in the Nonsymmetric Gravitational Theory
---
Introduction
============
The present work examines gravitational radiation in the Nonsymmetric Gravitational Theory (NGT) (for a recent detailed review see [@Moff91]). We probe the NGT asymptotic behaviour in the wave zone using an [*exact*]{} solution. A complementary analysis based on a DeWitt style background field expansion has already been published, announcing the main result of this current work [@pla].
The motivation for this work is twofold. Firstly, the nature of gravitational radiation is an important physical question which must be addressed in any candidate gravitational theory. Secondly, the literature already contains several incorrect treatments of gravitational radiation in NGT [@Krisher; @DDM1; @DDM2], and since more incorrect treatments are currently being published [@Dam], it is important that the record be set straight. The reasons why erroneous results were arrived at by these other authors are explained in [@CorMoff].
In General Relativity (GR) gravitational radiation from bounded sources has been studied not only through the linearized theory but also with the use of exact solutions. The latter was done for the general case of a bounded source in asymptotically flat spacetime [@Sachs]. It was found that confining the arguments to the axially symmetric case did not cause any essential loss of generality. Since even the relevant GR calculations are very tedious and the level of technical difficulty in the case of NGT increases considerably, we limit ourselves to the axi–symmetric case. The GR gravitational waves from isolated axially symmetric reflexion symmetric systems were studied in detail in [@BBM]. Since our treatment of the axi–symmetric NGT case is rather parallel, familiarity with this analysis is strongly recommended.
Since NGT was introduced [@Moff79-1] there have been few analytic solutions of the field equations published. The exact solutions known to date include the spherically symmetric vacuum case [@Moff79-2], the spherically symmetric interior case [@Sav89; @Sav90] and Bianchi type I cosmological solutions with and without matter [@Kunst79; @Kunst80]. This, at least in part, follows from the fact that deriving NGT field equations relevant for particular cases of interest is not as technically simple as may be suggested by its superficial similarity to the corresponding GR situations. Firstly, since the underlying geometry is non-Riemannian, neither the fundamental metric tensor $g_{\mu \nu}$ nor the affine connection is symmetric. This does not constitute a serious problem for the choice of the form of $g_{\mu \nu}$, since we can always assume that its nonsymmetric part takes on the isometries of the symmetric part, which in turn has a well defined GR limit. On the other hand, calculating the connection coefficients proves to be a tedious and time consuming exercise, independent of the method chosen. Secondly, the resultant formulae for the nonsymmetric connection coefficients are extremely complicated for all but the simplest forms of the metric, thus becoming unwieldy to use in the derivation of still more complicated field equations.
The NGT quantities presented in this paper were derived with the use of symbolic algebraic computation procedures. To this end, we have used the symbolic computation system [*Maple*]{}.
In Section \[fielde\], we briefly summarize the necessary fundamentals of NGT. Section \[coord\] deals with the coordinate system and generalization of the GR metric to the NGT case. Then in Section \[res\], we expand the metric in negative powers of a suitably chosen radial coordinate and analyze the field equations. The closing section contains our conclusions.
Throughout this paper we use units in which $G=c=1$.
NGT Vacuum Field Equations {#fielde}
==========================
The NGT Lagrangian without sources takes the form: $$\label{lagr}
{\cal L} = \sqrt{-g}g^{\mu \nu} R_{\mu \nu}(W),$$ with $g$ the determinant of $g_{\mu \nu}$. The NGT Ricci tensor is defined as: $$\begin{aligned}
\label{ricciw}
R_{\mu \nu}(W) &=& W^{\beta}_{\mu \nu , \beta}- \frac{1}{2}
(W^{\beta}_{\mu \beta , \nu}+W^{\beta}_{\nu \beta , \mu}) \nonumber\\
&& \hspace{.2in} -
W^{\beta}_{\alpha \nu}W^{\alpha}_{\mu \beta}+W^{\beta}_{\alpha
\beta}W^{\alpha}_{\mu \nu},\end{aligned}$$ and $W^{\lambda}_{\mu \nu}$ is an unconstrained nonsymmetric connection : $$\label{connw}
W^{\lambda}_{\mu \nu}=W^{\lambda}_{(\mu \nu)}+W^{\lambda}_{[\mu
\nu]}.$$ (Throughout this paper parentheses and square brackets enclosing indices stand for symmetrization and antisymmetrization, respectively.) The contravariant nonsymmetric tensor $g^{\mu \nu}$ is defined in terms of the equation: $$\label{inverse}
g^{\mu \nu} g_{\sigma \nu}=g^{\nu \mu} g_{\nu
\sigma}=\delta^{\mu}_{\sigma}.$$ If we define the torsion vector as: $$W_{\mu} \equiv W^{\nu}_{[\mu \nu]} = \frac{1}{2} \left(
W^{\nu}_{\mu \nu}- W^{\nu}_{\nu \mu} \right),$$ then the connection: $$\label{conng}
\Gamma^{\lambda}_{\mu \nu} = W^{\lambda}_{\mu \nu} + \frac{2}{3}
\delta^{\lambda}_{\mu} W_{\nu}$$ is torsion free: $$\label{gtors}
\Gamma_{\mu} \equiv \Gamma^{\alpha}_{[\mu \alpha]} = 0.$$ Defining now: $$\begin{aligned}
\label{riccig}
R_{\mu \nu}(\Gamma) &=& \Gamma^{\beta}_{\mu\nu,\beta} -
\frac{1}{2}(\Gamma^{\beta}_{(\mu\beta),\nu} + \Gamma^{\beta}_{(\nu
\beta) , \mu}) \nonumber\\
&& \hspace{.2in} -\Gamma^{\beta}_{\alpha\nu}\Gamma^{\alpha}_{\mu
\beta}+\Gamma^{\beta}_{(\alpha\beta)}\Gamma^{\alpha}_{\mu\nu},\end{aligned}$$ we can write: $$\label{ricciw=g}
R_{\mu \nu}(W) = R_{\mu \nu}(\Gamma) + \frac{2}{3} W_{[\mu ,\nu]},$$ where $W_{[\mu,\nu]}=\frac{1}{2}(W_{\mu,\nu}-W_{\nu,\mu})$. Finally, the NGT vacuum field equations can be expressed as: $$\label{fensgamma}
g_{\mu\nu,\sigma} - g_{\rho\nu} {\Gamma}^{\rho}_{\mu\sigma} -
g_{\mu\rho} {\Gamma}^{\rho}_{\sigma\nu} = 0 ,$$ $$\label{fensdiver}
{(\sqrt{-g}g^{[\mu \nu]})}_{ , \nu} = 0 ,$$ $$\label{fensricci}
R_{\mu \nu}(\Gamma) = \frac{2}{3} W_{[\nu , \mu]}.$$ For the purpose of the analysis of Section \[res\], it is convenient to decompose $R_{\mu\nu}$ into standard symmetric and antisymmetric parts: $R_{(\mu\nu)}$, $R_{[\mu\nu]}$, and then rewrite the field equation (\[fensricci\]) in the following form: $$\label{sym}
R_{(\mu \nu)}(\Gamma) = 0,$$ $$\label{asym}
R_{[\mu \nu , \rho]}(\Gamma) = 0,$$ where we used equations (\[conng\]), (\[gtors\]) and the notation: $$R_{[\mu\nu,\rho]} = {R_{[\mu\nu]}}_{,\rho} + {R_{[\nu\rho]}}_{,\mu}
+ {R_{[\rho\mu]}}_{,\nu} .$$
The Metric {#coord}
==========
Similarly to GR, the simplest NGT field due to a bounded source would be spherically symmetric. However, the NGT equivalent of Birkhoff’s theorem (see e.g. [@Moff91]) shows that a spherically symmetric gravitational field in an empty space must be static. Hence, no gravitational radiation escapes into empty space from a pulsating spherically symmetric source.
Following [@BBM] we consider the next simplest case: the field which was initially static and spherically symmetric and eventually becomes such, but undergoes an intermediate non–spherical wave emitting period. Also, spacetime is assumed to be axially symmetric and reflexion–symmetric at all times. Because of the complexity of the field equations, we are forced to use the method of expansion to examine the problem. This approach, namely expanding in negative powers of a radial coordinate, was also used in the GR analysis [@BBM] and naturally suits a wave problem.
Due to the physical picture sketched above and to the fact that we are interested in the asymptotic behaviour of the field at null infinity, ${\cal I}$, (in an arbitrary direction from our isolated source) polar coordinates $x^{0}=u, {\bf x} = (r,\theta,\phi)$ are the natural choice. The “retarded time” $u=t-r$ has the property that the hypersurfaces $u=\mbox{constant}$ are light–like. Detailed discussion of the coordinate systems permissible for investigation of outgoing gravitational waves from isolated systems can be found in [@Sachs; @BBM].
The covariant GR metric tensor corresponding to the situation described above is: $$\label{grmetric}
g_{\mu\nu}\! = \!\left( \begin{array}{cccc}
Vr^{-1}e^{2\beta}-U^{2}r^{2}e^{2\gamma} & e^{2\beta} &
Ur^{2}e^{2\gamma} & 0 \\
e^{2\beta} & 0 & 0 & 0 \\
Ur^{2}e^{2\gamma} & 0 & -r^{2}e^{2\gamma} & 0 \\
0 & 0 & 0 & -r^{2}e^{-2\gamma}\sin^{2}\theta \end{array} \right)$$ with $U,V,\beta,\gamma$ being functions of $u,r$ and $\theta$ was first given in [@Bondi60].
For any metric in polar coordinates, form conditions must be imposed in the neighbourhood of the polar axis, $\sin\theta=0$, to ensure regularity. In the case under consideration we have that, as $\sin\theta\rightarrow0$, $ V,\beta,U/\sin\theta,\gamma/\sin^{2}\theta$ each is a function of $\cos\theta$ regular at $ \cos\theta = \pm
1 $.
In order to find the NGT generalization of the metric tensor (\[grmetric\]) we require that the symmetric part of the NGT metric tensor be formally the same as the GR metric tensor. We then impose the spacetime symmetries of the symmetric metric onto the antisymmetric sector. This is achieved by enforcing $\pounds_{ {\vec{\xi}_{(i)}} }g_{[\mu\nu]}=0$, where the Killing vector field, $\vec{\xi}_{(i)}$, is obtained from $\pounds_{ {\vec{\xi}_{(i)}} }
g_{(\mu\nu)}=0$. The solution to this equation for the metric (\[grmetric\]) yields the single Killing vector field $\vec{\xi}_{(1)}=\xi^{3}_{(1)}
\partial_{\phi}=\sin^{2}\theta \partial_{\phi}$. Imposing $\pounds_{{\vec{\xi}_{(1)}} }g_{[\mu\nu]}=0$ yields: $$\xi^{3}_{(1)}\partial_{\phi} g_{[\mu\nu]} +g_{[\mu 3]}\partial_{\nu}
\xi^{3}_{(1)} +g_{[3\nu]}\partial_{\mu}\xi^{3}_{(1)}=0 \; .$$ This equation gives $\partial_{\phi}g_{[\mu\nu]}=0$, but does not exclude any antisymmetric components. This is markedly different from the static spherically symmetric case where the above procedure excludes four of the six antisymmetric components. Without further simplification, the NGT calculation would involve ten independent functions and ten independent non-linear differential equations. This would constitute a huge increase in complexity from the system of four equations and functions found in the GR case.
To make the problem tractable, we need to determine which antisymmetric functions can be set to zero. To accomplish this we note that the imposition of axi-symmetry splits the antisymmetric field equations (\[fensdiver\]), (\[asym\]) into two sets of three independent equations. (This can be seen directly from the block-diagonal form of the GR metric). The first set explicitly involves the three skew functions $g^{[01]}\, , g^{[02]}\, , g^{[12]}$: $$\begin{aligned}
\left(\sqrt{-g}g^{[\mu\nu]}\right)_{,\nu}&=&0 \hspace{1in}
(\mu=0,1,2) \; , \nonumber \\ \nonumber \\
R_{[01,2]}&=&0 \; .\end{aligned}$$ These four equations are not independent due to the one identity: $$\left(\sqrt{-g}g^{[\mu\nu]}\right)_{,\nu,\mu}=0 \hspace{1in} (\mu,\nu=0,1,2)
\; .$$ The second set of four equations explicitly involves the three skew functions $g^{[30]}\, ,g^{[31]}\, ,g^{[32]}$: $$\begin{aligned}
\left(\sqrt{-g}g^{[3 \nu]}\right)_{,\nu}&=&0 \; , \nonumber \\
\nonumber \\
R_{[3 \mu ,\nu]}&=&0 \; .\end{aligned}$$ These four equations are also not independent due to the one identity $$\epsilon^{3 \mu\nu\rho}R_{[3 \mu ,\nu],\rho}=0 \; .$$ We may now choose to work with either set of three equations and three functions, noting that eliminating one set of three functions simultaneously eliminates the three corresponding equations. We choose to work with the first set of functions and equations as they reproduce the usual static spherically symmetric solution when the $u$ and $\theta$ dependence is suppressed. The components that we discard correspond to the magnetic monopole-like solutions of NGT. These components can be discarded without loss of generality as an extension of the background field analysis of Ref.[@pla] to include all six skew functions does not alter our conclusions [@endnote].
In view of the above, the NGT generalization of the metric tensor (\[grmetric\]) is: $$\label{genmetric}
g_{\mu\nu} = \left( \begin{array}{cccc}
Vr^{-1}e^{2\beta}-U^{2}r^{2}e^{2\gamma} & e^{2\beta} + \omega &
Ur^{2}e^{2\gamma} + \lambda & 0 \\
e^{2\beta} - \omega & 0 & \sigma & 0 \\
Ur^{2}e^{2\gamma} - \lambda & -\sigma & -r^{2}e^{2\gamma} & 0 \\
0 & 0 & 0 & -r^{2}e^{-2\gamma}\sin^{2}\theta \end{array} \right),$$ where $\omega ,\lambda$ and $\sigma$ are functions of $u,r$ and $\theta$. The contravariant metric tensor is given by: $$\label{contrmetric}
g^{\mu\nu} = \left( \begin{array}{cccc}
-\sigma^{2}e^{-2\gamma}A & g^{01}
& g^{02} & 0 \\
g^{10} & (re^{2\beta}V-\lambda^{2}e^{-2\gamma})A
& g^{12} & 0 \\
g^{20} &
g^{21} & -(e^{4\beta}-\omega^{2})A e^{-2\gamma} & 0 \\
0 & 0 & 0 & -r^{-2}e^{2\gamma}\sin^{-2}\theta \end{array} \right),$$ where $$\begin{aligned}
A&=&{r^{2}\sin^{2}\theta \over g} , \nonumber \\
g&=&r^{2}\sin^{2}\theta [ -r^{2}(e^{4\beta}-(\omega-\sigma U)^{2})
+\sigma e^{2\beta -2\gamma}(2\lambda -\sigma {V \over r}) ] , \nonumber \\
g^{01}&=&[r^{2}(\omega-\sigma U-e^{2\beta})+\sigma\lambda e^{-2\gamma}]
A , \nonumber \\
g^{02}&=&(e^{2\beta}-\omega)\sigma e^{-2\gamma}A ,\nonumber \\
g^{12}&=& [ Ur^{2}(\sigma-\omega-e^{2\beta})+e^{2\beta-2\gamma}(\lambda
-\sigma Vr^{-1})+\lambda\omega e^{-2\gamma} ]A , \nonumber \\
g^{\mu\nu}&=&g^{\nu\mu}[(\omega, \sigma, \lambda) \rightarrow
(-\omega, -\sigma, -\lambda)]. \nonumber\end{aligned}$$
The Field Equations {#res}
===================
The analysis determining the form of the functions $U,V,\beta,\gamma,\omega,\lambda,\sigma$ in our case is a natural extension of that given in detail by Bondi [*et. al.*]{}[@BBM] for finding the forms of $U,V,\beta,\gamma$. The requirement that the field contain only outgoing radiation at large distances from the source gives the form of $\gamma$: $$\gamma=\frac{f(u,\theta)}{r}+\frac{g(u,\theta)}{r^{3}}+...$$ Demanding that the solution have the correct static limit (or equilibrium configuration) leads to the following forms for $U,\, V,\, \beta$ and $\gamma$ (unless otherwise stated, all coefficients in the general expansions are functions of both $u$ and $\theta$): $$\begin{aligned}
U&=&{U_{1} \over r}+{U_{2} \over r^2} +\dots \, , \\
V&=& r-2M+{V_{1} \over r} +\dots \, , \\
\beta&=& {B_{1} \over r}+{B_{2} \over r^2} +\dots \, , \\
\gamma&=& {c \over r}+{C-{1\over 6}c^3 \over r^3} +\dots \, ,\end{aligned}$$
The skew functions, $\omega,\;\lambda$ and $\sigma$, are constrained by the requirement that the spacetime is asymptotically Lorentzian and admits inhomogeneous orthochronous Lorentz transformations [@Sachs]. This requirement demands that $ g_{[\mu\nu]}g^{[\mu\nu]}
\rightarrow 0$ as $r \rightarrow \infty$. In our present coordinates this condition is satisfied if $\omega,\, \lambda$ and $\sigma$ have the following forms: $$\begin{aligned}
\omega&=& {W_{1} \over r}+{W_{2} \over r^2}+\dots \, ,\\
\lambda&=& L_{0}+{L_{1} \over r}+ \dots \, ,\\
\sigma&=& S_{0}+{S_{1} \over r}+ \dots \, .\end{aligned}$$ The functions $M,\, c,\, C,\, L_{0},$ and $W_{2}$ are all functions of integration. Bondi [*et. al.*]{} refer to $c$ as the “news function” as it controls the form of the gravitational radiation in the symmetric sector. In an analogous way, $L_{0}$ is the “news function” for the antisymmetric sector. Consistent with these identifications, we shall see that the solution reduces to the static, non-radiative case when both $c$ and $L_{0}$ are set to zero. The static limit tells us that only $M$ and $W_{2}$ can be non-zero when the system passes through its equilibrium position and these coefficients will be identified as the mass and NGT charge of the body, respectively.
We begin our analysis of the field equations by considering the simplest set of field equations – the skew divergence equations $(\sqrt{-g}
g^{[\mu\nu]})_{,\nu}=0$. The $\theta$ component of this set becomes: $$0=S_{0\; ,u} - {2 (S_{0}c)_{,u}-S_{1\; ,u} \over r} +\dots \, ,$$ and since $\sigma$ must equal zero when passing through equilibrium, $S_{0}=
S_{1}=0$ always. Inserting this information into the $u$ component directly returns $W_{1}=0$ from the lowest order term. The remaining, $r$, component yields: $$0= L_{0}\cot\theta+L_{0\; ,\theta}-W_{2\; ,u} +\dots \, .$$
At this stage of the calculation, it is not profitable to continue to work with the skew divergence equations, as the next orders also contain unknown coefficients from the symmetric functions. Somewhat surprisingly, however, we are already in a position to calculate the NGT charge of the body, and to prove that it is conserved. The NGT charge, $l^2$, is defined by the Gaussian surface integral $$l^2\equiv {1 \over 4\pi} \int (\sqrt{-g} g^{[0\nu]})_{,\nu} d^3 x=
{1 \over 2} \int_{0}^{\pi} W_{2} \sin\theta\, d\theta = <W_{2}> , \label{ngtc}$$ where the brackets $<>$ denote the angular average. The charge is conserved since $$\label{ngtcc}
l^2_{,u}={1 \over 2}\int_{0}^{\pi} W_{2\; ,u} \sin\theta\, d\theta=
{1 \over 2}\int_{0}^{\pi} (L_{0}\sin\theta)_{,\theta}\, d\theta=0 \; .$$ This follows from the fact that $L_{0}$ must be regular on the polar axis.
We now turn our attention to the set of field equations $R_{(\mu\nu)}=0$. The affine connections that we require to construct these generalized Ricci tensor components are obtained by solving the system of 64 equations (\[fensgamma\]). The closed form expressions for the non-zero connection components are extremely lengthy. For brevity, we only display the connections expanded in inverse powers of $r$, and only to the order required for our present analysis. The list of expanded non-zero components is given in the appendix.
Rather than provide an exhaustive list of the $R_{(\mu\nu)}=0$ equations, we shall simply exhibit the components to the order that we require to obtain the solution. We begin with the $(rr)$ component which demands: $$0={2B_{1} \over r^3}+{c^2+4B_{2} \over r^4}+{6B_{3} \over r^5}+\dots \, ,$$ this gives $B_{1}=B_{3}=0$ and $B_{2}=-{c^2/4}$. The $(r \theta)$ component then gives $$0={U_{1} \over r} +{U_{2}+U_{1}c+c_{,\theta}+2c \cot\theta \over r^2}+\dots
\, ,$$ which yields $U_{1}=0$ and $U_{2}=-(c_{,\theta}+2c \cot\theta)$. Inserting these expressions into the $(uu)$ component results in the important condition $$\label{massderiv}
M_{,u}= -{c_{,u}}^{2} + \frac{1}{2} {(c_{,\theta\theta}
+3c_{,\theta}\cot\theta -2c)}_{,u} \; .$$ We can now use the $(ur)$ components of $R_{(\mu\nu)}=0$ at next order to solve for $V_1$, but first we must choose how we wish to write the function of integration contained in $U_3$. Following Bondi [*et. al.*]{} we shall write $U_{3}$ as $$U_{3}=2N + 3cc_{,\theta} +4c^{2}\cot\theta \; ,$$ where $N$ is the additional function of integration (the reason that $U_{3}$ is written in this way rather than as $U_{3}=\widetilde{N}$, and that the second function of integration in $\gamma$ is written as $C-{1\over 6}c^3$ rather than as $\widetilde{C}$, is that we can then identify $N$ and $C$ as the dipole and quadrupole moments of the source, respectively). Now the $(ur)$ equation at next order: $$\begin{aligned}
0&=& \sin^{2}\theta\left(11c^2 -5(c_{,\theta})^2+4c^{2}c_{,u}
-6cc_{,\theta\theta}\right) \nonumber \\
&&+U_{3}\sin{\theta}\cos{\theta} -8c^2-
19cc_{,\theta}\cos{\theta}\sin{\theta} \nonumber\\
&&+\sin^{2}\theta\left(U_{3\, ,\theta}-2L_{0}^2+2V_{1}\right) \; ,\end{aligned}$$ can be solved to give $$\begin{aligned}
V_{1}&=&-N_{,\theta} -N\cot\theta +{c_{,\theta}}^{2} +
4cc_{,\theta}\cot\theta \nonumber\\
&&+\frac{1}{2}c^{2}(1+8\cot^{2}\theta)+L_{0}^2 \; .\end{aligned}$$ It is interesting to note that $V_{1}$ contains the first explicit difference between the symmetric functions found for GR and those found for NGT. Substituting the above results into the $(\theta\theta)$ and $(u \theta)$ components of $R_{(\mu\nu)}=0$ produces the following auxiliary conditions on the multipole moments $C,\, N$: $$\begin{aligned}
4C_{,u}&=& 2c^{2}c_{,u}+2cM+N\cot\theta-N_{,\theta}-L_{0}^2 \; ,\\
-3N_{,u}&=& M_{,\theta} +3cc_{,u\theta} +4cc_{,u}\cot\theta
+c_{,u}c_{,\theta} \; .\end{aligned}$$ We see that the quadrupole moment of a source will decrease more rapidly in NGT than in GR due to the $L_{0}^2$ term.
For completeness, we shall now return to the antisymmetric sector where the skew divergence equation can now be used to obtain the additional relations: $$\begin{aligned}
W_{3}&=& 2S_{2\, ,\theta}+S_{2}\cot{\theta} \; , \\
S_{2\, , u}&=& L_{1}-2cL_{0} \; .\end{aligned}$$ The only remaining antisymmetric field equation (\[asym\]): $$\label{anti}
R_{[01,2]}=R_{[01],2}+R_{[12],0}+R_{[20],1}=0 ,$$ gives to lowest order: $$\left(W_{2\, ,\theta}+L_{1}+S_{2\, ,u}\right)_{,u}=0 \; .$$ This equation yields the additional relation: $$W_{2\, ,\theta}=2cL_{0}-2L_{1} \; .$$
Analysis of the solution {#analysis-of-the-solution .unnumbered}
------------------------
To demonstrate the physical interpretation of $M$, we consider the static limit. We can scale the function $c$ for either one of the static periods to be $c=0$ (forsaking the $\theta$ dependence of $c$ limits us here to a static spherically symmetric system). We now remove the terms containing the functions $N$ and $C$ since they correspond to multipole moments. Since there is no radiation during the static period, $N=C=0$. The metric (\[genmetric\]) tends now to its static spherically symmetric limit: $$\begin{aligned}
\label{expstmetric00}
g_{00}&=&1-\frac{2M_{s}}{r} +\frac{l_{s}^{4}}{r^{4}} -
\frac{2M_{s}l_{s}^{4}}{r^{5}} , \\
\label{expstmetric01s}
g_{(01)}&=&1+\frac{l_{s}^{4}}{2r^{4}} , \\
\label{expstmetric01as}
g_{[01]}&=&\frac{l_{s}^{2}}{r^{2}} , \\
\label{expstmetric02}
g_{(02)}&=&g_{[02]}=0 , \\
\label{expstmetric22}
g_{22}&=& -r^{2} , \\
\label{expstmetric33}
g_{33}&=& -r^{2} \sin^{2}\theta ,\end{aligned}$$ where by $M_{s},\, l^{2}_{s}$ we denote the static limit of $M$ and $l^{2}$, respectively.
Now a coordinate transformation from the retarded time $u$ to the usual time coordinate $t=u+r$ converts the above metric into the NGT static spherically symmetric metric [@Moff91; @Moff79-2]: $$\begin{aligned}
\label{NGTSchwarz}
{ds}^2 &=& (1 + \frac{l_{s}^4}{r^4})(1 - \frac{2M_{s}}{r}){dt}^2 - {(1 -
\frac{2M_{s}}{r})}^{-1}{dr}^2 \nonumber \\
&& - r^2 \left( {d\theta}^2 + {\sin}^2\theta{d\phi}^2 \right),\end{aligned}$$ and $$g_{[01]} = \frac{{l_{s}^2}}{{r^2}}.$$ Thus, the static spherically symmetric limit, $M_{s}$, of the “mass aspect” $M(u,\theta)$ can only be interpreted as the mass of the system. Similarly, the static spherically symmetric limit, $l_{s}^{2}$, of the “charge aspect” $W_{2}(u,\theta)$ is identically the NGT charge of the system as shown by equations (\[ngtc\]) and (\[ngtcc\]).
If we define the mass $m(u)$ of the system as the mean value of $M(u,\theta)$ over the sphere: $$\label{mass}
m(u)=\frac{1}{2} \int^{\pi}_{0} M(u,\theta)\sin\theta d\theta =
<M(u,\theta)> ,$$ then $c(u,\theta)$ completely determines the time evolution of the mass $m(u)$. Integrating (\[massderiv\]) and noticing that the second term does not contribute to the integral due to the condition that $c$ be regular on the polar axis, we get $$\label{centralresult}
m_{,u}=\frac{dm}{du}=-\frac{1}{2} \int^{\pi}_{0} {c_{,u}}^{2}
\sin\theta d\theta .$$ Since we discussed here systems whose initial and final states are static, the physical interpretation of $m(u)$ as the mass of the system is unambiguous. Analogously to the GR case the main result is as follows:
[*The mass of an axially symmetric NGT system is constant only if the system remains static. If the system evolves in time (emits waves), the mass decreases monotonically.*]{}
Since radiation is the only energy loss mechanism available to the system, the above proves that gravitational waves emitted by an axi–symmetric reflexion symmetric NGT source compatible with the metric (\[genmetric\]) carry positive energy or, in other words, the flux of gravitational energy in NGT is positive.
Conclusions {#con .unnumbered}
===========
We have proved that an NGT axi–symmetric system emitting gravitational waves has the usual GR-like asymptotic behaviour in the wave zone. The NGT contributions to the physical quantities decay rapidly with the distance from the source and the energy flux at spatial infinity is necessarily positive.
While we concentrated on an axi-symmetric source to simplify computations, the validity of our result is not confined to this particular symmetry. This contention can be made concrete by adapting the analysis made by Sachs [@Sachs], where it was shown that the axi-symmetric solution contains all the important features of any isolated, radiative system in GR. Sachs’ result follows from considerations of the asymptotic nature of the spacetime, and these considerations are unchanged in NGT. Physically, we can argue that an axi-symmetric source provides a complete range of multipole moments to act as a source of gravitational waves, and thus provides a general test of the wave sector of any gravitational theory. Moreover, in the wave zone, the superposition principle may be used to construct the radiation pattern of any isolated body from a suitable sum of axi-symmetric solutions. Our result totally refutes the recent claims that NGT has bad wave behaviour [@DDM1; @Dam], and shows that aesthetically unappealing, phenomenological modifications to NGT [@DDM2] are not necessary.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the Natural Sciences and Engineering Research Council of Canada. One of the authors (NJC) is grateful for the support provided by a Canadian Commonwealth Scholarship. We thank P. Savaria for helpful discussions.
{#section .unnumbered}
$$\begin{aligned}
\hspace*{-2cm}\Gamma^{0}_{00}=& & 2{B_{1}}_{,u}r^{-1}-\left(U_{1}U_{2}+
{U_{1}}^2c+ M-2{B_{2}}_{,u}-B_{1} \right)r^{-2} \nonumber \\
&+& \left[ 2B_{2}-2MB_{1}+V_{1}-2{L_{0}}^{2}-{U_{2}}^{2}
+2{B_{3}}_{,\theta}\right. \nonumber \\ & & \left. +2U_{1}\left(
U_{2}B_{1}-U_{3}-2cU_{2}\right)+2c{U_{1}}^{2}\left(B_{1}-c\right)
\right]r^{-3} + ...,\\
\Gamma^{0}_{(01)}=&-& \left( S_{2}L_{0}+2W_{2}^2 \right)r^{-5} + ...,\\
\Gamma^{0}_{[01]}=&-& 2W_{2}r^{-3} - \left( 3W_{3} +\frac{1}{2}
S_{2}U_{1} -6B_{1}W_{2} \right)r^{-4}+ ...,\\
\Gamma^{0}_{(02)}=&-& \frac{1}{2}U_{1} + \left( B_{1}U_{1} +
{B_{1}}_{,\theta} \right)r^{-1} \nonumber \\& &+ \left[ cU_{2}+
{B_{2}}_{,\theta} +\frac{1}{2}U_{3}+U_{1}\left(B_{2}-{B_{1}}^2+c^2\right)
\right]r^{-2} + ..., \\
\Gamma^{0}_{[02]}=&-& L_{0}r^{-1} + \left[L_{0}(c+2 B_{1}) +
\frac{1}{2}\left({W_{2}}_{,\theta}+{S_{2}}_{,u}+U_{1}W_{2}-3L_{1}
\right)\right]r^{-2}+ ...,\\
\Gamma^{0}_{(12)}=& & 2W_{2}S_{2}r^{-5}+...,\\
\Gamma^{0}_{[12]}=&-& 3S_{2}r^{-3} + S_{2}\left(8B_{1}+c\right)r^{-4}..., \\
\Gamma^{0}_{22}=& &c +r - 2B_{1} -2\left(B_{2}+cB_{1}-{B_{1}}^{2}\right)
r^{-1} \nonumber \\ &+&\left[2c{B_{1}}^{2}+4B_{2}B_{1}-2B_{2}c-C
-\frac{1}{2}c^{3}-2B_{3}-\frac{4}{3}{B_{1}}^{3}\right]r^{-2} + ..., \\
\Gamma^{0}_{33}/\sin^{2}\theta=& & r - c - 2B_{1} +2\left( cB_{1}-B_{2}
+{B_{1}}^{2}\right)r^{-1} \nonumber \\
&+& \left[4B_{2}B_{1}-2c{B_{1}}^{2}+2B_{2}c+C+\frac{1}{2}c^{3}
-2B_{3}-\frac{4}{3}{B_{1}}^{3}\right]r^{-2} + ..., \\
\Gamma^{1}_{00}=& & \left[ {U_{1}}^2\left({U_{1}}_{,\theta}+c_{,u}
\right)-{B_{1}}_{,u}-M_{,u}\right]r^{-1} \nonumber
\\&+& \left[ {U_{1}}^2 \left(2(c-B_{1})(c_{,u}+{U_{1}}_{,\theta})+
{U_{2}}_{,\theta}+c+c_{,\theta}U_{1} \right) + \frac{1}{2}{V_{1}}_{,u}
-{B_{2}}_{,u}-B_{1} \right. \nonumber \\& & \left. +M(1+2{B_{1}}_{,u})
+U_{1}U_{2}\left(1+2c_{,u}+2{U_{1}}_{,\theta}\right)+U_{1}
{(M-B_{1})}_{,\theta} \right]r^{-2} + ..., \nonumber \\ & & \\
\Gamma^{1}_{(01)}=& & \frac{1}{2}{U_{1}}^2r^{-1} + \left[{U_{1}}^2
(c-B_{1})+M-B_{1}+U_{1}\left(\frac{3}{2}U_{2}-{B_{1}}_{,\theta}\right)
\right]r^{-2} \nonumber \\
&+& \left[{U_{1}}^{2}\left(c^{2}+{B_{1}}^{2}-B_{2}-2cB_{1}\right)
+3U_{1}U_{2}\left(c-B_{1} \right)-V_{1}-2B_{2} \right. \nonumber \\ & &
\left. +2MB_{1}+{U_{2}}^{2}
-U_{1}{B_{2}}_{,\theta}-U_{2}{B_{1}}_{,\theta}+2U_{1}U{3}+{L_{0}}^{2}
\right]r^{-3} + ..., \\
\Gamma^{1}_{[01]}=& & \left({W_{2}}_{,u} -\frac{1}{2}L_{0}U_{1}\right)r^{-2}
\nonumber \\&+& \left[W_{2}\left(2-2{B_{1}}_{,u}+\frac{1}{2}{U_{1}}^2\right)
-L_{0}\left(U_{2}+{B_{1}}_{,\theta}\right) - 2B_{1}{W_{2}}_{,u} \right.
\nonumber \\& & \left.+U_{1} \left(\frac{1}{2}{W_{2}}_{,\theta}-\frac{1}{2}
{S_{2}}_{,u}-L_{1}+L_{0}B_{1} \right) +{W_{3}}_{,u}-S_{2}{U_{1}}_{,u}\right]
r^{-3}+ ..., \\
\Gamma^{1}_{(02)}=& & \frac{1}{2}U_{1} \left(1-2{U_{1}}_{,\theta}-
2c_{,u}\right) \nonumber \\&+& \left[U_{1}\left(B_{1}\left(2c_{,u}-1\right)
+2{U_{1}}_{,\theta}(B_{1}-c)-{U_{2}}_{,\theta}-M-2cc_{,u}\right) \right.
\nonumber \\& & \left.-U_{2} \left(c_{,u}+{U_{1}}_{,\theta}\right)-M_{,\theta}
-{U_{1}}^2c_{,\theta} \right]r^{-1} \nonumber \\
&+& \left[ W_{2}{L_{0}}_{,u}-L_{0}
{L_{0}}_{,\theta} +\frac{1}{2}{V_{1}}_{,\theta}
-U_{3}\left(\frac{1}{2}+{U_{1}}_{,\theta}+c_{,u}\right) \right. \nonumber \\
& &+U_{2}\left(2B_{1}c_{,u}-2cc_{,u}+2B_{1}{U_{1}}_{,\theta}
-2c{U_{1}}_{,\theta}-{U_{2}}_{,\theta}-c \right) \nonumber \\
& & +U_{1}\left( 2MB_{1}-2U_{2}c_{,\theta}-\frac{3}{2}c^{2}c_{,u}-c^{2}
-B_{2}+{B_{1}}^{2}-C_{,u}-{U_{3}}_{,\theta} \right. \nonumber \\
& & +\frac{1}{2}V_{1}-\frac{3}{2}{L_{0}}^{2}
+2\left(B_{1}-c\right)\left({U_{2}}_{,\theta}+U_{1}c_{,\theta}\right)
-2c^{2}{U_{1}}_{,\theta} \nonumber \\ & & \left. \left.
+2\left(2cB_{1}+B_{2}-{B_{1}}^{2}\right)
\left(c_{,u}+{U_{1}}_{,\theta}\right)\right)\right] r^{-2} + ..., \\
\Gamma^{1}_{[02]}=& & {L_{0}}_{,u} + \left[{L_{1}}_{,u} + L_{0}\left(
1-c_{,u}-2{B_{1}}_{,u}\right)+{L_{0}}_{,\theta}U_{1}-2B_{1}{L_{0}}_{,u}
\right]r^{-1} + ..., \\
\Gamma^{1}_{11}=& & -2B_{1}r^{-2}-4B_{2}r^{-3}-6B_{3}r^{-4}
+2\left(S_{2}L_{0}+2{W_{2}}^{2}\right)r^{-5} + ...,\\
\Gamma^{1}_{(12)}=&-& \frac{1}{2}U_{1}+\left[{B_{1}}_{,\theta}-U_{2}+
U_{1}(B_{1}-c)\right]r^{-1} \nonumber \\
&+&\left[ U_{1}\left(B_{2}-{B_{1}}^{2}-c^{2}+2cB_{1}\right)
+2U_{2}\left(B_{1}-c\right)-\frac{3}{2}U_{3}+{B_{2}}_{,\theta}
\right]r^{-2} + ..., \\
\Gamma^{1}_{[12]}=& & \frac{1}{2}\left(U_{1}W_{2}-L_{1}-{W_{2}}_{,\theta}
+{S_{2}}_{,u}\right)r^{-2} + ..., \\
\Gamma^{1}_{22}=&-& r(1-c_{,u}-{U_{1}}_{,\theta}) \nonumber \\
&+& \left[2M +2c_{,u}(c-B_{1})
+2B_{1}(1-{U_{1}}_{,\theta}) -c(1-2{U_{1}}_{,\theta})+
{U_{2}}_{,\theta} +U_{1}c_{,\theta}\right] \nonumber \\
&+& \left[ 2B_{2}-2{B_{1}}^{2}+2cB_{1}-4MB_{1}-V_{1}+2{L_{0}}^{2}
+2Mc +C_{,u} \right. \nonumber \\ & &+c_{,u}\left(2{B_{1}}^{2}-2B_{2}-4cB_{1}
+\frac{3}{2}c^{2}\right)
+ c_{,\theta}\left(U_{2}+2cU_{1}-2U_{1}B_{1}\right) \nonumber \\
& & \left. +2{U_{1}}_{,\theta} \left(c^{2}+{B_{1}}^{2}-2cB_{1}-B_{2}\right)
+2{U_{2}}_{,\theta}\left(c-B_{1}\right) +{U_{3}}_{,\theta}
\right] r^{-1} + ..., \\
\Gamma^{1}_{33}/\sin^{2}\theta=&-& r(1+c_{,u}-U_{1}\cot\theta)
\nonumber \\ &+& \left[ 2M + c
-c_{,\theta}U_{1}+ 2c_{,u}(c+B_{1}) + 2B_{1} - \left(2U_{1}
(c+B_{1})-U_{2} \right)\cot\theta \right] \nonumber \\
&+& \left[ 2B_{2}-2{B_{1}}^{2}-2cB_{1}-4MB_{1}-V_{1}
-2Mc-C_{,u} \right. \nonumber \\ & &-c_{,u}\left(2{B_{1}}^{2}-2B_{2}+4cB_{1}
+\frac{3}{2}c^{2}\right)
-c_{,\theta}\left(U_{2}-2cU_{1}-2U_{1}B_{1}\right) \nonumber \\
& & \left. +\left(2U_{1} \left(c^{2}+{B_{1}}^{2}+2cB_{1}-B_{2}\right)
-2U_{2}\left(c+B_{1}\right) +U_{3}\right)\cot\theta
\right] r^{-1} + ..., \\
\Gamma^{2}_{00}=&-& {U_{1}}_{,u}r^{-1} +\left[U_{1}\left( 2{B_{1}}_{,u}
-2c_{,u}-{U_{1}}_{,\theta}\right)-{U_{2}}_{,u}\right]r^{-2} \nonumber \\
&+& \left[ {B_{1}}_{,\theta}-M_{,\theta}-{U_{3}}_{,u}-{U_{1}}^{2}
\left(U_{2}+cU_{1}-c_{,\theta}\right)+U_{2}\left(2{B_{1}}_{,u}-2c_{,u}
-{U_{1}}_{,\theta}\right) \right. \nonumber \\
& & \left. +U_{1}\left(B_{1}-{U_{2}}_{,\theta}+2{B_{2}}_{,u}\right) \right]
r^{-3} + ..., \\
\Gamma^{2}_{(01)}=&-& \frac{1}{2}U_{1}r^{-2} + \left({B_{1}}_{,\theta}+
U_{1}c\right)r^{-3}\nonumber \\ &+& \left[ cU_{2}+\frac{1}{2}U_{3}
+{B_{2}}_{,\theta}+2{B_{1}}_{,\theta}\left(B_{1}-c\right) \right]r^{-4}+ ... ,
\\
\Gamma^{2}_{[01]}=& & L_{0}r^{-3}-\left[3(U_{1}W_{2}+cL_{0})+\frac{1}{2}
\left({W_{2}}_{,\theta}-{S_{2}}_{,u}-3L_{1}\right)\right]r^{-4} + ..., \\
\Gamma^{2}_{(02)}=& & \left(c_{,u}-\frac{1}{2}{U_{1}}^2\right)r^{-1}
+U_{1}\left(B_{1}U_{1}+{B_{1}}_{,\theta}-\frac{1}{2}U_{2}\right)r^{-2}
\nonumber \\
&+& \left[ C_{,u}-\frac{1}{2}c^{2}c_{,u}+{L_{0}}^{2}+U_{1}{B_{2}}_{,\theta}
+U_{2}{B_{1}}_{,\theta}\right. \nonumber \\
& & \left. +{U_{1}}^{2}\left(B_{2}+c^{2}-{B_{1}}^{2}\right)
+U_{1}U_{2}\left(B_{1}+c\right) \right]r^{-3}
+ ..., \\
\Gamma^{2}_{[02]}=&-& \left({L_{0}}_{,\theta}+\frac{1}{2}U_{1}L_{0}
\right)r^{-2} + ..., \\
\Gamma^{2}_{(11)}=& &4W_{2}S_{2} r^{-7} + ..., \\
\Gamma^{2}_{(12)}=& & r^{-1} -cr^{-2} +\left(\frac{1}{2}c^{3}-3C\right)
r^{-4} + ..., \\
\Gamma^{2}_{[12]}=&-& W_{2}r^{-3}+\left(2W_{2}B_{1}-W_{3}+cW_{2}
-\frac{5}{2}S_{2}U_{1}-{S_{2}}_{,\theta}\right)r^{-4} + ..., \\
\Gamma^{2}_{22}=& & U_{1} + \left[U_{1}(c-2B_{1})+U_{2}+c_{,\theta}\right]
r^{-1} \nonumber \\
&+& \left[ U_{3}+U_{2}\left(c-2B_{1}\right)+2U_{1}\left({B_{1}}^{2}
-B_{2}-cB_{1}\right)\right]r^{-2} + ..., \\
\Gamma^{2}_{33}/\sin^2\theta=& & U_{1} - \cot\theta +\left(U_{2}+c_{,\theta}
-cU_{1}-2B_{1}U_{1}+4c\cot\theta\right)r^{-1} \nonumber \\
&+& \left[ U_{3}-U_{2}\left(c+2B_{1}\right)+2U_{1}\left({B_{1}}^{2}
-B_{2}+cB_{1}\right)\right. \nonumber \\
& & \left. -4c\left(c_{,\theta}+2c\cot\theta\right)\right]r^{-2}
+ ..., \\
\Gamma^{3}_{(03)}=&-&c_{,u}r^{-1} +\left(\frac{1}{2}c^2c_{,u}-C_{,u}\right)
r^{-3} + ..., \\
\Gamma^{3}_{[03]}=& & L_{0}(U_{1}-\cot\theta)r^{-2} \nonumber \\&+& \left[
W_{2}(U_{1}\cot\theta-1-c_{,u})+ L_{1}(U_{1}-\cot\theta) \right. \nonumber \\
& & \left.+L_{0}(c_{,\theta}-2B_{1}U_{1}+2c\cot\theta +cU_{1}+U_{2})
\right]r^{-3} + ..., \\
\Gamma^{3}_{(13)}=& & r^{-1} + cr^{-2} + \left(3C-\frac{1}{2}c^{3}\right)
r^{-4} + ..., \\
\Gamma^{3}_{[13]}=&-& W_{2}r^{-3}+ \left[W_{2}\left(2B_{1}-c\right)
+S_{2}\left(U_{1}-\cot\theta\right) -W_{3}\right]r^{-4} + ..., \\
\Gamma^{3}_{(23)}=& & \cot\theta - c_{,\theta}r^{-1} +
\left(\frac{1}{2}c^{2}c_{,\theta}-C_{,\theta}\right)r^{-3} + ..., \\
\Gamma^{3}_{[23]}=&-& L_{0}r^{-1} +\left[L_{0}(2B_{1}-c)-L_{1}\right]r^{-2}
+ ...,\end{aligned}$$
Moffat J. W. in [*Gravitation — A Banff Summer Institute*]{}, eds. Mann R. B. & Wesson P., World Scientific, Singapore, 1991, p. 523. Cornish N. J., Moffat J. W. & Tatarski D. C., [*Phys. Lett.*]{} A [**173**]{}, 109, 1993. Krisher T., [*Phys. Rev.*]{} D [**32**]{}, 329, 1985. Damour T., Deser S. & McCarthy J., [*Phys. Rev.*]{} D [**45**]{}, R3289, 1992. Damour T., Deser S. & McCarthy J., [*Phys. Rev.*]{} D [**47**]{}, 1541, 1993. Damour T., To appear in [*Proceedings of the International Colloquium in Honour of Yvonne Choquet-Bruhat*]{}, eds. Flato M., Kerner R. & Lichnérowicz A., Kluwer, Dordrecht, Netherlands. Cornish N. J. & Moffat J. W., [*Phys. Rev.*]{} D [**47**]{}, 4421, 1993. Sachs R. K., [*Proc. Roy. Soc.*]{} A, [**270**]{}, 103, 1962. Bondi H., van der Burg M. G. J. & Metzner A. W. K., [*Proc. Roy. Soc.*]{} A, [**269**]{}, 21, 1962. Moffat J. W., [*Phys. Rev.*]{} D [**19**]{}, 3554, 1979. Moffat J. W., [*Phys. Rev.*]{} D [**35**]{}, 3733, 1987. Savaria P., [*Class. Quantum Grav.*]{} [**6**]{}, 1003, 1989. Savaria P., [*Class. Quantum Grav.*]{} [**9**]{}, 1349, 1992. Kunstatter G., Moffat J. W. & Savaria P., [*Phys. Rev.*]{} D [**19**]{}, 3559, 1979. Kunstatter G., Moffat J. W. & Savaria P., [*Can. J. Phys.*]{} [**58**]{}, 729, 1980. Bondi H., [*Nature*]{}, [**186**]{}, 535, 1960. The full static spherically symmetric solution in NGT admits a term $g_{[23]}\sim r^2 \sin\theta$, which is not compatible with asymptotic inhomogeneous orthochronous Lorentz invariance. It is important to impose such an invariance on the wave solutions so as to facilitate a definition of energy. One may keep the three functions $g_{[03]},\; g_{[13]},
\; g_{[23]}$ in the calculation and maintain asymptotic invariance, if: $g_{[03]}=D_{0}+D_{1}/r+\dots$, $g_{[13]}=E_{0}+E_{1}/r+\dots$, and $g_{[23]}=r F_{-1}+F_{0}+\dots$, where $D_{i},\; E_{i},\; F_{i}$ are functions of $u$ and $\theta$. Repeating the calculation in ref. 2 with all six skew functions gives the following: At lowest order the skew divergence equation, $(\sqrt{-g}g^{[3 \nu]})_{,\nu}=0$, gives $E_{0,\, u}=0 \Rightarrow E_{0}=0$. The lowest order in $R_{\{[02],3]\}}=0$ gives $F_{-1\, ,uu}=0 \Rightarrow F_{-1}=0$ (as demanded by the quasi-periodic boundary conditions). The next order in the skew divergence equations then gives $E_{1\, ,u}=0 \Rightarrow E_{1}=0$. Putting all this into $R_{\{[01],3\}}=0$ yields $D_{0\, ,u}=0
\Rightarrow D_{0}=0$. Since the two sets of three skew functions have remained uncoupled to this order, the solutions for $g_{[01]},\; g_{[02]},
\; g_{[12]}$ are unchanged from those in ref. 2. At this stage our work is done since all leading terms ($1/r$ terms in cartesian coordinates) have vanished. It is thus a trivial exercise to show that $g_{[03]},\; g_{[13]}, \; g_{[23]}$ can play no role in determining the flux of gravitational radiation.
|
---
abstract: 'We derive formulas for the matrix elements of the two dimensional square lattice Green function along the diagonal, and along the coordinate axes. We also give an asymptotic formula for the diagonal elements.'
author:
- Stefan Hollos
- Richard Hollos
bibliography:
- '../lgfint/gf.bib'
- 'lgfmath1.bib'
title: Some Square Lattice Green Function Formulas
---
introduction
============
In some recent papers [@cserti00; @cserti02] Cserti showed how a lattice Green function (LGF) can be used to find the resistance between two points in an infinite lattice of resistors. Cserti gives an expression for the matrix elements of the LGF in the form of an integral. In this paper we will show how to solve this integral for the case of a two dimensional square lattice along the diagonal and the coordinate axes. This allows any arbitrary diagonal or coordinate axis LGF matrix element to be calculated directly. Formulas for these elements were first derived by McCrea and Whipple [@comment][@mccrea40] using a different procedure than that presented here. We will also give an asymptotic formula for the diagonal matrix elements that converges to Cserti’s asymptotic limit formula for large values of $n$.
The LGF with which we are concerned here can in general be used to solve the discrete two dimensional Poisson equation with boundary conditions at infinity. Therefore it will be useful in solving two dimensional electrostatics problems [@exstrom2005] as well as many other problems that can be modeled by a Poisson equation.
diagonal matrix elements
========================
The matrix elements of the two dimensional square lattice Green function can be expressed in terms of an integral as$$g(n,m)=\frac{1}{2\pi^{2}}\int_{0}^{\pi}d\phi\int_{0}^{\pi}d\theta\frac{1-\cos n\theta\cos m\phi}{2-\cos\theta-\cos\phi}\label{eq:1}$$ This is essentially the same as Cserti’s [@cserti00] eq. B1. We will begin by looking at the diagonal elements where $m=n$. First note that for $m=n$ eq. \[eq:1\] can be rewritten in the following form$$g(n,n)=\frac{1}{2\pi^{2}}\int_{0}^{\pi}d\phi\int_{0}^{\pi}d\theta\frac{1-\frac{1}{2}\cos n(\phi-\theta)-\frac{1}{2}\cos n(\phi+\theta)}{2-2\cos\frac{1}{2}(\phi-\theta)\cos\frac{1}{2}(\phi+\theta)}\label{eq:2}$$ By symmetry the two cosine terms in the numerator of the integrand can be combined to give$$g(n,n)=\frac{1}{2\pi^{2}}\int_{0}^{\pi}d\phi\int_{0}^{\pi}d\theta\frac{1-\cos n(\phi+\theta)}{2-2\cos\frac{1}{2}(\phi-\theta)\cos\frac{1}{2}(\phi+\theta)}\label{eq:3}$$ In terms of new variables $\phi'=\frac{1}{2}(\phi-\theta)$, $\theta'=\frac{1}{2}(\phi+\theta)$ eq. \[eq:3\] becomes$$g(n,n)=\frac{1}{4\pi^{2}}\int_{0}^{\pi}d\phi\int_{0}^{\pi}d\theta\frac{1-\cos2n\theta}{1-\cos\phi\,\cos\theta}\label{eq:4}$$ The integration over $\phi$ can now be done to give$$g(n,n)=\frac{1}{4\pi}\int_{0}^{\pi}d\theta\,\frac{1-\cos2n\theta}{\sin\theta}\label{eq:5}$$ Using the identity $1-\cos2n\theta=2\sin^{2}n\theta$, eq. \[eq:5\] can be written as$$g(n,n)=\frac{1}{2\pi}\int_{0}^{\pi}d\theta\,\frac{\sin^{2}n\theta}{\sin\theta}=\frac{1}{2\pi}\int_{0}^{\pi}d\theta\,\left(\frac{\sin n\theta}{\sin\theta}\right)^{2}\sin\theta\label{eq:6}$$ Now if we let $x=\cos\theta$ then this becomes the integral of a type II Chebyshev polynomial [@mason2003]$$g(n,n)=\frac{1}{2\pi}\int_{-1}^{1}U_{n-1}^{2}(x)\, dx=\frac{1}{\pi}\int_{0}^{1}U_{n-1}^{2}(x)\, dx\label{eq:7}$$ The square of a type II Chebyshev polynomial can be expressed as$$U_{n}^{2}(x)=\sum_{k=0}^{n}U_{2k}(x)\label{eq:8}$$ To prove this identity it is sufficient to show that $U_{n}^{2}(x)-U_{n-1}^{2}(x)=U_{2n}(x)$. With $x=\cos\theta$ we have $U_{n}(x)=\sin(n+1)\theta/\sin\theta$ and $$U_{n}(x)-U_{n-1}(x)=\frac{\sin(n+1)\theta-\sin n\theta}{\sin\theta}=\frac{\cos(n+\frac{1}{2})\theta}{\cos\frac{1}{2}\theta}$$ $$U_{n}(x)+U_{n-1}(x)=\frac{\sin(n+1)\theta+\sin n\theta}{\sin\theta}=\frac{\sin(n+\frac{1}{2})\theta}{\sin\frac{1}{2}\theta}$$ $$U_{n}^{2}(x)-U_{n-1}^{2}(x)=\frac{\cos(n+\frac{1}{2})\theta\;\sin(n+\frac{1}{2})\theta}{\cos\frac{1}{2}\theta\;\sin\frac{1}{2}\theta}=\frac{\sin(2n+1)\theta}{\sin\theta}=U_{2n}(x)$$
So using eq. \[eq:8\], eq. \[eq:7\] can be written as$$g(n,n)=\frac{1}{\pi}\sum_{k=0}^{n-1}\int_{0}^{1}U_{2k}(x)\, dx\label{eq:9}$$ Letting $x=\cos\theta$, the integrals in eq. \[eq:9\] become$$\int_{0}^{1}U_{2k}(x)\, dx=\int_{0}^{\frac{\pi}{2}}\sin(2k+1)\theta\, d\theta=\frac{1}{2k+1}\label{eq:10}$$ Substituting this into eq. \[eq:9\] and changing the summation index gives$$g(n,n)=\frac{1}{\pi}\sum_{k=1}^{n}\frac{1}{2k-1}\label{eq:11}$$ Note that $g(n,n)$ also obeys a difference equation and that the solution for $g(n,n)$ given in eq. \[eq:11\] could also be arrived at by solving the difference equation. The difference equation for $g(n,n)$ is the same as that given by Cserti [@cserti00] eq. 32 for the resistances along the diagonal$$(2n+1)\, g(n+1,n+1)-4n\, g(n,n)+(2n-1)\, g(n-1,n-1)=0\label{eq:11A}$$ Since the coefficients of this equation add up to zero, if we substitute $g(n,n)=\sum_{k=1}^{n}f(k)$, $g(n-1,n-1)=g(n,n)-f(n)$, $g(n+1,n+1)=g(n,n)+f(n+1)$ into the equation, we will get a first order equation for $f(n)$.$$(2n+1)\, f(n+1)-(2n-1)\, f(n)=0\label{eq:11B}$$ With the initial condition $f(1)=\frac{1}{\pi}$ the solution to this equation is $f(k)=\frac{1}{\pi}\,\frac{1}{2k-1}$ which once again gives us eq. \[eq:11\] as the solution for $g(n,n)$.
We will now derive an asymptotic formula for $g(n,n)$. First note that the $g(n,n)$ elements are proportional to the partial sums of a generalized harmonic series. They can also be expressed in terms of the standard harmonic series as follows$$g(n,n)=\frac{1}{\pi}\left(\sum_{k=1}^{2n}\frac{1}{k}-\frac{1}{2}\sum_{k=1}^{n}\frac{1}{k}\right)=\frac{1}{\pi}\left(H_{2n}-\frac{1}{2}H_{n}\right)\label{eq:12}$$ where we have introduced the notation$$H_{n}=\sum_{k=1}^{n}\frac{1}{k}\label{eq:13}$$ for the $n$th partial sum of the standard harmonic series. The asymptotic formula for the $n$th partial sum of the harmonic series is [@arfken1985] p. 338$$H_{n}=\ln n+\gamma+\frac{1}{2n}-\sum_{k=1}\frac{B_{2k}}{2kn^{2k}}\label{eq:13A}$$ where $\gamma=0.5772156649\ldots$ is the Euler-Mascheroni constant and $B_{2k}$ is a Bernoulli number. Using this in eq. \[eq:12\] results in the following asymptotic formula for $g(n,n)$
$$g(n,n)=\frac{1}{2\pi}\left[\ln(n)+\gamma+2\ln(2)+\sum_{k=1}^{\infty}\frac{B_{2k}(2^{2k-1}-1)}{k(2n)^{2k}}\right]\label{eq:14}$$
Without the Bernoulli sum, this is essentially the same as Cserti’s [@cserti00] eq. 33 for the asymptotic limit of the resistance.
the on-axis elements
====================
We now turn to the on-axis elements where $m=0$ and eq. \[eq:1\] becomes$$g(n,0)=\frac{1}{2\pi^{2}}\int_{0}^{\pi}d\phi\int_{0}^{\pi}d\theta\frac{1-\cos n\theta}{2-\cos\theta-\cos\phi}\label{eq:15}$$ The integral with respect to $\phi$ can be carried out to give$$g(n,0)=\frac{1}{2\pi}\int_{0}^{\pi}d\theta\frac{1-\cos n\theta}{\sqrt{(2-\cos\theta)^{2}-1}}\label{eq:16}$$ Now note that $1-\cos n\theta=2\sin^{2}(\frac{n\theta}{2})$ and $2-\cos\theta=1+2\sin^{2}(\frac{\theta}{2})$ so that the denominator of the integrand in eq. \[eq:16\] becomes $\sqrt{(1+2\sin^{2}(\frac{\theta}{2}))^{2}-1}=2\sin(\frac{\theta}{2})\sqrt{1+\sin^{2}(\frac{\theta}{2})}$. Eq. \[eq:16\] then becomes$$g(n,0)=\frac{1}{2\pi}\int_{0}^{\pi}d\theta\frac{\sin^{2}(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})\sqrt{1+\sin^{2}(\frac{\theta}{2})}}\label{eq:18}$$ Making the change in variable $\theta'=\frac{\theta}{2}$ , we write eq. \[eq:18\] as$$g(n,0)=\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}d\theta\left(\frac{\sin n\theta}{\sin\theta}\right)^{2}\frac{\sin\theta}{\sqrt{1+\sin^{2}\theta}}\label{eq:19}$$ With $x=\cos\theta$ we then have an integral involving the Chebyshev polynomial $U_{n-1}(x)$$$g(n,0)=\frac{1}{\pi}\int_{0}^{1}\frac{U_{n-1}^{2}(x)}{\sqrt{2-x^{2}}}\, dx\label{eq:20}$$ Now if we let $x=\sqrt{1-\cos\theta}$ then$$g(n,0)=\frac{1}{2\pi}\int_{0}^{\frac{\pi}{2}}U_{n-1}^{2}(\sqrt{1-\cos\theta})\, d\theta\label{eq:21}$$ $$g(n,0)=\frac{1}{2\pi}\sum_{k=0}^{n-1}\int_{0}^{\frac{\pi}{2}}U_{2k}(\sqrt{1-\cos\theta})\, d\theta\label{eq:22}$$ If the type I and type II Chebyshev polynomials are expressed in the following forms [@mason2003]$$\begin{aligned}
T_{n}(x) & = & \frac{1}{2}\left[\left(x+\sqrt{x^{2}-1}\right)^{n}+\left(x-\sqrt{x^{2}-1}\right)^{n}\right]\label{eq:23}\\
U_{n}(x) & = & \frac{\left(x+\sqrt{x^{2}-1}\right)^{n+1}-\left(x-\sqrt{x^{2}-1}\right)^{n+1}}{2\sqrt{x^{2}-1}}\nonumber \end{aligned}$$ then it is straightforward to prove the identity$$U_{2k}(\sqrt{1-x})=\frac{(-1)^{k}T_{2k+1}(\sqrt{x})}{\sqrt{x}}\label{eq:24}$$ Using this identity eq. \[eq:22\] becomes$$g(n,0)=\frac{1}{2\pi}\sum_{k=0}^{n-1}(-1)^{k}\int_{0}^{\frac{\pi}{2}}\frac{T_{2k+1}(\sqrt{\cos\theta})}{\sqrt{\cos\theta}}\, d\theta\label{eq:25}$$ $T_{2k+1}(x)$ can be expressed in series form as$$T_{2k+1}(x)=\sum_{j=0}^{k}(-1)^{k-j}2^{2j}\frac{2k+1}{2j+1}\left(\begin{array}{c}
k+j\\
2j\end{array}\right)x^{2j+1}\label{eq:26}$$ so that we have$$(-1)^{k}\frac{T_{2k+1}(\sqrt{x})}{\sqrt{x}}=\sum_{j=0}^{k}(-4)^{j}\frac{2k+1}{2j+1}\left(\begin{array}{c}
k+j\\
2j\end{array}\right)x^{j}\label{eq:27}$$ and eq. \[eq:25\] can be written as$$g(n,0)=\frac{1}{2\pi}\sum_{k=0}^{n-1}\,\sum_{j=0}^{k}a(k,j)b(j)\label{eq:28}$$ where$$\begin{aligned}
a(k,j) & = & (-4)^{j}\frac{2k+1}{2j+1}\left(\begin{array}{c}
k+j\\
2j\end{array}\right)\label{eq:29}\\
b(j) & = & \int_{0}^{\frac{\pi}{2}}\cos^{j}\theta\, d\theta=\left\{ \begin{array}{cc}
\frac{\pi}{2}\frac{(j-1)!!}{j!!} & j=0,2,4,6,\ldots\\
\frac{(j-1)!!}{j!!} & j=1,3,5,7,\ldots\end{array}\right.\nonumber \end{aligned}$$ Eq. \[eq:28\] can be used to directly calculate $g(n,0)$ for arbitrary values of $n$.
conclusion
==========
We have derived equations by which $g(n,n)$ and $g(n,0)$ can be calculated for arbitrary values of $n$. For the case of $g(n,n)$ we have an asymptotic formula eq. \[eq:14\] that allows for a quick and efficient calculation. In the case of $g(n,0)$ we have eq. \[eq:28\] whose evaluation can be optimized for large values of $n$. A complete asymptotic formula for $g(n,0)$ has not yet been found. A formula very similar to eq. \[eq:14\] for $g(n,n)$ has been found for $g(n,0)$ but only the first few terms in the Bernoulli sum have so far been determined. Formulas for the general matrix elements $g(n,m)$ have been found by us. These formulas are found by solving the partial difference equation for $g(n,n)$. This equation can only be solved after a formula for the diagonal elements, $g(n,n)$ has been found.
The authors acknowledge the generous support of Exstrom Laboratories and its president Istvan Hollos.
|
---
abstract: 'Matchgates are a restricted set of two-qubit gates known to be classically simulable when acting on nearest-neighbor qubits on a path, but universal for quantum computation when the qubits are arranged on certain other graphs. Here we characterize the power of matchgates acting on arbitrary graphs. Specifically, we show that they are universal on any connected graph other than a path or a cycle, and that they are classically simulable on a cycle. We also prove the same dichotomy for the XY interaction, a proper subset of matchgates related to some implementations of quantum computing.'
author:
- 'Daniel J. Brod'
- 'Andrew M. Childs'
title: |
The computational power of matchgates\
and the XY interaction on arbitrary graphs
---
Introduction {#sec:intro}
============
Studying the computational power of restricted sets of operations can shed light on the nature of quantum speedup. From a theoretical perspective, such studies can determine what resources are necessary and/or sufficient for universal quantum computation. This issue is also relevant in experimental settings, where available operations or resources may be restricted.
In this paper, we focus on the class of operations known as matchgates. Matchgates are a class of $2$-qubit gates defined by Valiant [@Valiant02] that are closely related to noninteracting fermions [@Terhal02]. To define matchgates, let $G(A,B)$ denote the unitary gate that acts as unitaries $A$ and $B$, respectively, on the even- and odd-parity subspaces of a 2-qubit Hilbert space: $$\label{eq:Matchgate}
G(A,B) = \begin{pmatrix}
A_{11} & 0 & 0 & A_{12} \\
0 & B_{11} & B_{12} & 0 \\
0 & B_{21} & B_{22} & 0 \\
A_{21} & 0 & 0 & A_{22}
\end{pmatrix}.$$ The gate $G(A,B)$ is a *matchgate* if $\det A = \det B$.
As originally shown by Valiant [@Valiant02], and soon after by Terhal and DiVincenzo [@Terhal02] in the setting of fermionic linear optics, a quantum computation composed only of (i) qubits (arranged on a path) initially prepared in a product state, (ii) a circuit of nearest-neighbor matchgates, and (iii) a final measurement in the computational basis can be efficiently simulated on a classical computer. Curiously, the computational power of matchgates varies greatly with seemingly small changes in spatial restrictions: by relaxing the nearest-neighbor condition and allowing matchgates to also act on next-nearest neighbors, they become universal for quantum computation, as shown by Kempe, Bacon, DiVincenzo, and Whaley [@Kempe01b; @Kempe02]. Both regimes were revisited and extended by Jozsa and Miyake [@Jozsa08b], who also provided simpler proofs.
More generally, one can consider matchgates restricted to act on pairs of qubits joined by the edges of any graph. In [@Brod12] it was shown that matchgates can implement universal quantum computation on many graphs, such as a complete binary tree, a star, or a path with one extra vertex appended to some point, and the authors suggested that the path might be a pathological instance where matchgates are classically simulable. The authors also left as an open question whether there is a regime of intermediate computational power, between that of classical and quantum computers, such as in recent proposals using commuting operators [@Bremner10] or linear optics [@Aaronson11].
Here we use ideas from [@Brod12] to prove that matchgates are universal on any connected graph other than a path or a cycle. We also adapt previous results [@Terhal02; @Jozsa08b] to show that matchgates are classically simulable on a cycle. Thus we completely characterize the power of matchgates on arbitrary graphs, resolving the two open questions from [@Brod12].
Furthermore, we consider the computational power of the XY (or anisotropic Heisenberg) interaction acting on the edges of a graph. The XY interaction generates matchgates, so it is non-universal on paths [@Terhal02] and cycles. In fact, even this restricted class of matchgates is universal when acting on next-nearest neighbors [@Kempe01b; @Kempe02]. Here we show that the XY interaction is also universal on any connected graph other than a path or a cycle, so it is as powerful as general matchgates.
This paper is organized as follows. In we review the proof of universality of matchgates acting on nearest and next-nearest neighbors on a path, focusing on ideas used in our first main result. In we present two instructive examples from [@Brod12] that lead to the proof, in , that matchgates are universal on any connected graph other than a path or a cycle. In we review the classical simulation of matchgates on a path, and in we show how this result can be adapted to provide an equivalent result for matchgates acting on a cycle. Finally, in we specialize the result of and show that the subset of matchgates known as the XY interaction is also universal on any graph other than a path or cycle. Although this latter result implies the first, we present the results separately as the first proof is easier and develops tools that are useful later, while the simulation using the XY interaction is less explicit.
**Notation and terminology.** Throughout this paper we consider matchgates acting on the edges of a graph, unless stated otherwise, and we refer to “universal graphs” as those on which such matchgates are universal. We restrict our attention to connected graphs without loss of generality, as qubits in different components of a general graph cannot interact, so the components can be treated separately. By a universal gate set we mean a set that can simulate a universal quantum computer with at most polynomial overhead in number of operations and number of qubits. We extensively use the concept of encoded universality (see, e.g., [@Kempe01b; @Kempe02; @DiVincenzo00]), where one logical qubit is encoded in two or more physical qubits, so we occasionally denote logical basis states or logical operators that act on an encoded space by a subscript $L$ when there is risk of ambiguity. We also interchangeably refer to a set of quantum gates by their unitary matrices or their generating Hamiltonians, as we will not consider the case of discrete sets of unitaries.
Universality for arbitrary graphs {#sec:match_arbit}
=================================
Matchgates acting on nearest and next-nearest neighbors {#sec:univ_Jozsa}
-------------------------------------------------------
We begin by giving a simple proof, along the lines of [@Jozsa08b], that matchgates are universal on a path when supplemented by the 2-qubit ${\textsc{swap}}$ gate. Consider an encoding of a logical qubit into two physical qubits, given by $$\begin{aligned}
{\left| 0 \right>}_L = {\left| 00 \right>}, \notag \\
{\left| 1 \right>}_L = {\left| 11 \right>}. \label{eq:evenencoding}\end{aligned}$$
Clearly an encoded single-qubit gate $A_L$ can be implemented simply by applying the matchgate $G(A,A)$ to the pair of physical qubits that encode the logical qubit.
The other requirement for a universal set is an entangling 2-qubit gate, such as the controlled-$Z$ (${\textsc{cz}}$) gate. Consider two adjacent logical qubits encoded in physical pairs labeled $\{1,2\}$ and $\{3,4\}$, respectively. Then a ${\textsc{cz}}_L$ between the logical qubits can be implemented simply by a ${\textsc{cz}}$ between the neighboring qubits 2 and 3. Note that this is not a matchgate—indeed, no nearest-neighbor matchgate can generate entanglement while preserving the encoding of [@Brod11], as this would contradict that matchgates are classically simulable when acting on a path. Therefore the entangling gate must be implemented with the aid of some non-matchgate operation. One such example is the sequence $$\label{eq:CZL}
{\textsc{cz}}= {\textrm{f-}{\textsc{swap}}}\cdot {\textsc{swap}}.$$ Here ${\textsc{swap}}= G(I,X)$ is not a matchgate. The closely related gate $$\label{eq:fS}
{\textrm{f-}{\textsc{swap}}}:= G(Z,X) = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & -1
\end{pmatrix}$$ is a matchgate that swaps the two qubits and induces a minus sign when both are in the ${\left| 1 \right>}$ state (so we call it the fermionic ${\textsc{swap}}$). In we can interpret the ${\textsc{swap}}$ as undoing an undesired interchange of the qubits induced by the ${\textrm{f-}{\textsc{swap}}}$ during the entangling operation.
We thus conclude that matchgates, when supplemented by the ${\textsc{swap}}$, form a universal set. Furthermore, the ${\textsc{swap}}$ is only applied on disjoint sets of physical qubits (i.e., $\{2i, 2i+1\}$ for $1 \leq i \leq n$, where $n$ is the total number of logical qubits), so no qubit is swapped more than one position away from its original place [@Jozsa08b]. Thus the ${\textsc{swap}}$ gate in this construction can be replaced by allowing matchgates to also act on second and third neighbors on the path. In fact, matchgates between only nearest and next-nearest neighbors are already universal, as shown in [@Kempe02] and [@Jozsa08b] using alternative encodings, where each logical qubit is encoded into $3$ and $4$ physical qubits, respectively.
Matchgates acting on arbitrary graphs {#sec:examples}
-------------------------------------
Now suppose that the qubits are arranged on the vertices of a more general (connected) graph, and matchgates can act between every neighboring pair of qubits. Henceforth, we restrict ourselves to interactions between nearest neighbors in these general graphs. In this setting, the result mentioned at the end of the previous section straightforwardly translates to the universality of the “triangular ladder” graph of [@Kempe02].
![In a triangular ladder graph, vertices have a one-to-one correspondence to vertices of a path such that nearest neighbors on the triangular ladder correspond to nearest and next-nearest neighbors on the path.[]{data-label="fig:triangladder"}](triangladder.pdf){width="30.00000%"}
In a previous paper [@Brod12], it was proven that matchgates are also universal on many other graphs. Here we extend this result to show that matchgates are universal on any graph that is not a path or a cycle.
Before giving the proof for the most general case, it is instructive to work through two cases that exemplify the main ideas. Both examples are taken from [@Brod12] with small adaptations.
\[ex:path\] Suppose the qubits are arranged according to a graph of the form shown in , which is obtained by joining a new vertex to some degree-2 vertex of a path. To prove that such a graph is universal, we use two tricks from [@Brod12].
![An $n$-vertex graph obtained from an $(n-1)$-vertex path by joining a new vertex to some degree-$2$ vertex of the original path.[]{data-label="fig:appendedline"}](appendedline.pdf){width="30.00000%"}
First, suppose we have a logical qubit in an arbitrary state ${\left| \Psi \right>}_L=\alpha {\left| 00 \right>} + \beta {\left| 11 \right>}$ and a third physical qubit in any state ${\left| \phi \right>}$. We then have the identity $$\label{eq:logicswap}
{\textrm{f}\textsc{s}}_{12} {\textrm{f}\textsc{s}}_{23} {\left| \Psi \right>}_L {\left| \phi \right>} = {\left| \phi \right>} {\left| \Psi \right>}_L,$$ where ${\textrm{f}\textsc{s}}$ is shorthand for the ${\textrm{f-}{\textsc{swap}}}$ gate, and subscripts denote the pair being acted on. The above identity follows from the trivial observation that the logical qubit is always a superposition of ${\left| 00 \right>}$ and ${\left| 11 \right>}$, so the ${\textrm{f-}{\textsc{swap}}}$ gate either does not induce a minus sign, or does so twice. Thus, the ${\textrm{f-}{\textsc{swap}}}$ can replace the ${\textsc{swap}}$ provided it exchanges a complete logical qubit. Note that, by linearity, this holds even if the logical state of qubits $1$ and $2$ is entangled with other logical qubits, as long as it is a physical state of even parity.
The second trick is the identity $$\label{eq:0swap}
{\textrm{f}\textsc{s}}{\left| 0 \right>} {\left| \psi \right>} = {\left| \psi \right>} {\left| 0 \right>}$$ where ${\left| \psi \right>}$ is the state of any physical qubit. This follows simply because when either of the qubits is in the ${\left| 0 \right>}$ state, the ${\textrm{f-}{\textsc{swap}}}$ does not induce a minus sign, behaving exactly as the ${\textsc{swap}}$. We will use this fact to initialize some ancilla qubits in the ${\left| 0 \right>}$ state and move them around as necessary.
We now prove universality for . First note that the graph of is guaranteed to appear as a subgraph of the one in if the number of vertices is greater than $6$. We refer to the degree-$3$ vertex in that graph—and more generally, to any vertex of degree greater than $2$ in a tree—as a branching point. We initialize two ancilla qubits near the branching point (specifically, at vertices $\alpha$ and $\beta$ in ) as ${\left| 0 \right>}$ and encode the logical qubits using pairs of adjacent qubits as in . Depending on the number of vertices and the location of the branching point, some physical qubits might be unpaired, in which case one or two qubits at the endpoints may not be used.
![Close-up view of the degree-$3$ vertex of the graph in . Vertices labeled $\alpha$ and $\beta$ correspond to ancillas initialized in the ${\left| 0 \right>}$ state. Vertex pairs $\{1,2\}$ and $\{3,4\}$ correspond to the two logical qubits on which we want to implement a logical ${\textsc{cz}}$ gate. The $\alpha$ and $\beta$ ancillas are used to change the order of the state of the other qubits, as per .[]{data-label="fig:branching"}](branching){width="30.00000%"}
As discussed above, a logical single-qubit gate $A$ can be implemented simply by a $G(A,A)$ matchgate between adjacent qubits. Now say we want to implement a logical ${\textsc{cz}}$ gate between two (not necessarily adjacent) logical qubits. We first use the identity of to place the two desired pairs near the branching point, as in . In the previous section, we mentioned that the logical ${\textsc{cz}}$ can be implemented by a physical ${\textsc{cz}}$ between two of the four qubits (e.g., $1$ and $3$, as labeled in ), which in turn is equal to ${\textsc{swap}}$ followed by ${\textrm{f-}{\textsc{swap}}}$. We can implement this sequence by swapping qubit $2$ with both qubits of the pair ($3$,$4$), which is possible by , and then using the fact that $\alpha$ and $\beta$ are ancillas in the ${\left| 0 \right>}$ state to switch the order of the qubits placed in vertices $1$ and $2$. This effectively implements the ${\textsc{swap}}$ of . If we follow this with an ${\textrm{f-}{\textsc{swap}}}$ again between qubits $1$ and $2$, the final result is the desired ${\textsc{cz}}$ gate. We can then use to return all qubits to their original places. The explicit sequence is $$\label{eq:switch}
{\textrm{f}\textsc{s}}_{23} \, {\textrm{f}\textsc{s}}_{34} \, {\textrm{f}\textsc{s}}_{12} \, {\textrm{f}\textsc{s}}_{\beta 1} \, {\textrm{f}\textsc{s}}_{12} \, {\textrm{f}\textsc{s}}_{\alpha 1} \, {\textrm{f}\textsc{s}}_{\beta 1} \, {\textrm{f}\textsc{s}}_{12} \, {\textrm{f}\textsc{s}}_{\alpha 1} \, {\textrm{f}\textsc{s}}_{34} \, {\textrm{f}\textsc{s}}_{23}.$$
This sequence uses only matchgates to implement a ${\textsc{cz}}$ between the logical qubits which, together with the single-qubit gates mentioned previously, gives a universal set. Since any logical qubit can be moved to any desired location using $O(n)$ ${\textrm{f-}{\textsc{swap}}}$ gates, the overhead in the number of such gates grows polynomially with the number of 2-qubit gates in the original circuit.
\[ex:leaves\] Now suppose the qubits are arranged on a complete binary tree of $m$ levels, as in . This graph has $n = 2^{m+1}-1$ vertices. Since the longest path contains only $2m-1=O(\log n)$ vertices, the strategy of cannot be trivially adapted to this case: the number of available logical qubits would not be sufficient.
![An $n$-vertex complete binary tree. White vertices represent ${\left| 0 \right>}$ ancillas and black vertices are used in pairs to store computational qubits. This arrangement enables universal computing with matchgates.[]{data-label="fig:binarytree"}](binarytree.pdf){width="35.00000%"}
Instead, we store logical qubits using the $2^m=(n+1)/2$ leaves as shown in . By using the leaves as the computational qubits and filling the paths that connect them with ${\left| 0 \right>}$ ancillas, we can use the identity of to move the state of any qubit to a vertex adjacent to any other desired qubit in less than $2 \log(n/2)$ steps, apply the desired matchgate between them, and return them to their initial positions. This means we can use the ${\textrm{f-}{\textsc{swap}}}$ to implement an effective interaction between any pair among the $(n+1)/2$ computational qubits, which clearly is sufficient for universal computation, as per the construction of . The overhead of this approach is modest: it requires twice the number of qubits and uses $2 \log(n)$ ${\textrm{f-}{\textsc{swap}}}$ operations per 2-qubit gate in the original circuit. Note that this approach works for any pairing of physical into logical qubits.
Main result {#sec:univ_arbit_graphs}
-----------
The two examples of the previous section provide the main ideas for a complete characterization of the power of matchgates on arbitrary graphs. To obtain this result, we first prove the following lemma:
\[lem:graph\] Let $T$ be an $n$-vertex tree with $l$ leaves and a longest path of length $p$. Then either (i) $l > \sqrt{n}$ or (ii) $p > \sqrt{n}$.
Choose any leaf $v$ of $T$. Delete every vertex on the path from $v$ to the nearest branching point, not including the branching point (see ). Since, by hypothesis, this path has length smaller than $p$, the result is a subtree of $T$ where one leaf and at most $p-1$ vertices are removed. Repeat this procedure until only a path remains (i.e., $l-2$ times). Finally, delete the remaining path, removing the last two leaves and at most $p$ vertices. This process deletes every vertex in $T$. Therefore $n \leq (l-2)(p-1)+p < lp$, so $\max\{l, p\} > \sqrt{n}$ as claimed.
![A tree. The dashed rectangle indicates the vertices in the path from $v$ to the nearest branching point, which are deleted in the proof of . Upon deletion of these vertices, the remaining tree has one fewer leaf, and at most $p-1$ vertices have been removed.[]{data-label="fig:arbitree"}](arbitree.pdf){width="30.00000%"}
The main result follows straightforwardly from and the examples of the previous section:
\[thm:maintheo\] Let $G$ be any $n$-vertex connected graph, other than a path or a cycle, where every vertex represents a qubit and we can implement arbitrary matchgates between neighbors in $G$. Then it is possible to efficiently simulate (i.e., with polynomial overhead in the number of operations) any quantum circuit on $\Omega(\sqrt{n})$ qubits.
Since $G$ is not a path or a cycle, it has some spanning tree $T$ that is not a path. This holds because $G$ necessarily contains a vertex of degree more than 2 and one can construct a spanning tree that includes all edges adjacent to this vertex. It suffices to show that universal computation can be implemented in $T$, since all edges of $T$ are edges of $G$. By , either (i) the longest path of $T$ or (ii) the set of all its leaves must have more than $\sqrt{n}$ vertices.
First, suppose (i) holds. Assign each qubit of a longest path of $T$ as a computational qubit, with the exception of one qubit at a branching point. We also use one qubit adjacent to the branching point and not in the path as an ancilla. All other qubits are ignored. We implement the circuit as shown in of the previous section. Since the longest path has more than $\sqrt{n}$ vertices by hypothesis, this allows the simulation of an arbitrary quantum circuit on $\lfloor (\sqrt{n}-1)/2 \rfloor$ qubits. This simulation uses $O(n)$ ${\textrm{f-}{\textsc{swap}}}$ operations for each two-qubit gate.
Otherwise (ii) holds, so $T$ has more than $\sqrt{n}$ leaves. Proceed by assigning every qubit at a leaf as a computational qubit and initializing every other qubit as a ${\left| 0 \right>}$ ancilla. The intermediate vertices on the (unique) path between any two leaves represent qubits in the ${\left| 0 \right>}$ state. As in , we can use the identity of to move the state of any qubit to a vertex adjacent to any other, implement a matchgate, and move it back. Thus we can effectively implement any matchgate between any pair of logical qubits. Since the longest path has length less than $\sqrt{n}$, this simulation uses $O(\sqrt{n})$ ${\textrm{f-}{\textsc{swap}}}$ operations for each gate in the original circuit.
Classical simulation of matchgates on the path and cycle {#sec:simul}
========================================================
In the previous section, we proved the universality of matchgates on any connected graph that is not a path or a cycle. We now show that this is also a necessary condition (assuming that quantum computers cannot be efficiently simulated classically).
As mentioned in , it is well-known that matchgates on a path can be simulated classically for any product state input and computational basis measurement [@Valiant02; @Terhal02; @Jozsa08b]. We briefly review the proof of this fact as shown in [@Jozsa08b] and then generalize the proof to the case of a cycle.
The Jordan-Wigner transformation and classical simulation of nearest-neighbor matchgates {#sec:simul_line}
----------------------------------------------------------------------------------------
We begin by defining the Jordan-Wigner operators [@Jordan28] acting on $n$ qubits: $$\begin{aligned}
\label{eq:JW}
c_{2j-1} &:= \left( \prod_{i=1}^{j-1} Z_i \right) X_j \notag \\
c_{2j} &:= \left( \prod_{i=1}^{j-1} Z_i \right) Y_j\end{aligned}$$ for $j \in \{1,\ldots,n\}$, where $X_i,Y_i,Z_i$ denote the Pauli $X$, $Y$, and $Z$ operators, respectively, acting on qubit $i$. Using this transformation, we can write $$\begin{aligned}
c_{2k-1} c_{2k} & = i Z_k \label{eq:JWtransf1}\end{aligned}$$ for $k \in \{1,\ldots,n\}$ and $$\begin{aligned}
c_{2k} c_{2k+1} & = i X_k X_{k+1} \notag \\
c_{2k-1} c_{2k+2} & = -i Y_k Y_{k+1} \notag \\
c_{2k-1} c_{2k+1} & = -i Y_k X_{k+1} \notag \\
c_{2k} c_{2k+2} & = i X_k Y_{k+1} \label{eq:JWtransf2}\end{aligned}$$ for $k \in \{1,\ldots,n-1\}$. These two-qubit Hamiltonians are precisely the generators of the group of nearest-neighbor matchgates [@Terhal02].
Suppose that the circuit being simulated has an initial product state input ${\left| \psi \right>}={\left| \psi_1 \right>} {\left| \psi_2 \right>}\ldots{\left| \psi_n \right>}$, a sequence of nearest-neighbor matchgates, and a final measurement in the computational basis. To simulate the final measurement of qubit $k$, it suffices to calculate of the expectation value $\langle Z_k \rangle$ = $-i \langle c_{2k-1} c_{2k} \rangle = -i {\left< \psi \right|} U^{\dagger} c_{2k-1} c_{2k} U {\left| \psi \right>}$, where $U$ is the unitary corresponding to the action of the matchgate circuit. To show that this can be calculated efficiently, we invoke the following (cf. [@Knill01; @Terhal02; @Jozsa08b], as stated in [@Jozsa08b]):
\[thm:quadratic\] Let H be any Hamiltonian quadratic in the operators $c_i$ and let $U = e^{iH}$ be the corresponding unitary. Then, for all $\mu \in \{1,\ldots,2n\}$, $$U^{\dagger} c_\mu U = \sum_{\nu=1}^{2n} R_{\mu,\nu} c_\nu,$$ where $R\in\operatorname{SO}(2n)$, and we obtain all of $\operatorname{SO}(2n)$ this way.
The straightforward proof of this theorem appears in [@Jozsa08b]. Observe that, according to [Eqs. and ]{}, the Hamiltonians that generate nearest-neighbor matchgates are quadratic in the operators $c_i$, so $$\begin{aligned}
\label{eq:expectedZ}
\langle Z_k \rangle & = -i {\left< \psi \right|} U^{\dagger} c_{2k-1} c_{2k} U {\left| \psi \right>} \\ \notag
& = -i \sum_{a,b=1}^{n} R_{2k-1, a} R_{2k, b} {\left< \psi \right|} c_a c_b {\left| \psi \right>}.\end{aligned}$$
If $t$ is the number of matchgates in the circuit, $R \in \operatorname{SO}(2n)$ can be calculated in $\operatorname{poly}(n,t)$ time as the product of the rotations corresponding to each matchgate. Also notice that the sum in has only $O(n^2)$ terms. Finally, note that ${\left| \psi \right>}$ is a product state, and any monomial $c_a c_b$ is a tensor product of Pauli matrices, as is clear from . Thus, each term in the sum factors as a product of the form $\prod_{i=1}^{n} {\left< \psi_i \right|} \sigma_i {\left| \psi_i \right>}$, which involves $n$ efficiently computable terms. Since $\langle Z_k \rangle$ is a sum of a polynomial number of efficiently computable terms, it can be computed efficiently, which completes the proof of classical simulability of matchgates on a path.
Classical simulation of matchgates on a cycle {#sec:simul_cyc}
---------------------------------------------
The result of the previous section does not immediately apply to the case of a cycle, which corresponds to applying periodic boundary conditions to a path, because a matchgate between the first and last qubits does not translate into a Hamiltonian that is quadratic in the $c_i$s, and vice versa. For example, $$\label{eq:notnnmatch}
c_1 c_{2n} = i X_1 X_n \prod_{i=1}^{n} Z_i,$$ which is clearly not a matchgate, as it is a unitary operation acting on every qubit in the circuit.
Note that still applies to the Hamiltonian in even though it does not correspond to a matchgate. However, we do not have a straightforward way of writing the operators we need, such as $X_1 X_n$, in terms of these quadratic operators.
To show that matchgates are simulable in this case nonetheless, first consider the case where the input state ${\left| \psi \right>}$ is a computational basis state. Suppose that ${\left| \psi \right>}$ has even parity (e.g., ${\left| 000\ldots0 \right>}$). Matchgates preserve parity, so the state at any point in the computation has a well-defined (even) parity. Now notice that $\prod_{i=1}^{n} Z_i$ is the operator that measures overall parity, so it acts as the identity on the even-parity subspace. This means that for any even-parity input we have the correspondence $$X_1 X_n = X_1 X_n \prod_{i=1}^{n} Z_i = - i c_1 c_{2n} \quad \text{(even parity)},$$ where the second equality is just . The equivalent equations for $Y_1 Y_n$, $X_1 Y_n$, and $Y_1 X_n$ are straightforward. Since we have recovered a correspondence between matchgates on qubits $1$ and $n$ and quadratic Hamiltonians, the simulation can be carried out exactly as in . The case of an odd-parity input state (e.g., ${\left| 100\ldots0 \right>}$) is analogous, except that the operator $\prod_{i=1}^{n} Z_i$ now acts as minus the identity, and we write $$X_1 X_n = - X_1 X_n \prod_{i=1}^{n} Z_i = i c_1 c_{2n} \quad \text{(odd parity)}$$ and its equivalents for $Y_1 Y_n$, $X_1 Y_n$, and $Y_1 X_n$.
Now consider a general product input state ${\left| \psi \right>}$. Let ${\left| \psi_{\pm} \right>}$ denote the projections of ${\left| \psi \right>}$ onto the even- and odd-parity subspaces, respectively. The expectation value $\langle Z_K \rangle$, analogous to , is $$\begin{aligned}
\label{eq:simulationcycle}
\langle Z_k \rangle = & -i {\left< \psi \right|} U^{\dagger} c_{2k-1} c_{2k} U {\left| \psi \right>} \notag \\
= & -i \sum_{a,b=1}^{n} ( R_{2k-1, a} R_{2k, b} {\left< \psi_{+} \right|} c_a c_b {\left| \psi_{+} \right>} \notag \\
& \qquad\quad + R'_{2k-1, a} R'_{2k, b} {\left< \psi_{-} \right|} c_a c_b {\left| \psi_{-} \right>} ).\end{aligned}$$ Here $R$ and $R'$ correspond to two sets of rotations, where $R'$ includes an extra minus sign for every matchgate applied between qubits $1$ and $n$. The expression above does not contain cross terms such as ${\left< \psi_{-} \right|} c_{a} c_{b} {\left| \psi_{+} \right>}$ because $c_a c_b$ preserves parity.
The sum in contains a polynomial number of terms, just as in , but now each term may not be easy to compute, since ${\left| \psi_{\pm} \right>}$ are not product states in general. However, we have $$\begin{aligned}
{\left< \psi \right|} c_{a} c_{b} {\left| \psi \right>} & = {\left< \psi_{+} \right|} c_{a} c_{b} {\left| \psi_{+} \right>} + {\left< \psi_{-} \right|} c_{a} c_{b} {\left| \psi_{-} \right>}, \\
{\left< \psi \right|} c_{a} c_{b} \prod_{i=1}^{n} Z_i {\left| \psi \right>} & = {\left< \psi_{+} \right|} c_{a} c_{b} {\left| \psi_{+} \right>} - {\left< \psi_{-} \right|} c_{a} c_{b} {\left| \psi_{-} \right>}.\end{aligned}$$ We can invert these equations to obtain $$\begin{aligned}
{\left< \psi_{+} \right|} c_{a} c_{b} {\left| \psi_{+} \right>} & = \frac{1}{2} \left[ {\left< \psi \right|} c_{a} c_{b} {\left| \psi \right>}+{\left< \psi \right|} c_{a} c_{b} \prod_{i=1}^{n} Z_i {\left| \psi \right>} \right ], \notag \\
{\left< \psi_{-} \right|} c_{a} c_{b} {\left| \psi_{-} \right>} & = \frac{1}{2} \left[ {\left< \psi \right|} c_{a} c_{b} {\left| \psi \right>}-{\left< \psi \right|} c_{a} c_{b} \prod_{i=1}^{n} Z_i {\left| \psi \right>} \right ]. \label{eq:expectedparity}\end{aligned}$$
The left-hand sides are precisely the two terms of $\langle Z_k \rangle$ that we need, while the right-hand sides are combinations of terms that can be efficiently computed, as both are expected values of products of Pauli operators on product states. Explicitly, if ${\left| \psi \right>}={\left| \psi_1 \right>}{\left| \psi_2 \right>}\ldots{\left| \psi_n \right>}$ and $c_a c_b = \sigma_1 \sigma_2 \ldots \sigma_n$, we have $$\begin{aligned}
{\left< \psi \right|} c_{a} c_{b} {\left| \psi \right>} &= \prod_{i=1}^{n} {\left< \psi_i \right|} \sigma_i {\left| \psi_i \right>}, \\
{\left< \psi \right|} c_{a} c_{b} \prod_{i=1}^{n} Z_i {\left| \psi \right>} &= \prod_{i=1}^{n} {\left< \psi_i \right|} \sigma_i Z_i {\left| \psi_i \right>}.\end{aligned}$$
Plugging into , we recover an expression that can be efficiently computed in the same manner as , with only four times as many terms. This gives an efficient classical simulation for matchgates acting on a cycle, as claimed.
Note that the simulation scheme of was recently exploited [@Jozsa10] to show that circuits of nearest-neighbor matchgates on $n$ qubits are equivalent to general quantum circuits on $O(\log n)$ qubits, and subsequently [@Kraus11; @Boyajian13] to show a protocol for “compressed” simulations (i.e., with quantum circuits on $O(\log n)$ qubits) of the Ising and XY models of spin systems with open boundary conditions. We leave it as an open question whether the observations made in this section lead to analogous results for systems with periodic boundary conditions.
Universality of the XY interaction on arbitrary graphs {#sec:XY}
======================================================
In and , we investigated the computational power of the set of all matchgates on arbitrary graphs. We now consider the computational power of a restricted set of matchgates corresponding to the XY (or anisotropic Heisenberg) interaction on arbitrary graphs. This interaction corresponds to a subset of matchgates generated by the Hamiltonian $H := X \otimes X + Y \otimes Y$ (recall from that matchgates are generated by the two-qubit Hamiltonians $X \otimes X$, $X \otimes Y$, $Y \otimes X$, $Y \otimes Y$ together with the single-qubit Hamiltonian $Z$). It is easy to see that these interactions form a proper subset of matchgates as, e.g., they act non-trivially only on the odd-parity subspace of the $2$-qubit Hilbert space.
The Hamiltonian $H$ is an idealized model of the interactions present in several proposed physical implementations of quantum computing, such as quantum dots [@Imamoglu99; @Quiroga99], atoms in cavities [@Zheng00], and quantum Hall systems [@Mozyrsky01]. A comparison of these proposals can be found in [@Lidar01].
Despite being a proper subset of matchgates, the XY interaction is also known [@Kempe02] to be universal for quantum computation when acting on the graph of (i.e., nearest and next-nearest neighbor interactions on a path). It also follows trivially from that the XY interaction is classically simulable on paths and cycles. This prompts the question of whether our results from can be adapted for the XY interaction on arbitrary graphs.
In fact, we now show that the XY interaction alone is universal for quantum computation on any connected graph that is not a path or a cycle. Since these operations are a subset of matchgates, this result subsumes the one of . However, the argument we give for the XY interaction is less explicit, and the simulation is less efficient in general.
First observe that the XY interaction acts trivially on the even-parity subspace, so the encoding of cannot be used. A suitable alternative (as used in [@Kempe02]) is $$\begin{aligned}
{\left| 0 \right>}_L & = {\left| 01 \right>}, \notag \\
{\left| 1 \right>}_L & = {\left| 10 \right>}, \label{eq:oddencoding}\end{aligned}$$ which is simply the corresponding encoding on the odd-parity subspace.
We also need to adapt some of the identities used in . The fermionic ${\textsc{swap}}$ gate is not available, so instead we use the following similar gate, which we call the ${\textrm{i-}{\textsc{swap}}}$ (and denote by the shorthand ${\textrm{i}\textsc{s}}$): $$\label{eq:i-SWAP}
{\textrm{i}\textsc{s}}:= \exp( i \tfrac{\pi}{4} H ) = G(I, iX) = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & i & 0 \\
0 & i & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}.$$
For an arbitrary logical state ${\left| \Psi \right>}_L = \alpha {\left| 10 \right>} + \beta {\left| 01 \right>}$ in the encoding of , and for any physical qubit in an arbitrary state ${\left| \phi \right>}$, we have the following identity (already used implicitly in [@Kempe02]): $$\label{eq:logicswap2}
{\textrm{i}\textsc{s}}_{12} \, {\textrm{i}\textsc{s}}_{23} {\left| \Psi \right>}_L {\left| \phi \right>} = i {\left| \phi \right>} {\left| \Psi \right>}_L.$$ Thus these states can be swapped up to an irrelevant global phase.
Another useful identity, akin to , is given by $$\label{eq:0iswap}
{\textrm{i}\textsc{s}}_{12}{\left| 0 \right>} {\left| \psi \right>} = \left ( P {\left| \psi \right>} \right ) {\left| 0 \right>},$$ where ${\left| \psi \right>}$ is any state and $P := \operatorname{diag}(1,i)$. This identity has a familiar operational interpretation: once more any state can be “swapped through” a ${\left| 0 \right>}$ ancilla, but now with the caveat that the state suffers an unwanted $P$ gate. We must take this into account when using in a simulation, but one can already see that if we only need to swap states through an even number of ancillas at a time, we can cancel out the $P$ gates by alternating ${\textrm{i-}{\textsc{swap}}}$ and ${\textrm{i-}{\textsc{swap}}}$$^\dagger$ swapping operations. In fact, a trivial adaptation of gives a proof of universality for those graphs that have an odd cycle (i.e., non-bipartite graphs), since then there is always an even-length path between any two vertices. We state this without proof, as the details are not instructive and the result is implied by the general case. Note however that for non-bipartite graphs, one can obtain a universal set of unitary matrices, whereas for general graphs we will only obtain a universal set of orthogonal matrices.
\
We first show how to implement a particular set of one- and two-qubit gates on the two 5-vertex graphs of , similar to the simulation in (cf. ). Suppose the two logical qubits can be initialized as in a, according to the encoding of , together with one ${\left| 0 \right>}$ ancilla.
Since $$H = \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 2 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix},$$ a logical $X$ rotation on the logical qubit stored in physical qubits $\{1,2\}$ can be implemented by a simple XY interaction: $$\exp( i a X_L ) = \exp( i \tfrac{a}{2} H_{12} ) = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & \cos{a} & i \sin{a} & 0 \\
0 & i \sin{a} & \cos{a} & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}.$$
We can also implement the two-qubit gate $R_{XZ}(a) := \exp(i a \, X \otimes Z)$ on the logical qubits $\{1,2\}$ and $\{3,4\}$ by the following sequence: $$\label{eq:2qubitg}
{\textrm{i}\textsc{s}}_{25} \, {\textrm{i}\textsc{s}}_{23} \, {\textrm{i}\textsc{s}}_{34} \, \left[ {\textrm{i}\textsc{s}}_{25}^\dagger \, \exp( i \tfrac{a}{2} H_{12} ) \, {\textrm{i}\textsc{s}}_{25} \right ] \, {\textrm{i}\textsc{s}}_{34}^\dagger \, {\textrm{i}\textsc{s}}_{23}^\dagger \, {\textrm{i}\textsc{s}}_{25}^\dagger.$$ This sequence works as follows. The first three ${\textrm{i-}{\textsc{swap}}}$ gates use to swap the qubits and place them as in b. Notice that the first logical qubit suffers a $P$ gate during this operation. The sequence inside the square brackets implements an effective unitary with Hamiltonian $Y \otimes Z$. This can be verified by explicit multiplication, but can also be understood as follows: the ${\textrm{i}\textsc{s}}_{25}$ and ${\textrm{i}\textsc{s}}_{25}^{\dagger}$ swap qubits $2$ and $5$, leaving the first logical qubit encoded in pair $\{1,2\}$, up to some phases that depend upon the states of both qubits. The $H_{12}$ Hamiltonian then acts as a logical $X$ rotation on the first qubit. Keeping track of the dependence of the relative phases on the states of both qubits, we see that the overall operation is $Y \otimes Z$. Finally, the last three ${\textrm{i-}{\textsc{swap}}}$ gates return the states of all qubits to their original positions, while inducing a $P^{\dagger}$ gate on the first logical qubit. Since $P^{\dagger} Y P=X$, the overall operation on the encoded states is $X \otimes Z$, as claimed.
We now make a brief digression to explain why the set of Hamiltonians $${\mathcal{A}}:= \{X \otimes I, I \otimes X, X \otimes Z, Z \otimes X, X \otimes Y, Y \otimes X \}$$ is universal for quantum computation in the usual circuit model. First notice that the Hamiltonians $X \otimes Y$ and $Y \otimes X$ are included; this is without loss of generality, as they can be obtained as simple sequences of the remaining interactions, e.g., $X \otimes Y = U(X \otimes Z)U^{\dagger}$ where $U=\exp[ i \tfrac{\pi}{4} (I \otimes X)]$. By conjugating every element in ${\mathcal{A}}$ by $P$, we obtain the set $${\mathcal{B}}:= \{Y \otimes I, I \otimes Y, Y \otimes Z, Z \otimes Y, X \otimes Y, Y \otimes X \}.$$ These are exactly the generators of the special orthogonal group $\operatorname{SO}(4)$. This can be seen by writing them down explicitly, but also understood by a counting argument, as ${\mathcal{B}}$ contains six linearly independent, purely imaginary $4 \times 4$ matrices.
Now we recall the well-known fact (see, e.g., [@Bernstein93] and [@Rudolph02]) that universal quantum computation is possible using only orthogonal, rather than general unitary, matrices, with the overhead of one extra ancilla qubit and a polynomial number of operations. Furthermore, any special orthogonal matrix on $n$ qubits \[i.e., in $\operatorname{SO}(2^n)$\] can be decomposed in terms of $\operatorname{SO}(4)$ gates acting non-trivially only on pairs of qubits, so the set ${\mathcal{B}}$ is universal for quantum computation. But this means that the set ${\mathcal{A}}$ is also universal, since we can assume that initialization and measurements are done in the computational basis, so the initial and final single-qubit $\{P,P^{\dagger}\}$ gates do not affect the outcomes.
While the graph in a may not appear as a subgraph of the given graph, the sequence can be easily adapted to the graph of c. In that case, we can just use to swap the ancilla with any of the other qubits and obtain a similar arrangement to that of b. The corresponding sequence is $$\label{eq:2qubitgb}
{\textrm{i}\textsc{s}}_{24} \, \left[ {\textrm{i}\textsc{s}}_{25}^\dagger \, \exp( i \tfrac{a}{2} H_{12} ) \, {\textrm{i}\textsc{s}}_{25} \right ] \, {\textrm{i}\textsc{s}}_{24}^\dagger.$$ In this case, every operation described before is obtained up to conjugation by $P$, and the set of available operations is ${\mathcal{B}}$, rather than ${\mathcal{A}}$. However, as described above, this still suffices for universal computation.
It remains to show that, for any graph other than a path or cycle, we can assign sufficiently many vertices as computational qubits and swap them around to one of the arrangements of with a polynomial number of operations.
\[thm:maintheo2\] Let $G$ be any $n$-vertex connected graph, other than a path or a cycle, where every vertex represents a qubit and we can implement the interaction $H=X \otimes X + Y \otimes Y$ between any nearest neighbors in $G$. Then it is possible to efficiently simulate any quantum circuit on $\Omega(\sqrt{n})$ qubits.
As in , it suffices to prove the universality of $H$ on any $n$-vertex tree $T$ that is not a path.
By , either (i) the longest path of $T$ or (ii) the set of all its leaves must have more than $\sqrt{n}$ vertices. Suppose first that (i) holds. Then the universal construction is directly analogous to case (i) of . Simply assign pairs of adjacent vertices on the longest path as logical qubits, and every other as a ${\left| 0 \right>}$ ancilla. Then, by using , we can swap any two logical qubits to the closest degree-3 vertex, where we use sequence to implement the $X \otimes Z$ Hamiltonian as per a. As explained previously, this together with the logical $X$ Hamiltonian on any qubit (given by $H$ on adjacent qubits) enables universal computation with overhead of at most $O(n)$ ${\textrm{i-}{\textsc{swap}}}$ operations per orthogonal matrix in the original circuit of [@Rudolph02].
Otherwise, (ii) holds. Then, first suppose that $T$ is not a star. Any such $T$ contains the graph of a as a subgraph, so we assign those $5$ vertices as ${\left| 0 \right>}$ ancillas, together with all non-leaves, and pair the remaining leaves arbitrarily into computational qubits. We can now use to bring the states of any two logical qubits to the structure of a, but with one caveat: this process may induce an overall $P$ gate on some logical qubits, depending on whether an odd or even number of ${\left| 0 \right>}$ ancillas is traversed. This separates the logical qubits into two disjoint sets, namely those that suffer an overall $P$ gate and those that do not (there is no need to single out the case where the qubits suffer an overall $P^{\dagger}$, as this can be prevented by using ${\textrm{i-}{\textsc{swap}}}$$^{\dagger}$, rather than ${\textrm{i-}{\textsc{swap}}}$, as the swapping operation). We then take the larger of these two sets, which has at least $\sqrt{n}/4$ logical qubits, and for simplicity we disregard the rest. On the remaining qubits, as argued previously, we can either implement the set of operations ${\mathcal{A}}$ or its conjugated-by-$P$ version ${\mathcal{B}}$. Since either set is universal, this gives an universal construction with an overhead of $O(\sqrt{n})$ operations for each gate in the original circuit.
Finally, for the star graph, we replace sequence , corresponding to a, by the equivalent sequence corresponding to c. This enables us to implement the set of Hamiltonians mentioned in the previous paragraph, and concludes the proof.
Final remarks
=============
We have completely characterized the computational power of nearest-neighbor matchgates when the qubits are arranged on an arbitrary graph. This continues a line of research started in [@Brod12], where the authors showed that matchgates are universal on many different graphs. Here we proved that the only connected graphs for which matchgates are classically simulable are paths and cycles, whereas on any other connected graph they are universal for quantum computation. Furthermore, we have shown that the same dichotomy holds when we restrict matchgates to the proper subset described by the XY interaction. This further expands the exploration of quantum computation with a single physical interaction [@DiVincenzo00; @Kempe01b; @Kempe02], and could have applications for a variety of physical systems where the XY interaction arises naturally, if the placement of the qubits is subject to geometrical constraints.
This dichotomy excludes the possibility that these two sets of interactions (general matchgates and the XY interaction), acting on graphs, could exhibit intermediate computational power such as that displayed by circuits of commuting observables (IQP) [@Bremner10] or noninteracting bosons [@Aaronson11]. However, this does not rule out such a result for other subsets of matchgates. As one example, consider the set generated by the $X \otimes X$ Hamiltonian acting on some graph. All such operations commute, and this set corresponds to a proper subclass of IQP. Furthermore, it was recently shown [@Hoban13] that the set of two-qubit $X \otimes X$ and single-qubit $X$ Hamiltonians are hard to simulate classically, modulo plausible complexity-theoretic assumptions, in the same way as IQP. We leave as open questions whether an analogous result can be obtained by further restricting the operation to only the $X \otimes X$ Hamiltonian, or possibly some other proper subset of matchgates, and how the power of such a model depends on the underlying interaction graph.
In our investigation we have not considered the use of non-trivial measurements to implement other unitary operations—it has been shown, for example, that noninteracting fermions (i.e., matchgates on a path) become universal if nondestructive charge measurements are allowed [@Beenakker04]. These charge measurements clearly cannot be implemented by combining matchgates and computational basis measurements. It might be interesting to consider other measurements and/or input states, beyond those obtainable by matchgates, and understand whether they change the computational capabilities of restricted subsets of matchgates on graphs.
While we have established universality of matchgates on any connected graph that is not a path or a cycle, it should be possible to improve the efficiency of our constructions. We have taken an operational approach, where each ${\left| 0 \right>}$ is seen as an “empty space” through which we can move logical qubits, allowing for a simple and unified proof of universality for all graphs. In some cases, such as for the star graph, where all vertices but one are leaves, this construction is optimal. But in many others, our construction could ignore many vertices and/or edges, making it far from optimal. One such case is the binary tree of , where we could have filled most of the non-leaves with logical qubits and used rather than whenever it was necessary to “move” two logical qubits through each other. Since the bounds of are tight (e.g., consider the graph obtained from a $\sqrt{n}$-leaf star by subdividing each edge $\sqrt{n}$ times), an optimal simulation presumably requires a more efficient assignment of logical qubits than in . We believe that, while being markedly non-optimal in some cases, our construction nevertheless provides powerful tools for case-by-case optimization. We leave it as an open question whether there is a way to systematically obtain a more efficient construction, and in particular, whether in every case only a constant fraction of the qubits must be discarded as non-computational.
We thank Robin Kothari and Laura Mančinska for a helpful discussion of the proof of , and Ernesto Galvão for helpful discussions. D.B. would like to acknowledge financial support by Brazilian funding agency CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico). This work was also supported in part by NSERC, the Ontario Ministry of Research and Innovation, and the US ARO/DTO.
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|
---
author:
- Alejandro Lara
- Nat Gopalswamy
- Tatiana Niembro
- 'Román Pérez-Enríquez'
- Seiji Yashiro
bibliography:
- 'draft.bib'
title: 'Space, time and velocity association of successive coronal mass ejections'
---
[Through a statistical study of the main characteristics of 27761 CMEs observed by SOHO/LASCO during the past 20 years.]{} [We found the waiting time (WT) or time elapsed between two consecutive CMEs is $< 5$ hrs for 59% and $< 25$ hrs for 97% of the events, and the CME WTs follow a Pareto Type IV statistical distribution. The difference of the position-angle of a considerable population of consecutive CME pairs is less than $30^\circ$, indicating the possibility that their source locations are in the same region. The difference between the speed of trailing and leading consecutive CMEs follows a generalized Student t-distribution. The fact that the WT and the speed difference have heavy-tailed distributions along with a detrended fluctuation analysis shows that the CME process has a long-range dependence. As a consequence of the long-range dependence, we found a small but significative difference between the speed of consecutive CMEs, with the speed of the trailing CME being higher than the speed of the leading CME. The difference is largest for WTs < 2 hrs and tends to be zero for WTs > 10 hrs, and it is more evident during the ascending and descending phases of the solar cycle. We suggest that this difference may be caused by a drag force acting over CMEs closely related in space and time.]{} [Our results show that the initiation and early propagation of a significant population of CMEs cannot be considered as a “pure” stochastic process; instead they have temporal, spatial, and velocity relationship.]{}
Introduction {#sec:intro}
============
Coronal mass ejections (CMEs) were discovered in the 1970s by the observations of the white-light coronograph experiment on board the OSO-7 mission [@1973SoPh...33..265T]. Since then, a large number of data have been accumulated that help us understand the basic characteristics of CMEs, but many questions still remain unanswered [see e. g., @2004ASSL..317..201G; @2012LRSP....9....3W; @2016GSL.....3....8G and references therein]. In particular, the Large Angle and Spectrometric Coronograph [LASCO, @1995SoPh..162..357B] on board the Solar and Heliospheric Observatory (SOHO) spacecraft has been very successful in contributing considerably to this understanding of CME characteristics. In this work, we use the high number of LASCO observations and the CME characteristics, measured and made available online in the CDAW database , to explore the statistical properties of CMEs. In particular, we are interested in consecutive CMEs, which may help in the understanding of the energy storage and its release via a triggering mechanism, and the early stages of the CME dynamics. Coronal mass ejections and flares consume only part of the free magnetic energy available in active regions [@2013ApJ...765...37F]. The released energy (considering the flare and CME energy) may be, on average, a third of the free magnetic energy [@2012ApJ...759...71E]. Furthermore, CMEs carry on average 7% of the total dissipated energy during the eruptive event [@2017ApJ...836...17A]. Therefore, in terms of the stored energy, more than one CME can be ejected from the same active region [@2005JGRA..110.9S15G; @2013AdSpR..52..521M; @2017arXiv170903165G], and more importantly, within short waiting times (WT), which is the time elapsed between the launch of two successive CMEs (see Section \[sec:wt\]). The triggering mechanisms remain unsolved [@2000JGR...10523153F]. Recently, it has been proposed that CMEs may destabilize nearby active regions and trigger “sympathetic” CMEs [see @2017SoPh..292...64L and references therein], in such cases short WT are expected, although, to support this causality, the WT distribution should be different from one of a Poisson random process. Therefore, the WT distribution gives valuable information about the so-called “memory” of the system. For memory-less systems (stochastic process) the distribution is exponential or gamma. On the other hand, the so called “fat-tailed” distributions (e.g., Power Law, Pareto, Weibull, etc) are often associated with memory systems [see @2005Natur.435..207B; @Samorodnitsky2007; @SHENG2011 for a general discussion of this subject]. As evidenced by coronal EUV dimmings, the footprints of CMEs [@2000GeoRL..27.1431T], the corona behind CMEs is “evacuated” [@2017ApJ...839...50K] during a few hours [@2017SoPh..292....6L]. Therefore, a second CME launched in the same region within a short WT encounters a different coronal medium than the first CME, and its dynamics are different according to the drag force models of CME propagation . If the second CME is faster than the first one, and the source region is similar, there is a high probability of interaction at any heliospheric distance. This interaction causes CME cannibalism [@2001ApJ...548L..91G], and the merged regions have a different dynamics than a single CME [@2015ApJ...811...69N], making the prediction of the travel time and arrival speed at 1 AU difficult [@2013SpWea..11..661G]. The paper is organized as follows: The SOHO/LASCO data used in this work is presented in Section \[sec:db\]. We determine the waiting time (WT) between consecutive CMEs, and present statistical characteristics in Section \[sec:wt\]. The WT has a strong dependence on the solar cycle phase, which we describe in Section \[sec:phases\]. We study the source region of consecutive CMEs, analyzing the statistical behavior of the difference of position angle (PA) (Section \[sec:pa\]) and speed (Section \[sec:speed\]) between consecutive CME pairs. Given that both WT and speed have heavy-tailed distributions, which are associated with long-range dependence, we perform a detrended fluctuation analysis to corroborate the long-range dependence in Section \[sec:dfa\]. This dependence is clearly seen in the difference between the speed of trailing and leading CMEs (Section \[sec:speed\_rel\]). Finally, our discussion and conclusions are presented in Sections \[sec:disc\] and \[sec:conclusions\], respectively.
CME database {#sec:db}
============
The CME database is maintained by the CDAW team at the CDAW Data Center, NASA Goddard Space Flight Center (https://cdaw.gsfc.nasa.gov). The database includes the basic sky-plane measurements of CMEs such as speed, angular width, central position angle, acceleration, mass, and kinetic energy. The first-appearance time of each CME in the LASCO field of view (FOV) is also given in the catalog, which we use for determining the WT. The observations have been continuously available since 1996, with a major three-month gap in 1998 when the SOHO spacecraft was temporarily disabled . In order to avoid possible instrumental errors during the first stage of the LASCO mission, from the 29577 CMEs recorded from January 1996 until the end of December 2018, in this work we consider only 28270 CMEs observed after the second major data gap, starting with the CME observed on February 2, 1999 at 16:35, and finishing with the CME observed on December 31, 2018 at 09:48 UT. Taking into account that our analysis is performed over pairs of consecutive CMEs, using the second CME of each pair as a reference, the number of pairs is equal to the number of CMEs minus one. Furthermore, we skip the first CME after a data gap (which corresponds to the second CME of the registered pair during the data gap), this way discarding the data gaps from the analysis (it is important to note that we do not apply any restriction to select the leading-trailing CME pairs, so that a leading one can be a trailing one of a previous event, and these are treated as an independent pair). During the entire period (1996-2018), 1003 data gaps longer than two hours were reported, from these data gaps, 673 occurred during our period of study. Although, in some cases (specifically, 164), more than one data gap took place in between two observed CMEs. Therefore, our final set contains 27761 CMEs.
Waiting time {#sec:wt}
============
We define the CME WT as the time elapsed between the first observation of two consecutive CMEs in the LASCO FOV. It is important to note that the CME initial time, as reported when using coronographic observations, is not exactly the launching time, but the time when a CME first appears in the coronograph FOV above the occulting disk. Therefore, there is a small non-uniform error in assigning the initial time. This error is smaller for limb CMEs and increases when the source region of the CME approaches the center of the disk. This effect can be seen in Figure \[fig:tdelay\], where we show the delay time of a CME observed by a coronograph with a two solar radii ($R_\odot$) occulting disk. This is equivalent to the time that it takes for a CME with given constant speed (from 200 to 2000 km s$^{-1}$, marked by colors in the figure) to travel a distance of 2 $R_\odot$ projected in the plane of the sky, as a function of the helio-longitude of the CME source region. The delay time ($t$) is given as $t= \left[ \sin(\theta) + \tan(w/2) \cos(\theta) \right]^{1/2} R_\odot/v$, where $\theta$ is the helio-longitude, and $w$ and $v$ are the CME width and speed, respectively. This analysis shows that for very slow CMEs ($V \leq 200$ km s$^{-1}$), the delay time is $< 1$ hr when the source region is located beyond $45^\circ$ of helio-longitude, whereas for very fast CMEs ($V > 1000$ km s$^{-1}$), the delay time may be considered constant for helio-longitudes higher than $30^\circ$. We note that this basic analysis assumes similar visibility for front- and back-sided CMEs. The delay time is also shorter for wider CMEs for a given speed.
As shown in Figure \[fig:tdelay\], narrow and slow CMEs launched close to the center of the disk (lon $\ge \pm30^\circ$) have a delay of the observed initial time longer than two hours. Although, CMEs may measure any width and may be launched in any longitude (the center/limb launching position is an observational effect, and thus, the probability that a CME of a given speed is launched inside this longitude range is $\sim 1/3$). Therefore, as we have a large number of events (launched with different velocities, widths and longitudes), by selecting a minimum of two hours for the WT, we are able to neglect the error introduced by the slow-narrow CMEs launched close to the center of the disk.
![Delayed time of first observation of a CME with constant speed, occulted by a 2 $R_\odot$ disk as a function of the helio-longitude of the source region. Continuous and dashed lines correspond to CME width of $30^\circ$ and $60^\circ$, respectively.[]{data-label="fig:tdelay"}](ang_vel.png){width="\columnwidth"}
Another source of systematic errors is the cadence of the instrument. In the case of LASCO, this is between 12 and 40 minutes (depending on the observation mode). Therefore, WTs $< 1$ hr are difficult to quantify properly. On the other hand, only slow ($V < 600$ km s$^{-1}$) and central source region ($|\theta| < 15^\circ$) CMEs have delays $ > 2$ hrs (Figure \[fig:tdelay\]). Therefore, for the majority of CMEs the delay is under two hours, and this can be taken as the WT resolution. As a consequence, the bin size of the distributions in the next section has been taken as two hours. We note that the cadence of LASCO was doubled after August 2010, this caused a steep elevation of the CME rate reported in automated (CACTus and SEEDS), but not in CDAW catalogs [@2014ApJ...784L..27W; @2017ApJ...836..134H]. This is aside from the fact that the real number of CMEs was higher for cycle 24 than for cycle 23 [@2015ApJ...812...74P]. The cadence change affected only very narrow CMEs, while wide CMEs such as halos were not affected [@2013SpWea..11..661G].
The statistical distributions are useful for understanding the physical nature of the underlying process. In particular, the CME WT may help to determine whether the CME process is purely stochastic or if there is a dependence or physical connection between consecutive CMEs. This is an important question in terms of CME triggering, and therefore a major issue in terms of prediction of space weather.
Figure \[fig:rep-rate\] shows the distribution of the WT (black circles) of 27761 CMEs observed by LASCO from February 1999 up to December 2018.
![Relative frequency (divided by the bin size of 2 hrs) of observed WT distribution (black circles), along the fit exponential (blue) and Pareto (red) distributions. The inner plot is the so-called “Q-Q plot,” which graphically illustrates that the Pareto distribution (red line) is closer to the “true” distribution (represented by the gray line) than the exponential distribution.[]{data-label="fig:rep-rate"}](rr_fin.png){width="\columnwidth"}
The main characteristics of the WT distribution are: i) Almost all CMEs (98%) occurred inside a time interval of 25 hours. Only 567 CMEs have a time difference longer than 25 hours. ii) $\sim$ 85% of the events have a time difference shorter than 10 hours. iii) The WT is under five hours for $\sim 61\%$ of the events.
Using a maximum likelihood method, we fit an exponential and a Pareto Type IV distribution to the observed WT. These distributions are plotted with blue and red continuous lines, respectively, in Figure \[fig:rep-rate\]. Their probability density functions (PDF) and best fit parameters found through a maximum likelihood method are:
- Exponential distribution (blue curve): $$f(x)=\lambda \exp^{-x \lambda} , \label{eq:expdist}$$ with $\lambda = 0.17 $ and a mean value of 5.75.
- Pareto Type-IV distribution (red curve): $$f(x)= \frac{ \kappa^{-1/\gamma} \alpha}{\gamma } \left[ 1 + \left( \frac{\kappa}{
x - \mu} \right)^{-1/\gamma}\right]^{-1 - \alpha} (x - \mu)^{-1 +
1/\gamma} , \label{eq:paretodist}$$ with $\kappa= 10.49$, $\alpha=2.84$ , $\gamma=0.81,$ and $\mu=0.07$, and a mean value of 5.79.
The Exponential distribution is memory-less and arises from a “simple” stochastic process like the waiting times of a Poisson process [@marshall2007life]; the memory-less property implies that there is no dependence or precondition between consecutive events. On the other hand, the Pareto distribution has a heavy-tail[^1] [@marshall2007life] and has been associated with long-range dependence or long-memory processes in a large variety of systems [@Samorodnitsky2007]. It is a very interesting property often associated with nonstationary processes, and the scaling and fractal behavior of the system and phase transitions [see @Samorodnitsky2007 for a survey of different points of view of the long-range dependence]. As shown in Figure \[fig:rep-rate\], the Pareto distribution (red line) better follows the observed WT (black circles) in the entire range. This fact is better demonstrated by the quantile-quantile or “Q-Q plot” on the inner frame of Figure \[fig:rep-rate\]. Q-Q plots are used to graphically assess the quality of the fit between the model and the empirical distribution, which is represented as a straight gray line in this case. To quantify the differences between the proposed and the observed distributions, we use the Bayesian information criterion [BIC, @1978Schwarz], which uses the maximum likelihood to determine the deviation between the empirical and the proposed distributions, with a penalty term for the number of parameters of the distributions. The distribution favored by BIC ideally corresponds to the candidate model which is a posteriori Therefore, the relevant indicator of the BIC is the relative change. The fitting results are shown in the last row of Table \[table:wt-dist\], where the Pareto - exponential $\Delta$BIC is shown in the last column. The fit parameters of the Pareto Type IV and exponential are shown in columns 2-5 and 7, respectively. As expected and clearly seen in Figure \[fig:rep-rate-ssn\], the WT changes appreciably during the different phases of the solar cycle. This causes the nonstationary aspect of the process. Therefore, an analysis taking into account the changes on each phase of the solar cycle is necessary.
Waiting time during the solar cycle {#sec:phases}
===================================
The WT varies with time and follows the solar cycle, as seen in Figure \[fig:rep-rate-ssn\], where we have plotted the CME WT as a function of time (gray plus symbols); its smoothed version (running average of 60 points, black dots) and a 30-day WT average (colored plus symbols), which shows a clear differentiation of the WT during the different solar cycle phases. In order to compare these changes with other solar cycle parameters, we plotted the sunspot number with a cyan line (taken from WDC-SILSO, Royal Observatory of Belgium, Brussels). In general, the shortest WTs correspond to periods of maximum solar activity and viceversa. In fact, during the maximum of activity of solar cycle 24, the mean WT was $ \leq 4$ hrs, whereas the mean WT reaches $\sim 11$ hrs during the descending phase of cycle 24.
![WT between every two consecutive CMEs (gray plus symbols) as a function of time during the analyzed period. The black dots correspond to a smoothed running average of the WT, whereas the cyan line corresponds to the Sunspot number. The different phases of the solar cycle (upper right) are marked with colored plus symbols, which correspond to the 30-day WT average.[]{data-label="fig:rep-rate-ssn"}](t_diff.png){width="\columnwidth"}
There are clear differences of the CME WT rate evolution during the solar cycle, the 30-day average of the WT shows major changes along the time during low activity phases, whereas for high activity phases, it remains relatively constant. Following these changes, we divided the observed data in seven periods as marked in the upper right of Figure \[fig:rep-rate-ssn\] with different colors. The ascending phase of solar cycle 23 corresponds to the first stage of LASCO, is somehow less stable than the other phases (as seen by the magenta symbols in Figure \[fig:rep-rate-ssn\]) and also contains the two major LASCO data gaps. Therefore, we did not take into account this phase for the rest of the analysis. The observed WT distributions for each selected phase of the cycle are plotted in Figure \[fig:rr-cycle\] with colored circles. Similarly to the entire time range WT distribution, we fit Pareto Type IV (Eq. \[eq:paretodist\]) and an exponential (Eq. \[eq:expdist\]) distributions.
There are important differences between the Pareto Type IV distribution parameters at each phase of the solar cycle. The wider WT distribution (mean = 11.23 hrs) is associated with the descending phase of cycle 24. The minimum phase of the cycle also has a wide WT distribution (mean = 7.88 hrs). On the other hand, the maximum of 24 has the narrowest distribution (with a mean of 3.90 hrs). This is in accordance with the increase of the number of CMEs observed in cycle 24 [@2015ApJ...812...74P]. Table \[table:wt-dist\] shows the parameters of the Pareto Type IV (columns 2 to 5) and exponential (column 7) distributions that best fit the WT during each phase of the solar cycles.
--------- ---------- ---------- ---------- ------- ------- ----------- ------- --------------
Phase $\kappa$ $\alpha$ $\gamma$ $\mu$ mean $\lambda$ mean $\Delta$ BIC
Max 23 18.87 5.11 0.84 0.07 5.43 0.184542 5.42 -259.40
Desc 23 18.13 4.00 0.80 0.10 6.91 0.144912 6.90 -218.20
Min 182.73 23.34 1.02 0.20 7.88 0.126992 7.87 -104.30
Asc 24 19.94 5.03 0.87 0.07 5.65 0.177439 5.64 -99.20
Max 24 11.46 4.52 0.79 0.07 3.90 0.256769 3.89 -632.80
Desc 24 68.34 7.25 0.99 0.17 11.23 0.0890623 11.23 -73.80
P3-P8 10.49 2.84 0.81 0.07 5.79 0.173847 5.75 -1727.00
--------- ---------- ---------- ---------- ------- ------- ----------- ------- --------------
![Similar to Figure \[fig:rep-rate\] but for different phases of solar cycles 23 and 24, marked by colors. The continuous and dash lines correspond to Pareto Type IV and exponential distributions, respectively. []{data-label="fig:rr-cycle"}](rr_per_fin_log_exp.png){width="\columnwidth"}
The BIC shows that the observed WT distributions follow better the Pareto Type IV distribution. The differences between these distributions are smaller during the phases of low activity (minimum and descending phase of cycle 24), pointing towards a simple stochastic WT process. On the other hand, during high activity periods, the BIC differences clearly show that the WT follows the heavy-tailed Pareto Type IV distribution pointing towards a long-range dependence or memory process. This long-range dependence of the WT time maybe reflected in other characteristics of the CME phenomena. Therefore, we explore the source region and speed of consecutive CMEs in the following sections.
Angular (PA) difference of consecutive CMEs {#sec:pa}
===========================================
The CME source region may be used to explore the possible relationship between consecutive CMEs. In this analysis, we use the CME position angle (PA) as a proxy of the CME source region [@2008ApJ...688..647L]. In particular, we use the central position angle (CPA), which is the mid-angle between the CME edges[^2]. Out of the total number of observed CMEs, we excluded halo CMEs (which, by definition, do not have CPAs) and performed the present analysis over 26086 CMEs. The rationale is that if there is long-range dependence in the CME WT, it should be statistically reflected in the distribution of the angular difference between the PA of consecutive CMEs ($PA_{diff}$). If the source regions of both consecutive events are relatively close, then the $PA_{diff}$ must approach zero.
![Distribution of the difference of the position angle between any two consecutive CMEs with a bin size of $10^\circ$. The green, pink, red, and blue continuous lines represent the Gaussian distributions added up (black curve) to fit the observations (black circles). The orange diamonds represent the observed distribution but for narrow (W $< 50^\circ$) CMEs.[]{data-label="fig:diff-cpa"}](diff_cpa.png){width="\columnwidth"}
The normalized distribution of the observed $PA_{diff}$ between consecutive CMEs is marked with black circles in Figure \[fig:diff-cpa\]. Over-plotted are the Gaussian PDFs, $$f(x)=\frac{1}{\sqrt{2\pi}\sigma} \exp\left(- \frac{(x-\mu)^2}{2 \sigma^2}\right), \label{eq:normal}$$ which fit the data. We note that the three wide Gaussian distributions are centered at $\mu_{1,3} = \sim \pm 180^\circ$ (green and blue curves), and $\mu_2 \sim 0^\circ$ (pink curve) representing the $PA_{diff}$ of randomly distributed events. What is unexpected for a random distribution of PAs is the narrow peak at the center of the distribution (around $0^\circ$), which can also be fit by a Gaussian (but narrower) distribution shown by the red line in Figure \[fig:diff-cpa\]. This large peak suggests that an important number of consecutive CMEs have less than $10^\circ$ of PA difference, meaning that these CMEs are produced in a very close source region (even assuming that half of the events in the central peak of Figure \[fig:diff-cpa\] were produced on opposite sides of the Sun, this peak is $\sim 3$ times larger than the others ). It is worth noting that we have conserved the PA reference point (north pole) and the sign of the $PA_{diff}$ to facilitate the fitting process and to make the symmetries of the PA difference distribution clearer. Of course, the real angular distance ranges from $0^\circ$ to $180^\circ$. We are aware of the difficulty of using coronographs to give information about the real position, in the low corona, of the source region of CMEs. As the observations are projected in the plane of the sky, it is difficult to characterize the actual position of the CME source region. Nevertheless, the PA is a good indicator of the radial direction of the CME, which in turn, extrapolated backwards to the solar surface, indicates the CME source region [see also @2003ApJ...598L..63G; @2008ApJ...688..647L; @2018SoPh..293...60M].
To diminish the effect of the projection, we performed the same analysis using narrow (width $<50^\circ$) CMEs only (we note that wherever we restrict the number of events, the imposed condition has to be fulfilled by the two events of the leading-trailer pair of CMEs, if one of them does not meet the condition, the pair is discarded for that particular part of the analysis). Obviously, the number of events is lower (12477), but the statistical characteristics do not change, as shown by the observed $PA_{diff}$ distribution of narrow CMEs plotted with diamonds and the associated Gaussian distributions (orange color) in Figure \[fig:diff-cpa\]. In fact, the inference of a close source region of consecutive CMEs is more accentuated when we take into account only the narrow CMEs, as suggested by the larger height of the central peak. These results suggest that the subset of CMEs with small $PA_{diff}$ may be spatially-related, in concordance with the scenario where new emerging flux frequently occurs in an active region which has already emerged [@1985SoPh...97...51L] or in its vicinity forming nests or clusters of flux emergence known as active longitudes [see @vanDriel-Gesztelyi2015 and references therein].
Speed difference of consecutive CMEs {#sec:speed}
====================================
Taking into account the short time difference ($ WT \le 10$ hrs) between a vast majority of consecutive CMEs (note that after 10 hrs, a CME with speed of 500 km s$^{-1}$ has traveled 25 $R_\odot,$ and therefore remains inside the LASCO C3 field of view) and the close spatial relation between the source region ($PA_{diff} \le 30^\circ$) of a group of them, a natural question arises: Is there any relationship between the speed of the leading ($V_{lead}$) and trailing ($V_{trail}$ ) CMEs? To explore this possibility, we tested several distributions and found that the $V_{diff}= V_{trail} - V_{lead}$ best follows a generalized student T-distribution of the form: $$f(x) = \frac{1}{\sqrt{\nu}
\sigma \textrm{Beta} (\frac{\nu}{2},\frac{1}{2})} \left[ \frac{\nu}{\nu +
\left( \frac{x-\mu}{\sigma}\right)^2 }\right]^{\frac{1+\nu.}{2}}$$ \[eq:tstudent\] Figure \[fig:vdiff\] shows the observed (open circles) and the fitted T-distributions (continuous lines) of the $V_{diff}$ during the whole period (teal color) and during the individual periods of the solar cycle, plotted with different colors as marked in the Figure. The corresponding parameters are shown in Table \[table:student\]. It is important to note that the student T-distribution is also a heavy-tailed distribution. In this case, the proper equivalent for an exponential distribution is given by the Laplace distribution [often known as the double-exponential distribution, @9780470390634] of the form:
$$f(x)= \begin{cases}
\frac{1}{2\beta}\exp\left(\frac{-(x-\mu)}{\beta}\right) & \text{if } x \geq 0 \\
\frac{1}{2\beta}\exp\left(\frac{-(\mu-x)}{\beta}\right) & \text{if } x < 0.
\end{cases}
\label{eq:laplace}$$
The BIC shows that the student T-distribution better fits the observed $V_{diff}$ than the double exponential distribution (the parameters of both distributions as well as the $\Delta$ BIC = BIC$_{Student}$ - BIC$_{Laplace}$ are shown in Table \[table:student\]) and allows us to conclude that $V_{diff}$ distribution is heavy-tailed, and therefore, the associated CMEs may have long-range dependence.
--------- --------- ---------- ------- --------- --------- --------------
Phase $\mu_S$ $\sigma$ $\nu$ $\mu_L$ $\beta$ $\Delta$ BIC
Asc 23 7.18 195.90 3.41 5.00 203.66 -4.40
Max 23 -0.04 271.86 5.82 -2.00 249.77 -129.20
Desc 23 -2.62 189.33 4.01 -3.00 187.78 -32.80
Min -0.44 133.01 5.31 -3.00 124.46 -53.10
Asc 24 -0.76 165.08 3.96 2.00 164.79 -41.30
Max 24 -1.73 180.24 4.44 -2.00 173.72 -23.00
Desc 24 1.28 149.26 4.50 2.00 143.99 -22.80
Total -1.13 186.15 3.91 -1.00 185.37 -62.00
--------- --------- ---------- ------- --------- --------- --------------
![Observed (circles) and fit generalized student T- (continuous lines) and Laplace (dotted lines) distributions of the $V_{diff}$ during different phases of the solar cycle and the entire period of study (teal color). []{data-label="fig:vdiff"}](dist_vel_dif.png){width="\columnwidth"}
Long-range dependence {#sec:dfa}
=====================
The heavy-tailed distributions, such as the WT and the speed difference of consecutive CMEs, suggest that the underlying process may not be associated with a simple Poisson process, instead, some long-range dependence between the events may be present. In this section, we apply a test, often used to determine the presence of long-range dependence, on the CME variables. Historically, the long-range dependence has been associated with time series where correlation decays slowly (decay exponentially in terms of the lag) or with a spectral density pole at zero frequency (i.e., at the origin). Although, to apply these criteria, the time series under consideration must be stationary, and unfortunately in our case, this condition is not fulfilled. To overcome this restriction, the detrended fluctuation analysis (DFA) was proposed by @PhysRevE.49.1685 to study the long-range dependence of nucleoids of DNA chains. Basically, the goal is the determination of the scaling exponent $\alpha_s$ of the (log-log) dependence between the detrended fluctuation function $F(s)$ and the size $s$ of the window where $F(s)$ was computed. The computational details can be found in @2001PhyA..295..441K.
We applied the DFA to the WT, $PA_{diff,}$ and the speed difference between the trailing and leading CME ($V_{diff}$) over the entire time range of study and the resulting DFA are shown in Figure \[fig:dfa\_tot\], in black, blue, and green, respectively. Valuable information about the range dependence of the data is given by the scaling exponent. For a random process such as Brownian motion, $\alpha_s = 0.5$. This is the case of the $PA_{diff}$ and the randomly re-arranged WT and $V_{diff}$. The former is in accordance with its (sum of) Gaussian distribution.
On the other hand, $\alpha_s > 0.5$ implies long-range dependence. In this case, the $\alpha_s$ is 0.67 for the speed difference and 0.77 for the WT. To ensure these results, we randomly shifted the values of those series and applied the DFA analysis, the results are plotted with open circles and dashed lines in Figure \[fig:dfa\_tot\] and for all cases $\alpha_s \sim 0.5$.
![ Detrended fluctuation as a function of the window size or scale (s) for the entire time period (upper left) of the WT (black), $V_{diff}$ (blue) and $PA_{diff}$ (green). The filled circles and continuous lines correspond to the actual data, whereas the empty circles and dashed lines correspond to the same, but randomly re-arranged (shuffled), data. The same analysis but for each phase of the cycle applied to the $PA_{diff}$ (upper right), $V_{diff}$ (lower left) and WT (lower right). The $\alpha_s$ value during each phase of the cycle is marked with the corresponding color. Open circles and dashed lines correspond to the DFA applied to a random vector created for comparison purposes.[]{data-label="fig:dfa_tot"}](rr_dfa.png){width="\columnwidth"}
Similar to the probability distributions of the WT, $PA_{diff,}$ and $V_{diff}$, the scaling exponent changes with the phase of the solar cycle. These changes are relatively small, for $PA_{diff}$, $\alpha_s \approx 0.5$ during all phases (upper-right panel of Figure \[fig:dfa\_tot\]). Similarly to during the minimum activity period, $\alpha_s \approx 0.5$ for both $V_{diff}$ and WT (bottom-left and right panels, respectively). Conversely, the $\alpha_s$ reaches larger values during the maximum and descending phases of the solar cycle.
In summary, the DFA shows that there is long-range dependence on both the WT and $V_{diff}$ differences between successive CMEs. This is evident during the maximum and declining phase of the solar cycle.
Trailing and leading CME speed relationship {#sec:speed_rel}
===========================================
As suggested by the precedent analysis, consecutive CMEs have some dependence. In this section, we explore the possibility of finding a relationship between the speed of these events. In particular, the possibility that the speed of a CME may be affected by a previous event. For instance, it may be surmised that the leading CME changes the ambient medium where the trailing CME is going to travel, in such a way that the leading CME has an influence on the dynamics of the trailing CME as predicted by drag force models . At this point, it is important to recall that we do not apply any restriction to select the leading-trailing CME pairs, so that a leading can be a trailing of a previous event, and these are treated as an independent pair. In order to investigate the possible statistical differences between the speed of the leading and trailing CMEs, we constructed subsets of spatial and temporally related pairs of CMEs with the following restrictions: i) A group with no restriction, meaning all events, hereinafter called $All_{CMEs}$. ii) A group of spatially-related consecutive CMEs where the PA difference ($PA_{diff}$) between the leading and trailing CMEs must be less than $30^\circ$. iii) For both groups, the temporal association was established creating subsets of time windows where the WT is less than or equal to two to 25 hours.
Then, we computed the mean speed value ($\overline{V}$) of each subset of leading and trailing CMEs and found a small but persistent difference between the $\overline{V}$ of the trailing and leading CMEs. This is shown in the left panel of Figure \[fig:vdif\_hist\] where we plotted the (blue) histogram of the difference ($\overline{V_{diff}}$) between trailing minus the leading CME means (i. e., $\overline{V_{diff}} = \overline{V_{trail}} - \overline{V_{lead}}$) of the $All_{CMEs}$ subsets. The main part of distribution is inside the -5 to 10 km s$^{-1}$ range, but with an asymmetrical component that extends towards positive differences up to $\sim 17$ km s$^{-1}$. The asymmetry is evident for the spatially-related CMEs groups (red histogram), where the difference attains higher values. @2004JGRA..10912105G found larger differences for the case of CMEs associated with solar energetic particles (SEP), although these were CMEs from the same active region.
![Differences of trailing minus leading CME mean (left) and median (right) speed of $All_{CMEs}$ (blue) and spatially-related (red) subsets for all the considered WTs.[]{data-label="fig:vdif_hist"}](speed_diff_hist.png){width="\columnwidth"}
The CME speed is well-fit by a lognormal distribution [@2006JGRA..111.6107L], which is also a heavy-tailed, and therefore the standard statistical tools, such as sample mean and sample standard deviation, are highly unstable [@pisarenko2010]. In this case, the standard deviations of the speed distributions are $\sim 200$ km s$^{-1,}$ and the standard errors of the mean are of the order of the differences. Then, a value of a few km s$^{-1}$ used to characterize the differences of the $\overline{V}$ has low statistical significance if we use the mean to characterize the speed distributions. Therefore, we have to use other parameters to quantify the observed differences. The quartiles (particularly the second quartile or median) are better suited to characterize skewed distributions. So, we computed the quartiles of all subsets: for example, Figure \[fig:quantil-vel\] shows the quartiles[^3] of the speed distributions as a function of the phase of the cycle for four different WT subsets. Blue and red lines correspond to the trailing and leading spatially-related CMEs; whereas pink and light-blue correspond to the trailing and leading $All_{CMEs}$ groups, respectively. As expected, the solar cycle variations are clear in the quartile behavior (Figure \[fig:quantil-vel\]), showing higher values during the maxima, and the lowest values are reached during the minimum activity period. We note that the quartiles clearly show the differences between the distributions of the leading and trailing CMEs.
For comparison (with the $\overline{V_{diff}}$), the right panel of Figure \[fig:vdif\_hist\] shows the differences between the medians of the trailing minus the leading speed distributions (i. e., $\widetilde{V_{diff}} = \widetilde{V_{trail}} - \widetilde{V_{lead}}$). In this case, the maximum of the $\widetilde{V_{diff}}$ for the spatially-related CMEs is shifted towards negative values, but the high asymmetry is conserved and attains higher values than the $\overline{V_{diff}}$. The most important fact shown by both $\overline{V_{diff}}$ and $\widetilde{V_{diff}}$ is the asymmetry towards positive values, implying that the $V_{trail}$ tends to be higher than the $V_{lead}$ for consecutive CMEs. In order to quantify these differences and their statistical significance, we performed a sign test with the null hypothesis $H_0$: $ \widetilde{V_{trail}} - \widetilde{V_{lead}} \le 0$. It is unlikely that $H_0$ is true, and therefore, the alternative hypothesis, in this case $H_a$: $ \widetilde{V_{trail}} - \widetilde{V_{lead}} > 0$, meaning $\widetilde{V_{trail}} > \widetilde{V_{lead}}$ is likely to be true.
The $\widetilde{V_{diff}}$ and $\overline{V_{diff}}$ (upper- and bottom-right panels) as well as the differences of the lower and upper quartiles (upper and bottom-left panels) are shown in Figures \[fig:vdif\_23\] and \[fig:vdif\_24\] for cycles 23 and 24, respectively. For the sake of clarity, we selected the subsets whose median sign-test probability is lower than 0.15, 0.1, and 0.05, marked with small, medium, and large symbols, respectively. Moreover, the number of events in each subset with probabilities lower than 0.05 are shown in the lower-right panel. The error bars in these Figures correspond to the median absolute deviation divided by the square root of the number of elements of each subset. Here, it is clear that the positive $V_{diff}$ is statistically significative, and that there are excesses of up to $\sim 100$ km s$^{-1}$ for $WT \le 2$ hrs during the ascending and maximum phases of cycle 23, and a more modest excess of $\sim 20$ km s$^{-1}$ during the decreasing phase of cycle 24. These $V_{trail}$ excesses decrease rapidly when the WT increases reaching $\sim 20$ km s$^{-1}$ for cycle 23, and $\sim 0$ km s$^{-1}$ for cycle 24 when $WT < 10$ hrs.
![Quartiles of speed distributions of the leading (light-blue) and trailing (pink) $All_{CMEs}$ consecutive pairs, as well as spatially-related leading (blue) and trailing (red) consecutive CME pairs, as a function of the phase of the solar cycle for four different WTs (as marked in each panel). []{data-label="fig:quantil-vel"}](vel_quantil_periodos_horas.png){width="\columnwidth"}
The spatially-related CMEs (marked with circles in Figures \[fig:vdif\_23\] and \[fig:vdif\_24\]) show the largest speed differences. Overall, the behavior of the $V_{diff}$ is clearly seen in the bottom panels of Figures \[fig:vdif\_23\] and \[fig:vdif\_24\], where the upper, meaning the high velocity quartile (left), and the $\overline{V_{diff}}$ (right) differences, are shown. The speed difference has higher values for WTs $\le 2$ hrs and decreases in a quasi-exponential way up to WTs $\sim 7 - 8$ hrs. Then, it slowly decreases towards zero. Finally, two important facts that maintain the CME speed bounded: (i) the speed difference reaches high values when we consider spatially-related CMEs, whereas the values remain low for all CMEs; and (ii) the speed difference tends to zero for large WTs during the minimum activity period.
![Difference between location parameters of the $V_{trail} - V_{lead}$ distributions as a function of the WT. The three quartiles and the mean values during the minimum and two phases of solar cycle 23 are marked by different colors. The size of the circles depends on the probability given by the sign test as: probability $\le$ 0.15 (small), 0.1 (medium), and 0.05 (large), i. e., when the null hypothesis ($H_0 : \widetilde{V_{trail}} - \widetilde{V_{lead}} \le 0$) is rejected at the 15, 10, and 5 % level based in the sign test. The events considered in each subset with probability $\le$ 0.05 are also shown in the lower right panel.[]{data-label="fig:vdif_23"}](speed_diff_wt_23.png){width="\columnwidth"}
![Same as Figure \[fig:vdif\_23\] but for solar cycle 24 and for the entire time range considered in this work (blue color).[]{data-label="fig:vdif_24"}](speed_diff_wt_24.png){width="\columnwidth"}
Discussion {#sec:disc}
==========
The statistical characterization of the properties of consecutive CMEs is important because it sheds light on the generation processes and early stages of the CME phenomena. In particular, the WT statistical distribution has been addressed by several authors [see @2014ApJ...781L...1T and references therein]. Although, as far as we know, the associated PA and speed differences between successive CMEs have not been explored until now. A major subject of study is the nature of the CME initiation process: Is this a “pure” stochastic process? Or is there some kind of relationship between consecutive events? This possible relationship is called “memory” of the system or “long-term dependence,” and there are different tools to investigate if the events of a given system have this dependence. If the process and its time series are stationary, the most common approaches used to look for the existence of long-range dependence are the auto-correlation function and the power spectrum behavior at the origin. The CME process is not stationary, so we approached the problem through the shape of the statistical distribution tail and the DFA [see @pisarenko2010; @Beran2013 for extensive reviews of this subject]. In this way, a “pure” stochastic process has an exponential distribution and is considered as a memory-less system. This is the case for the waiting times of a Poisson process [@LEE20051], for example. In contrast, processes where the distribution tail decreases more slowly than the exponential, the so called heavy-tailed distributions, such as the Pareto family and the generalized student T- distributions, are often associated with long-range dependence or memory processes [@pisarenko2010; @Beran2013].
The main question in this statistical study can be formulated as: Is the CME process “purely” stochastic? Or does the CME process have a long-range dependence or “memory” in such a way that an event is influenced by the occurrence of past events?
To address this question, @2003SoPh..214..361W analyzed a set of CMEs observed by LASCO during the ascending phase and part of the maximum of cycle 23 (1996 - 2001). The author fit the tail of the WT distribution (WT $> 10$ hrs) by a power law function (with index $\gamma \approx -2.36$). The author also applied a Bayesian statistical analysis and concluded that the observed WT distribution corresponds to a time-dependent Poisson process. Performing a similar analysis but using LASCO observations in a restricted time range around the maximum of cycle 23 (1999 - 2001), @2003ApJ...588.1176M reached the same conclusion that the WT distribution of 3187 CMEs may be represented by a nonstationary Poisson process.
A power law distribution, using only 113 CMEs observed by the Solar Maximum Mission, during the minimum phase of the cycle (1984 - 1989), was also found by , with $\gamma = 1.41$. In this case, the authors associated the WT with a characteristic length $\Lambda$, (between two consecutive CMEs) as $\Lambda \propto V_a ~ WT$, where $V_a$ is the Alfvén speed (this implies causality), and conclude that the WT power law distribution is consistent with their picture of characteristic length.
It is clear that a power law may be used to fit the tail of the WT distribution, but as mentioned earlier, 83% of the total events have a WT under 10 hrs. Therefore, the tail after this WT time is not necessarily representative of the whole process (only the extreme events fall in the extended tail of the distribution). In fact, a power law distribution is heavy-tailed and may represent a scaling system, which, in turn, may have self-organized criticality or processes with long-range dependence or “memory” [@pisarenko2010; @Beran2013]. For instance, @Takahashi_2016 found a scaling relationship between CMEs and proton events. Also, @2014ApJ...781L...1T found that the WT distribution may be fit by a Weibull distribution (a heavy-tailed distribution), implying some degree of correlation between CMEs in opposition to a purely stochastic process. Our findings are in agreement with @2009JGRA..11410105J (who found that SEP events appear to have some memory indicating that events are not completely random) and with the scaling relationship between CMEs and SEP found by @Takahashi_2016.
The possible correlation between consecutive CMEs is supported by the remarkable peak at short $PA_{diff}$, which shows that the source region of a large number of CMEs is located within close distance (Figure \[fig:diff-cpa\]). This result confirms the finding of @2003ApJ...588.1176M of an excess of events within an angular difference of $10^\circ$, these authors interpreted it as evidence of CMEs that occurred in the same active region. This result is also in agreement with the hypothesis that many CMEs (and flares) are produced in the so-called active longitudes [e.g., @2017ApJ...838...18G and references therein].
The WT distribution is telling us that, on average, the reconfiguration of the magnetic field after the launch of the leading CME, in order to replenish the free energy and launch the trailing CME, takes place in $\sim 3.7$ hrs during the maximum phase of cycle 24 and $\sim 8.5$ hrs during the minimum phase. This is an important fact from the point of view of the CME energetics, in particular in the case where two or more consecutive CMEs originated in the same source region. It is worth to note that the maximum fraction of free energy exhausted by one CME may be 25% [@2005JGRA..110.9S15G; @2012ApJ...759...71E; @2013ApJ...765...37F].
Finally, the speed distributions of all leading/trailing CMEs show a small but systematic tendency to have higher mean and median velocities for shorter WTs. The mean and median speed of the trailing CME are a few km s$^{-1}$ higher than that of the leading CMEs, for WTs shorter than 7 hrs, after a WT of 10 hrs the differences remain constant and negligible. Nevertheless, these differences are higher (tens of km s$^{-1}$) when we only take into account spatially-related CMEs (see Figure \[fig:vdif\_hist\]).
As the mean is not the best location indicator for highly skewed distributions, we also analyzed the quartiles (which are better suited to characterizing these distributions), the results are plotted in Figure \[fig:quantil-vel\]. To assess the accuracy of the analysis, we applied the sign test to show that the differences between the medians of trailing and leading CMEs are statistically significant. The phases of the solar cycle where the sign-test probability (with the null hypothesis $H_0 : \widetilde{V_{trail}} - \widetilde{V_{lead}} \le 0$) is less than 0.1 are shown in Figures \[fig:vdif\_23\] and \[fig:vdif\_24\].
These observations may be explained by the drag force models. It is clear that the magnetic field provides the necessary energy to launch and first acceleration of CMEs. Although this energy does not play an important role in the CME dynamics after few solar radii, at least for fast CMEs (such as those that form the tail of the distributions). Recently, @2017SoPh..292..118S investigated CME dynamics using coronographic observations, and they concluded that for fast CMEs, the Lorentz force has its maximum effect at heights lower than 2.5 $R_{\odot}$, and then, at heights of 3.5 - 4 $R_{\odot}$ becomes negligible compared with the drag force. In the case of slow CMEs, the Lorentz force may be significant at larger distances (up to 50 $R_{\odot}$, but slow CMEs do not contribute to the tail of the distributions). The drag force $F_D = - \frac{1}{2} C_D \rho A (V_{CME} - V_{SW})^2$ \[eq:dragforce\] is proportional to the drag coefficient ($C_D$), the transverse area of the CME ($A$), the speed difference between the CME ($V_{CME}$) and the solar wind ($V_{SW}$), and the ambient density ($\rho$) .
In a statistical analysis of a large number of events, the main values of $A$ and $V_{SW}$ are similar for both leading and trailing CMEs. On the other hand, the leading CME may sweep the ambient medium, and the trailing CME encounters a less dense medium and therefore the drag force decreases, resulting in a faster CME. In this case, the shorter the WT, the higher the CME speed. This effect is clearly seen in Figures \[fig:vdif\_23\] and \[fig:vdif\_24\].
Conclusions {#sec:conclusions}
===========
We performed a statistical study of 27761 CMEs observed by SOHO/LASCO from the ascending phase of cycle 23 to the declining phase of cycle 24, looking for long-range or memory signatures in the genesis and the first stages of evolution of CMEs. The main findings of this work are:
- The time elapsed between consecutive CMEs or WT for 98% (61%) of all events is less than 25 (5) hrs.
- We found that a Pareto Type IV probability density function fits the observed WT distribution during the entire period (1996 - 2018), and during the different phases of the solar cycle.
- As expected, the WT varies with the solar cycle, with large mean values ($\sim 10$ hrs) and high rates of change over time at low activity phases, and short mean ($\sim 4$ hrs) and $\sim 0$ rates of change over time when the activity is high. These mean values are consistent with recurrence times in large active regions reported before [see Figure 3 in @2005JGRA..110.9S15G].
- The distribution of the PA difference of consecutive CMEs (as a proxy to the CME source region) shows four components: three wide Gaussian distributions centered at $0^\circ$ and $\pm 180^\circ$ related to the spatially uncorrelated CMEs; the fourth is a narrow Gaussian centered at $0^\circ$ related to CMEs with very close source regions.
- The difference between the trailing and leading CME speed follows a generalized student T-distribution.
- The heavy-tailed distributions as well as the DFA show that there is a long-range dependence or memory in the WT and speed difference of consecutive CMEs.
- The long-range dependence of the CME speed is confirmed by the strong statistical evidence that the speed of the trailing CMEs is slightly higher than that of leading CMEs where the WTs are $< 10$ hrs.
- The trailing or leading speed difference is higher for the CMEs with close source-region association.
These findings point towards the fact that leading CMEs modify (sweep up) the ambient medium, causing the trailing CMEs to encounter less opposition to their movement. This difference is clear for CMEs produced by individual active regions [see Figure 9 of @2004JGRA..10912105G].
In summary, by analyzing a large number of consecutive CMEs, we find that their WT distribution follows a Pareto Type IV distribution, and the associated speed difference follows a generalized student T-distribution. These heavy-tailed distributions along the DFA suggest long-term dependence in the CME process. The position-angle difference distribution shows a large Gaussian peak centered at zero, indicating a close spatial relationship between consecutive CMEs. Furthermore, there is a small but statistically significant difference between the speed of consecutive CMEs indicating that the trailing CME is a few km s$^{-1}$ faster than the leading CME when the WT is $< 10$ hrs. All these findings suggest a physical connection between a considerable population of consecutive CMEs.
The CME catalog is generated and maintained at the CDAW Data Center by NASA and The Catholic University of America in cooperation with the Naval Research Laboratory. SOHO is a project of international cooperation between ESA and NASA. This work was partially supported by CONACyT (179588) and UNAM-PAPIIT (IN111716-3). N. Gopalswamy is supported by NASA’s Heliophysics LWS and GI programs. We thank the anonymous referee for his/her useful and constructive comments.
[^1]: Heavy-tailed distributions are characterized by the slow decreasing tail as compared with the exponential distribution
[^2]: The CDAW catalog also includes the measurement position angle (MPA), where the height-time measurements were taken [@2004JGRA..109.7105Y]: we do not use this angle in the present study.
[^3]: This presentation is similar to the “box-plot”, an often-used graphical method to illustrate the differences between distributions where the quartiles (and the minimum and maximum values or “whiskers”) of the distributions that one wants to compare, are plotted side by side. The difference is that in this case, we do not plot the whiskers.
|
---
abstract: |
Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly.
For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the the reducible ones, the $s$-powerful ones (divisible by the $s$-th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size.
Furthermore, a univariate polynomial $f$ is decomposable if $f = g
\circ h$ for some nonlinear polynomials $g$ and $h$. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic $p$ of $\Fq$ does not divide $n = \deg f$, is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where $p$ does divide $n$, the bounds are less satisfactory, in particular when $p$ is the smallest prime divisor of $n$ and divides $n$ exactly twice. The crux of the matter is to count the number of collisions, where essentially different $(g, h)$ yield the same $f$. We present a classification of all collisions at degree $n = p^{2}$ which yields an exact count of those decomposable polynomials.
author:
- |
Joachim von zur Gathen & Konstantin Ziegler\
B-IT, Universität Bonn\
D-53113 Bonn, Germany\
\
<http://cosec.bit.uni-bonn.de/>
bibliography:
- 'journals.bib'
- 'refs.bib'
- 'lncs.bib'
nocite: '[@grakal03]'
title: |
Survey on counting\
special types of polynomials
---
=1
counting special polynomials, finite fields, combinatorics on polynomials, generating functions, analytic combinatorics, asymptotic behavior, multivariate polynomials, polynomial decomposition, Ritt’s Second Theorem
00B25, 11T06, 12Y05
Introduction
============
Most integers are composite and most univariate polynomials over a finite field are reducible. The classical results of the Prime Number Theorem and a theorem of Gauß present approximations saying that randomly chosen integers up to $x$ or polynomials of degree up to $n$ are prime or irreducible with probability about $1/\ln x$ or $1/n$, respectively.
Concerning special classes of univariate polynomials over a finite field, [@zsi94] counts those with a given number of distinct roots or without irreducible factors of a given degree. In the same situation, [@art24] counts the irreducible ones in an arithmetic progression and [@hay65] generalizes these results. [@coh69a] and [@car87] count polynomials with certain factorization patterns and [@wil69] those with irreducible factors of given degree. Polynomials that occur as a norm in field extensions are studied by [@goglut81].
In two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. [@car63a] provides the first count of irreducible multivariate polynomials. In [@car65], he goes on to study the fraction of irreducibles when bounds on the degrees in each variable are prescribed; see also [@coh68]. In this survey, we opt for bounding the total degree because it has the charm of being invariant under invertible linear transformations. [@gaolau02] consider the counting problem in yet another model, namely where one variable occurs with maximal degree. The natural generating function (or zeta function) for the irreducible polynomials in two or more variables does not converge anywhere outside of the origin. [@wan92] notes that this explains the lack of a simple combinatorial formula for the number of irreducible polynomials. But he gives a $p$-adic formula, and also a (somewhat complicated) combinatorial formula. For further references, see @mulpan13 [Section 3.6].
In the bivariate case, [@gat08-incl-gat07] proves precise approximations with an exponentially decreasing relative error. extend those results to multivariate polynomials and give further information such as exact formulas and generating functions. [@bod08] gives a recursive formula for the number of irreducible bivariate polynomials and remarks on a generalization for more than two variables; he follows up with [@bod10].
We present exact formulas for the numbers of reducible (Sections \[sec:gen\]-\[sec:red\]), $s$-powerful (Section \[sec:powerful\]), and relatively irreducible polynomials (Section \[sec:rel\_irr\]). The formulas also yield simple, yet precise, approximations to these numbers, with rapidly decaying relative errors.
Geometrically, a single polynomial corresponds to a hypersurface, that is, to a cycle in affine or projective space of codimension 1. This correspondence preserves the respective notions of reducibility. Thus, Sections \[sec:gen\]-\[sec:red\] can also be viewed as counting reducible hypersurfaces, in particular, planar curves, and Section \[sec:powerful\] those with an $s$-fold component. From a geometric perspective, these results say that almost all hypersurfaces are irreducible. Can we say something similar for other types of varieties? [@cesgat13] give an affirmative answer for curves in $\PP^{r}$ for arbitrary $r$. A first question is how to parametrize the curves. Moduli spaces only include irreducible curves, and systems of defining equations do not work except for complete intersections. The natural parametrization is by the Chow variety $C_{r,n}$ of curves of degree $n$ in $\PP^{r}$, for some fixed $r$ and $n$. The foundation of this approach is a result by [@eishar92], who identified the irreducible components of $C_{r,n}$ of maximal dimension. We present the counting results in Section \[sec:reducible-curves\].
It is intuitively clear that the decomposable polynomials form a small minority among all multivariate polynomials over a field. gives a quantitative version of this intuition (see Section \[sec:multi-decomp\]). The number of multivariate decomposable polynomials is also studied by [@boddeb09].
This concludes the first half () of our survey, dealing with multivariate polynomials. The second half () is devoted to counting univariate decomposable polynomials.
Some of the results in this survey are from joint work with Raoul Blankertz, Eda Cesaratto, Mark Giesbrecht, Guillermo Matera, and Alfredo Viola.
A version of this paper is to appear in [@RICAM2013]. The final publication will be available at Springer after publication.
Counting multivariate polynomials {#sec:exact}
=================================
We work in the polynomial ring $F[x_{1}, \dots ,x_{r}]$ in $r\geq 1$ variables over a field $F$ and consider polynomials with total degree equal to some nonnegative integer $n$: $${P_{r,n}^{\text{all}}}(F) = \{ f \in F[x_{1},\dots,x_{r}] \colon \deg f = n\}.$$ The polynomials of degree at most $n$ form an $F$-vector space of dimension $\binom{r+n}{r}$.
The property of a certain polynomial to be reducible, squareful, relatively irreducible, or decomposable is shared with all polynomials associated to the given one. For counting them, it is sufficient to take one representative. We choose an arbitrary monomial order, say, the degree-lexicographic one, so that the monic polynomials are those with leading coefficient 1, and write $${P_{r,n}}(F) = \{f \in {P_{r,n}^{\text{all}}}(F) \colon f \text{ is monic}\}.$$
We use two different methodologies to obtain such bounds: generating functions and combinatorial counting. The usual approach, see [@flased09], of analytic combinatorics on series with integer coefficients leads, in our case, to power series that diverge everywhere (except at $0$). We have not found a way to make this work. Instead, we use power series with symbolic coefficients, namely rational functions in a variable representing the field size. Several useful relations from standard analytic combinatorics carry over to this new scenario. In a first step, this yields in a straightforward manner an exact formula for the number under consideration (). This formula is, however, not very transparent. Even the leading term is not immediately visible.
In a second step, coefficient comparisons yield easy-to-use approximations to our number (). The relative error is exponentially decreasing in the bit size of the data. Thus, gives a “third order” approximation for the number of reducible polynomials, and thus a “fourth order” approximation for the irreducible ones. The error term is in the big-Oh form and thus contains an unspecified constant.
In a third step, a different method, namely some combinatorial counting, yields “second order” approximations with explicit constants in the error term ().
The results of Sections \[sec:gen\]-\[sec:rel\_irr\] are from @gatvio13 unless otherwise attributed, those of Section \[sec:reducible-curves\] are from [-@cesgat13], and those of Section \[sec:multi-decomp\] are from @gat10a.
Exact formula for reducible polynomials {#sec:gen}
---------------------------------------
To study reducible polynomials, we consider the following subsets of ${P_{r,n}}(F)$: $$\begin{aligned}
{I_{r,n}}(F) & = \{ f \in {P_{r,n}}(F) \colon f \text{ is irreducible} \}, \\
{R_{r,n}}(F) & = {P_{r,n}}(F) \mysetminus {I_{r,n}}(F).\end{aligned}$$ In the usual notions, the polynomial $1$ is neither reducible nor irreducible. In our context, it is natural to have ${R_{r,0}}(F) =
\{ 1 \}$ and $ {I_{r,0}}(F) = \varnothing$.
The sets of polynomials $$\begin{aligned}
{{{\mathcal P}_{r}}}& = \bigcup_{n \geq 0} {P_{r,n}} (\FF_{q}), \\
{{{\mathcal I}_{r}}}& = \bigcup_{n \geq 0} {I_{r,n}} (\FF_{q}), \\
{{{\mathcal R}_{r}}}& = {{{\mathcal P}_{r}}}\mysetminus {{{\mathcal I}_{r}}},\end{aligned}$$ are combinatorial classes with the total degree as size functions and we denote the corresponding generating functions by ${{{\mathrm P}_{r}}}, {{{\mathrm I}_{r}}}, {{{\mathrm R}_{r}}}\in
\ZZ_{\geq 0} \bbracket{z}$, respectively. Their coefficients are $$\begin{aligned}
{{{\mathrm P}}_{r,n}} & = \# {P_{r,n}} (\FF_{q}) = q^{\binom{r+n}{r}-1} \frac{1-q^{-\binom{r+n-1}{r-1}}}{1-q^{-1}}, \label{eq:44} \\
{{{\mathrm R}}_{r,n}} & = \# {R_{r,n}} (\FF_{q}), \\
{{{\mathrm I}}_{r,n}} & = \# {I_{r,n}} (\FF_{q}), \label{eq:13}\end{aligned}$$ respectively, dropping the finite field $\FF_{q}$ with $q$ elements from the notation. By definition, ${{{\mathcal P}_{r}}}$ equals the disjoint union of ${{{\mathcal R}_{r}}}$ and ${{{\mathcal I}_{r}}}$, and therefore $$\label{eq:42}
{{{\mathrm R}_{r}}}= {{{\mathrm P}_{r}}}- {{{\mathrm I}_{r}}}.$$ By unique factorization, every element in ${{{\mathcal P}_{r}}}$ corresponds to an unordered finite sequence of elements in ${{{\mathcal I}_{r}}}$, where repetition is allowed, and therefore $${{{\mathrm I}_{r}}}= \sum_{k \geq 1} \frac{\mu (k)}{k} \log {{{\mathrm P}_{r}}}(z^{k}) \label{eq:28}$$ by @flased09 [Theorem I.5], where $\mu$ is the number-theoretic Möbius-function. A resulting algorithm is easy to program and returns exact results with lightning speed.
This approach quickly leads to explicit formulas. A *composition* of a positive integer $n$ is a sequence $j = (j_{1}, j_{2}, \dots, j_{\abs{j}})$ of positive integers $j_{1}, j_{2}, \dots, j_{\abs{j}}$ with $j_{1} +
j_{2} + \dots + j_{\abs{j}} = n$, where $\abs{j}$ denotes the length of the sequence. We define the set $$\label{eq:10}
M_{n} = \{ \text{compositions of $n$} \}.$$ This standard combinatorial notion is not to be confused with the composition of polynomials, which we discuss in Sections \[sec:multi-decomp\] and \[sec:univ-decomp\].
\[pro:R\_exact\_by\_recursion\] Let $r \geq 1$, $q \geq 2$, ${{{\mathrm P}}_{r,n}}$ as in , and ${{{\mathrm I}}_{r,n}}$ the number of irreducible monic $r$-variate polynomials of degree $n$ over $\Fq$. Then we have $$\label{eq:12}
\begin{split}
{{{\mathrm I}}_{r,0}} & = 0, \\
{{{\mathrm I}}_{r,n}} & = - \sum_{k \,\mid\, n} \frac{\mu (k)}{k} \sum_{j \in
M_{n/k}} \frac{(-1)^{\abs{j}}}{\abs{j}} {{{\mathrm P}}_{r,j_{1}}} {{{\mathrm P}}_{r,j_{2}}} \cdots {{{\mathrm P}}_{r,j_{\abs{j}}}},
\end{split}$$ for $n \geq 1$, and therefore for the number ${{{\mathrm R}}_{r,n}}$ of reducible monic $r$-variate polynomials of degree $n$ over $\Fq$ $$\begin{aligned}
{{{\mathrm R}}_{r,0}} & = 1, \\
{{{\mathrm R}}_{r,n}} & = {{{\mathrm P}}_{r,n}} + \sum_{k \,\mid\, n} \frac{\mu (k)}{k} \sum_{j \in M_{n/k}} \frac{(-1)^{\abs{j}}}{\abs{j}} {{{\mathrm P}}_{r,j_{1}}} {{{\mathrm P}}_{r,j_{2}}} \cdots {{{\mathrm P}}_{r,j_{\abs{j}}}},
\end{aligned}$$ for $n \geq 1$.
The formula of is exact but somewhat cumbersome. The following two sections provide simple yet precise approximations, with rapidly decaying error terms.
Symbolic approximation for reducible polynomials {#sec:symb-appr-reduc}
------------------------------------------------
For $r \geq 2$, the power series ${{{\mathrm P}_{r}}}$, ${{{\mathrm I}_{r}}}$, and ${{{\mathrm R}_{r}}}$ do not converge anywhere except at 0, and the standard asymptotic arguments of analytic combinatorics are inapplicable. We now deviate from this approach and move from power series in ${\QQ \bbracket{z}}$ to power series in ${{\QQ ({\mathbf{q}})}\bbracket{z}}$, where ${\mathbf{q}}$ is a symbolic variable representing the field size. For $r \geq 2$ and $n \geq 0$ we let $$\label{eq:19}
{{{\mathsf P}}_{r,n}}({\mathbf{q}}) = {\mathbf{q}}^{\binom{r+n}{r}-1}\frac{1-{\mathbf{q}}^{-\binom{r+n-1}{r-1}}}{1-{\mathbf{q}}^{-1}} \in \ZZ [{\mathbf{q}}]$$ in analogy to . We define the power series ${{{\mathsf P}_{r}}}, {{{\mathsf I}_{r}}}, {{{\mathsf R}_{r}}}\in {{\QQ ({\mathbf{q}})}\bbracket{z}}$ by $$\begin{aligned}
{{{\mathsf P}_{r}}}({\mathbf{q}}, z) & = \sum_{n \geq 0} {{{\mathsf P}}_{r,n}}({\mathbf{q}}) z^{n}, \label{eq:5} \\
{{{\mathsf I}_{r}}}({\mathbf{q}}, z) & = \sum_{k \geq 1} \frac{\mu (k)}{k} \log {{{\mathsf P}_{r}}}({\mathbf{q}}, z^{k}), \label{eq:relationIP} \\
{{{\mathsf R}_{r}}}({\mathbf{q}}, z) & = {{{\mathsf P}_{r}}}({\mathbf{q}}, z) - {{{\mathsf I}_{r}}}({\mathbf{q}}, z). \label{eq:70}\end{aligned}$$ Then ${{{\mathsf R}}_{r,n}}({\mathbf{q}})$ denotes the coefficient of $z^{n}$ in ${{{\mathsf R}_{r}}}$ and counts symbolically the reducible monic $r$-variate polynomials of degree $n$.
For nonzero $f \in \QQ({\mathbf{q}})$, ${\deg_{{\mathbf{q}}}}f$ is the degree of $f$, that is, the numerator degree minus the denominator degree. The appearance of $O({\mathbf{q}}^{-m})$ with a positive integer $m$ in an equation means the existence of some $f$ with degree at most $-m$ that makes the equation valid. If a term $O({\mathbf{q}}^{-m})$ appears, then we may conclude a numerical asymptotic result for growing prime powers $q$.
\[thm:R-from-gen\] Let $r\geq 2$ and $$\rho_{r,n}({\mathbf{q}})
= {\mathbf{q}}^{\binom{r+n-1}{r}+r-1}
\frac{1-{\mathbf{q}}^{-r}}{(1-{\mathbf{q}}^{-1})^2} \in \QQ({\mathbf{q}}). \label{eq:rho}$$ Then the symbolic formula ${{{\mathsf R}}_{r,n}}({\mathbf{q}})$ for the number of reducible monic $r$-variate polynomials of degree $n$ over $\Fq$ satisfies $$\begin{aligned}
{{{\mathsf R}}_{r,0}}({\mathbf{q}}) & = 1, \quad {{{\mathsf R}}_{r,1}} ({\mathbf{q}})= 0, \quad {{{\mathsf R}}_{r,2}} ({\mathbf{q}})= \frac{ \rho_{r,2}({\mathbf{q}}) }{2} \cdot (1-{\mathbf{q}}^{-r-1}), \\
{{{\mathsf R}}_{r,3}} ({\mathbf{q}})& = \rho_{r,3}({\mathbf{q}}) \Bigl( 1-{\mathbf{q}}^{-r(r+1)/2} +
{\mathbf{q}}^{-r(r-1)/2}
\frac{1-2{\mathbf{q}}^{-r}+2{\mathbf{q}}^{-2r-1}-{\mathbf{q}}^{-2r-2}}{3(1-{\mathbf{q}}^{-1})}\Bigr),
\\
{{{\mathsf R}}_{r,4}} ({\mathbf{q}})& = \rho_{r,4} ({\mathbf{q}}) \cdot \Bigl( 1 +
{\mathbf{q}}^{-\binom{r+1}{3}} \cdot \frac{1 +
O({\mathbf{q}}^{-r(r-1)/2})}{2 (1-{\mathbf{q}}^{-r})} \Bigr), \label{eq:48} \\
\intertext{and for $n \geq 5$}
{{{\mathsf R}}_{r,n}} ({\mathbf{q}})& = \rho_{r,n} ({\mathbf{q}}) \cdot \Bigl( 1 + {\mathbf{q}}^{-\binom{r+n-2}{r-1}+r(r+1)/2} \cdot \frac{1+ O({\mathbf{q}}^{-r(r-1)/2})}{1-{\mathbf{q}}^{-r}} \Bigr). \label{eq:26}\end{aligned}$$
[@ale06] lists $(\#
I_{r,n}(\Fq))_{n \geq 0}$ as A115457–A115472 in The On-Line Encyclopedia of Integer Sequences, for $2 \leq r \leq 6$ and prime $q \leq 7$. @bod08 [Theorem 7] states (in our notation) $$1-\frac{\# I_{r, n}}{ \# P_{r, n} } \sim q^{-\binom{n+r-1}{r-1}-r}
\frac{1-q^{-r}}{1-q^{-1}}.$$ [@houmul09] provide results for $\# I_{r,n} (\FF_q)$. These do not yield error bounds for the approximation of $\# R_{r,n} (\FF_q)$. [@bod10] also uses to claim a result similar to .
Explicit bounds for reducible polynomials {#sec:red}
-----------------------------------------
The third approach by “combinatorial counting” is somewhat more involved. The payoff of this additional effort is an explicit relative error bound. However, the calculations are sufficiently complicated for us to stop at the first error term. Thus we replace the asymptotic $1 +
O({\mathbf{q}}^{-r(r-1)/2})$ in by $1/(1-q^{-1})$.
\[thm:red\] Let $r, q \geq 2$, and $\rho_{r,n}$ as in . For the number $\#
{R_{r,n}}(\Fq)$ of reducible monic $r$-variate polynomials of degree $n$ over $\Fq$ we have $$\begin{gathered}
\# {R_{r,0}} (\FF_{q}) = 1, \quad \# {R_{r,1}} (\FF_{q}) = 0,
\quad \#
{R_{r,2}} (\FF_q) = \frac{ \rho_{r,2}(q)}{2} \cdot (1-q^{-r-1}), \\
\begin{split}
\left| \# {R_{r,3}} (\FF_q)- \rho_{r,3}(q) \right| & = \rho_{r,3}(q) \cdot q^{-r(r-1)/2}
\frac{1-2q^{-r}+2q^{-2r-1}-q^{-2r-2}}{3(1-q^{-1})} \\
& \leq
\rho_{r,3}(q) \cdot q^{-r(r-1)/2}, \\
\intertext{and for $n \geq 4$}
\left| \# {R_{r,n}} (\FF_q)-
\rho_{r,n}(q) \right| & \leq \rho_{r,n}(q) \cdot
\frac{q^{-\binom{r+n-2}{r-1}+r(r+1)/2}}{(1-q^{-1})(1-q^{-r})} \label{eq:16} \\
& \leq \rho_{r,n}(q) \cdot 3 q^{-\binom{r+n-2}{r-1}+r(r+1)/2}.
\end{split}\end{gathered}$$
\[rem:exp\_decay\] How close is our relative error estimate to being exponentially decaying in the input size? The usual dense representation of a polynomial in $r$ variables and of degree $n$ requires $b_{r,n} =
\binom{r+n}{r}$ monomials, each of them equipped with a coefficient from $\FF_q$, using about $\log_2 q$ bits. Thus the total input size is about $\log_2 q \cdot b_{r,n}$ bits. This differs from $\log_2 q
\cdot (b_{r-1,n-1}-b_{r-1,2})$ by a factor of $$\frac{b_{r,n}}{b_{r-1,n-1}-b_{r-1,2}} <
\frac{b_{r,n}}{\frac{1}{2}b_{r-1,n-1}} = \frac{2(n+r)(n+r-1)}{nr}.$$ Up to this polynomial difference (in the exponent), the relative error is exponentially decaying in the bit size of the input, that is, $(\log q)$ times the number of coefficients in the usual dense representation. In particular, it is exponentially decaying in any of the parameters $r$, $n$, and $\log_2
q$, when the other two are fixed.
Powerful polynomials {#sec:powerful}
--------------------
For an integer $s \geq 2$, a polynomial is called *$s$-powerful* if it is divisible by the $s$th power of some nonconstant polynomial, and *$s$-powerfree* otherwise; it is *squarefree* if $s=2$. Let $$\begin{aligned}
{Q_{r,n,s}} (F) & = \{f\in {P_{r,n}} (F) \colon f \text{ is
$s$-powerful}\}, \\
{S_{r,n,s}} (F) & = {P_{r,n}}(F) \mysetminus {Q_{r,n,s}}(F).\end{aligned}$$ As in the previous section, we restrict our attention to a finite field $F= \FF_q$, which we omit from the notation.
For the approach by generating functions, we consider the combinatorial classes ${\mathcal{Q}_{r,s}}= \bigcup_{n \geq 0} {Q_{r,n,s}}$ and ${\mathcal{S}_{r,s}}= {{{\mathcal P}_{r}}}\mysetminus {\mathcal{Q}_{r,s}}$. Any monic polynomial $f$ factors uniquely as $f=g\cdot h^s$ where $g$ is a monic $s$-powerfree polynomial and $h$ an arbitrary monic polynomial, hence $$\label{eq:61}
{{{\mathrm P}_{r}}}= {\mathrm{S}_{r,s}}\cdot {{{\mathrm P}_{r}}}( z^{s})$$ and by definition ${\mathrm{Q}_{r,s}}= {{{\mathrm P}_{r}}}- {\mathrm{S}_{r,s}}$ for the generating functions of ${\mathcal{S}_{r,s}}$ and ${\mathcal{Q}_{r,s}}$, respectively. For univariate polynomials, [@car32] derives directly from generating functions to prove the counting formula for $r=1$. @flagou01 [Section 1.1] use for $s=2$ to count univariate squarefree polynomials, see also @flased09 [Note I.66].
As in , this approach quickly leads to explicit formulas.
\[pro:Q\_exact\_by\_recursion\] For $r \geq 1$, $q,s \geq 2$, ${{{\mathrm P}}_{r,n}}$ as in , and $M_{n}$ as in , we have for the number ${{{\mathrm Q}}_{r,n,s}} = \#
{Q_{r,n,s}} (\FF_{q})$ of $s$-powerful monic $r$-variate polynomials of degree $n$ over $\Fq$ $${{{\mathrm Q}}_{r,n,s}} = - \sum_{\substack{1 \leq i
\leq n/s \\ j \in M_{i}}} (-1)^{\abs{j}} {{{\mathrm P}}_{r,j_{1}}}
{{{\mathrm P}}_{r,j_{2}}} \cdots {{{\mathrm P}}_{r,j_{\abs{j}}}} {{{\mathrm P}}_{r,n-is}} \label{eq:71}.$$
To study the asymptotic behavior of ${{{\mathrm Q}}_{r,n,s}}$ for $r \geq 2$ we again deviate from the standard approach and move to power series in ${{\QQ ({\mathbf{q}})}\bbracket{z}}$. With ${{{\mathsf P}_{r}}}$ from , we define ${\mathsf{S}_{r,s}}, {\mathsf{Q}_{r,s}}\in {{\QQ ({\mathbf{q}})}\bbracket{z}}$ by $$\begin{aligned}
{{{\mathsf P}_{r}}}& = {\mathsf{S}_{r,s}}\cdot {{{\mathsf P}_{r}}}(z^{s}), \label{def:S} \\
{\mathsf{Q}_{r,s}}& = {{{\mathsf P}_{r}}}- {\mathsf{S}_{r,s}}.\end{aligned}$$
The approach by generating functions now yields the following result. Its “general” case is \[item:13\]. We give exact expressions in special cases, namely for $ n < 3s$ in \[item:17\] and for $(n,s)=(6,2)$ in \[item:18\], which also apply when we substitute the size $q$ of a finite field $\FF_{q}$ for ${\mathbf{q}}$.
\[thm:Q\_by\_gen\] Let $r, s \geq 2$, $n\geq 0$, and $$\begin{aligned}
\eta_{r,n,s} ({\mathbf{q}}) & = {\mathbf{q}}^{\binom{r+n-s}{r} + r-1}
\frac{(1-{\mathbf{q}}^{-r}) (1-{\mathbf{q}}^{-\binom{r+n-s-1}{r-1}})}{(1-{\mathbf{q}}^{-1})^2} \in
\QQ ({\mathbf{q}}), \label{eq:89} \\
\delta & = \binom{r+n-s}{r}-\binom{r+n-2s}{r} - \frac{r(r+1)}{2}.
\end{aligned}$$ Then the symbolic formula ${{{\mathsf Q}}_{r,n,s}} ({\mathbf{q}})$ for the number of $s$-powerful monic $r$-variate polynomials of degree $n$ over $\Fq$ satisfies the following.
\[item:4\] If $n \geq 2s$, then $\delta \geq r$.
\[item:17\] $$\label{eq:50}
{{{\mathsf Q}}_{r,n,s}}({\mathbf{q}}) = \begin{cases}
0 & \text{for $n<s$,}\\
\eta_{r,n,s}({\mathbf{q}}) & \text{for $s \leq n<2s$,} \\
\eta_{r,n,s} ({\mathbf{q}}) \biggl( 1 + {\mathbf{q}}^{-\delta} \cdot \frac{1-{\mathbf{q}}^{-\binom{n+r-2s-1}{r-1}}}{1-{\mathbf{q}}^{-\binom{n+r-s-1}{r-1}}} & \\
\quad \cdot \Bigl( \frac{1-{\mathbf{q}}^{-r(r+1)/2}}{1-{\mathbf{q}}^{-r}} - {\mathbf{q}}^{-r(r-1)/2}\frac{1-{\mathbf{q}}^{-r}}{1-{\mathbf{q}}^{-1}}\Bigr)\biggr) & \text{for $2s \leq n<3s$.}
\end{cases}$$
\[item:18\] For $(n,s)=(6,2)$, we have $$\begin{aligned}
{{\mathsf Q}}_{r,6,2} ({\mathbf{q}}) & = \eta_{r,6,2} ({\mathbf{q}}) \bigl( 1 + {\mathbf{q}}^{-\delta+(r-2)(r-1)(r+3)/6}(1+O({\mathbf{q}}^{-1}))\bigr). \label{eq:86}
\end{aligned}$$
\[item:13\] For $n \geq 2s$ and $(n,s)\neq (6,2)$, we have $$\label{eq:49}
{{{\mathsf Q}}_{r,n,s}} ({\mathbf{q}}) = \eta_{r,n,s}({\mathbf{q}}) \big(1 + {\mathbf{q}}^{-\delta}
(1 + O({\mathbf{q}}^{-1}))\big).$$
For $r\geq 3$, we can replace $1+O({\mathbf{q}}^{-1})$ in by ${\mathbf{q}}^{-1}+O({\mathbf{q}}^{-2})$. The combinatorial approach replaces the asymptotic $1+O({\mathbf{q}}^{-1})$ for $n \geq 3s$ with an explicit bound. For $n<3s$ the exact formula of \[item:17\] applies.
\[thm:Q\_by\_map\] Let $r,s,q\geq 2$, $\# {Q_{r,n,s}}(\Fq)$ the number of $s$-powerful monic $r$-variate polynomials of degree $n$ over $\Fq$, and $\eta_{r,n,s}$ and $\delta$ as in .
\[item:8\] For $(n,s) = (6,2)$, we have $\delta = r(r+1)(r^{2}+9r+2)/24$ and $$\label{eq:53}
\left| \# {Q_{r,6,2}} (\FF_q) - \eta_{r,6,2}(q) \right| \leq
\eta_{r,6,2}(q) \cdot 2 q^{-\delta+(r-2)(r-1)(r+3)/6}.$$
\[item:9\] For $n \geq 3s$ and $(n,s) \neq (6,2)$, we have $$\label{eq:18}
\left| \# {Q_{r,n,s}} (\FF_q) - \eta_{r,n,s}(q) \right| \leq
\eta_{r,n,s}(q) \cdot 6 q^{-\delta}.$$
As noted in for reducible polynomials, the relative error term is (essentially) exponentially decreasing in the input size, and exponentially decaying in any of the parameters $r$, $n$, $s$, and $\log_2 q$, when the other three are fixed.
Relatively irreducible polynomials {#sec:rel_irr}
----------------------------------
A polynomial over $F$ is *absolutely irreducible* if it is irreducible over an algebraic closure of $F$, and *relatively irreducible* (or *exceptional*) if it is irreducible over $F$ but factors over some extension field of $F$. We define $$\begin{aligned}
{A_{r,n}} (F) & = \{ f \in {P_{r,n}}(F) \colon f \text{ is absolutely irreducible}\} \subseteq {I_{r,n}}(F),\\
{E_{r,n}} (F) & = {I_{r,n}} (F) \mysetminus {A_{r,n}}(F) \label{eq:66}.\end{aligned}$$ As before, we restrict ourselves to finite fields and recall that all our polynomials are monic. We relate the generating function ${{{\mathrm A}_{r}}}(\FF_{q})$ of $\# {A_{r,n}} (\FF_{q})$ to the generating function ${{{\mathrm I}_{r}}}(\FF_{q})$ of irreducible polynomials as introduced in Section \[sec:gen\] and obtain $$\begin{aligned}
[z^{n}] \, {{{\mathrm I}_{r}}}(\FF_{q}) & = \sum_{k \,\mid\, n } \frac{1}{k} \sum_{s
\,\mid\, k} \mu (k/s) \cdot [z^{n/k}] \, {{{\mathrm A}_{r}}}(\FF_{q^{s}}), \\
[z^{n}] \, {{{\mathrm A}_{r}}}(\FF_{q}) & = \sum_{k \,\mid\, n} \frac{1}{k} \sum_{s \,\mid\, k} \mu (s)
\cdot [z^{n/k}] \, {{{\mathrm I}_{r}}}(\FF_{q^{s}}) \label{eq:43}\end{aligned}$$ with Möbius inversion. For an explicit formula, we combine the expression for ${{{\mathrm I}}_{r,n}} (\FF_{q})$ from with .
\[pro:E\_exact\_by\_recursion\] For $r,n \geq 1$, $q \geq 2$, $M_{n}$ as in , ${{{\mathrm P}}_{r,n}}$ as in , and ${{{\mathrm I}}_{r,n}}$ as in , we have for the number ${{{\mathrm E}}_{r,n}}$ of relatively irreducible monic $r$-variate polynomials of degree $n$ over $\Fq$ $$\begin{aligned}
{{{\mathrm E}}_{r,0}} (\FF_{q})& = 0, \\
{{{\mathrm E}}_{r,n}} (\FF_{q}) & = -\sum_{1 < k \,\mid\, n} \frac{1}{k} \sum_{s \,\mid\, k} \mu (s) {{{\mathrm I}}_{r,n/k}} (\FF_{q^{s}}) \label{eq:60} \\
& = \sum_{1< k \,\mid\, n} \frac{1}{k} \sum_{\substack{s \,\mid\, k \\
m \,\mid\, n/k }} \frac{\mu (s) \mu (m)}{m} \\
& \quad \quad \quad \quad \cdot\sum_{j \in M_{n/(km)}} \frac{(-1)^{\abs{j}}}{\abs{j}} {{{\mathrm P}}_{r,j_{1}}} (\FF_{q^{s}}) {{{\mathrm P}}_{r,j_{2}}} (\FF_{q^{s}}) \cdots {{{\mathrm P}}_{r,j_{\abs{j}}}} (\FF_{q^{s}}).
\end{aligned}$$
The approach by generating functions gives the following result.
\[thm:E\_by\_gen\] Let $r, n \geq 2$, let $\ell$ be the smallest prime divisor of $n$, and $$\begin{aligned}
\epsilon_{r,n}({\mathbf{q}}) & = \frac{{\mathbf{q}}^{\ell ( \binom{r+n/\ell}{r} - 1)}}{\ell(1-{\mathbf{q}}^{-\ell})} \in \QQ({\mathbf{q}}), \label{eq:epsilon} \\
\kappa & = (\ell-1) (\binom{r-1+n/\ell}{r-1}-r ) + 1.
\end{aligned}$$ Then the symbolic formula ${{{\mathsf E}}_{r,n}}({\mathbf{q}})$ for the number of relatively irreducible monic $r$-variate polynomials of degree $n$ over $\Fq$ satisfies the following.
\[enum3:i\] ${{{\mathsf E}}_{r,1}} ({\mathbf{q}})= 0$.
\[enum3:ii\] If $n$ is prime, then $$\begin{aligned}
{{{\mathsf E}}_{r,n}} ({\mathbf{q}}) & = \epsilon_{r,n}({\mathbf{q}}) (1-{\mathbf{q}}^{-nr}) \Bigl( 1 -
{\mathbf{q}}^{-r(n-1)}\frac{(1-{\mathbf{q}}^{-r})(1-{\mathbf{q}}^{-n})}{(1-{\mathbf{q}}^{-1})(1-{\mathbf{q}}^{-nr})}\Bigr).
\end{aligned}$$
\[enum3:iii\] If $n$ is composite, then $\kappa \geq 2$ and $${{{\mathsf E}}_{r,n}} ({\mathbf{q}}) = \epsilon_{r,n}({\mathbf{q}}) (
1 + O ({\mathbf{q}}^{- \kappa}) ).$$
While \[enum3:i\] and \[enum3:ii\] yield explicit bounds, the combinatorial approach does this for \[enum3:iii\].
\[thm:E\_complete\] Let $r,q \geq 2$, and $\epsilon_{r,n}$ and $\kappa$ as in , and $n$ be composite. Then for the number $\# {E_{r,n}}(\Fq)$ of relatively irreducible monic $r$-variate polynomials of degree $n$ over $\Fq$ we have $$\left| \# {E_{r,n}}(\FF_q) - \epsilon_{r,n}(q) \right| \leq
\epsilon_{r,n}(q) \cdot 3 q^{-\kappa}.$$
Reducible space curves {#sec:reducible-curves}
----------------------
The *Chow variety* of curves of degree $n$ in the $r$-dimensional projective space $\PP^{r} = \PP^{r}(\Fqbar)$ over an algebraic closure $\Fqbar$ is denoted by $C_{r,n}$. Each point of the Chow variety $C_{r,n}$ actually corresponds to a unique *effective cycle* in $\PP^{r}$ of dimension $1$ and degree $n$, that is, to a formal linear combination $\sum a_{i} C_{i}$, where each $C_{i}$ is an irreducible curve in $\PP^{r}$, each $a_{i}$ is a positive integer and $\sum a_{i} \deg
(C_{i}) = n$.
For a subfield $F\subseteq \Fqbar$, an effective $F$-cycle $C$ is called *$F$-reducible* if there exist $m \geq 2$ and effective $F$-cycles $C_{1}, \dots, C_{m}$ such that $C = \sum_{i=1}^{m} C_{i}$ holds. Let $C_{r,n}(\Fq)$ denote the Chow variety of effective $\Fq$-cycles and ${R_{r,n}^{*}}(\Fq)$ its closed subvariety of $\Fq$-reducible $\Fq$-cycles. Methods of algebraic geometry yield the following bounds on the probability that a random curve of degree $n$ in $\PP^{r}(\Fq)$ is $\Fq$-reducible.
\[thr:1\] Let $r \geq 3$ and $$\begin{aligned}
g_{r,n} & = {\binom{r+n-2}{n}}^{2} \cdot
\frac{r+n-1}{(r-1)(n+1)}, \label{eq:34} \\
c_{r,n} & =(2en)^{r(r+1)(n^{2}+1)+4rg_{r,n}},\end{aligned}$$ where $e$ denotes the basis of the natural logarithm. For the number $\# {R_{r,n}^{*}} (\Fq)$ of $\Fq$-reducible cycles of degree $n$ we have the following.
If $n \geq \min\{ 4r-7, 7\}$, then $$\label{eq:4}
\frac{1}{4c_{r,n}}q^{-(n-2r+3)} \leq \frac{\# {R_{r,n}^{*}} (\Fq)}{\#
C_{r,n}(\Fq)} \leq c_{r,n} q^{-(n-2r+3)}.$$
If $n = 4r-8$, then $$\frac{1}{2n!\,c_{r,n}} q^{-r+2} \leq \frac{\# {R_{r,n}^{*}} (\Fq)}{\#
C_{r,n}(\Fq)} \leq c_{r,n} q^{-r+2}.$$
We call an $\Fqbar$-reducible cycle *absolutely reducible*. An $\Fq$-cycle can be absolutely reducible for two reasons: either it is $\Fq$-reducible, as treated above, or *relatively $\Fq$-irreducible*, that is, is $\Fq$-irreducible and $\Fqbar$-reducible. The set of relatively $\Fq$-irreducible (or *exceptional*) $\Fq$-curves of degree $n$ in $\PP^{r}$ is denoted by ${E_{r,n}^{*}} (\Fq)$.
\[thr:2\] Let $r \geq 3$, $n \geq 4r - 8$, let $\ell$ denote the smallest prime divisor of $n$, and $$\begin{aligned}
b_{r,n} & = 3(r-2)+n(n+3)/2, \\
d_{\ell,n,r} & = (en/\ell)^{r(r+1)(n^{2}/\ell^{2}+1)+4rg_{r,n/\ell}}.\end{aligned}$$ For the number $\# {E_{r,n}^{*}}(\Fq)$ of relatively $\Fq$-irreducible cycles of degree $n$ we have $$\begin{aligned}
q^{2n(r-1)}(1-4q^{2(1-n)(r-1)}) \leq \# {E_{r,n}^{*}} (\Fq) \leq
2d_{\ell,n,r}q^{2n(r-1)} \text{ for } n/\ell \leq 4r-7, \\
q^{\ell b_{r,n/\ell}}(1-16q^{\ell-n}) \leq \# {E_{r,n}^{*}} (\Fq) \leq 3
d_{\ell,n,r}q^{\ell b_{r,n/\ell}} \text{ for } n/\ell \geq 4r-8.\end{aligned}$$
Decomposable polynomials {#sec:multi-decomp}
------------------------
For monic univariate $g\in F[y]$ and $h\in {P_{r,n}}$, we define their *composition* $$\label{eq:9}
f = g \circ h = g(h) \in {P_{r,n}}.$$ If $\deg g \geq 2$ and $\deg h \geq 1$, then $(g,h)$ is a *decomposition* of $f$. A polynomial $f \in {P_{r,n}}$ is *decomposable* if there exist such $g$ and $h$. There are other notions of decompositions. The present one is called uni-multivariate in [@gatgut03]. Another one is studied in [@fauper08] for cryptanalytic purposes. In the context of univariate polynomials $\deg h \geq 2$ is also required, see Section \[sec:univ-decomp\].
It is sufficient to concentrate on polynomials with vanishing constant term, see , and we denote by $D_{r,n}(F)$ the set of all decomposable polynomials $f \in {P_{r,n}}(F)$ with $f(0,\dots,0) = 0$.
Let $\Fq$ be a finite field with $q$ elements, $r\geq 2$, and $\ell$ the smallest prime divisor of the composite integer $n \geq 2$. Let $$\begin{aligned}
\label{eq:pm}
m & = \begin{cases*}
n & if $r=2$, $n/\ell$ is prime, and $n/\ell \leq 2\ell - 5$, \\
\ell & otherwise,
\end{cases*} \\
\alpha_{r,n} &=
q^{\binom{r+n/m}{r}+m-3} \frac{1-q^{-\binom{r-1+n/m}{r-1}}}{1-q^{-1}}, \\
\beta_{r,n} & = \frac{2q^{-\frac{1}{2} \binom{r-1+n/\ell}{r-1} + 1}}{1-q^{-1}}.
\end{aligned}$$ Then for the number $\# D_{r,n}(\Fq)$ of decomposable monic $r$-variate polynomials with vanishing constant term of degree $n$ over $\Fq$ we have $$\label{eq:3}
\left|{\#D_{r,n}(\Fq)}- \alpha_{r,n}\right| \leq
\alpha_{r,n} \cdot \beta_{r,n}.$$
Counting univariate decomposable polynomials {#sec:univ-decomp}
============================================
The *composition* of two univariate polynomials $g,h \in F[x]$ over a field $F$ is denoted as $f= g \circ h= g(h)$, and then $(g,h)$ is a *decomposition* of $f$, and $f$ is *decomposable* if $g$ and $h$ have degree at least $2$. In the 1920s, Ritt, Fatou, and Julia studied structural properties of these decompositions over $\mathbb{C}$, using analytic methods. Particularly important are two theorems by Ritt on the uniqueness, in a suitable sense, of decompositions, the first one for (many) indecomposable components and the second one for two components, as above. [@eng41] and [@lev42] proved them over arbitrary fields of characteristic zero using algebraic methods.
The theory was extended to arbitrary characteristic by [@frimac69], [@dorwha74], [@sch82c; @sch00c], [@zan93], and others. Its use in a cryptographic context was suggested by [@cad85]. In computer algebra, the decomposition method of [@barzip85] requires exponential time. A fundamental dichotomy is between the *tame case*, where the characteristic $p$ does not divide $\deg g$, and the *wild case*, where $p$ divides $\deg g$, see [@gat90d; @gat90c]. ([@sch00c], § 1.5, uses *tame* in a different sense.) A breakthrough result of [@kozlan89] was their polynomial-time algorithm to compute tame decompositions; see also [-@gatkoz87]; [-@kozlan96]; [@gutsev06], and the survey articles of [@gat02c] and [@gutkoz03] with further references. Schur’s conjecture, as proven by [@tur95], offers a natural connection between the tame indecomposable polynomials in this section and certain absolutely irreducible bivariate polynomials, as studied in Section \[sec:rel\_irr\]. More precisely, a tame polynomial $f$ is indecomposable if $(f(x)-f(y))/(x-y)$ is absolutely irreducible. Aside from natural exceptions, the converse is also true.
In the wild case, considerably less is known, both mathematically and computationally. [@zip91] suggests that the block decompositions of [@lanmil85] for determining subfields of algebraic number fields can be applied to decomposing rational functions even in the wild case. A version of Zippel’s algorithm in [@bla14] computes in polynomial time all decompositions of a polynomial that are minimal in a certain sense. [@avazan03] study ambiguities in the decomposition of rational functions over $\mathbb{C}$. On a different but related topic, [@ziemue08] found interesting characterizations for Ritt’s First Theorem, which deals with complete decompositions, where all components are indecomposable.
We have seen fairly precise estimates for the number of multivariate decomposable polynomials in Section \[sec:multi-decomp\]. It is intuitively clear that the univariate decomposable polynomials also form only a small minority among all univariate polynomials over a field, and this second part of our survey confirms this intuition. The task is to approximate the number of decomposables over a finite field, together with a good relative error bound. One readily obtains an upper bound. The challenge then is to find an essentially matching lower bound.
A set of distinct decompositions of $f$ is called a *collision*. The number of decomposable polynomials of degree $n$ is thus the number of all pairs $(g,h)$ with $\deg g \cdot \deg h = n$ reduced by the ambiguities introduced by collisions. An important tool for estimating the number of collisions is Ritt’s Second Theorem. The first algebraic versions of this in positive characteristic $p$ required $p>\deg(g\circ h)$. [@zan93] reduced this to the milder and more natural requirement $g' \neq 0$ for all $g$ in the collision. His proof works over an algebraic closed field, and Schinzel’s [-@sch00c] monograph adapts it to finite fields. In Section \[sec:ritt-2\], we provide a precise quantitative version of Ritt’s Second Theorem, by determining exactly the number of such collisions in the tame case, assuming that $p \nmid n/\ell$, where $n$ is the degree of the composition and $\ell$ is the smallest prime divisor of $n$. This is based on a unique normal form for the polynomials occurring in Ritt’s Second Theorem.
[@gie88b] was the first to consider this counting problem. He showed that the decomposable polynomials form an exponentially small fraction of all univariate polynomials. General approximations to the number of univariate decomposable polynomials are shown in Section \[sec:cdup\]. They come with satisfactory (rapidly decreasing) relative error bounds except when $p$ divides $n = \deg f$ exactly twice. [@zie14] provides an exact count of tame univariate polynomials. In Section \[sec:prelim\], we determine exactly the number of decomposable polynomials in one of the difficult wild cases, namely when $n = p^{2}$.
[@zan08] studies a different but related question, namely compositions $f= g \circ h$ in $ \mathbb{C} [x]$ with a *sparse* polynomial $f$, having $t$ terms. The degree is not bounded. He gives bounds, depending only on $t$, on the degree of $g$ and the number of terms in $h$. Furthermore, he gives a parametrization of all such $f$, $g$, $h$ in terms of varieties (for the coefficients) and lattices (for the exponents). @boddeb09 also deal with counting.
Unless otherwise attributed, the results of Section \[sec:ritt-2\] are from [@gat12a], those of Section \[sec:cdup\] from [@gat08c], and those of Section \[sec:prelim\] from [-@blagat13].
Notation {#sec:notation}
--------
A nonzero polynomial $f\in F[x]$ over a field $F$ of characteristic $p \geq 0$ is *monic* if its leading coefficient $\operatorname{lc}(f)$ equals $1$. We call $f$ *original* if its graph contains the origin, that is, $f(0)=0$. For $g, h \in F[x]$, $$\label{defComp:f}
f = g \circ h = g(h) \in F[x]$$ is their *composition*. If $\deg g, \deg h \geq 2$, then $(g,h)$ is a *decomposition* of $f$. A polynomial $f \in F[x]$ of degree at least $2$ is *decomposable* if there exist such $g$ and $h$, otherwise $f$ is *indecomposable*. A decomposition is *tame* if $p\nmid \deg g$, and $f$ is *tame* if $p\nmid \deg f$.
Multiplication by a unit or addition of a constant does not change decomposability, since $$\label{eq:11}
f = g \circ h \Longleftrightarrow a f+b = (a g+b) \circ h$$ for all $f$, $g$, $h$ as above and $a,b \in F$ with $a\neq 0$. In other words, the set of decomposable polynomials is invariant under this action of $F^{\times} \times F$ on $F[x]$.
Furthermore, any decomposition $(g,h)$ can be normalized by this action, by taking $a = \operatorname{lc} (h)^{-1} \in F^{\times}$, $b=-a \cdot h(0)
\in F$, $g^{*} = g((x-b)a^{-1}) \in F[x]$, and $h^{*} = ah+b$. Then $g\circ h = g^{*} \circ h^{*}$ and $g^{*}$ and $ h^{*}$ are monic original.
It is therefore sufficient to consider compositions $f = g \circ h$ where all three polynomials are monic original. In such a tame decomposition, $g$ and $h$ are uniquely determined by $f$ and $\deg
g$. For $n \geq 1$ and any proper divisor $e$ of $n$, we write $$\begin{aligned}
P_{n}(F) & = \{ f \in F[x] \colon \text{$f$ is monic original of degree $n$}\}, \label{eq:45} \\
D_{n}(F) & = \{ f \in P_{n}(F) \colon \text{$f$ is
decomposable} \}, \label{eq:46} \\
D_{n,e}(F) & = \{ f \in P_{n}(F) \colon \text{$f = g \circ h$
for some $(g,h) \in P_{e}(F) \times P_{n/e}(F)$} \}.\end{aligned}$$ Thus $P_{n}(F)$ and $D_{n}(F)$ are the subsets of original polynomials in the sets $P_{1,n}(F)$ and $D_{1,n}(F)$, respectively, as defined in the context of multivariate polynomials () but with right component $h$ of degree at least $2$. We sometimes leave out $F$ from the notation when it is clear from the context and have over a finite field $\mathbb{F}_{q}$ with $q$ elements $$\begin{aligned}
\#P_{n} &= q^{n-1},\\
\#D_{n,e} &\leq q^{e+n/e-2}.\end{aligned}$$ The set $D_{n}$ of all decomposable polynomials in $P_{n}$ satisfies $$\label{substack}
D_{n}= \bigcup_{\substack{e\mid n\\1<e<n}} D_{n,e}.$$ In particular, $D_{n} = \varnothing$ if $n$ is prime and $x \in P_{1}$ is neither decomposable nor indecomposable. For the resulting inclusion-exclusion formula for $\# D_{n}$, we have to determine the *collisions* (or nonuniqueness) of decompositions, that is, different components $(g,h)\neq(g^{*},h^{*})$ with equal composition $g\circ h=g^{*}\circ h^{*}$.
It is useful to single out a special case of wild compositions when $p
> 0$.
\[rem:coll\] We call an $f \in P_{n} \cap F[x^{p}]$ a *Frobenius composition*, since then $f = g^{*} \circ x^{p}$ for some $g^{*}
\in P_{n/p}$, and any decomposition $(g,h)$ of $f = g \circ h$ is a *Frobenius decomposition*. We denote by $\varphi \colon F
\longrightarrow F$ the Frobenius endomorphism over a field $F$ of characteristic $p$, with $\varphi(a)= a^{p}$ for all $a \in F$, and extend it to an $\mathbb{F}_{p}$-linear map $\varphi \colon P_{n}
\longrightarrow P_{n}$ with $\varphi(x)=x$. For $h \in P_{n/p}
\mysetminus \{x^{p}\}$, this provides the collision $$\label{eq:frob}
x^{p} \circ h = \varphi (h) \circ x^{p}.$$
If $F$ is perfect – in particular if $F$ is finite or algebraically closed – then $\varphi$ is an automorphism on $F$ and every Frobenius composition except $x^{p^{2}}$ is a collision as in . Over $F = \mathbb{F}_{q}$, this yields $q^{p - 1} -
1$ collisions in $D_{p^{2}}$ and $q^{n/p-1}$ collisions in $D_{n}$ for $p \mid n \neq p^{2}$, called *Frobenius collisions*. This example is noted in @sch82c [Section I.5, page 39].
For $f \in P_{n} (F)$ and $a \in F$, the *original shift* of $f$ by $a$ is $$\label{eq:14}
f^{[a]} = (x-f(a)) \circ f \circ (x+a) \in P_{n}(F).$$ Original shifting defines a group action of the additive group of $F$ on $P_{n}(F)$. Shifting respects decompositions in the sense that for each decomposition $(g,h)$ of $f$ we have a decomposition $(g^{[h(a)]}, h^{[a]})$ of $f^{[a]}$, and vice versa. We denote $(g^{[h(a)]}, h^{[a]})$ as $(g, h)^{[a]}$.
Normal form for Ritt’s Second Theorem {#sec:ritt-2}
-------------------------------------
Ritt presented two types of essential collisions: $$\begin{aligned}
x^{\ell}\circ x^{k}w(x^{\ell}) & =
x^{k\ell}w^{\ell}(x^{\ell})=x^{k}w^{\ell}\circ
x^{\ell}, \label{al:colli} \\
T_{m}(x,z^{\ell})\circ T_{\ell}(x,z) & = T_{\ell m}(x,z)=T_{\ell}(x,z^{m})\circ
T_{m}(x,z),\end{aligned}$$ where $w\in F[x]$, $z\in F^{\times} = F \mysetminus \{0\}$, and $T_{m}$ is the $m$th Dickson polynomial of the first kind. And then he proved that these are all possibilities up to composition with linear polynomials. This involved four unspecified linear functions, and it is not clear whether there is a relation between the first and the second type of example.
presents a normal form for the decompositions in Ritt’s Theorem under Zannier’s assumption $g'(g^{*})'\neq 0$ and the standard assumption $\gcd(\ell,m)=1$, where $m=k+\ell\deg w$ in . This normal form is unique unless $p\mid m$.
\[th:fifi\] Let $F$ be a field of characteristic $p \geq 0$, let $m>\ell \geq 2$ be integers with $\gcd(\ell, m) = 1$ and $n=\ell m$. Furthermore, we have monic original $f, g, h,
g^{*}, h^{*} \in F[x]$ satisfying $$\begin{gathered}
f = g \circ h = g^{*} \circ h^{*}, \label{eq:2} \\
f, g, h, g^{*}, h^{*} \text{ are monic original}, \label{eq:6} \\
\deg g = \deg h^{*} = m, \deg h = \deg g^{*} = \ell, \label{eq:8} \\
g'(g^{*})' \neq 0, \label{eq:15}
\end{gathered}$$ where $g'= \partial g/ \partial x$ is the derivative of $g$. Then either \[th:fifi-1\] or \[th:fifi-2\] hold, and \[th:fifi-3\] is also valid.
\[th:fifi-1\] (First Case) There exists a monic polynomial $w \in F[x]$ of degree $s$ and $a \in F$ so that $$\label{eq:mopo}
f= (x^{k\ell}w^{\ell}(x^{\ell}))^{[a]} ,$$ where $m=s\ell+k$ is the division with remainder of $m$ by $\ell$, with $1 \leq k < \ell$. Furthermore, we have $$\begin{gathered}
\begin{split}
(g,h) & = (x^{k}w^{\ell},x^{\ell})^{[a]}, \\
(g^{*}, h ^{*}) & = (x^{\ell}, x^{k} w (x^{\ell}))^{[a]} ,
\end{split} \label{eq:unidet-1} \\
kw+\ell xw' \neq 0 \text{ and } p \nmid \ell. \label{eq:unidet}
\end{gathered}$$
Conversely, any $(w,a)$ as above for which holds yields a collision satisfying through , via . If $p\nmid m$, then $(w,a)$ is uniquely determined by $f$ and $\ell$.
\[th:fifi-2\] (Second Case) There exist $z,a \in F$ with $z \neq 0$ so that $$\label{eq:TN}
f = T_{n}(x,z)^{[a]} .
$$ Now $(z,a)$ is uniquely determined by $f$. Furthermore, we have $$\begin{gathered}
\begin{split}
(g,h) & = (T_{m}(x,z^{\ell}) ,
T_{\ell}(x,z))^{[a]}, \\
(g^{*}, h^{*}) & = (T_{\ell}(x,z^{m}),
T_{m}(x,z))^{[a]},
\end{split} \label{eq:ab} \\
p \nmid n. \label{eq:ab-5}
\end{gathered}$$ Conversely, if holds, then any $(z,a)$ as above yields a collision satisfying through , via .
\[th:fifi-3\] When $\ell \geq 3$, the First and Second Cases are mutually exclusive. For $\ell=2$, the Second Case is included in the First Case.
If $p \nmid n$, then the case where $\gcd (\ell,m) \neq 1$ is reduced to the previous one by a result of [@tor88a]. This determines $D_{n,\ell} \cap D_{n,m}$ exactly if $p \nmid n =
\ell m$.
\[thm:FFC\] Let $\Fq$ be a finite field of characteristic $p$, let $\delta$ denote Kronecker’s delta function, and let $m > \ell \geq 2$ be integers with $p \nmid n = \ell m$, $i =
\gcd(\ell,m)$ and $s=\lfloor m/\ell \rfloor$. For the number of monic original polynomials of degree $n$ over $\Fq$ with left components of degree $\ell$ and $m$ we have $$\#(D_{n,\ell}(\Fq) \cap
D_{n,m}(\Fq)) = \begin{cases}
q^{2\ell+s-3} & \text{if } \ell \mid m,\\
q^{2i}(q^{s-1}+(1- \delta_{\ell,2})
(1-q^{-1})) & \\
\quad \quad \quad \leq q^{2\ell+s-3} & \text{otherwise}.
\end{cases}$$
In the remaining case where $p \mid n$, the Frobenius collisions are easily counted and therefore excluded. We have the following upper bounds.
\[cor:ffchar\] Let $\Fq$ be a finite field of characteristic $p$ and $\ell$, $m$, $n \geq 2$ be integers with $p \mid n = \ell m$, and let $c =\#(D_{n, \ell}(\Fq) \cap D_{n,m} (\Fq) \mysetminus F[x^{p}])$ be the number of monic original polynomials of degree $n$ over $\Fq$ with left components of degree $\ell$ and $m$ that are not Frobenius collisions. Then the following hold.
\[cor:ffchar-2\] If $p \nmid \ell$, then $$c \leq q^{m+\lceil \ell/p \rceil-2}.$$
\[cor:ffchar-3\] If $p\mid \ell$ and $\ell < m$, we set $b=\lceil (m-\ell+1)/\ell\rceil$. Then $$c\leq q^{m+\ell-b+\lceil b/p\rceil-2}.$$
For perspective, we also note the following lower bounds on $c$ from [@gat08c; @gat13]. Unlike the exact result of , there is a substantial gap between the upper and lower bounds.
\[cor:ffcharb\] Let $\Fq$ be a finite field of characteristic $p$, $\ell$ a prime number dividing $m >\ell$, assume that $p \mid n = \ell m$, and let $c =\#(D_{n, \ell} (\Fq) \cap D_{n,m} (\Fq) \mysetminus F[x^{p}])$ be the number of monic original polynomials of degree $n$ over $\Fq$ with left components of degree $\ell$ and $m$ that are not Frobenius collisions. Then the following hold.
\[cor:ffcharb-4\] If $p=\ell \mid m$ and each nontrivial divisor of $m / p$ is larger than $p$, then $$c \geq q^{2p+m/p-3}(1-q^{-1})(1-q^{-p+1}).$$
\[cor:ffcharb-1\] If $p \neq \ell$ divides $m$ exactly $d \geq 1$ times, then $$\label{eq:ffcharb-1}
c \geq q^{2\ell+m/\ell-3}(1-q^{-m/\ell})(1-q^{-1}(1+q^{-p+2}
\frac{(1-q^{-1})^{2}}{1-q^{-p}}))$$ if $\ell \nmid p^{d}-1$. Otherwise we set $\mu=\gcd(p^{d}-1,\ell)$, $r=(p^{d}-1)/\mu$ and have $$\begin{aligned}
\label{al:ffcharb-1}
\begin{aligned}
c & \geq q^{2\ell+m/\ell-3}\bigl((1-q^{-1}(1+q^{-p+2}
\frac{(1-q^{-1})^{2}}{1-q^{-p}}))(1-q^{-m/\ell})\\
& \quad-q^{-m/\ell-r+2}
\frac{(1-q^{-1})^{2}(1-q^{-r(\mu-1)})}{1-q^{-r}}
(1+q^{-r(p-2)})\bigr).
\end{aligned}\end{aligned}$$
The number of decomposable univariate polynomials {#sec:cdup}
-------------------------------------------------
The basic statement is that $\alpha_{n}$ as in is an approximation to the number of monic original decomposable polynomials of degree $n$, with relative error bounds of varying quality. The following is a condensed version of the more precise bounds in [@gat08c].
\[cor:Fq\] Let $\Fq$ be a finite field with $q$ elements and characteristic $p$, let $\ell$ be the smallest prime divisor of the composite integer $n \geq 2$, and $$\label{eq:17}
\alpha_{n} =
\begin{cases}
2q^{\ell+n/\ell-2}
& \text{if } n \neq \ell^{2}, \\
q^{2\ell-2}
& \text{if } n = \ell^{2}.\\
\end{cases}$$ Then the following hold for the number $\# D_{n} (\Fq)$ of decomposable monic original polynomials of degree $n$ over $\Fq$, where $p \parallel n$ means that $p$ divides $n$ exactly twice.
\[cor:Fq-6\] $ q^{2\sqrt{n}-2} \leq \alpha_{n}
\leq 2q^{n/2}$.
\[cor:Fq-1\] $ \alpha_{n}/2 \leq \# D_{n} (\Fq) \leq
\alpha_{n}(1+q^{-n/3\ell^{2}}) < 2\alpha_{n} \leq 4q^{n/2}$.
\[cor:Fq-2\] If $n\neq p^{2}$ and $q> 5$, then $\# D_{n} (\Fq) \geq
(3-2q^{-1})\alpha_{n}/4 \geq q^{2\sqrt{n}-2}/2$.
\[cor:Fq-4\] Unless $p = \ell \parallel n$ and , we have $\#D_{n} (\Fq) \geq \alpha_{n}(1-2q^{-1})$.
\[cor:Fq-5\] If $p \nmid n$, then $| \# D_{n} (\Fq) -
\alpha_{n} | \leq \alpha_{n}\cdot q^{-n/3\ell^{2}}$.
The relative error in \[cor:Fq-5\] is exponentially decreasing in the input size $n \log q$, in the tame case and for growing $n/3\ell^{2}$. In \[cor:Fq-4\], the factor is $1 + O(q^{-1})$ over $\Fq$. When $p = \ell \parallel n$, then we have a factor of about $2$ in \[cor:Fq-1\], which is improved to about $4/3$ in \[cor:Fq-2\]. The case $n = p^{2}$ is settled in .
Beyond the previous precise bounds, without asymptotics or unspecified constants, we now derive some conclusions about the asymptotic behavior. There are two parameters: the field size $q$ and the degree $n$. When $n$ is prime, then $\#
D_{n} (\Fq) = 0$, and prime values of $n$ are excepted in the following. We consider the asymptotics in one parameter, where the other one is fixed, and also the special situations where $\gcd(q,n)=1$. Furthermore, we denote as “$q,n\longrightarrow
\infty $” any infinite sequence of pairwise distinct $(q,n)$. The cases $n=4$ and $p^{2}\parallel n \neq p^2$ for some prime $p$ are the only ones where our methods do not show that $\# D_{n} (\Fq)/\alpha_{n}\longrightarrow 1$.
\[thm:consid\] Let $\# D_{n}(\Fq)$ be the number of decomposable monic original polynomials of degree $n$ over $\Fq$, $\alpha_{n}$ as in , and $\nu_{q,n} = \# D_{n}(\Fq)/\alpha_{n}$. We only consider composite $n$.
\[thm:consid-1\] For any $q$, we have $$\label{eq:25}
\lim_{\substack{n\to\infty \\ \gcd(q,n)=1}}{\nu_{q,n}}=1,$$ $$\label{eq:24}
\underset{n\to\infty}{\lim\sup} ~ {\nu_{q,n}}=1,$$ $$\begin{aligned}
\frac 1 2 &\leq \nu_{q,n} \text{ for any }
n,\\
\frac{3-2q^{-1}}{4}&\leq \nu_{q,n} \text{ for any }n \text{ if }
q > 5.
\end{aligned}$$
\[thm:consid-2\] Let $n$ be a composite integer and $\ell$ its smallest prime divisor. Then $$\label{eq:30}
\lim_{\substack{q\to\infty\\ \gcd(q,n)=1}}{\nu_{q,n}}=1,$$ $$\label{eq:29}
\underset{q\to\infty}{\lim\sup} ~{\nu_{q,n}}=1,$$ $$\underset{{q\to\infty}}{\lim\inf}
~{\nu_{q,n}}\begin{cases}
= 2/3 & \text{ if } n = 4,\\
\geq\frac 1 4 (3+\frac{1}{\ell+1})\geq \frac 5 6 &\text{ if }
\ell^{2}\parallel n \text{ and }n\neq \ell^{2},\\
=1 &\text{ otherwise.}
\end{cases}$$
\[thm:consid-3\] For any sequence $q,n \rightarrow
\infty$, we have $$\label{eq:32}
\lim_{\substack{q,n\to\infty\\ \gcd(q,n)=1}}{\nu_{q,n}}=1,$$ $$\label{eq:31}
\frac 1 2 \leq \underset{q,n\to\infty}{\lim\inf} ~{\nu_{q,n}}\leq
\underset{q,n\to\infty}{\lim\sup}
~{\nu_{q,n}}=1.$$
Collisions at degree $p^{2}$ {#sec:prelim}
----------------------------
The previous section gives satisfactory estimates for the number of decomposable polynomials at degree $n$ unless $p^{2} \parallel n$. The material of this section determines the number in the easiest of these open cases, namely for $n = p^{2}$.
First, we present two classes of explicit collisions at degree $r^{2}$, where $r$ is a power of the characteristic $p>0$ of the field $F$. The collisions of consist of additive and subadditive polynomials. A polynomial ${A}$ of degree $r^{k}$ is *$r$-additive* if it is of the form ${A}= \sum_{0 \leq i
\leq k} a_i x^{r^i}$ with all $a_i \in F$. We call a polynomial *additive* if it is $p$-additive. A polynomial is additive if and only if it acts additively on an algebraic closure $\overline{F}$ of $F$, that is ${A}(a + b) = {A}(a)+
{A}(b)$ for all $a$, $b \in \overline{F}$; see @gos96 [Corollary 1.1.6]. The composition of additive polynomials is additive, see for instance Proposition 1.1.2 of the cited book. The decomposition structure of additive polynomials was first studied by [@ore33b]. @dorwha74 [Theorem 4] show that all components of an additive polynomial are additive. [@gie88b] gives lower bounds on the number of decompositions and algorithms to determine them.
For a divisor $m$ of $r -1$, the *$(r,m)$-subadditive* polynomial associated with the $r$-additive polynomial ${A}$ is ${S}= x(\sum_{0 \leq i \leq k} a_i x^{(r^i - 1)/m})^m$ of degree $r^{k}$. Then ${A}$ and ${S}$ are related as $x^m \circ {A}= {S}\circ x^m$. @dic97 notes a special case of subadditive polynomials, and [@coh85] is concerned with the reducibility of some related polynomials. [@coh90b; @coh90c] investigates their connection to exceptional polynomials and coins the term “sub-linearized”; see also [@cohmat94]. @couhav04 derive the number of indecomposable subadditive polynomials and present an algorithm to decompose subadditive polynomials.
@ore33b [Theorem 3] describes exactly the right components of degree $p$ of an additive polynomial. [@henmat99] relate such additive decompositions to subadditive polynomials, and in their Theorems 3.4 and 3.8 describe the collisions of below. shows that together with those of and the Frobenius collisions of , these examples and their shifts comprise all collisions at degree $p^{2}$.
\[thm:nonadd\] Let $r$ be a power of $p$, $u, s\in F^{\times}$, $\varepsilon \in \{0,1\}$, $m$ a positive divisor of $r-1$, $\ell = (r-1)/m$, and $$\label{eq:7}
\begin{split}
f &= {S({u,s,\varepsilon,m})} = x (x^{\ell(r+1)}-\varepsilon u
s^{r}x^{\ell} + us^{r+1})^{m} \in P_{r^{2}}(F), \\
T &= \{t \in F \colon t^{r+1} -\varepsilon ut + u = 0\}.
\end{split}$$ For each $t \in T$ and $$\label{eq:80}
\begin{split}
g & = x (x^{\ell}-us^{r}t^{-1})^{m}, \\
h & = x (x^{\ell}-st)^{m},
\end{split}$$ both in $P_{r}(F)$, we have $f = g \circ h$. Moreover, $f$ has a $\# T$-collision.
The polynomials $f$ in are “simply original” in the sense that they have a simple root at $0$. This motivates the designation $S$. The second construction of collisions goes as follows.
\[thm:constmulti\] Let $r$ be a power of $p$, $b \in F^\times$, $a \in F \mysetminus \{0, b^{r}\}$, $a^* = b^r - a$, $m$ an integer with $1 < m <r-1$ and $p \nmid m$, $m^* = r - m$, and $$\label{eq:3normal}
\begin{split}
f = {M({a, b, m})} &= x^{m m^*} (x - b)^{m m^*} \left(x^m + a^* b^{-r} ((x-b)^{m} - x^m)\right)^m \\
&\quad\quad\quad \cdot
\left(x^{m^*} + a b^{-r} ((x-b)^{m^*} - x^{m^*})
\right)^{m^*},\\
g &= x^m (x - a)^{m^*}, \\
h &= x^{r} + a^{*}b^{-r}(x^{m^*}(x-b)^m - x^{r}), \\
g^* &= x^{m^*} (x - a^*)^m, \\
h^* &= x^{r} + ab^{-r}(x^m (x-b)^{m^*} - x^{r}).
\end{split}$$ Then $f = g \circ h = g^{*} \circ h^{*} \in P_{r^{2}}(F)$ has a 2-collision.
The polynomials $f$ in are “multiply original” in the sense that they have a multiple root at $0$. This motivates the designation $M$. The notation is set up so that ${}^{*}$ acts as an involution on our data, leaving $b$, $f$, $r$, and $x$ invariant.
[@zie11] points out that the rational functions of case (4) in Proposition 5.6 of [@avazan03] can be transformed into . Zieve also mentions that this example already occurs in unpublished work of his, joint with Robert Beals.
\[thm:normal\] Let $F$ be a perfect field of characteristic $p$ and $f \in P_{p^{2}}(F)$. Then $f$ has a collision $\{(g,h), (g^*, h^*)\}$ if and only if exactly one of the following holds.
- The polynomial $f$ is a Frobenius collision as in .
- The polynomial $f$ is simply original and there are $u$, $s$, $\varepsilon$, and $m$ as in and $w \in F$ such that $$f^{[w]} = {S({u,s,\varepsilon,m})}$$ and the collision $\{(g,h)^{[w]}, (g^{*},h^{*})^{[w]}\}$ is contained in the collision described in , with $\# T \geq 2$.
- The polynomial $f$ is multiply original and there are $a$, $b$, and $m$ as in and $w \in F$ such that $$f^{[w]} = {M({a,b,m})}$$ and the collision $\{(g,h)^{[w]}, (g^*,h^*)^{[w]}\}$ is as in .
In particular, the collisions in case \[class:1normal\] and case \[class:3normal\] have exactly $\# T$ and $2$ distinct decompositions, respectively. Inclusion-exclusion now yields the following exact formula for the number of decomposable polynomials of degree $p^{2}$ over $\Fq$.
\[cor:main\] Let $\Fq$ be a finite field of characteristic $p$, $\delta$ Kronecker’s delta function, and $\tau$ the number of positive divisors of $p-1$. Then for the number $\# D_{p^{2}} (\Fq)$ of decomposable monic original polynomials of degree $p^{2}$ over $\Fq$ we have $$\label{eq:1}
\begin{split}
\# D_{p^{2}} (\Fq) & = q^{2p-2} -q^{p-1} + 1 - \frac{(\tau q -q +1)(q-1)(qp-p-2)}{2(p+1)} \\
& \quad - (1- \delta_{p, 2}) \frac{q(q-1)(q-2)(p-3)}{4}.
\end{split}$$
In particular, we have $$\begin{aligned}
\# D_{4} (\Fq) & = q^{2} \cdot \frac{2+q^{-2}}{3} && \text{for $p =
2$}, \\
\# D_{9} (\Fq) & = q^{4} \bigl( 1 - \frac{3}{8} ( q^{-1} + q^{-2} - q^{-3} - q^{-4})\bigr) && \text{for $p = 3$},\\
\# D_{p^{2}} (\Fq) & = q^{2p-2} \bigl( 1 - q^{-p+1} + O(q^{-2p+5+1/d}) \bigr) &&\text{for $q = p^{d}$ and $p \geq 5$}.\end{aligned}$$
We have two independent parameters $p$ and $d$, and $q= p^d$. For two eventually positive functions $f, g \colon \mathbb{N}^{2} \rightarrow \mathbb{R}$, here $g \in O(f)$ means that there are constants $b$ and $c$ so that $g(p,d) \leq c \cdot f(p,d)$ for all $p$ and $d$ with $p+d \geq b$. We have the following asymptotics.
Let $p \geq 5$, $d \geq 1$, $q= p^d$, and $k \geq
1$. Then the number $c_{k}$ of decomposable monic original polynomials of degree $p^{2}$ over $\Fq$ with exactly $k$ decompositions is as follows $$\begin{aligned}
c_{1} & = q^{2p-2} (1 - 2q^{-p+1} + O(q^{-2p+5+1/d})), \\
c_{2} & = q^{p-1} (1 + O(q^{-p+4+1/d})), \\
c_{p+1} & = (\tau - 1) q^{3-3/d} \bigl(1 + O(q^{-\max\{2/d,1-1/d\}})\bigr) \\
& \subseteq O\bigl(q^{3-3/d+1/(d \loglog p)} \bigr), \\
c_{k} & = 0 \text{ if } k \notin \{1,2,p+1\}.\end{aligned}$$
leads to $\lim_{q\to\infty} \nu_{q,\ell^2} = 1$ for any prime $\ell > 2$ in \[thm:consid-2\]. For $n=4$, the sequence has no limit, but oscillates between values close to $\lim\inf_{q\to\infty} \,\nu_{q,4} = 2/3$ and to $\lim\sup_{q\to\infty}
\,\nu_{q,4} = 1$, and these are the only two accumulation points of the sequence $\nu_{q,4}$.
Open problems
=============
Further types of multivariate polynomials that are examined from a counting perspective include singular bivariate ones [@gat08-incl-gat07] and pairs of coprime polynomials [@houmul09]. It remains open to extend the methods of to singular multivariate ones and achieve exponentially decreasing error bounds for coprime multivariate polynomials.
For univariate decomposable polynomials, the question of good asymptotics for $\nu_{q,n}$ when $q$ is fixed and $n \to\infty$ is still open. More work is needed to understand the case where the characteristic $p$ is the smallest prime divisor of the degree $n$, divides $n$ exactly twice, and $n \neq p^{2}$. Ritt’s Second Theorem covers distinct-degree collisions, even in the wild case, see [@zan93]; it would be interesting to see a parametrization even for $p \mid m$ and obtain a similar classification for general equal-degree collisions.
Finally, this survey deals with polynomials only and the study of rational functions with the same methods remains open.
Acknowledgments
===============
This work was funded by the B-IT Foundation and the Land Nordrhein-Westfalen.
|
---
abstract: 'The magnetic phase diagrams, and other physical characteristics, of the hole–doped [*La$_{2-x}$Sr$_x$CuO$_4$*]{} and electron–doped [*Nd$_{2-x}$Ce$_x$CuO$_4$*]{} high–temperature superconductors are profoundly different. Given that it is envisaged that the simplest Hamiltonians describing these systems are the same, viz. the $t - t^\prime - J$ model, this is surprising. Here we relate these physical differences to their ground states’ single–hole quasiparticles, the spin distortions they produce, and the spatial distribution of carriers for the multiply–doped systems. As is well known, the low doping limit of the hole–doped material corresponds to $\vec k = ({\pi\over 2}, {\pi\over 2})$ quasiparticles, states that generate so–called Shraiman–Siggia long–ranged dipolar spin distortions via backflow. These quasiparticles have been proposed to lead to an incommensurate spiral phase, an unusual scaling of the magnetic susceptibility, as well as the scaling of the correlation length defined by $\xi^{-1} (x,T) = \xi^{-1} (x,0)~+~\xi^{-1} (0,T)$, all consistent with experiment. We suggest that for the electron–doped materials the single–hole ground state corresponds to $\vec k = (\pi, 0)$ quasiparticles; we show that the spin distortions generated by such carriers are short–ranged. Then, we demonstrate the effect of this single–carrier difference in many–carrier ground states via exact diagonalization results by evaluating $S(\vec q)$ for up to 4 carriers in small clusters. Consistent with experiment, for the hole–doped materials short–ranged incommensurate spin orderings are induced, whereas for the electron–doped system only commensurate spin correlations are found. Further, we propose that there is an important difference between the spatial distributions of mobile carriers for these two systems: for the hole–doped material the quasiparticles tend to stay far apart from one another, whereas for the electron–doped material we find tendencies consistent with the clustering of carriers, and possibly of low–energy fluctuations into an electronic phase separated state. Phase separation in this material is consistent with the mid–gap states found by recent ARPES studies. Lastly, we propose the extrapolation of an approach based on the $t - t^\prime - J$ model to the hole–doped 123 system.'
address:
- |
Dept. of Physics, Queen’s University,\
Kingston, Ontario CANADA K7L 3N6
- |
Dept. of Physics, Hong Kong University of Science and Technology,\
Clear Water Bay, Hong Kong\
author:
- 'R.J. Gooding and K.J.E. Vos'
- 'P.W. Leung'
date: 'To be published in Phys. Rev. B, Nov. 1, 1994.'
title: 'Theory of electron–hole asymmetry in doped [*CuO$_2$*]{} planes'
---
\#1[[$\backslash$\#1]{}]{}
Keywords: Weakly doped high $T_c$ superconductors; $t - t^{\prime} - J$ model; electronic phase separation.
Introduction: {#sec:intro}
=============
The [*CuO$_2$*]{} plane based high–temperature superconductors have anomalous normal state properties. One part of the normal state puzzle involves the spin orderings, and for the majority of this paper we focus on the differing magnetic phase diagrams for the hole–doped Bednorz–Müller [*La$_{2-x}$Sr$_x$CuO$_4$*]{} compounds [@bednorz] in comparison to the electron–doped [*Nd$_{2-x}$Ce$_x$CuO$_4$*]{} materials [@tokura]. In particular, for both of these systems the $x=0$ phase possesses three–dimensional antiferromagnetic long–ranged order [@birgeneau; @electronmag]. However, one must heavily dope the $Nd$ compound to destroy this order, say $\sim$ 12 %, while only a small doping of $\sim$ 2 % is required to kill long–ranged antiferromagnetic order in the $La$ cuprate material. A schematic of the contrasting phase diagrams is given in Fig. 1.
One important aspect of the magnetic orderings found in these systems involves the kinds of spin correlations that these systems exhibit when doped to levels greater than 2 %. As mentioned above, for the electron–doped material antiferromagnetic order persists until $\sim 12 \%$ doping levels are reached. However, when the hole–doped system has say 7 % holes in a [*CuO$_2$*]{} plane, short–ranged incommensurate magnetic ordering is found. This appears experimentally in the dynamic magnetic response, and has been found via inelastic neutron scattering for a number of intermediate doping levels [@aeppli; @mason; @thurston2]. Consequently, an alternate way in which one can phrase the question concerning the differing magnetic phase diagrams is: why does the electron–doped system maintain commensurate antiferromagnetic order at doping levels that induce incommensurate correlations in the hole–doped system?
Several theories have addressed the phase diagrams of these doped antiferromagnetic insulators. When the [*La*]{}-based system is doped, the carriers predominantly occupy [*O*]{} sites, in particular as [*O$^-$*]{} states, whereas for the [*Nd*]{}-based material the carriers are believed to be associated with the addition of an electron to a [*Cu 3d $^9$*]{} ion, thus forming [*Cu$^+$*]{}. Then, a common explanation of these differing magnetic phase diagrams follows from the assumption of the complete localization of the carriers, and the ensuing spin distortions generated by such static defects. Namely: (i) The dramatic reduction of the Néel temperature $T_N (x)$ in the hole–doped system results from frustrating ferromagnetic bonds being embedded in an antiferromagnetic background, the so–called Aharony model [@aharony]. Such frustrated bonds generate a long–ranged spin distortion with dipolar symmetry, and numerical studies [@polarons] have shown that a version of this perturbation renormalized by quantum fluctuations [@polarons; @frenkel] indeed alters the spin–spin correlation length in a fashion consistent with experiment [@keimer]. (ii) Fixed [*Cu$^+$*]{} sites in a background antiferromagnetic lattice act like static vacancies, and subsequently diminish the spin–wave stiffness [@manousakis] in a manner consistent with experiment [@thurston1]. Monte Carlo studies of this quantum dilution problem [@manousakis; @behre] do not suffer from the minus sign problem, and thus are capable of accurately characterizing the spin correlation length.
Unfortunately, these cannot be complete explanations of the differing magnetic phase diagrams. Near the antiferromagnetic phase boundary (temperature vs. doping) the carriers are mobile in both of these systems [@batlogg; @uji], and thus one must come to an understanding of the differing reductions of the spin orderings for mobile not just localized carriers. This is one of the focuses of this paper [@second].
Here we put forward a proposal explaining the electron–hole asymmetry of the magnetic phase diagrams shown in Fig. 1 for mobile carriers. Our work builds on our earlier study [@GVL] that showed that for the $t - J$ model and the hole–doped system, a tendency towards short–ranged incommensurate order vs. hole doping arises. The results of [@GVL] are consistent with experiment [@aeppli; @mason; @thurston2], and in part is further verification of earlier exact diagonalization work of Moreo et al. [@moreo1], as well as Monte Carlo work [@moreo2; @furukawa]. However, the important point of [@GVL] was that this incommensurability only occurred for the single–hole system having a ground state momentum of $\vec k = \pm ({\pi\over 2}, \pm {\pi\over 2})$ (since $\vec k =({\pi\over 2}, {\pi\over 2})$ quasiparticles generate long–ranged spin distortions similar to those produced by a ferromagnetic bond, the success of the above–mentioned Aharony model bodes well for any theory predicting that these carriers exist in the hole–doped 214 systems). If the momentum of the single–hole ground state is $\vec k = (\pi, 0)$, something that may be accomplished through the use of the $t - t^\prime - J$ model (see below) with $t^\prime / t > 0$, then the tendency towards incommensurability is eliminated and only commensurate ordering remains. Clearly, this is similar to the above–mentioned behaviour of the hole and electron–doped materials, and in this paper we will make this comparison complete.
Very shortly after [@GVL], Tohyama and Maekawa [@tohyama1] also studied the $t - t^\prime - J$ model, now for the electron and hole–doped systems. They suggested, consistent with a considerable amount of electronic structure work [@hybertsen; @eskes; @tohyama2], as well as angle–resolved photoemission spectroscopy results for the electron–doped system [@allen], that the [*Nd*]{} system corresponded to hopping integrals $t$ and $t^\prime$ with (approximately) just flipped signs in comparison to the same hopping integrals believed to be appropriate for the hole–doped materials (we elaborate on this relation in the next section). Then, by examining the energy to the first excited states, they demonstrated that the commensurate antiferromagnetic order was much more stable for the electron–doped system than for the hole–doped systems. As we show below, this is complimentary to our work in [@GVL], since the single hole ground state momentum for their electron–doped $t - t^\prime - J$ Hamiltonian is actually $\vec k = (\pi, 0)$! We then strengthen the arguments of Tohyama and Maekawa by evaluating the magnetic structure factor $S(\vec q)$ for the many–carrier ground states for these two systems. Indeed, using the relevant material parameters, we find incommensurate ordering tendencies for the hole–doped material, while for the electron–doped material the magnetic ordering is always commensurate, in complete agreement with experiment.
We shall also be concerned with other physical properties of the hole and electron–doped materials. For the $La$ cuprates, some experiments can be explained beginning with the assumption that at low doping levels Shraiman–Siggia [@SS] dipolar quasiparticles are present (implicit in these theories is the existence of hole pockets, or a “small" fermi surface, something that so far has not been observed). One such magnetic problem involves the scaling of the correlation length with doping and temperature, viz. $\xi^{-1} (x,T) = \xi^{-1} (x,0)~+~\xi^{-1} (0,T)$, and this has been found to be reproduced by a model (based on the existence of these dipolar quasiparticles) proposed by one of us [@nskyrmions] for both mobile and localized carriers, and is in close agreement with experiment [@keimer]. Further, a theory [@commentmillis] of the temperature and doping dependent magnetic susceptibility $\chi (x,T)$ has been found to be consistent with experiment [@johnston]. Lastly, the same theory [@subir] also predicts the existence of a pseudogap, something that may be consistent with experiment [@timusk].
We are proposing that these dipolar quasiparticles do not exist in the weakly–doped $Nd$ cuprates, and thus, e.g., $\xi (x,T),~\chi (x,T)$, and the optical properties of this system, may be quite different from those of the $La$ cuprates. Unfortunately, the collection of experiments performed on the electron–doped materials is not as complete as the set performed for the hole–doped system, and so a full comparison is not possible. However, there is one striking difference between the normal state properties of these systems, and that is the resistivities at optimal doping. The $La$ system shows a linear–$T$ resistivity [@batlogg], whereas the $Nd$ system shows a clear fermi–liquid–like $T^2$ dependence [@hikada]. Another intriguing experimental result for the electron–doped material corresponds to the mid–gap states seen in the angle–integrated photoemission work of Anderson $et~al.$ [@allen]. It has been proposed [@emery2] that this is a signal of phase separation, and to this end we have examined the spatial distributions of many carriers, as well measured the interaction energies between carriers. Our results tend to support a phase separation hypothesis, but [*only*]{} for the electron–doped system.
Our paper is organized as follows. In § \[sec:Ham\] we discuss the formal aspects of the $t -t^\prime - J$ model and display the electron–hole symmetries and asymmetries that can be identified when these materials are doped away from half filling. In § \[sec:numerics\] we summarize previously published and our recent exact diagonalization results for the one–carrier ground states for these two systems, and show that these two systems have different one–carrier ground states. Then, the spin correlations found in $S(\vec q)$ as a function of doping are shown to be in agreement with the phase diagram of Fig. 1. In § \[sec:analysis\] we examine the spin distortions produced by each of these quasiparticles via our semi–classical analysis of this Hamiltonian for a single carrier; this will be shown to agree with the exact diagonalization results. Quantum fluctuations are shown to not affect these conclusions. Then we examine the spatial distribution of carriers pointing out the acute difference between the hole and electron–doped systems — from this we relate the tendency towards clustering of carriers of the electron–doped system to the mid–gap states seen in photoemission. Finally, in § \[sec:discussion\] we discuss the relevance of our results to the transport properties of both kinds of systems, and as a prelude to possible future work we consider the extrapolation of this approach to other systems, and in particular consider the commensurability that persists in the hole–doped $YBaCuO$ system.
$\lowercase{t} -\lowercase{t}^\prime - J$ model: {#sec:Ham}
================================================
Electron–Hole Asymmetry induced by $\lowercase{t}^\prime$: {#sect:asymm1D}
----------------------------------------------------------
We begin our discussion of the $t - t^\prime - J$ model by first considering a simpler model involving only hopping terms to the nearest and next nearest neighbour sites along a one–dimensional chain — this model provides the most direct demonstration of the electron–hole asymmetry about half filling that $t^\prime$ introduces into the problem.
Consider a linear chain with a single orbital per site, and define the vacuum to be the state with all orbitals devoid of electrons. Introduce $$H_{hop} = -~t\sum_{<i,j>_{nn}~\sigma} \Big( c_{i\sigma}^\dagger
c_{j\sigma} + h.c. \Big)
-~t^\prime\sum_{<i,k>_{nnn}~\sigma} \Big( c_{i\sigma}^\dagger
c_{k\sigma} + h.c. \Big)
\label{equation:hopping}$$ where $<~>_{nn}$ denotes nearest neighbours, $<~>_{nnn}$ next nearest neighbours, and $c_{i\sigma}$ is the electron annihilation operator for site $i$ and spin $\sigma$ (note that these operators are [*not*]{} constrained to prohibit double occupancy — see § \[sect:sch\]). Then imagine that the system has an up spin at every site except one, at which no electron is present. The energy eigenvalues for this hole plus spin polarized background are given by $$E (k) = - 2 t \cos (k) + 2 t^\prime \cos (2k)~~~~~~~({\rm hole}).
\label{equation:holettp}$$ Now consider a system with up spins at every site, and that in addition to these electrons one down–spin electron is placed in some orbital. Then the energy eigenvalues of this composite spin–polarized plus mobile electron system are $$E (k) = - 2 t \cos (k) - 2 t^\prime \cos (2k)~~~~~~~({\rm electron}).
\label{equation:electronttp}$$
From these results it is to be noted that while the near–neighbour hopping leads to identical energies for both the hole and electron–doped systems, the next–nearest–neighbour hopping has different signs. This reflects the breaking of electron–hole symmetry that $t^\prime$ induces, and thus its inclusion in our starting Hamiltonian is crucial. It is to be stressed that this asymmetry is found in any dimension, for any spin background, but only for doping levels close to half filling.
Strong–Coupling Hamiltonian: {#sect:sch}
----------------------------
Equation (\[equation:hopping\]) is not a good starting point for describing the physics of the $CuO_2$ planes in either $LaSrCuO$ or $NdCeCuO$, since it only describes free carriers. If instead one begins with an extended Hubbard model description [@emery1] of the strong correlation effects known to be present in these systems, and then examines the systems at close to half filling, one can obtain the so–called ${t} -{t}^\prime - J$ model [@dagotto]. To be specific, one considers $$H =
-~t\sum_{<i,j>_{nn}~\sigma} \Big( \tilde c_{i\sigma}^\dagger
\tilde c_{j\sigma} + h.c. \Big)
-~t^\prime\sum_{<i,k>_{nnn}~\sigma} \Big( \tilde c_{i\sigma}^\dagger
\tilde c_{k\sigma} + h.c. \Big)
+ J \sum_{<i,j>_{nn}}
(\vec S_i \cdot \vec S_j - {1\over 4} n_i n_j)
\label{equation:ttpJ}$$ where $J$ is the Heisenberg superexchange constant coupling spins $\vec S_i$ between nearest–neighbour sites. Also, in Eq. (\[equation:ttpJ\]) constrained electron operators are used (constrained to disallow double occupancy at any site), viz. $$\tilde c_{i\sigma} = c_{i\sigma} ( 1 - c^\dagger_{i-\sigma}c_{i-\sigma} )~~.
\label{equation:contrainedcs}$$
We wish to stress that our use of the ${t} - {t}^\prime - J$ model is based on beginning with a three–band extended Hubbard model description [@emery1] for [*both*]{} the $LaSrCuO$ and $NdCeCuO$ systems. While it is well known [@zhangrice; @shastry; @goodingelser] that for the hole–doped systems one may approximately map the three-band Hubbard model onto the one–band $t - t^\prime - J$ model (N.B. — not necessarily the one–band Hubbard model), it may be argued that this is also the correct starting point for the electron–doped systems. For example, the dominant contribution to the $t^\prime$ hopping integral for the electron doped system is [*not*]{} due to a direct $Cu - Cu$ overlap, but rather follows from a third–order $Cu \rightarrow O \rightarrow O \rightarrow Cu$ hopping [@schlueter]. Thus, in what follows we consider (i) the $t$ and $t^\prime$ parameters that are to be [*fitted*]{} via quantum cluster studies (see below), and not due to a strong–coupling mapping of the one–band Hubbard model onto a Hibert space prohibiting double occupancy [@trugman], and (ii) the superexchange constant $J$ to be found from either quantum cluster studies, or from comparison to experiments probing the magnetic structure and/or excitations.
In the previous subsection we were able to treat hole and electron–doped systems with the same starting Hamiltonian. However, now it is clear that the above Hamiltonian, for a vacuum corresponding to no electrons at any site, cannot be used to describe both hole and electron doping of the half–filled state. For example, a vacancy can easily be added to a spin–polarized ferromagnetic state, and is completely mobile, while a down spin electron added to such a spin background is completely localized. Consequently, with this Hamiltonian and this vacuum, only hole–doped $CuO_2$ planes may be investigated.
To investigate the electron–doped system it is usual to introduce an electron–hole transformation, viz. $$c_{i\sigma} \rightarrow a^\dagger_{i\sigma}
\label{equation:eh1}$$ where $a_{i\sigma}$ is the annihilation operator of a hole at site $i$ of spin $\sigma$. (With this transformation it follows that one now considers a vacuum with no holes at any site.) However, this does not lead to the desired Hamiltonian as one finds that the new hopping term does not describe constrained (viz. two holes are never allowed on the same site) hopping. Instead, one must utilize a more general model than Eq. (\[equation:ttpJ\]), one that includes so–called doublon hopping processes [@clarke], and from this starting Hamiltonian it is straightforward to derive the required transformation. The result of this transformation can then be shown to be equivalent to $$\tilde c_{i\sigma} \rightarrow \tilde a^\dagger_{i\sigma}
\label{equation:eh2}$$ where the $\tilde a_{i\sigma}$ also satisfy fermionic anti–commutation relations. Thus, one eventually finds that the appropriate $t - t^\prime - J$ model for the electron–doped system is the same as Eq. (\[equation:ttpJ\]) but with the $\tilde a_{i\sigma}$ replacing the $\tilde c_{i\sigma}$ [*and*]{} with the minus signs in front of $t$ and $t^\prime$ changed to be plus signs.
Summarizing the above discussion, the $t - t^\prime - J$ Hamiltonian will be used to model the physical systems that we are interested in. We consider this model to possess the bare minimum number of processes through which both the hole and electron–doped $CuO_2$ planes can be represented. To be specific, (i) Eq. (\[equation:ttpJ\]) with a vacuum of zero electrons at every site of a square lattice will describe the hole–doped system, and (ii) for the electron–doped systems we use $$H =
~t\sum_{<i,j>_{nn}~\sigma} \Big( \tilde a_{i\sigma}^\dagger
\tilde a_{j\sigma} + h.c. \Big)
+~t^\prime\sum_{<i,k>_{nnn}~\sigma} \Big( \tilde a_{i\sigma}^\dagger
\tilde a_{k\sigma} + h.c. \Big)
+ J \sum_{<i,j>_{nn}}
(\vec S_i \cdot \vec S_j - {1\over 4} n_i n_j)
\label{equation:ttpJel}$$ For this latter Hamiltonian we use a vacuum of zero holes at every site of a square lattice. Thus, note that, quite simply, in comparison to the hole–doped Hamiltonian the signs of the $t$ and $t^\prime$ hopping terms are flipped and the $\tilde a$ operators replace the $\tilde c$ operators. Thus, from now on we simply use Eq. (\[equation:ttpJ\]) for both systems, specifying $t=1$ for hole doping, and $t=-1$ for electron doping [@tohyama1].
We now consider the numerical values of the hopping and superexchange parameters. A consequence of the holes residing on the $O$ sites in the $La$ material is that for a quantitative determination of the hopping parameters $t$ and $t^\prime$, something very different from overlap integrals must be evaluated. We refer the interested reader to the summary of this problem given by Tohyama and Maekawa [@tohyama1], and simply state the range of accepted values [@tohyama1; @hybertsen; @eskes]. For the hole–doped $LaSrCuO$ system, using Eq. (\[equation:ttpJ\]), one may scale all energies such that $t=1$. Then one has $J \approx .3 \rightarrow .4$ and $t^\prime \approx -.2 \rightarrow -.4$. For the electron–doped $NdCeCuO$ system we again use Eq. (\[equation:ttpJ\]) but now with $t= - 1$, and $J \approx .3 \rightarrow .4$ and $t^\prime \approx + .2 \rightarrow + .4$. Thus, it is interesting to note that even though a Zhang–Rice singlet is not exactly equivalent to a $Cu~3d^8$ ion, the material parameters for these two systems are such that the electron–hole mapping given in Eq. (\[equation:eh2\]) predicts reasonably well the numerical values that one must employ for one system given the parameters of the other.
Numerical Results: {#sec:numerics}
==================
Stephan and Horsch [@stephan] have proposed that knowledge of the single–hole system does not lead to valuable information on the multiply–doped state. In [@GVL] we showed that this is not always the case, and that, in particular, knowledge of the one–hole ground state can aid in predicting the presence/absence of incommensurate correlations. Thus, we begin a discussion of our numerical work with a summary of the single–hole ground state.
There has already been a comprehensive exact diagonalization study of the ground state of one hole in the $t - t^\prime - J$ model for a $4 \times 4$ square cluster in Ref. [@gagliano] (since they use a bosonic representation, their sign of $t^\prime$ is flipped in comparison to ours). For the physically relevant ratio of $J/t$ they find a $\vec k
= \pm ({\pi\over 2}, \pm {\pi\over 2})$ ground state when the $t^\prime$ corresponding to hole–doped materials is used (unless $\mid t^\prime \mid$ becomes too large), and a $\vec k = \pm (\pi,0), \pm (0,\pi)$ ground state for the $t^\prime$ associated with the electron–doped materials.
We have used a Lanczos routine and have also performed exact diagonalization evaluations [@leung] of the ground states of the $t - t^\prime - J$ model for $J=.4$, with $t = \pm 1$ for $t^\prime = \pm .1, \pm .2$ and $ \pm .3$. For the conventional $4 \times 4$ 16–site square cluster our work agrees with that of Ref. [@gagliano]. We have also studied the $\sqrt {8} \times \sqrt {32}$ 16–site cluster introduced in [@GVL], as well as some results generated using a $\sqrt {18} \times \sqrt {32}$ 24–site cluster [@finitesize]. For the hole–doped material, viz. when $t^\prime < 0$, we always find a ground state of $\vec k = \pm ({\pi\over 2}, \pm {\pi\over 2})$, while for the electron–doped material, viz. when $t^\prime > 0$, we always [@24site] find a ground state of $\vec k = (\pm\pi, 0),~(0,\pm\pi)$. Our results for these two different clusters are entirely consistent with the phase diagram of Ref. [@gagliano] found using the $4 \times 4$ cluster.
Our numerical results also lead us to expect that the band structures for the hole and electron–doped materials will be quite different. We find that the band structure of the hole–doped material is similar to Fig. 4 of [@GVL] (which is a $t^\prime = 0$ curve), although with the increased ratio of $\mid t^\prime / t \mid$ that we are using here the band along $\vec k = (0,0) \rightarrow (\pi,0)$ becomes quite flat. This is similar to recent photoemission work of the valence band of undoped $Sr_2CuO_2Cl_2$ [@wells], and provides some experimental support for a value of $t^\prime$ around $-0.3~t$. We have listed the single carrier band structure energies for our ${\sqrt 18} \times {\sqrt 32}$ 24–site cluster in Table I for both the hole and electron–doped systems having used $t^\prime / t~=~-0.3.$
The single–hole ground states of the hole and electron–doped materials are different — so what? As mentioned above, the important fact contained in Ref. [@GVL] is that a knowledge of the single–hole ground state can aid in understanding whether or not any short–ranged incommensurate correlations are introduced into the ground state of the multiply–doped system. This followed from (i) if the single–hole ground state is not a $\vec k = \pm ({\pi\over 2}, \pm {\pi\over 2})$ state, no tendencies towards incommensurate correlations occurred, consistent with the spiral phase prediction of Shraiman and Siggia [@spiral], and (ii) the electron momentum distribution function for 2, 3 and 4 holes was found to be composed of the half–filled fermi surface with dimples at those momenta corresponding to the single–hole ground states, thus suggesting that some form of rigid band filling [@GVL] is in effect. Combining this with the differing single–hole ground states mentioned above, it is clear that we can now extrapolate these abstract results to the physical systems under consideration.
Figures 2 and 3 show the magnetic structure factor for 0, 1, 2, 3, and 4 holes (electrons) for $t^\prime = \pm .3$ evaluated with the $\sqrt {8} \times \sqrt {32}$ cluster introduced in [@GVL] — similar results are obtained for other $\pm t^\prime$ pairs. For the hole–doped material, Fig. 2 displays that the peak in the structure factor shifts from $(\pi,\pi)$ for 0 and 1 hole, to $({3\pi\over 4},{3\pi\over 4})$ for 2 holes, to $({\pi\over 2},{\pi\over 2})$ for 3 and 4 holes. This behaviour is similar to the behaviour displayed in Fig. 5 of Ref. [@GVL] for $t^\prime = 0$. However, in Fig. 3 it is seen that no shifts of the maximum of $S(\vec q)$ away from $(\pi,\pi)$ occurs for any doping concentration away from half filling for the electron–doped system. These results are entirely consistent with Fig. 1, and, in particular, are consistent with the lack of any tendency of the electron–doped materials to display the kind of incommensurate correlations that are present in the hole–doped system [@incomdirection]. Clearly, these numerical results suggest that we have found a reasonable starting point from which one can hope to be able to describe the hole and electron–doped 214 systems. We now try to come to an understanding of these numerical results.
Analysis: {#sec:analysis}
=========
As mentioned in the introduction, the existence of Shraiman–Siggia quasiparticles in weakly doped $LaSrCuO$ is supported by the theories [@polarons; @nskyrmions; @commentmillis; @subir; @noha] that assume their existence, and are subsequently able to produce results consistent with experiment. To be specific, it seems that a scenario in which these quasiparticles are weakly interacting and tend to stay very far apart from one another (thus making it likely that some form of rigid band filling (e.g. with hole pockets around $\pm({\pi\over 2},\pm{\pi\over 2})$) is applicable.
Now we study the $(\pi,0)$ quasiparticles: we are proposing that these are the single–hole ground state constituents that exist in the weakly doped $NdCeCuO$ system. We wish to characterize the spin distortions that each such quasiparticle produces, as well as come to an understanding of the spatial distribution for a state which is multiply doped. This will allow us to (i) understand why the $(\pi,0)$ and $({\pi\over 2}, {\pi\over 2})$ quasiparticles lead to differing magnetic structure factors vs. doping, and (ii) examine some of the other physical features that are known to be specific to the electron–doped system.
According to the semiclassical theory of Shraiman and Siggia [@SS], for ground states with these wave vectors, differing distortions of the spin background occur. For the $\vec k = ({\pi\over 2},
{\pi\over 2})$ states, long–ranged spin distortions with dipolar symmetry are produced via backflow. For the $\vec k = (\pi,0)$ ground states, the hydrodynamic theory of Shraiman and Siggia [@SS] leads to the prediction that the spin distortion induced by one $(\pi,0)$ quasiparticle is short ranged. These ideas are verified in our Figs. 4 and 5 where our results from evaluating the energies at this wave vector for the $t - t^\prime - J$ model with the electron–doped material parameters using a semiclassical variational wave function, such as that described in [@SS], are displayed. To be specific, we utilize a product state of classical spins in an infinite lattice, and incorporate a broken antiferromagnetic symmetry in calculations of the hopping matrix elements for a single hole (see [@SS] for the details of such a variational wave function). Then, to display the range of the spin distortions for this ground state we have calculated the minimum energy as a function of the number of spins away from the hole that are allowed to be distorted from their undoped Neel configuration — these sites are shown in Fig. 4, where they are labeled according to the equivalent sites around a single carrier. In Fig. 5 we show the minimum energies found from the variational principle as a function of these site labels for both the hole–doped $\vec k = ({\pi\over 2},{\pi\over 2})$ one hole ground state, and the $\vec k = (\pi,0)$ electron–doped one carrier ground state. The long–ranged spin distortion of the hole–doped ground state is clear, whereas the short–ranged distortion of the electron–doped ground state is made manifest by the insensitivity of the minimum energy when one allows more distant neighbours to be distorted from their Neel state.
The above procedure may be repeated for all wave vectors for both hole and electron–doped Hamiltonians. We find that providing $t^\prime$ satisfies certain inequalities (which necessarily depend on the ratio of $J/t$) that the semiclassical band structure largely reproduces the band structures determined by exact diagonalization. Specifically, the ground states are reproduced when (i) $t^\prime > -.12$ for the hole–doped material, and (ii) $t^\prime > +.12$ for the electron–doped materials (these inequalities will be affected by quantum fluctuations — see below). This agreement with the quantum cluster studies provides further evidence that the renormalized classical description of the spin background ([*not*]{} including the charge carriers) of the weakly doped systems is valid at least at low temperatures [@birgeneau].
This semiclassical work ignores quantum fluctuations. However, as shown by Reiter [@reiter], for an infinite lattice when quantum fluctuations are included in such variational wave functions one still finds the Shraiman–Siggia dipolar quasiparticles with the long–ranged dipolar spin distortion intact. We have found [@unpublished1] that within this lowest–order self–consistent approximation [@reiter] the $(\pi,0)$ states still only involve short–ranged spin distortions, the same as the semiclassical prediction. Thus, it is not unexpected that our quantum cluster and semiclassical analyses agree.
Now let us consider the effect of the short–ranged spin distortions that this kind of quasiparticle produces, viz. why do these quasiparticles not introduce any incommensurate correlations into the spin texture. Given the short–ranged nature of the spin distortion, it is clear that they could lead to a disturbance of the spin texture similar to that found in the quantum dilution problem [@manousakis], since static vacancies also produce a short–ranged disturbance of the spin texture. However, the quantum dilution problem begins with the assumption that the dilutants are randomly distributed throughout a plane. We now show that this is probably not the case for the mobile carriers in the electron–doped materials.
In Ref. [@GVL] we presented some numerics for the hole–hole correlation function for a pair of holes doped into a $\sqrt {8} \times \sqrt {32}$ cluster described by the $t - J$ model. We found that the holes tended to stay as far apart as possible. (It is interesting to note that this is entirely consistent with experimental work purporting to see charge–rich walls, separated by distances scaling like $1/x$, arising from finite–size striped magnetic domains at low $x$ [@cho]. Thus, our work demonstrates a magnetic mechanism for such stripes, and is different than the Coulomb–interaction generated carrier–rich walls of Emery and Kivelson’s [@emery2] phase separation ideas.) Since this numerical result is in direct contradiction to the results of Poilblanc [@poilblanc2], who studied pairs of holes in a variety of square clusters, we have further scrutinized this behaviour. We have found that in (i) Monte Carlo simulations of the $t - U$ model, for an average of 2 holes in an $8 \times 8$ lattice, the holes still tend to want to be as far apart as possible, and (ii) the same behaviour is found for the larger $\sqrt {18} \times \sqrt {32}$ 24–site cluster with 2 holes — these results will be published in a future comment [@chen].
We now present our results for similar calculations for the electron–doped system. Figure 6 shows the carrier–carrier correlation function, defined analogously to the hole–hole correlation function of Eq. (6.1) of [@GVL], by $$P_{cc} (| i - j|) = {1\over N_T N_e} \sum_{i,j} <
(1 - \sum_{\sigma} n_{i, \sigma} ) (1 - \sum_{\sigma} n_{j, \sigma} ) >$$ where $n_{i, \sigma}$ is the hole number operator for electron–doped systems (electron number operator for hole–doped systems, as in Eq. (6.1) of [@GVL]) for spin $\sigma$ at site $i$, $N_T$ being the number of lattice sites, and $N_e$ is the number of equivalent sites a distance $|i - j|$ from site $i$ (note that we do not include the angular dependence which is actually present for our non-square lattice, for simplicity only, since we have found that this does not affect our conclusions). We evaluated this function for two carriers doped into a $\sqrt {8} \times \sqrt {32}$ cluster described by the $t - t^\prime - J$ model with $t=-1,~J=.4~,$ and $t^\prime = +.3$. Juxtaposed with this curve is the analogous hole-hole correlation function for $t=1,~J=.4,$ and $t^\prime = -.3$. The differences between the two different systems is striking: whereas pairs of holes tend to stay as far apart as possible, pairs of mobile electrons in the electron–doped system tend to cluster together.
To further elucidate this behaviour we have calculated the 2, 3 and 4 carrier interaction energies (analogous to the binding energy) defined relative to [*independent*]{} carriers, (denoted by $E_{2I},~E_{3I}$ and $E_{4I}$, where $E_n$ denotes the ground state for $n$ carriers), via $$\begin{aligned}
E_2 &= E_0 + 2 (E_1 - E_0) + E_{2I} \nonumber \\
E_3 &= E_0 + 3 (E_1 - E_0) + E_{3I} \nonumber \\
E_4 &= E_0 + 4 (E_1 - E_0) + E_{4I}
\label{equation:binding}\end{aligned}$$ and our results for both hole and electron–doped systems are presented in Table II. One sees that for the hole–doped system, the tendency for holes to cluster together is very small — this is consistent with the binding energy study of Riera and Young [@riera] (if one uses their definition for the binding energy one finds results very similar to those of Table II). However, for the electron–doped material one finds that there is both a stronger binding of a single pair of holes, as well as a very low interaction energy, at least relative to the hole–doped material, as the number of carriers increases. This shows that the mobile electrons have a much greater propensity to cluster together than do the mobile holes.
Summarizing the numerical work of the last two sections: (i) The single–hole quasiparticles of the hole–doped system are Shraiman–Siggia dipolar quasiparticles. They tend to remain far from each other, and are thus consistent with some form of superposition of these many–body quantum states in the low–doping limit [@GVL]. (ii) The single–carrier ground state of the electron–doped material is different than that of the hole–doped system, viz. it is a $(\pi,0)$ state as opposed to a $({\pi\over 2}, {\pi\over 2})$ state, and a non–zero density of the former tend to exhibit a much stronger tendency to cluster together than do the holes of the hole–doped material. The spin distortions for these quasiparticles are long (short) ranged for the hole (electron) doped systems.
Considering the data contained in Fig. 6 and Table II, we have reason to believe that for the electron–doped materials the carriers have a strong tendency towards phase separation — certainly the tendency is much stronger than in the hole–doped system. However, since Table II shows that there are not purely attractive interactions between the mobile electron carriers, this tendency will probably lead to the presence of low–energy excited states corresponding to fluctuations into a locally phase separated state. Then, perhaps these phase separated states are the cause of the mid–gap states seen in the photoemission work [@allen], as was suggested elsewhere [@emery2].
As espoused by Emery and Kivelson [@emery2], phase separation in the large hopping limit is likely to be frustrated, the frustration being due to the presence of long–ranged Coulomb interactions. We have investigated the effect of including a near–neighbour Coulomb repulsion between carriers on neighbouring sites, and find that no qualitative changes take place until very large Coulomb interactions (say of order much larger than $t$) are present. However, we do not know how long–ranged Coulomb interactions will effect our conclusions, and this is presently being studied. Further, Trugman [@trugman] noted that tendencies towards the binding of a pair of holes can be greatly reduced by next–nearest neighbour three–site spin–dependent hopping. While this form of hopping will be of much lower order than the next–nearest neighbour hopping that we are studying (since we begin with a three-band, not a one–band (as does Trugman), description), and thus we do not expect any qualitative changes in our analysis, we feel it appropriate to see in what quantitative ways our observed tendency towards a phase separated state in electron–doped planes changes with such a hopping term present. Thus, we are also presently studying the effects of such terms.
Consequently, we propose that the commensurate antiferromagnetic spin background of the electron–doped materials do not have incommensurate correlations introduced into them because (i) the quasiparticles introduced into the weakly doped planes are not Shraiman–Siggia dipolar quasiparticles, and (ii) the mobile electrons found in these materials exhibit a form of phase separation, and no incommensurate correlations are expected for such a state of matter. We have completed a study of the electron/hole momentum distribution functions, similar to those presented in Ref. [@GVL], including an evaluation of the density of states for both the hole and electron–doped compounds, and this work (to be presented later) further strengthens our assertion that some form of carrier clustering is taking place in the electron–doped materials.
Discussion: {#sec:discussion}
===========
Our numerical results and semiclassical studies have led to the proposal that the single–hole ground states in the hole and electron–doped materials are different, and that this has a profound effect on (i) their magnetic phase diagrams, and (ii) the tendency towards phase separation that these carriers display. The Shraiman–Siggia dipolar quasiparticle picture has been shown [@nskyrmions; @commentmillis; @subir] to be consistent with many different experiments studying the hole–doped $LaSrCuO$ system, and it will be interesting to see if our $(\pi,0)$ quasiparticle picture of phase separation in the $NdCeCuO$ can be used to explain more than just the ARPES results associated with the mid–gap states found in ARPES studies [@allen].
One other experiment that may possibly find an explanation in the frustrated phase separation scenario for the electron–doped material involves the resistivity vs. temperature results of Hikada $et~al.$ [@hikada]. To be specific, if the mobile electrons tend to cluster together for sufficiently long periods of time, the dominant interaction between carriers is electronic as opposed to magnetic, and thus a fermi–liquid–like $\rho~\sim~T^2$ is not that surprising. Of course, such a qualitative explanation requires rigorous calculations to be carried out before it is to be taken seriously.
We have addressed the magnetic and other material characteristics of the two 214 high $T_c$ systems. However, the approach of utilizing the $t - t^\prime - J$ system does not seem to be limited to these single–layer high $T_c$ materials. Recently, it has been proposed [@chub123] that hole pockets appear in the $YBaCuO$ material as one dopes away from the antiferromagnetic insulator, and that these pockets are centred at $({\pi\over 2},{\pi\over 2})$. This is entirely consistent with the “small fermi surface" observed in ARPES studies of $YBa_2Cu_3O_{6.3}$ [@liu]. In order to model such a system one must include a $t^\prime$ parameter that rotates the fermi surface by 45$^\circ$ with respect to that predicted by the $t - J$ model, and to this end we have used $t=1,~t^\prime\sim-.5,$ and $J=.4$. Then, we find (i) that indeed the single–hole ground state occurs at $\vec k = (\pi,\pi)$ (this was also found in the study of Ref. [@gagliano]), and (ii) that pairs of such holes tend to stay as far apart as possible, with a hole–hole correlation function similar to that shown for the hole–doped material in Fig. 6. Thus, weakly interacting quasiparticles forming small hole pockets at $(\pi,\pi)$ is consistent with our numerics, and are not expected to lead to any incommensurability [@chub123]. Noting that both our semiclassical work (similar to that shown in Fig. 5) and the spin–wave analysis of Reiter [@reiter] predict that the spin distortions produced by $(\pi,\pi)$ quasiparticles are long ranged, it is very interesting that in the two systems in which the holes tend to stay as far apart as possible, the single–carrier quasiparticles produce long–ranged spin distortions, and we are presently exploring this coincidence.
**[ACKNOWLEDGEMENTS:]{}**
We are very grateful to Elbio Dagotto for providing us a copy of Fig. 1, and for sending us a copy of Ref. [@dagotto] prior to publication. We also wish to thank Andrey Chubukov for discussions concerning the 123 material, and Liang Chen for helpful comments on the hole–hole correlation function for the $t -U$ model. Lastly, we wish to thank Vic Emery and Barry Wells for conveying to us important information on the $Sr_2CuO_2Cl_2$ photoemission data. This work was supported by the NSERC of Canada.
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------------------------------------- ------------------------------------------ -----------------------------------------------
$\vec k$ Energy Energy
(Hole doped: $t = 1 {,}~t^\prime = -.3$) (Electron doped: $t = -1 {,}~t^\prime = +.3$)
$(0,0)$ -12.824 -12.153
$({\pi\over 4},{\pi\over 4})$ -12.961 -12.338
$({\pi\over 2},{\pi\over 2})$ -13.207 -12.686
$({3\pi\over 4},{3\pi\over 4})$ -13.063 -12.375
$(\pi,\pi)$ -12.853 -12.177
$({-\pi\over 3},{\pi\over 3})$ -12.997 -12.471
$({-2\pi\over 3},{2\pi\over 3})$ -13.127 -12.363
$({-\pi\over 12},{7\pi\over 12})$ -12.896 -12.746
$({-5\pi\over 12},{11\pi\over 12})$ -12.980 -12.679
$({\pi\over 6},{5\pi\over 6})$ -12.821 -13.037
------------------------------------- ------------------------------------------ -----------------------------------------------
: Single carrier minimum energies for both hole and electron singly–doped planes, described by Eq. (2.4), for the allowed wave vectors of our non–square 24–site cluster. All energies are in units of $t$.
6.0 truecm
---------- ---------------------------- -----------------------------
Hole Doped Electron Doped
$t = 1 {,}~t^\prime = -.3$ $t = -1 {,}~t^\prime = +.3$
E$_{2I}$ -.12 -.21
E$_{3I}$ +.45 +.12
E$_{4I}$ +1.5 +.26
---------- ---------------------------- -----------------------------
: Interaction energies, defined in Eq. (4.1), for 2, 3 and 4 carriers in the $t - t^\prime -J$ model, using representative material parameters for the hole and electron–doped compounds based on Eq. (2.4). As long as the single–hole ground states are those that we predict for the hole and electron–doped systems, these interaction energies do not sensitively depend on the ratio of $\mid t^\prime / t \mid$.
|
---
abstract: |
We study for a class of symmetric Lévy processes with state space ${{{\mathds R}^n}}$ the transition density $p_t(x)$ in terms of two one-parameter families of metrics, $(d_t)_{t>0}$ and $(\delta_t)_{t>0}$. The first family of metrics describes the diagonal term $p_t(0)$; it is induced by the characteristic exponent $\psi$ of the Lévy process by $d_t(x,y)=\sqrt{t\psi(x-y)}$. The second and new family of metrics $\delta_t$ relates to $\sqrt{t\psi}$ through the formula $$\exp\left(-\delta_t^2(x,y)\right)
= {\mathcal{F}}\left[\frac{e^{-t\psi}}{p_t(0)}\right](x-y)$$ where ${\mathcal{F}}$ denotes the Fourier transform. Thus we obtain the following “Gaussian” representation of the transition density: $p_t(x)=p_t(0) e^{-\delta_t^2(x,0)}$ where $p_t(0)$ corresponds to a volume term related to $\sqrt{t\psi}$ and where an “exponential” decay is governed by $\delta_t^2$. This gives a complete and new geometric, intrinsic interpretation of $p_t(x)$.
*MSC 2010: Primary: 60J35. Secondary: 60E07; 60E10; 60G51; 60J45; 47D07; 31E05.*
*Key Words: transition function estimates; Lévy processes; metric measure spaces; heat kernel bounds; infinitely divisible distributions; self-reciprocal distributions.*
author:
- 'N. Jacob, V. Knopova, S. Landwehr'
- 'R.L. Schilling'
title: A geometric interpretation of the transition density of a symmetric Lévy Process
---
[^1] [^2] [^3] [^4]
Introduction {#intro}
============
We start with a simple example. Let $(C_t)_{t{\geqslant}0}$ be the one-dimensional Cauchy process. Its transition function $p_t$ has a density with respect to Lebesgue measure in ${\mathds R}$ and we denote the density again by $p_t$, $$\label{intro-e01}
p_t(x,y) = \frac{t}{\pi\,(t^2 + |x-y|^2)}.$$ Since $(C_t)_{t{\geqslant}0}$ has stationary and independent increments, $p_t(x,y)$ depends only on the increment $x-y$, i.e. $$ p_t(x) := p_t(x,0) = \frac{t}{\pi\,(t^2 + |x|^2)}.$$ Let us introduce two one-parameter families of metrics, $d_{C,t}(\cdot,\cdot)$, $t>0$, and $\delta_{C,t}(\cdot,\cdot)$, $t>0$, on ${\mathds R}$ defined by $$ d_{C,t}(x,y) := \sqrt{t\,|x-y|}$$ and $$ \delta_{C,t}(x,y) := \sqrt{\ln\left[\frac{|x-y|^2+t^2}{t^2}\right]}.$$ With $$ B^{d_{C,t}}(0,1)
:= \big\{ x\in{\mathds R}\::\: d_{C,t}(x,0)<1\big\}
= \big\{ x\in{\mathds R}\::\: |x|<1/t\big\}$$ we find $$ \lambda\Big(B^{d_{C,t}}(0,1)\Big) = \frac 2t$$ and, therefore, $$ p_t(0) = \frac 1{2\pi}\,\lambda\Big(B^{d_{C,t}}(0,1)\Big).$$ Thus, we find $$\label{intro-e08}
p_t(x) = \frac 1{2\pi}\,\lambda\Big(B^{d_{C,t}}(0,1)\Big)\,e^{-\delta_{C,t}^2(x,0)}.$$ Let us compare with the Gaussian, i.e. with the density ${\nu}_t$ of the transition function of a one-dimensional Brownian motion $(B_t)_{t{\geqslant}0}$ where $$ {\nu}_t(x) = \frac 1{\sqrt{2\pi t}}\,e^{-|x|^2/2t}.$$ We may introduce the following two one-parameter families of metrics on ${\mathds R}$ $$ d_{G,t}(x,y) = \sqrt t\,|x-y|$$ and $$ \delta_{G,t}(x,y) = \frac 1{\sqrt{2t}}\,|x-y|.$$ For $$B^{d_{G,t}}(0,1)
:= \big\{ x\in{\mathds R}\::\: d_{G,t}(x,0)<1\big\}
= \big\{ x\in{\mathds R}\::\: |x|<1/\sqrt t\,\big\}$$ it holds $$ \lambda\Big(B^{d_{G,t}}(0,1)\Big) = \frac 2{\sqrt t}$$ and, consequently, $$ {\nu}_t(0) = \frac 1{\sqrt{8\pi}}\,\lambda\Big(B^{d_{G,t}}(0,1)\Big).$$ Hence, we find $$\label{intro-e14}
{\nu}_t(x) = \frac 1{\sqrt{8\pi}}\,\lambda\Big(B^{d_{G,t}}(0,1)\Big)\,e^{-\delta^2_{G,t}(x,0)}.$$ Comparing and we note that $p_t$ and ${\nu}_t$ have the same structure.
The central purpose of this paper is to work out that we should expect many symmetric Lévy processes to have a density which is completely determined by two one-parameter families of metrics by a formula which is completely analogous to or . We will provide large classes of examples and discuss some consequences.
Throughout this paper $(X_t)_{t{\geqslant}0}$ will be a symmetric Lévy process with state space ${{{\mathds R}^n}}$. It is well known that its characteristic function is given by $$ {\mathds E}e^{i\xi\cdot X_t} = e^{-t\psi(\xi)}$$ where $\psi:{{{\mathds R}^n}}\to{\mathds C}$ is the characteristic exponent. The exponent $\psi$ has a Lévy-Khintchine representation, $$\psi(\xi)
= i\ell\cdot\xi + \frac 12 \sum_{j,k=1}^n q_{jk}\xi_j\xi_k + \int_{y\neq 0} \left(1-e^{iy\cdot\xi} + \frac{iy\cdot\xi}{1+|y|^2}\right)\nu(dy)$$ where $\ell\in{{{\mathds R}^n}}$, $(q_{jk})_{j,k}\in{\mathds R}^{n\times n}$ is a symmetric positive semidefinite matrix and $\nu$ is a Borel measure on ${{{\mathds R}^n}}\setminus\{0\}$ such that $\int_{y\neq 0} \left(1\wedge |y|^2\right)\,\nu(dy)<\infty$. This means that $\psi$ is a continuous negative definite function in the sense of Schoenberg. If $(X_t)_{t{\geqslant}0}$ is symmetric, then $\psi(\xi){\geqslant}0$, $\psi(\xi)=\psi(-\xi)$ and $\xi\mapsto\sqrt{\psi(\xi)}$ is subadditive, i.e. $$\sqrt{\psi(\xi+\eta)}{\leqslant}\sqrt{\psi(\xi)} + \sqrt{\psi(\eta)}.$$ Thus, if we require that $\psi(\xi)=0$ if, and only if, $\xi=0$, then $(\xi,\eta)\mapsto \sqrt{\psi(\xi-\eta)}$ generates a metric on ${{{\mathds R}^n}}$ and we can understand $\big({{{\mathds R}^n}},\sqrt{\psi},\lambda\big)$ as a metric measure space.
We will always assume that $e^{-t\psi}\in L^1({{{\mathds R}^n}})$ which implies that the probability distribution $p_t$ of each $X_t$, $t>0$, has a density with respect to Lebesgue measure $\lambda$; we will denote these densities again by $p_t(x)$.
The first important observation is Theorem \[pto-03\] which tells us that $$\label{intro-e16}
p_t(0)
= (2\pi)^{-n}\int_{{{\mathds R}^n}}e^{-t\psi(\xi)}\,d\xi
= (2\pi)^{-n}\int_0^\infty \lambda\Big(B^{d_\psi}\big(0,\sqrt{r/t}\big)\Big)\,e^{-r}\,dr$$ where $B^{d_\psi}(0,r)$ denotes the ball with centre $0$ and radius $r>0$ with respect to the metric $d_\psi(x,y)=\sqrt{\psi(x-y)}$. A first version of this result was already proved in [@KS1]. While is an exact formula, we get $$\label{intro-e17}
p_t(0) \asymp \lambda\Big(B^{d_\psi}\big(0, 1/\sqrt t\big)\Big)$$ whenever the metric measure space $\big({{{\mathds R}^n}},d_\psi,\lambda\big)$ has the volume doubling property. (By $f\asymp g$ we mean that there exists a constant $0<\kappa<\infty$ such that $\kappa^{-1} f(x) {\leqslant}g(x) {\leqslant}\kappa f(x)$ for all $x$.) Introducing the one-parameter family of metrics $d_{\psi,t}(\cdot,\cdot)$ by $$ d_{\psi,t}(\xi,\eta)
:= \sqrt{t\,\psi(\xi-\eta)}$$ we find, if holds, that $$\label{intro-e19}
p_t(0) \asymp \lambda\Big(B^{d_{\psi,t}(\cdot,\cdot)}\big(0,1\big)\Big).$$ In order to prove or we first need to understand the metric measure space $\big({{{\mathds R}^n}},d_\psi,\lambda\big)$. This is done in Section \[cndf\]. Following some basic definitions we provide conditions for the doubling property to hold and we discuss when $\big({{{\mathds R}^n}},d_\psi,\lambda\big)$ is a space of homogeneous type. Note that only in case of Brownian motion we can expect that $\big({{{\mathds R}^n}},d_\psi,\lambda\big)$ is a length space. A more detailed study is devoted to the case of subordination, i.e. when $\psi$ is the composition of a Bernstein function $f$ (the Laplace exponent of an increasing Lévy process) and a continuous negative definite function (characteristic function of a Lévy process) $\phi$. The most interesting case is $\psi(\xi) = f(|\xi|^2)$. We discuss several examples and these are used to illustrate .
In order to understand the behaviour of $p_t(x)$ for $x\neq 0$ we observe that $$ \frac{p_t(x)}{p_t(0)}
= \int_{{{\mathds R}^n}}e^{-ix\cdot\xi}\,\frac{e^{-t\psi(\xi)}}{p_t(0)}\,d\xi$$ is the Fourier transform of a probability measure. The question is whether we can write $p_t(x)/p_t(0)$ as $$\label{intro-e21}
\frac{p_t(x)}{p_t(0)} = e^{-\delta_{\psi,t}^2(x,0)}$$ with a suitable one-parameter family of metrics $\delta_{\psi,t}(x,y)$, $t>0$. Section \[ptx\] explains this idea in more detail and first examples are given. Our approach is not just an ‘educated guess’. A theorem of Schoenberg—in a formulation suitable for our discussion—states that a metric space $({{{\mathds R}^n}},d)$ can be isometrically embedded into an (in general infinite-dimensional) Hilbert space $\mathcal H$ if, and only if, $d(x,y)=\sqrt{\psi(x-y)}$ for some suitable continuous negative definite function $\psi:{{{\mathds R}^n}}\to{\mathds R}$. Using the Dirichlet form and the carré du champ associated with the Lévy process $(X_t)_{t{\geqslant}0}$ we outline the proof of the fact that the metric space $({{{\mathds R}^n}},d_\psi)$ can isometrically be embedded into a Hilbert space; this is the part of Schoenberg’s result which is important for our considerations. Our general guide for the investigations in this paper is the rough idea that Fourier transforms of Gaussians are Gaussians—also in Hilbert spaces. Thus, we might consider to obtain $p_t(x)$ or $p_t(x)/p_t(0)$ as pre-image of Fourier transforms of Gaussians in $\mathcal H$. So far, we did not succeed to formalize this idea, however, already during the *3rd Conference on Lévy processes: Theory and Applications* 2003 in Paris the first-named author launched this idea to use this correspondence to study Lévy processes picking up some work of P.A. Meyer [@meyer].
So far we have only partial answers for to hold. In Section \[dist\] we begin with the density of a single random variable, i.e. we will not take into account that they belong to the transition function of a process. However, we assume that they are infinitely divisible random variables, hence they can always be embedded into the transition function of a Lévy process. We introduce in Section \[dist\] the class ${\mathsf{N}}$ of infinitely divisible probability distributions consisting of those $p$ for which $\mathcal F^{-1}\big[\frac p{p(0)}\big]$ is again infinitely divisible. Thus, if for a Lévy process $(X_t)_{t{\geqslant}0}$ the density $p_{t_0}$ belongs to ${\mathsf{N}}$, then $p_{t_0}$ satisfies . We give large classes of examples including Fourier self-reciprocal densities, generalized hyperbolic distributions and more.
In Section \[sub\] and \[proc\] we return to our investigations on processes. As so often when dealing with Lévy processes, subordinate Brownian motion plays a distinguished role. In Section \[sub\] we present some results of general nature. We prove in Theorem \[sub-15\] (for $n=1$) that if ${\mathcal{F}}^{-1}[p_t] (\xi)=e^{-tf(|\xi|)}$ for some Bernstein function $f$ such that $f(0)=0$ and $\int_0^\infty e^{-tf(r)}dr<\infty$, then $p_t(x)=p_t(0)e^{-g_t(|x|^2)}$ for a suitable family of Bernstein functions $g_t$; of course, $\sqrt{g_t(x-y)}$ gives a metric on ${\mathds R}$. Although the theorem is proved only for $n=1$, its proof extends to $n=2$ and $n=3$.
In Section \[proc\] we discuss further examples of processes for which holds. These examples are processes with transition functions which are certain mixtures of Gaussians. While our examples already indicate the scope of our approach, their proofs depend essentially on the special structures of the underlying transition densities. So far we do not have a proof for our general
Let ${\mathcal{MCN}}({{{\mathds R}^n}})$ denote the continuous negative definite functions that induce a metric on ${{{\mathds R}^n}}$ which generates the Euclidean topology. If $\psi\in {\mathcal{MCN}}({{{\mathds R}^n}})$ and $e^{-t\psi}\in L^1({{{\mathds R}^n}})$, then there exists a one-parameter family of metrics $\delta_t(\cdot,\cdot)$ such that $$ p_t(x)=p_t(0)e^{-\delta^2_t(x,0)}$$ holds.
We emphasize that we are looking for a metric $\delta_t(\cdot,\cdot)$, and we do not require that it is of the type $\sqrt{\psi_t(x-y)}$ where $\psi_t(\xi)$ is a family of continuous negative definite functions indexed by $t>0$. An interesting remark was made by Rama Cont, namely to investigate whether the metric $\delta_t(\cdot,\cdot)$ can be related to a good rate function for large deviations as it is the case for diffusions.
In the final Section \[fel\] we give a brief outline of the situation when the Lévy process is replaced by a Feller process generated by a pseudo-differential operator with a negative definite symbol, compare [@H95]–[@H98b] and [@J94] as well as [@J2] for large classes of examples. We will have to work with metrics varying with the current position in space as it is the case in (sub-)Riemannian geometry. However, the fact that we cannot expect to work in length spaces causes serious problems when we try to understand the underlying geometry.
We would like to mention more recent work in which estimates for heat kernels are obtained when starting with a metric measure space having the volume doubling property: M. Barlow, A. Grigor’yan and T. Kumagai [@bar-gri-kum], Z.-Q. Chen and T. Kumagai [@che-kum; @che-kum08], A. Grigor’yan and J. Hu [@gri-hu], and A. Grigor’yan, J. Hu and K.-S. Lau [@gri-hu-lau; @gri-hu-lau10], to mention some of this work. Note the difference to our point of view. The metric measure space considered by us is induced by the characteristic exponent *on the Fourier space*, not on the state space. Our conjecture, here proved for many classes of processes, is that we can also find a *new metric on the state space* which will yield a Gaussian estimate when combined with the metric induced by the characteristic exponent which gives the diagonal term. For Lévy and Lévy-type processes this seems to be the natural approach.
In general, we follow our monographs [@J]–[@J3] and [@SSV]. In particular, we use ${\mathcal{F}}u(\xi) = \widehat u(\xi) = (2\pi)^{-n}\int_{{{\mathds R}^n}}e^{i\xi x} u(x)\,dx$ for the Fourier transform and we write $\mathscr S({{{\mathds R}^n}})$ for the Schwartz space. By $K_\lambda$ we denote the Bessel functions of the third kind, cf. [@erd-et-al vol. 2]. We write $X\sim Y$ if two random variables $X$ and $Y$ have the same probability distribution and $X\sim \mu$ means that $X$ has the probability distribution $\mu$. If $f$ and $g$ are functions, $f\asymp g$ means that there exists a constant $\kappa$ such that $\kappa^{-1} f(x){\leqslant}g(x){\leqslant}\kappa \,f(x)$ holds for all $x$, and $f\approx g$, $x\to a$, stands for $\lim_{x\to a} f(x)/g(x) = 1$. All other notations are standard or explained in the text.
We dedicate this paper to Professor Mu-Fa Chen and Professor Zhi-Ming Ma in appreciation of their outstanding contributions to mathematics and their remarkable success of building up in China one of the world’s finest centres in probability theory.
The authors would like to thank Björn Böttcher and Walter Hoh for comments made while working on this paper.
Auxiliary results {#aux}
=================
Fourier transforms and characteristic functions {#fourier-transforms-and-characteristic-functions .unnumbered}
-----------------------------------------------
The *Fourier transform* of a bounded Borel measure $\mu$ on ${{{\mathds R}^n}}$ is defined by $$\label{aux-e02}
{\mathcal{F}}\mu(\xi) = (2\pi)^{-n}\int_{{{\mathds R}^n}}e^{-ix\xi}\,\mu(dx),\quad\xi\in{{{\mathds R}^n}}.$$ By *Bochner’s theorem* the Fourier transform is a bijective and bi-continuous mapping from the cone of bounded Borel measures (equipped with the weak topology) to the cone of continuous positive definite functions (equipped with the topology of locally uniform convergence). By linearity we can extend to signed measures; for $u\in\mathscr S({{{\mathds R}^n}})$ we get the classical formulae for the (inverse) Fourier transform $$ {\mathcal{F}}u(\xi)
= (2\pi)^{-n}\int_{{{\mathds R}^n}}e^{-ix\xi} u(x)\,dx
\quad\text{and}\quad
{\mathcal{F}}^{-1}v(\eta) = \int_{{{\mathds R}^n}}e^{iy\eta} v(y)\,dy.$$ Obviously, ${\mathcal{F}}^{-1}$ extends canonically to the bounded Borel measures $$ {\mathcal{F}}^{-1}\mu(\eta) = \int_{{{\mathds R}^n}}e^{iy\eta}\,\mu(dy),\quad \eta\in{{{\mathds R}^n}}.$$ If $\mu$ is the probability law of a random variable $Y$, ${\mathcal{F}}^{-1} \mu(\eta)$ is the *characteristic function* $\chi_Y(\eta) = {\mathds E}e^{i\eta Y}$.
With our normalization of the Fourier transform *Plancherel’s theorem* becomes $$ {\|u\|}_{L^2}
= (2\pi)^n {\|{\mathcal{F}}u\|}_{L^2}
\quad\text{and}\quad
\int_{{{{\mathds R}^n}}} u(x)\,\mu(dx)
= (2\pi)^n \int_{{{{\mathds R}^n}}} {\mathcal{F}}u(\xi) \, \overline{{\mathcal{F}}\mu(\xi)}\,d\xi.$$ Whenever convolution and Fourier transforms of $u$ and $v$ are defined, the *convolution theorem* holds, i.e.$$ {\mathcal{F}}^{-1}(u\star v) = {\mathcal{F}}^{-1} u \cdot {\mathcal{F}}^{-1} v
\quad\text{and}\quad
{\mathcal{F}}(u\cdot v) = {\mathcal{F}}u\star{\mathcal{F}}v.$$
Infinite divisibility {#infinite-divisibility .unnumbered}
---------------------
A probability measure $\mu$ is called an *infinitely divisible probability distribution* if for every $n>0$ there exists some probability measure $\mu_n$ such that $\mu = \mu_n^{\star n} = \mu_n \star\ldots\star\mu_n$ ($n$ factors). Let $X$ be a random variable with law $X$. Then the following statements are equivalent to saying that $\mu$ is infinitely divisible:
1. the random variable satisfying $X\sim\mu$ is an *infinitely divisible random variable*, i.e. for every $n\in{\mathds N}$ there exist independent and identically distributed random variables $X_1, \ldots, X_n$ such that $X\sim X_1+\cdots+X_n$.
2. the characteristic function $\chi_X(\xi) = {\mathds E}e^{i\xi X} = {\mathcal{F}}^{-1}\mu(\xi)$ of the random variable $X$ is an *infinitely divisible characteristic function*, i.e. for every $t>0$ the function $(\chi_X)^t$ is again the characteristic function of some random variable.
It is a classical result that $X$ or $\mu$ are infinitely divisible if, and only if, the log-characteristic function $\psi(\xi) := -\ln\chi_X(\xi) = -\ln{\mathcal{F}}^{-1}\mu(\xi)$ is a *continuous negative definite function* (in the sense of Schoenberg). These functions have a unique Lévy-Khintchine representation, i.e.$$\label{aux-e13}
\psi(\xi)
= i\ell\cdot\xi + \frac 12 \sum_{j,k=1}^n q_{jk}\xi_j\xi_k + \int_{y\neq 0} \left(1-e^{iy\cdot\xi} + \frac{iy\cdot\xi}{1+|y|^2}\right)\nu(dy)$$ where $\ell\in{{{\mathds R}^n}}$, $(q_{jk})_{j,k}\in{\mathds R}^{n\times n}$ is a symmetric positive semidefinite matrix and $\nu$ is a Borel measure on ${{{\mathds R}^n}}\setminus\{0\}$ such that $\int_{y\neq 0} \left(1\wedge |y|^2\right)\,\nu(dy)<\infty$.
Convolution semigroups {#convolution-semigroups .unnumbered}
----------------------
A *convolution semigroup* $(\mu_t)_{t{\geqslant}0}$ on ${{{\mathds R}^n}}$ is a family of probability measures $\mu_t$ defined on ${{{\mathds R}^n}}$ satisfying $$\mu_s \star \mu_t = \mu_{s+t},\; s,t{\geqslant}0,
\quad\text{and}\quad
\mu_0 = \delta_0.$$ We will always assume that $(\mu_t)_{t{\geqslant}0}$ is *vaguely continuous*, i.e.$$\lim_{t\to 0} \int_{{{\mathds R}^n}}u(x)\,\mu_t(dx) = u(0)
\quad\text{for all}\quad u\in C_c({{{\mathds R}^n}}).$$ It follows from the definition that each measure $\mu_t$, $t>0$, is infinitely divisible. Therefore every (vaguely continuous) convolution semigroup $(\mu_t)_{t{\geqslant}0}$ on ${{{\mathds R}^n}}$ is uniquely characterized by a continuous negative definite function $\psi$ such that $$\label{aux-e14}
{\mathcal{F}}^{-1}\mu_t(\xi) = e^{-t\psi(\xi)},\quad t{\geqslant}0,\;\xi\in{{{\mathds R}^n}}.$$ Conversely, every continuous negative definite function $\psi$ with $\psi(0)=0$ determines, by , a unique (vaguely continuous) convolution semigroup $(\mu_t)_{t{\geqslant}0}$ on ${{{\mathds R}^n}}$.
If $\mu$ is an infinitely divisible probability distribution, there is a unique vaguely continuous convolution semigroup $(\mu_t)_{t{\geqslant}0}$ such that $\mu_1 = \mu$; indeed, ${\mathcal{F}}^{-1}\mu_t = ({\mathcal{F}}^{-1}\mu)^t$. Conversely, if $\mu_1$ is an element of $(\mu_t)_{t{\geqslant}0}$, then $\mu_1$ is infinitely divisible.
Subordination {#subordination .unnumbered}
-------------
Subordination in the sense of Bochner is a method to obtain new convolution semigroups from a given one. Let $(\eta_t)_{t{\geqslant}0}$ be a convolution semigroup on ${\mathds R}$ where all measures $\eta_t$, $t{\geqslant}0$, are supported in $[0,\infty)$. Since ${\operatorname{\mathrm{supp}}}\eta_t\subset[0,\infty)$ it is more convenient to describe $\eta_t$ in terms of the (one-sided) Laplace transform. Similar to we see that $$\label{aux-e18}
\mathcal L\eta_r(\lambda) = \int_{[0,\infty)} e^{-\lambda t}\,\eta_r(t) = e^{-rf(\lambda)}.$$ The characteristic (Laplace) exponent $f$ is a *Bernstein function*, i.e. $f\in C^\infty(0,\infty)$ such that $f{\geqslant}0$ and $(-1)^{k-1} f^{(k)}{\geqslant}0$ for all $k{\geqslant}1$. All Bernstein functions have a unique representation $$\label{aux-e20}
f(\lambda) = a + b\lambda + \int_{(0,\infty)} (1-e^{-\lambda t})\,\gamma(dt)$$ where $a,b{\geqslant}0$ and $\gamma$ is a Borel measure on $(0,\infty)$ satisfying $\int_{(0,\infty)} \left(1\wedge t\right)\,\gamma(dt) < \infty$. The triplet $(a,b,\gamma)$, the Bernstein function $f$ and the one-sided convolution semigroup $(\eta_t)_{t{\geqslant}0}$ are, because of and , in one-to-one correspondence.
Let $(\mu_t)_{t{\geqslant}0}$ and $(\eta_t)_{t{\geqslant}0}$, ${\operatorname{\mathrm{supp}}}\eta_t\subset [0,\infty)$, be convolution semigroups on ${{{\mathds R}^n}}$ and ${\mathds R}$, respectively. Then the following integrals (convergence in the vague topology) $$\label{aux-e22}
\mu_t^f := \int_{[0,\infty)} \mu_s\,\eta_t(ds),\quad t{\geqslant}0,$$ define a new convolution semigroup on ${{{\mathds R}^n}}$, $(\mu_t^f)_{t{\geqslant}0}$, which is called the *subordinate semigroup*. The characteristic function of the sub-probability measure $\mu_t^f$ is given by $$ {\mathcal{F}}^{-1}\mu_t^f(\xi) = e^{-tf(\psi(\xi))}.$$ In fact, $f\circ\psi$ is, for every Bernstein function $f$, again a continuous negative definite function. Note that the Bernstein functions are the only functions that operate on the continuous negative definite functions in the sense that $f\circ\psi$ is continuous negative definite whenever $\psi$ is, cf. [@J1].
Mixtures {#mixtures .unnumbered}
--------
The probability measure $\mu_t^f$ defined in the formula may be understood as a *mixture* of the probability measures $(\mu_s)_{s{\geqslant}0}$ under the mixing probability measure $\eta_t(ds)$. More generally, let $(\pi(s;\cdot))_{s\in{\mathds R}}$ be a family of probability measures on ${{{\mathds R}^n}}$ and assume that $\rho$ is a probability measure on the parameter space ${\mathds R}$. Then $$\label{aux-e26}
\pi^\rho(B) := \int_{\mathds R}\pi(s;B) \rho(ds),\quad B\subset{{{\mathds R}^n}}\text{\ \ Borel},$$ is again a probability measure on ${{{\mathds R}^n}}$.
Our standard references for the Fourier transform are the monographs [@J1] and Berg–Forst [@BF]; for Bernstein functions and related topics we refer to [@SSV]. Basic notions from probability theory can be found in Breiman [@B], mixtures of probability measures are discussed in Sato [@Sato] and in Steutel–van Harn [@SH04].
Metric measure spaces and negative definite functions {#cndf}
=====================================================
Recall that a *metric measure space* is a triple $(X,d,\mu)$ where $(X,d)$ is a metric space and $\mu$ is a measure on the Borel sets of the space $X$. A good introduction to the analysis on metric measure spaces is the book by Heinonen [@H].
We are mainly interested in metric measure spaces whose metric is induced by a negative definite function. Our basic reference for negative definite functions and their properties is [@J1]. Let $\psi:{{{\mathds R}^n}}\to{\mathds C}$ be a locally bounded negative definite function. Then $$ |\psi(\xi)| {\leqslant}c_\psi (1+|\xi|^2)
\quad\text{with}\quad
c_\psi = 2\sup_{|\eta|{\leqslant}1}|\psi(\eta)|$$ and $$\label{cndf-e04}
\sqrt{|\psi(\xi+\eta)|}
{\leqslant}\sqrt{|\psi(\xi)|} + \sqrt{|\psi(\eta)|}.$$ In particular, whenever $\psi(\xi_0)=0$ for some $\xi_0\neq 0$, $\psi$ is periodic with period $\xi_0$.
Since $\psi(-\xi) = \overline{\psi(\xi)}$, the map $\xi\mapsto |\psi(\xi)|$ is even; in view of it is easy to see that every locally bounded, non-periodic negative definite function with $\psi(0)=0$ induces a metric on ${{{\mathds R}^n}}$ by $$ d_\psi:{{{\mathds R}^n}}\times{{{\mathds R}^n}}\to [0,\infty),\quad d_\psi(\xi,\eta) := \sqrt{|\psi(\xi-\eta)|}.$$ The metric $d_\psi$ is invariant under translations, i.e. $$d_\psi(\xi+\zeta,\eta+\zeta) = d_\psi(\xi,\eta).$$ Therefore Lebesgue measure $\lambda$ is the canonical choice if we consider $({{{\mathds R}^n}},d_\psi,\lambda)$ as a metric measure space.
We denote by $$\begin{aligned}
B^{d_\psi}(\xi,r)
&:= \big\{\eta\in{{{\mathds R}^n}}\::\: d_\psi(\xi,\eta) < r \big\}
= \big\{\eta\in{{{\mathds R}^n}}\::\: |\psi(\xi-\eta)| < r^2 \big\}\\
K^{d_\psi}(\xi,r)
&:= \big\{\eta\in{{{\mathds R}^n}}\::\: d_\psi(\xi,\eta) {\leqslant}r \big\}
= \big\{\eta\in{{{\mathds R}^n}}\::\: |\psi(\xi-\eta)| {\leqslant}r^2 \big\}
\end{aligned}$$ the open and closed balls with radius $r>0$ and centre $\xi$ in the metric space $({{{\mathds R}^n}},d_\psi)$. Note that $B^{d_\psi}(\xi,r) = \xi + B^{d_\psi}(0,r)$. In general, $B^{d_\psi} \subsetneqq \overline{B^{d_\psi}} \subsetneqq K^{d_\psi}$. A typical counterexample can be constructed using continuous negative definite functions of Pólya-type. For example, if we set $\phi(\xi):=|\xi|\wedge 1$, $\xi\in{\mathds R}$, then $B^{d_\phi}(0,1) = (-1,1)$, $\overline{B^{d_\phi}}(0,1) = [-1,1]$ and $K^{d_\phi}(0,1)={\mathds R}$.
In order to compare the metric $d_\psi$ with the usual Euclidean metric we define $$\label{cndf-e10}\begin{aligned}
m(r) &:= \inf\left\{ |\eta| \::\: \sqrt{|\psi(\eta)|} = r\right\},
\\
M(r) &:= \sup\left\{ |\eta| \::\: \sqrt{|\psi(\eta)|} = r\right\}.
\end{aligned}$$ Clearly, $0{\leqslant}m(r){\leqslant}M(r){\leqslant}\infty$ are the maximal resp. minimal radii of Euclidean balls such that $$B(\xi,m(r)) \subset B^{d_\psi}(\xi,r) \subset B(\xi,M(r))$$ holds.
\[cndf-03\] Let $\psi:{{{\mathds R}^n}}\to{\mathds C}$ be a non-periodic continuous negative definite function with $\psi(0)=0$. Then $$0 < m(r) {\leqslant}M(r) < \infty
\quad\text{for all}\quad
0 < r < \liminf_{|\xi|\to\infty} \sqrt{|\psi(\xi)|}.$$ Moreover $m$ and $M$ are monotonically increasing, and the following assertions are equivalent:
1. $M(2r)/m(r) {\leqslant}c_2$ for all $r>0$;
2. $M(\gamma r)/m(r) {\leqslant}c_\gamma$ for all $r> 0$ and all $\gamma > 1$;
3. $M(\delta r)/m(r) {\leqslant}c_\delta$ for all $r>0$ and some $\delta > 1$.
Since $\psi$ is non-periodic and continuous, $0 < r < \liminf_{|\xi|\to\infty} \sqrt{|\psi(\xi)|}$ implies that the level sets $\left\{\xi\::\: |\psi(\xi)| = r^2\right\}$ are non-empty and compact (in the Euclidean topology). Therefore $0<m(r){\leqslant}M(r)<\infty$.
Now let $0<r<R<\liminf_{|\xi|\to\infty} |\psi(\xi)|$ and pick some $\xi_R\in{{{\mathds R}^n}}$ such that $|\psi(\xi_R)| = R^2$ and $m(R)=|\xi_R|$. Consider the curve $\gamma(t):=|\psi(t\xi_R)|$, $0{\leqslant}t{\leqslant}1$. By assumption, $t\mapsto\gamma(t)$ is continuous, $\gamma(0)=0$ and $\gamma(1) = R^2$. Therefore, there exists some $\theta=\theta_r\in (0,1)$ such that $\gamma(\theta) = r^2$. Since $|\theta\xi_R|<|\xi_R|$ and $|\psi(\theta\xi_R)|=r^2$, we conclude that $$\begin{gathered}
m(r) {\leqslant}|\theta\xi_R|<m(R).
\end{gathered}$$ The proof that $M$ is monotone is similar.
Let us now turn to the assertions a)–c). Clearly, b) implies a). Since $r\mapsto M(r)$ is increasing, it is enough to show that a) entails b) for $\gamma > 2$. If $\gamma > 2$ we can uniquely write it in the form $\gamma = 2^k \gamma_0$ where $k\in{\mathds N}$ and $1 {\leqslant}\gamma_0< 2$. Thus, $$\frac{M(\gamma r)}{m(r)}
= \prod_{j=1}^k \frac{M(2^j\gamma_0 r)}{M(2^{j-1}\gamma_0 r)} \frac{M(\gamma_0 r)}{m(r)}
{\leqslant}\prod_{j=1}^k \frac{M(2^j\gamma_0 r)}{m(2^{j-1}\gamma_0 r)} \frac{M(2 r)}{m(r)}
{\leqslant}c_2^{k+1}.$$ The direction b)$\Rightarrow$c) is obvious; the converse follows as in the case where $\delta=2$.
\[cndf-05\] Let $\psi:{{{\mathds R}^n}}\to{\mathds C}$ be a continuous negative definite function. Then the closed ball $K^{d_\psi}(0,r)$, $r>0$, is bounded in the Euclidean topology if, and only if, $r^2 < \liminf_{|\xi|\to\infty} |\psi(\xi)|$. Moreover, $d_\psi$ generates on ${{{\mathds R}^n}}$ the Euclidean topology if, and only if, $\liminf_{|\xi|\to\infty} |\psi(\xi)|>0$.
Assume that $r^2 < \liminf_{|\xi|\to\infty} |\psi(\xi)|$. Then $M(r)= \sup\left\{|\xi|\::\: \sqrt{|\psi(\xi)|} = r\right\}$ is finite, and the inclusion $K^{d_\psi}(0,r)\subset \overline{B(0,M(r))}$ shows that $K^{d_\psi}(0,r)$ is bounded in the Euclidean topology.
Conversely, assume that $K^{d_\psi}(0,r)$ is bounded. Then $K^{d_\psi}(0,r)\subset B(0,\rho)$ for some $\rho>0$. In particular, $|\psi(\xi)| > r^2$ for all $|\xi|>\rho$. This shows that $\liminf_{|\xi|\to\infty} |\psi(\xi)| > r^2$.
If $\liminf_{|\xi|\to\infty} |\psi(\xi)|>0$, then we have for all $0 < r < \liminf_{|\xi|\to\infty} \sqrt{|\psi(\xi)|}$ $$B(0,m(r)) \subset B^{d_\psi}(0,r) \subset \overline{B(0,M(r))}.$$ This proves that the neighbourhood basis induced by $d_\psi$ and the Euclidean neighbourhood basis are comparable, i.e. the topologies coincide.
If $\liminf_{|\xi|\to\infty}|\psi(\xi)|=0$ and if $\psi$ is not periodic, then the $d_\psi$ metric cannot distinguish between $0$ and the points at infinity. This means that the metric $d_\psi$ does not generate the Euclidean topology.
For our purposes it is helpful to assume that $d_\psi$ generates on ${{{\mathds R}^n}}$ the Euclidean topology. To simplify notation we introduce the following definition.
\[cndf-07\] Let $\psi:{{{\mathds R}^n}}\to{\mathds C}$ be a non-periodic, locally bounded negative definite function with $\psi(0)=0$. We call $\psi$ *metric generating* on ${{{\mathds R}^n}}$, if the metric $d_\psi(\xi,\eta):=\sqrt{|\psi(\eta-\xi)|}$ generates on ${{{\mathds R}^n}}$ the Euclidean topology. The set of all continuous metric generating negative definite functions on ${{{\mathds R}^n}}$ is denoted by ${\mathcal{MCN}}({{{\mathds R}^n}})$.
We will use the term ‘metric generating’ exclusively for $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$.
From Lemma \[cndf-05\] it follows that a continuous negative definite function is metric generating if, and only if, $\liminf_{|\xi|\to\infty}|\psi(\xi)|>0$.
Let $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$. We want to study the metric measure space $({{{\mathds R}^n}},d_\psi,\lambda)$. In the analysis on metric measure spaces the notion of volume doubling plays a central role.
\[cndf-09\] Let $(X,d,\mu)$ be a metric measure space. We say that $(X,d,\mu)$ or $\mu$ has the *volume doubling property* if there exists a constant $c_2$ such that $$\label{cndf-e20}
\mu(B^d(x,2r)) {\leqslant}c_2 \, \mu(B^d(x,r))$$ holds for all metric balls $B^d(x,r) = \{y\in X\::\: d(y,x)<r\}\subset X$. If holds only for all balls with radii $r<\rho$ for some fixed $\rho>0$, we say that $(X,d,\mu)$ (or $\mu$) is *locally* volume doubling.
\[cndf-11\] If $(X,d,\mu)$ is volume doubling with doubling constant $c_2>1$, then it follows for every $R{\geqslant}1$ that $$\label{cndf-e22}
\mu\big(B^d(x,R)\big)
{\leqslant}c_2^{\log_2 R} \mu\big(B^d(x,1)\big)
= R^{\log_2 c_2} \mu\big(B^d(x,1)\big).$$ Thus, volume doubling entails that balls have at most power growth of their volume.
Recall that a metric space $(X,d)$ is said to be of *homogeneous type* in the sense of Coifman and Weiss [@CW] if there exists some $N{\geqslant}1$ such that for all $x\in X$ and all radii $r>0$ the ball $B(r,x)$ contains at most $N$ points $x_1, \ldots, x_N$ such that $d(x_j,x_k)>\frac r2$ whenever $j\neq k$.
The following result is taken from [@SL].
\[cndf-13\] If $({{{\mathds R}^n}},d_\psi,\lambda)$, $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$, is volume doubling, it is of homogeneous type.
Let $c_2$ be the volume doubling constant as in and let $x_1,\ldots,x_N\in B^{d_\psi}(x,r)$ such that $d_\psi(x_j,x_k)>\frac r2$ for $j\neq k$. By the triangle inequality we see $$B^{d_\psi}(x_j,r/4)\cap B^{d_\psi}(x_k,r/4)= \emptyset
\quad\text{for all}\quad j\neq k.$$ Since $d_\psi$ is invariant under translations we get $\lambda\big(B^{d_\psi}(x_j,r/4)\big) = \lambda\big(B^{d_\psi}(x,r/4)\big)$. Moreover, $B^{d_\psi}(x_j,r/4) \subset B^{d_\psi}(x,2r)$ and applying three times yields $$N \lambda\left(B^{d_\psi}(x,r/4)\right)
=\lambda\left(\bigcup_{j=1}^N B^{d_\psi}(x_j,r/4)\right)
{\leqslant}\lambda\left(B^{d_\psi}(x,2r)\right)
{\leqslant}c_2^3 \lambda\left(B^{d_\psi}(x,r/4)\right).$$ This proves that $N{\leqslant}c_2^3$.
We are interested in the volume growth of balls in the metric measure space $({{{\mathds R}^n}},d_\psi,\lambda)$. The following result appears in in a weaker form in [@SL].
\[cndf-15\] Let $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$ and $m,M$ as in . Then the following inequality holds for all $0<r<R<\liminf_{|\xi|\to\infty}\sqrt{|\psi(\xi)|}$ $$\label{cndf-e24}
\lambda\big(B^{d_\psi}(0,R)\big)
{\leqslant}\left(\tfrac{M(R)}{m(r)}\right)^n \lambda\big(B^{d_\psi}(0,r)\big).$$
Note that does *not* imply the doubling property as $M(2r)/m(r)$ may depend on $r$.
For $0<r<R<\liminf_{|\xi|\to\infty}\sqrt{|\psi(\xi)|}{\leqslant}\infty$ the functions $m$ and $M$ are strictly positive and finite. By the very definition of the functions $m$ and $M$ we see that $$B^{d_\psi}(0,R) \subset B(0,M(R)) = \frac{M(R)}{m(r)} B(0,m(r)) \subset \frac{M(R)}{m(r)} B^{d_\psi}(0,r).$$ Taking Lebesgue measure in the above chain of inclusions yields .
Many of the most important and concrete continuous negative definite functions are related to subordination. Let $f$ be a (non-degenerate, i.e. non-constant) Bernstein function and $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$. If $f(0)=0$, then $f\circ\psi$ is again metric generating since every Bernstein function is strictly monotone increasing, cf. [@SSV Remark 1.5 and Definition 3.1]. For some subordinate negative definite functions we can calculate the volume growth constant appearing in explicitly. Assume that $\psi(\xi) = f(|\xi|^2)$. Since $f$ is strictly increasing, it is obvious that $$M(r) = m(r) = \sqrt{f^{-1}(r^2)}.$$ Therefore the volume growth constant is $(R/r)^{n/\alpha}$ if $\psi(\xi) = |\xi|^{2\alpha}$ with $0<\alpha<1$. In this case we do have volume doubling. If, however, $\psi(\xi) = \ln (1+|\xi|^2)$, the constant becomes $\left(\frac{\exp(R^2)-1}{\exp(r^2)-1}\right)^{n/2}$ and volume doubling clearly fails. Note that we also do not have power growth. Finally, if $\psi(\xi) = 1-\exp(-|\xi|^2)$, the inverse $f^{-1}$ is only defined on $(0,1)$. Therefore, the volume growth constant is only defined for radii $r<R<1$ and we get $\left(\frac{\ln(1-R^2)}{\ln(1-r^2)}\right)^{n/2}$. In this case we have local volume doubling.
Let us close this section with an observation from [@KS1 Lemma 9] where it is shown that the volume doubling property for large radii entails that $(1+\psi)^{-\kappa/2}\in L^2({\mathds R}^n,\lambda)$ for some suitable exponent $\kappa>0$. This, in turn, has implications for the smoothness of the transition densities, cf. [@KS1].
\[cndf-19\] Let $\phi : (0,\infty)\to (0,\infty)$ be an increasing function such that $$\liminf_{r\to\infty} \frac{\phi(Cr)}{\phi(r)} > 1
\quad\text{for some}\quad C > 1.$$ Then $\phi$ grows at least like a (fractional) power, i.e. there exist constants $c_0,r_0,\kappa>0$ such that $$\phi(r) {\geqslant}c_0\,r^\kappa\quad\text{for all}\quad r{\geqslant}r_0.$$
By assumption there exist some $\gamma>1$ and $r_0 > 0$ such that $$\phi(Cr){\geqslant}\gamma \phi(r)\quad\text{for all}\quad r{\geqslant}r_0.$$ Let $r\in [C^k r_0, C^{k+1}r_0)$. Then we find $$\phi(r) {\geqslant}\phi(C^k r_0) {\geqslant}\gamma^k \phi(r_0).$$ Let $\kappa$ be the unique solution of the equation $C = \gamma^{1/\kappa}$. Then we get $\gamma^k {\leqslant}r^\kappa/r_0^\kappa {\leqslant}\gamma\cdot \gamma^k$ and this entails that $$\begin{gathered}
\phi(r) {\geqslant}\gamma^k \phi(r_0) {\geqslant}\frac 1\gamma\,\frac{\phi(r_0)}{r_0^\kappa}\,r^\kappa
\quad\text{for all}\quad r{\geqslant}r_0.
\qedhere
\end{gathered}$$
\[cndf-21\] Let $\psi(\xi) = f(|\xi|^2)$ be a continuous negative definite function where $f$ is a Bernstein function with $f(0)=0$. Then $\psi$ has the volume doubling property if, and only if, $\liminf_{r\to\infty} f(Cr)/f(r) > 1$ and $\liminf_{r\to 0} f(Cr)/f(r) > 1$ for some $C>1$.
In particular, if $\psi$ has the volume doubling property, then $f(r)$ grows, as $r\to\infty$, at least like a fractional power.
Since $\psi(\xi) = f(|\xi|^2)$, we see that $M(r) = m(r) = \sqrt{f^{-1}(r^2)}$ and $B^{d_\psi}(0,r)=B(0,m(r))$. Therefore, by Lemma \[cndf-03\], the volume doubling property of $\psi$ is the same as $$M(\gamma r) {\leqslant}c_\gamma m(r) \quad\text{for all}\quad r>0, \; \gamma>1$$ which is equivalent to $$f^{-1}(\gamma^2 r) {\leqslant}c_\gamma^2 f^{-1}(r) \quad\text{for all}\quad r>0, \; \gamma>1.$$ This means that the Bernstein function $f$ has to be unbounded and, consequently, bijective. Substituting in this inequality $r=f(x)$ and applying on both sides $f$ we get $$\liminf_{x\to 0} \frac{f(Cx)}{f(x)} > 1
\quad\text{and}\quad
\liminf_{x\to\infty} \frac{f(Cx)}{f(x)} > 1$$ for some constant $C > 1$. Since $f$ is increasing, the second condition entails power growth, cf. Lemma \[cndf-19\].
Conversely, if $\liminf_{r\to\infty} f(Cr)/f(r) > 1$ for some $C>1$, $f$ is unbounded (otherwise the limit inferior would be $1$) and $f$ is bijective. Therefore we can reverse the above argument to deduce volume doubling from $\liminf_{r\to\infty} f(Cr)/f(r) > 1$ and $\liminf_{r\to 0} f(Cr)/f(r) > 1$.
\[cndf-23\] Let $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$ and consider the volume function $v_\psi(r):=\lambda(B^{d_\psi}(0,r))$. If $\psi$ has the volume doubling property, $v^{-1}_\psi$ grows at least like a (fractional) power.
Since $v_\psi$ is increasing, this is similar to the corresponding part of the proof of Proposition \[cndf-21\].
Note that $$v^{-1}_\psi(r) = \sup\big\{ t{\geqslant}0\::\: \lambda \{\xi\in{{{\mathds R}^n}}\::\: |\psi(\xi)|<t\}{\leqslant}r \big\}, \quad r>0.$$ This means that $v^{-1}_\psi(r)$ is actually the *increasing rearrangement* of $|\psi|$ which we denote by $\psi_*(r):= v^{-1}_\psi(r)$. With this notation Corollary \[cndf-23\] reads: $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$ has the volume doubling property if, and only if, $\liminf_{r\to\infty} \psi_*(Cr)/\psi_*(r) > 1$ and $\liminf_{r\to 0} \psi_*(Cr)/\psi_*(r) > 1$ for some $C>1$. If this is the case, then $\psi_*(r)$ grows, as $r\to\infty$, at least like a (fractional) power.
\[cndf-25\] Assume that $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$ enjoys the volume doubling property and that $f$ is a Bernstein function such that $$\liminf_{r\to 0} \frac{f(Cr)}{f(r)}>1
\quad\text{and}\quad
\liminf_{r\to\infty} \frac{f(Cr)}{f(r)}>1$$ for some $C>1$. Then $f\circ\psi\in{\mathcal{MCN}}$ and $f\circ\psi$ has the volume doubling property.
If $f\circ\psi$ is volume doubling, then $\liminf_{|\xi|\to\infty} |f(\psi(\xi))| > 0$ and we get that $f\circ\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$, cf. Lemma \[cndf-05\].
Assume that $\psi$ is volume doubling. Since $f$ is continuous, our assumptions ensure that $f^{-1}(C r){\leqslant}\gamma f^{-1}(r)$ for all $r>0$ and some $\gamma>1$. Then $$\begin{aligned}
\lambda\left(B^{d_{f\circ\psi}}\big(0,\sqrt C r\big)\right)
&= \lambda\left(B^{d_\psi}\big(0,\sqrt{f^{-1}( C r^2)}\big)\right)\\
&{\leqslant}\lambda\left(B^{d_\psi}\big(0,\sqrt{\gamma} \sqrt{f^{-1}(r^2)}\big)\right)\\
&{\leqslant}c_{\sqrt{\gamma}} \lambda\left(B^{d_\psi}\big(0,\sqrt{f^{-1}(r^2)}\big)\right)\\
&= c_{\sqrt{\gamma}} \lambda\left(B^{d_{f\circ\psi}}(0,r)\right).
\end{aligned}$$ Because of Lemma \[cndf-05\] we see that $f\circ\psi$ is volume doubling.
Understanding the role of $p_t(0)$ {#pto}
==================================
Let $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$ and denote by $(\mu_t)_{t{\geqslant}0}$ the corresponding convolution semigroup satisfying ${\mathcal{F}}^{-1}\mu_t = e^{-t\psi}$. We assume that $e^{-t\psi}\in L^1({{{\mathds R}^n}},\lambda)$, so that the measures $\mu_t$ are absolutely continuous with respect to Lebesgue measure. The probability densities are given by $$ p_t(x) = (2\pi)^{-n}\int_{{{{\mathds R}^n}}} e^{-ix\xi} e^{-t\psi(\xi)}\,d\xi = {\mathcal{F}}e^{-t\psi}(x),\quad t>0,$$ and, by the Riemann-Lebesgue lemma, we know that $x\mapsto p_t(x)$ is a continuous function vanishing at infinity.
In this section we will discuss the relation of $p_t(0)$ with the geometry induced by the metric measure space $({{{\mathds R}^n}},d_\psi,\lambda)$. **If not stated otherwise, we will assume that $\psi$ is real-valued.** This means, in particular, that $\psi(\xi){\geqslant}0$ and that $p_t(\cdot)$ is an even function.
Since $(2\pi)^n \, p_t(0) = \int_{{{{\mathds R}^n}}} e^{-t\psi(\xi)}\,d\xi$ and $d_\psi(\xi,0) = \sqrt{\psi(\xi)}$, we see that $$\sigma_t(d\xi) := \frac{e^{-td_\psi^2(\xi,0)}}{(2\pi)^n\,p_t(0)}\,d\xi,\quad t>0,$$ are probability measures on ${{{\mathds R}^n}}$. Recall that $B^{d_\psi}(\eta,r) = \left\{\xi\in{{{\mathds R}^n}}\::\: d_\psi(\xi,\eta) < r \right\}$. The following result is essentially contained in [@KS1]. Note that in [@KS1] the condition $e^{-t\psi}\in L^1({{{\mathds R}^n}})$ is substituted by a more general (Hartman-Wintner type) condition on the growth of $\psi$.
\[pto-03\] Let $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$ and assume that $e^{-t\psi}\in L^1({{{\mathds R}^n}},\lambda)$. Then $$\label{pto-e04}
p_t(0) = (2\pi)^{-n} \int_0^\infty \lambda \left(B^{d_\psi}\big(0,\sqrt{r/t}\big)\right)\,e^{-r}\,dr,\quad t>0.$$ If the metric measure space $({{{\mathds R}^n}},d_\psi,\lambda)$ has the volume doubling property, then $e^{-t\psi}\in L^1({{{\mathds R}^n}},\lambda)$ and $$\label{pto-e06}
p_t(0) \asymp
\lambda\left(B^{d_\psi}\big(0,1/\sqrt t\big)\right)
\quad\text{for all}\quad t>0.$$
Observe that $\lambda\left(B^{d_\psi}\big(0,\sqrt{\rho}\big)\right) = \lambda\left\{\xi\in{{{\mathds R}^n}}\::\: \psi(\xi) < \rho\right\}$ is the distribution function of $\psi$. Using Fubini’s theorem we get $$\begin{aligned}
(2\pi)^n\,p_t(0)
&= \int_{{{\mathds R}^n}}e^{-t\psi(\xi)}\,d\xi\\
&= t\int_0^\infty \lambda\left(B^{d_\psi}\big(0,\sqrt{\rho}\big)\right)\,e^{-t\rho}\,d\rho\\
&= \int_0^\infty \lambda\left(B^{d_\psi}\big(0,\sqrt{r/t}\big)\right)\,e^{-r}\,dr,
\end{aligned}$$ and follows.
We know from Corollary \[cndf-23\] that volume doubling implies power growth of the increasing rearrangement $\psi_* = v_\psi^{-1}$. By ‘d)$\Rightarrow$a)’ of [@KS1 Proposition 5] we get that $e^{-t\psi}\in L^1({{{\mathds R}^n}},\lambda)$. Using and the monotonicity of the function $r\mapsto \lambda\left(B^{d_\psi}(0,r)\right)$ we get $$\begin{aligned}
(2\pi)^n\,p_t(0)
&{\geqslant}\int_1^\infty \lambda\left(B^{d_\psi}\big(0,\sqrt{r/t}\big)\right) e^{-r}\,dr\\
&{\geqslant}\lambda\left(B^{d_\psi}\big(0,1/\sqrt{t}\big)\right) \int_1^\infty e^{-r}\,dr\\
&= \frac 1e\,\lambda\left(B^{d_\psi}\big(0,1/\sqrt{t}\big)\right).
\end{aligned}$$ This proves the first inequality of . The upper estimate requires that $\psi$ enjoys the volume doubling property. This means that $$\lambda\left(B^{d_\psi}(0,cr)\right)
{\leqslant}\gamma_0(c)\,\lambda\left(B^{d_\psi}(0,r)\right), \quad c>1,\; r>0,$$ for some function $\gamma_0$ such that $\gamma_0(c){\leqslant}\gamma_0(1)\, c^\alpha$ for all $c{\geqslant}1$ with some suitable constant $\alpha{\geqslant}0$. Combining this with gives $$\begin{aligned}
(2\pi)^n\,p_t(0)
&= \int_0^1 \lambda\left(B^{d_\psi}\big(0,\sqrt{r/t}\big)\right) e^{-r}\,dr
+ \int_1^\infty \lambda\left(B^{d_\psi}\big(0,\sqrt{r/t}\big)\right) e^{-r}\,dr\\
&{\leqslant}\left(1-e^{-1}\right) \lambda\left(B^{d_\psi}\big(0,1/\sqrt{t}\big)\right)
+ \lambda\left(B^{d_\psi}\big(0,1/\sqrt t\big)\right) \int_1^\infty \gamma_0(1) r^{\alpha/2}\,e^{-r}\,dr\\
&= \kappa_1\,\lambda\left(B^{d_\psi}\big(0,1/\sqrt{t}\big)\right);
\end{aligned}$$ this is the upper estimate in .
Let us illustrate Theorem \[pto-03\] with several examples. First, however, we note that the estimate indicates some kind of ‘Gaussian’ behaviour of $p_t(0)$—it is comparable to the Lebesgue volume of a metric ball. This is exactly what we see in the Gaussian setting where $\psi(\xi)=\frac 12 |\xi|^2$ and $
p_t(x) = (2\pi t)^{-n/2}\,e^{-\frac{|x|^2}{2t}}.
$ Clearly, $$p_t(0) = (2\pi t)^{-n/2}
\quad\text{hence}\quad
p_t(0) = c_n\,\lambda\left(B\big(0,1/\sqrt t\big)\right)$$ where $B(0,r)$ stands for the usual ball in the Euclidean topology.
\[pto-05\] Let $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$ and denote by $p_t$ the corresponding transition density function (which we assume to exist). Let $(\eta_t)_{t{\geqslant}0}$ be a convolution semigroup of measures on the half-line $[0,\infty)$ and denote by $f$ the corresponding Bernstein function $f$. The subordinate density $p_t^f$ is given by $$p_t^f(x) = \int_0^\infty p_s(x)\,\eta_t(ds),\quad t>0,$$ and, consequently, $p_t^f(0)=\int_0^\infty p_s(0)\eta_t(ds)$. By Theorem \[pto-03\] we see $$p_t^f(0)
\asymp \int_0^\infty \lambda\left(B^{d_\psi}\big(0,1/\sqrt{s}\big)\right)\eta_t(ds).$$
Moreover, if $f\circ\psi$ has the volume doubling property, see e.g. Corollary \[cndf-25\], gives $$ \frac{p_t^f(0)}{\lambda\left(B^{d_{f\circ \psi}}\big(0,1/\sqrt t\big)\vphantom{B^{d_\psi}\big(0,\sqrt{f^{-1}(1/t)}\big)}\right)}
= \frac{p_t^f(0)}{\lambda\left(B^{d_\psi}\big(0,\sqrt{f^{-1}(1/t)}\big)\right)}
\asymp 1$$ and so $$ \lambda\left(B^{d_{f\circ \psi}}\big(0,1/\sqrt t\big)\right)
\asymp \int_0^\infty \lambda\left(B^{d_\psi}\big(0,1/\sqrt{s}\big)\right)\eta_t(ds).$$
\[pto-07\] Let $f$ be a Bernstein function. By Lemma \[cndf-05\] we know that $\psi := f(|\cdot |^2)$ is in ${\mathcal{MCN}}({{{\mathds R}^n}})$, and from Proposition \[cndf-15\] it follows that $$ \lambda\left(B^{d_{f(|\cdot|^2)}}(0,cr)\right)
= \left(\frac{f^{-1}(c^2 r^2)}{f^{-1}(r^2)}\right)^{n/2} \lambda\left(B^{d_{f(|\cdot|^2)}}(0,r)\right).$$ In particular, we get for $f(s)=s^\alpha$, $0<\alpha<1$, that $$\lambda\left(B^{d_{|\cdot|^{2\alpha}}}(0,cr)\right)
= c^{n/\alpha} \lambda\left(B^{d_{|\cdot|^{2\alpha}}}(0,r)\right).$$ Since $\lambda\left(B^{d_{|\cdot|^{2\alpha}}}(0,r)\right) = c_{n,\alpha}\,r^{n/\alpha}$ we recover the well-known estimates $$ p_t^{(2\alpha)}(0) \asymp t^{-n/2\alpha}$$ where $p_t^{(2\alpha)}(x)$ is the transition density of the symmetric $2\alpha$-stable Lévy process. In fact, in this special case, we can calculate $p_t^{(2\alpha)}(0)$ exactly: $$p_t^{(2\alpha)}(0)
= \alpha \Gamma\left(\tfrac n\alpha + 1\right) \, \lambda\left(B^{d_{|\cdot|^{2\alpha}}}\big(0,1/\sqrt t\big)\right).$$
More generally, if $\psi\in{\mathcal{MCN}}$ then $f\circ\psi\in{\mathcal{MCN}}$ by Lemma \[cndf-05\]. If $f\circ\psi$ has the doubling property, we get that $e^{-f\circ\psi}\in L^1({{{\mathds R}^n}},\lambda)$, cf. Theorem \[pto-03\], and $$\label{pto-e22}
p_t(0)
\asymp
\lambda\left(B^{d_\psi}\big(0,\sqrt{f^{-1}(1/t)}\big)\right)$$ where $p_t(x) = p_t^{f\circ\psi}(x) = (2\pi)^{-n} \int e^{-ix\xi} e^{-t f(\psi(\xi))}\,d\xi$ is the transition density of the subordinate Lévy process.
\[pto-09\] Consider on ${\mathds R}^m\times{\mathds R}^n$ the function $\psi(\xi,\eta)=|\xi|^\alpha + |\eta|^\beta$ with $0<\alpha<\beta<2$. Then $\psi\in{\mathcal{MCN}}({\mathds R}^m\times{\mathds R}^n)$. It is shown in [@SL] that $$ \lambda\left(B^{d_\psi}(0,R)\right)
= \left(\frac Rr\right)^{2\left(\frac m\alpha + \frac n\beta\right)}
\lambda\left(B^{d_\psi}(0,r)\right).$$ Consequently, we get for $p_t^\psi(0) = (2\pi)^{-n-m}\iint_{{{{\mathds R}^m}}\times{{{\mathds R}^n}}} e^{-t(|\xi|^\alpha+|\eta|^\beta)}\,d\xi\,d\eta$ that $$p_t^\psi(0)
\asymp
t^{-\frac m\alpha - \frac n\beta}.$$ Note that controls both the growth of the singularity of $p_t^\psi(0)$ as $t\to 0$ and the decay of $p_t^\psi(0)$ as $t\to\infty$. Both controls are related to volume growth in the corresponding metric measure spaces. A further consequence of is that if any two $\psi_1,\psi_2\in{\mathcal{MCN}}({{{\mathds R}^n}})$ are comparable in the sense that $$ \tau_0\,d_{\psi_1}(\xi,\eta)
{\leqslant}\,d_{\psi_2}(\xi,\eta)
{\leqslant}\tau_1\,d_{\psi_1}(\xi,\eta),\quad \xi,\eta\in{{{\mathds R}^n}},$$ for suitable constants $0<\tau_0{\leqslant}\tau_1<\infty$, then $p_t^{\psi_1}(0)$ and $p_t^{\psi_2}(0)$ have comparable growth behaviour as $t\to 0$ and $t\to\infty$. Nevertheless, we cannot expect any obvious comparison of $p_t^{\psi_1}(\xi)$ and $p_t^{\psi_2}(\xi)$ if $\xi\neq 0$.
Take, for example, $\psi_1(\xi,\eta)=|\xi|+|\eta|$ and $\psi_2(\xi,\eta) = \sqrt{\xi^2+\eta^2}$ where $\xi,\eta\in{\mathds R}$. Then $$\frac 1{\sqrt 2}\,(|\xi|+|\eta|)
{\leqslant}\sqrt{|\xi|^2+|\eta|^2}
{\leqslant}|\xi|+|\eta|.$$ On the other hand, $\psi_1$ and $\psi_2$ have different smoothness properties near the origin and it is this property that determines the decay of the transition functions $p_t^{\psi_1}(x,y)$ and $p_t^{\psi_2}(x,y)$ as $|x|+|y|\to\infty$. In fact, we have $$p_t^{\psi_1}(x,y) = \frac 1{\pi^2}\,\frac{t^2}{(x^2+t^2)(y^2+t^2)}$$ and $$p_t^{\psi_2}(x,y) = \frac{1}{2\pi}\,\frac{t}{\big((x^2+y^2)+t^2\big)^{3/2}}$$ which gives, for example, for $|x|\to\infty$, $$p_1^{\psi_1}(x,0) \asymp |x|^{-2}
\quad\text{while}\quad
p_1^{\psi_2}(x,0) \asymp |x|^{-3}.$$
On the off-diagonal behaviour of $p_t(x)$ {#ptx}
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As in Section \[pto\] we assume that $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$ is real valued and that the associated convolution semigroup $(\mu_t)_{t>0}$ is absolutely continuous with respect to Lebesgue measure, $\mu_t(dx) = p_t(x)\,dx$. We have seen that $p_t(0)$ has a natural meaning in the metric measure space $({{{\mathds R}^n}},d_\psi,\lambda)$. We have $$\label{ptx-e04}
\frac{p_t(x)}{p_t(0)}
= \int_{{{{\mathds R}^n}}} e^{-ix\xi} \, \frac{e^{-t d_\psi^2(\xi,0)}}{(2\pi)^{n}\,p_t(0)}\,d\xi
= \int_{{{{\mathds R}^n}}} e^{-ix\xi} \, \frac{e^{-t \psi(\xi)}}{(2\pi)^{n}\,p_t(0)}\,d\xi,$$ and we want to understand $p_t(x)/p_t(0)$ better. For this we start with a few examples.
\[ptx-03\] **a)** Let $\psi(\xi)=\frac 12 |\xi|^2$. Then $p_t(x)$ is the Gauss kernel in ${{{\mathds R}^n}}$, $$ \frac{p_t(x)}{p_t(0)} = \exp\left(-\frac{|x|^2}{2t}\right).$$ and for the (almost Euclidean) metric $\phi_t(x,y) = |x-y|/\sqrt{2 t}$ we get $$ \frac{p_t(x)}{p_t(0)} = \exp\left(-\phi_t^2(x,0)\right).$$
**b)** Let $\psi(\xi)=|\xi|$. Then $p_t(x)$ is the density of the Cauchy process in ${{{\mathds R}^n}}$ and we find $$ \frac{p_t(x)}{p_t(0)} = \exp\left(-\phi_t^2(x,0)\right)$$ with $$ \phi_t(x,y) = \sqrt{ \frac{n+1}2\,\ln\left[\frac{|x-y|^2+t^2}{t^2}\right]}.$$ If we fix $t>0$, then $\phi_t(\cdot,\cdot):{{{\mathds R}^n}}\times{{{\mathds R}^n}}\to{\mathds R}$ is a translation invariant metric. Symmetry, positivity and definiteness are obvious. The triangle inequality follows from the fact that $r\mapsto\ln\left(1+\frac ra\right)$ is a Bernstein function and that $\phi_t^2(x,0)$ is a continuous negative definite function.
Thus we are naturally led to the following question.
\[ptx-05\] Let $p_t$ be the transition density of some symmetric Lévy process with characteristic exponent $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$. Does there exist a mapping $\delta_\psi:(0,\infty)\times{{{\mathds R}^n}}\times{{{\mathds R}^n}}\to{\mathds R}$ such that for every $t\in (0,\infty)$ the map $\delta_{\psi,t}(\cdot,\cdot):{{{\mathds R}^n}}\times{{{\mathds R}^n}}\to{\mathds R}$ is a (translation invariant) metric such that $$\label{ptx-e14}
\frac{p_t(x)}{p_t(0)} = \exp\left(-\delta_{\psi,t}^2( x,0)\right)\;?$$
Below we will see many more concrete examples for which Problem \[ptx-05\] can be answered in the affirmative. There is, however, a general result supporting the conjecture that should hold for all symmetric Lévy processes with characteristic exponent $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$.
The key observation is that the metric space $({{{\mathds R}^n}},d_\psi)$, $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$, can be isometrically embedded into some Hilbert space. The following result is originally due to Schoenberg [@Schoe1; @Schoe2] and it led to the notion of negative definite functions; a modern account is given in the monograph [@BL] by Benyamini and Lindenstrauss.
\[ptx-06\] Let $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$. Then the metric measure space $({{{\mathds R}^n}},d_\psi)$ can be isometrically embedded into some Hilbert space $(\mathcal H,{\langle \cdot,\cdot\rangle}_{\mathcal H})$.
Conversely, if $J:({{{\mathds R}^n}},d)\to(\mathcal H,{\langle \cdot,\cdot\rangle}_{\mathcal H})$ is an isometric embedding into some Hilbert space such that ${\langle J(x),J(y)\rangle}_{\mathcal H} = {\langle J(x-y),J(0)\rangle}_{\mathcal H}$, then $d=d_\psi$ for some negative definite function $\psi:{{{\mathds R}^n}}\to{\mathds R}$.
Let us give a *sketch of the proof* of the sufficiency using the theory of Dirichlet forms, see [@FOT] or [@J1]. Our proof will also reveal the structure of the embedding. In order to exclude trivial cases, we will assume that $\psi$ has no quadratic part. Therefore, the Lévy-Khintchine formula becomes $$\psi(\xi)
= \int_{y\neq 0} \left(1-\cos(y\xi)\right)\nu(dy).$$
We can associate with every continuous negative definite function $\psi:{{{\mathds R}^n}}\to{\mathds R}$ a pseudo-differential operator $\psi(D)$ on $\mathscr S({{{\mathds R}^n}})$ defined by $$ \psi(D)u(x)
= \int_{{{\mathds R}^n}}e^{ix\xi}\psi(\xi)\,{\mathcal{F}}u(\xi)\,d\xi.$$ Introducing the scale of Hilbert spaces $H^{\psi,s}({{{\mathds R}^n}})$, $s{\geqslant}0$, $$\begin{gathered}
H^{\psi,s}({{{\mathds R}^n}}) := \left\{u\in L^2({{{\mathds R}^n}}) \::\: {\|u\|}_{\psi,s}<\infty\right\},\\
{\|u\|}_{\psi,s}^2 := \int_{{{\mathds R}^n}}(1+\psi(\xi))^s |{\mathcal{F}}u(\xi)|^2\,d\xi,
\end{gathered}$$ it is easy to see that $\psi(D):H^{\psi,s+2}({{{\mathds R}^n}})\to H^{\psi,s}({{{\mathds R}^n}})$ is continuous and that the quadratic form $\mathcal E^\psi$ associated with $\psi(D)$ by $$ \mathcal E^\psi(u,v)
= \int_{{{\mathds R}^n}}\psi(D)^{1/2}u(x) \psi(D)^{1/2}v(x)\,dx
= (2\pi)^n \int_{{{\mathds R}^n}}\psi(\xi)\,{\mathcal{F}}u(\xi) \, \overline{{\mathcal{F}}v(\xi)}\,d\xi$$ is closed on $D(\mathcal E^\psi) = H^{\psi,1}({{{\mathds R}^n}})$. Using the Lévy-Khintchine representation we get $$ \mathcal E^\psi(u,v)
= \frac 12 \iint_{{{{\mathds R}^n}}\times{{{\mathds R}^n}}\setminus\{0\}} \big(u(x+y)-u(x)\big)\big(v(x+y)-v(x)\big)\,\nu(dy)\,dx.$$ We call the operator $(u,v)\mapsto\Gamma(u,v)$ where $$ \Gamma(u,v)(x)
= \frac 12 \int_{{{{\mathds R}^n}}\setminus\{0\}} \big(u(x+y)-u(x)\big)\big(v(x+y)-v(x)\big)\,\nu(dy)$$ the *carré du champ operator* associated with the Dirichlet form $\mathcal E^\psi$. For a comprehensive discussion of the carré du champ operator we refer to [@BH].
Consider the set $$ \mathcal C_\psi
:= \big\{ u\in C^2({{{\mathds R}^n}})\::\: \Gamma(u,u)(0)<\infty\big\} \Big/ \big\{ u\in C^2({{{\mathds R}^n}})\::\: \Gamma(u,u)(0)=0\big\}$$ and notice that all constants belong to the set $\big\{ u\in C^2({{{\mathds R}^n}})\::\: \Gamma(u,u)(0)=0\big\}$.
If the Lévy measure of $\psi$ has full support, ${\operatorname{\mathrm{supp}}}\nu = {{{\mathds R}^n}}$—e.g. if $\nu = g\lambda$ with an everywhere strictly positive density $g>0$—, then $\Gamma(u,v)(0)$ is a scalar product on $\mathcal C_\psi$. Let us denote this scalar product, for a moment, by $${\langle u,v\rangle}_{\mathcal H} := \Gamma(u,v)(0).$$ Then $\mathcal C_\psi = \big\{ u\in C^2({{{\mathds R}^n}})\::\: \Gamma(u,u)(0)<\infty\big\} \big/ \big\{ u\equiv\text{const}\big\}$. (Since $\nu$ has full support it is not hard to see that $\big\{ u\in C^2({{{\mathds R}^n}})\::\: \Gamma(u,u)(0)=0\big\}$ are exactly the constant functions). The completion of $\mathcal C_\psi$ with respect to ${\langle \cdot,\cdot\rangle}_{\mathcal H}$ gives a Hilbert space $(\mathcal H,{\langle \cdot,\cdot\rangle}_{\mathcal H})$. Each function $e_\xi$, $e_\xi(x):= e^{ix\xi}$, $\xi\in{{{\mathds R}^n}}$, represents some element of $\mathcal H$ and we get $$ \Gamma(e_\xi,e_\xi)(0) = \psi(\xi).$$ This shows that the map $J:{{{\mathds R}^n}}\to\mathcal H$, $\xi\mapsto e_\xi$, embeds the metric space $({{{\mathds R}^n}},d_\psi)$ isometrically into the Hilbert space $(\mathcal H, {\langle \cdot,\cdot\rangle}_{\mathcal H})$.
Thus, we can understand $e^{-t\psi(\xi)}$ as a *Gaussian in disguise*: Fourier transforms of Gaussians should be Gaussians, and Gaussians have obviously the proposed structure. It is tempting to find a representation of $p_t(x)$ as some image of an infinite dimensional Gaussian defined somehow on $(\mathcal H,{\langle \cdot,\cdot\rangle}_{\mathcal H})$. So far, however, such a result resists all of our attempts to prove it.
Before we return to Problem \[ptx-05\] and before we provide more examples, we want to give an interpretation of .
We have $$ p_t(x) = p_t(0) \exp\left(-\delta_{\psi,t}^2( x,0))\right)$$ and if $\psi$ satisfies the volume doubling condition, we get from Theorem \[pto-03\] $$ p_t(x) \asymp
\lambda\big(B^{d_\psi}(0,1/\sqrt t)\big) \exp\big(-\delta_{\psi,t}^2( x,0))\big)$$ with the balls $B^{d_\psi}(0,r)=\left\{y\in{{{\mathds R}^n}}\::\: d_\psi(y,0)<r \right\}$. Thus, if $({{{\mathds R}^n}},d_\psi,\lambda)$ has the volume doubling property, $p_t(x)$ is controlled by two geometric expressions. More precisely, for fixed $t>0$ we have two metrics $\delta_{\psi,t}(\cdot,\cdot)$ and $d_\psi(\cdot,\cdot)$ which describe the behaviour of $p_t(x)$. The situation becomes more transparent if we switch from $d_\psi(x,y)=\sqrt{\psi(x-y)}$ to $d_{\psi,t} : {{{\mathds R}^n}}\times{{{\mathds R}^n}}\to{\mathds R}$, $t>0$, $d_{\psi,t}(x,y)= \sqrt{t\cdot\psi(x-y)}$ since, in this new metric, $$\begin{aligned}
B^{d_\psi}\big( 0,\tfrac 1{\sqrt t}\big)
&= \left\{ y\in{{{\mathds R}^n}}\::\: d_\psi(y,0)<\tfrac 1{\sqrt t}\right\}\\
&= \left\{ y\in{{{\mathds R}^n}}\::\: d_{\psi,t}( y,0)<1\right\}
=: B^{d_{\psi,t}}(0,1).\end{aligned}$$ From now on we will adopt the point of view that $p_t(x)$ should be understood in terms of two families of metrics, $d_{\psi,t}( \cdot, \cdot)$ and $\delta_{\psi,t}(\cdot,\cdot)$, by the estimates $$ p_t(x) \asymp \lambda\left(B^{d_{\psi,t}}(0,1)\right) \exp\left(-\delta_{\psi,t}^2(x,0)\right).$$ Consequently, the understanding of $p_t(x)$ is reduced to the study of $d_{\psi,t}(\cdot,\cdot)$ and $\delta_{\psi,t}(\cdot,\cdot)$. For a Brownian motion $\psi(\xi)=\frac 12|\xi|^2$; in this case both metrics $\delta_{\psi,t}$ and $d_\psi$ are (essentially) Euclidean distances. This special situation is related to the fact that the Gaussian is, up to constants, a fixed point of the Fourier transform.
This interpretation allows us also to think about distributions of random variables or collections of densities $p_t$, $t\in I\subset (0,\infty)$. We will follow up this remark in the next section.
Of special interest is the case when holds with a metric $\delta_{\psi,t}$ such that $x\mapsto \delta_{\psi,t}^2(x,0)$ is a negative definite function. This is, e.g. the case in Example \[ptx-03\]. Here is a further example.
\[ptx-07\] The symmetric Meixner process on ${\mathds R}$ has the characteristic Lévy exponent $\psi(\xi) = \ln(\cosh \xi)$ and the transition density $$ p_t(x) = \frac{2^{t-1}}{\pi\,\Gamma(t)}\,\left|\Gamma\left(\frac{t+ix}{2}\right)\right|^2.$$ Using the representation of the Gamma function as an infinite product, we find $$ \frac{p_t(x)}{p_t(0)}
= \left|\frac{\Gamma\left(\frac{t+ix}{2}\right)}{\Gamma\left(\frac{t}{2}\right)}\right|^2
= \prod_{j=1}^\infty \left(1+\frac{x^2}{(t+2j)^2}\right)^{-1}.$$ Since for a sequence $(a_j)_{j{\geqslant}1}$ of positive numbers the convergence of $$\prod_{j=1}^\infty (1+a_j),\quad
\sum_{j=1}^\infty \ln(1+a_j)\quad\text{and}\quad
\sum_{j=1}^\infty a_j$$ is equivalent, we find that $$ \delta^2_t(x,0)
= - \ln \left|\frac{\Gamma\left(\frac{t+ix}{2}\right)}{\Gamma\left(\frac{t}{2}\right)}\right|^2
= \sum_{j=1}^\infty \ln\left(1+\frac{x^2}{(t+2j)^2}\right).$$ For every $j{\geqslant}1$ and $t>0$ the function $x\mapsto \ln\big(1+x^2(t+2j)^{-2}\big)$ is continuous and negative definite. Since the series $\sum_{j=1}^\infty \ln\big(1+x^2(t+2j)^{-2}\big)$ converges locally uniformly as a function of $x$, its sum $\delta^2_t(x,0)$ is again a continuous negative definite function. This shows that the transition density of a symmetric Meixner process satisfies $$\label{ptx-e36}
p_t(x) = p_t(0) \, \exp\left(-\delta^2_t(x,0)\right)$$ and $p_t(0)$ can be written, as before, as $$ p_t(0)
= (2\pi)^{-n}\int_{{{\mathds R}^n}}e^{-\frac t2 \ln\left(\cosh^2\xi\right)}\,d\xi
= (2\pi)^{-n}\int_{{{\mathds R}^n}}e^{- t \ln\left(\cosh\xi\right)}\,d\xi.$$
Although $x\mapsto\delta^2_t(x,0)$ is a continuous negative definite function, $e^{-\delta_t^2(x,0)}$ is *not* the characteristic function of an additive process, i.e. a stochastically continuous process with independent, but not necessarily stationary, increments, see [@Sato]. This can be seen from the the Lévy-Khintchine representation for $\delta_t^2(x,0)$: $$\begin{aligned}
\delta_t^2(x,0)
&=\sum_{j=0}^\infty \ln \left(1+\frac{x^2}{(t+2j)^2}\right)\\
&=\int_{{\mathds R}\setminus\{0\}} \frac{1-\cos(xz)}{|z|} \left( \sum_{j=0}^\infty e^{-|z|(t+2j)}\right) dz\\
&=\int_{{\mathds R}\setminus\{0\}} \bigl(1-\cos(xz)\bigr)\,g(t,z)\,dz\end{aligned}$$ with $g(t,z):=\sum_{j=0}^\infty e^{-|z|(t+2j)}$. In this calculation we used the Lévy-Khintchine representation for the continuous negative definite function $$\ln \left(1+\frac{\xi^2}{a^2}\right)
=\int_{{\mathds R}\setminus\{0\}} \bigl(1-\cos(\xi z)\bigr)\,\frac{e^{-|z|a }}{|z|}\,dz.$$ Since $g(t,z)$ is decreasing in $t$, we see that $e^{-\delta_t^2(x,0)}$ cannot be the characteristic function of an additive process.
In the next Sections \[dist\] and \[proc\] we will continue our investigation of . In fact, we will encounter a large class of processes with transition function $$p_t(x) = p_t(0) e^{-g(t;|x|^2)}$$ where for every fixed $t>0$ the function $g(t;\cdot)$ is a Bernstein function.
We want to collect some more information on $p_t$ and $p_t/p_t(0)$. Since $p_t$ is the Fourier transform of a measure, it is a positive definite function. The measure $\rho_t$ defined by $$ \rho_t(dx) := \frac{{\mathcal{F}}^{-1}p_t(x)}{p_t(0)}\,dx$$ is a probability measure and if $X_t$ is a random variable with distribution $\rho_t$, we have $${\mathds P}(X_t\in dx) = \rho_t(dx) = \frac{e^{-t\psi(x)}}{p_t(0)}\,dx.$$ Thus, we have a certain duality. Given a (symmetric) Lévy process $(Y_t^\psi)_{t{\geqslant}0}$ with characteristic exponent $\psi\in{\mathcal{MCN}}({{{\mathds R}^n}})$, then $Y_t^\psi$ is for every $t>0$ associated with $X_t$ and vice versa. Of particular interest should be the case where $(X_t)_{t{\geqslant}0}$ is itself a ‘nice’ process, say an additive process or even a Lévy process. In Section \[proc\] we will provide some examples for such pairings.
Our starting point was to understand the densities of Lévy processes in terms of two one-parameter families of metrics, $(d_{\psi,t}(\cdot,\cdot))_{t>0}$ and $(\delta_{\psi,t}(\cdot,\cdot))_{t>0}$. Although this is still the main aim of our study, it is often useful to fix $t=t_0$ and to consider a single probability density $p(x) = p_{t_0}(x)$ rather than the whole family $(p_t(x))_{t>0}$. This is no loss of generality since we can embed every infinitely divisible probability density $p(x)$ into a convolution semigroup $(\mu_t)_{t{\geqslant}0}$ such that $\mu_{t_0}(dx) = p(x)\,dx$. In the following section and in Section \[proc\] we will study examples of infinitely divisible probability densities.
Examples of class ${\mathsf{N}}$ distributions {#dist}
==============================================
The following definition covers all cases mentioned in Problem \[ptx-05\] where the exponent $\delta_{\psi,1}(x,0)$ appearing in is not only a metric but a metric induced by some negative definite function.
\[dist-03\] By ${\mathsf{N}}$ we denote the class of infinitely divisible probability densities $p(x)$ on ${{{\mathds R}^n}}$, such that ${\mathcal{F}}^{-1} p(\xi)/p(0)$ is again an infinitely divisible probability density.
Note that ${\mathcal{F}}^{-1} p(\xi) = e^{-\phi(\xi)}$ is always an *infinitely divisible characteristic function* with a continuous negative definite characteristic exponent $\phi:{{{\mathds R}^n}}\to{\mathds C}$. This is different from saying that ${\mathcal{F}}^{-1} p(\xi)/p(0)$ is an *infinitely divisible probability distribution*; $p\in {\mathsf{N}}$ means that $-\ln \bigl[p(x)/p(0)\bigr]$ is a continuous negative definite function. In particular, all distributions in the class ${\mathsf{N}}$ provide solutions to Problem \[ptx-05\].
In order to get examples of class ${\mathsf{N}}$ distributions we start with a simple class of examples related to a single infinitely divisible random variable $X$ on ${{{\mathds R}^n}}$ with an (infinitely divisible) probability density $p(x)$ which is symmetric. Then $${\mathds E}e^{i\xi X}
= {\mathcal{F}}^{-1} p(\xi)
= e^{-\phi(\xi)}$$ where $\phi:{{{\mathds R}^n}}\to{\mathds R}$ is a continuous negative definite function. Suppose that $p$ is *extended (Fourier) self-reciprocal* in the sense that there is a constant $\gamma>0$ and a non-degenerate matrix $C\in{\mathds R}^{n\times n}$, $\text{det}(C)>0$, such that $$ {\mathcal{F}}^{-1}p(\xi) = \frac 1\gamma\,p\big( C\xi\big)\quad\text{for all}\quad \xi\in{{{\mathds R}^n}}.$$ It follows that $p(0)=\gamma$ and $$ p(\xi) = p(0)\,e^{-d_\phi^2(C^{-1}\xi,0)}\quad\text{for all}\quad \xi\in{{{\mathds R}^n}},$$ where $d_\phi(\xi,\eta) := \sqrt{\phi(\xi-\eta)}$ is the metric induced by the continuous negative function $\phi$.
\[dist-05\] **a)** For every $\sigma>0$ the normal distribution $N(0,\sigma^2)$ is extended Fourier self-reciprocal and we have $$\frac 1{(2\pi\sigma^2)^{n/2}}\, \exp\left(-\frac{|\xi|^2}{2\sigma^2}\right)
= \exp\left(-\frac 12\,\sigma^2\,|\xi|^2\right)
\quad\text{for all}\quad \xi\in{{{\mathds R}^n}}.$$
**b)** The one-dimensional symmetric Meixner process $(X_t^M)_{t{\geqslant}0}$ has $\psi(\xi) = \ln(\cosh\xi)$, $\xi\in{\mathds R}$, as characteristic exponent. Denote by $(p_t(x)\,dx)_{t>0}$ the corresponding convolution semigroup. For $t=1$ we have, see [@PY03], $$p_1(x) = \frac{1}{2\cosh\big(\frac 12\,\pi x \big)}
\quad\text{and}\quad
{\mathcal{F}}^{-1} p_1(\xi) = \frac 1{\cosh \xi}.$$
**c)** Denote by $K_\lambda$ the modified Bessel function of the third kind, cf. [@erd-et-al vol. 2]. Let $Q>0$ and $\kappa>0$ be two positive numbers. Then $$p(x)
:= \frac{\kappa^{1/4}}{\sqrt{2\pi} \, K_{1/4}(\kappa^2)} \, \frac{K_{-1/4}\big(\kappa\sqrt{\kappa^2 + (x/Q)^2}\big) }{\big(\kappa^2 + (x/Q)^2\big)^{n/8}}$$ is the density of a generalized hyperbolic distribution on ${\mathds R}$ which is known to be infinitely divisible, see [@BNH] and [@BKS]. Then $${\mathcal{F}}^{-1} p(\xi)
= \frac{\kappa^{1/4}}{K_{1/4}(\kappa^2)} \, \frac{K_{1/4}\big(\kappa\sqrt{\kappa^2 + (Qx)^2}\big) }{\big(\kappa^2 + (Qx)^2\big)^{n/8}}.$$ Since $K_{1/4} = K_{-1/4}$, we see that generalized hyperbolic densities are extended Fourier self-reciprocal.
Examples \[dist-05\] a) and c) are special cases of so-called *normal variance-mean mixtures*. The following definition is taken from [@BKS].
\[dist-07\] An $n$-dimensional random vector $Z$ is called a *normal variance-mean mixture*, if $Z$ is an $n$-dimensional normal distribution with covariance $Y Q$ and mean vector $\mu+Y \beta$, where $Q\in{\mathds R}^{n\times n}$ is a symmetric positive definite matrix, $\mu$ and $\beta$ are $n$-dimensional matrices, and $Y$ is a positive random variable with probability law $\rho(ds)$ on $[0,\infty)$; $\rho$ is called the *mixing (probability) distribution*.
Since we are mainly interested in the symmetric case, we assume from now that $\mu=\beta=0$. Denote by $p^\rho(dx)$ the law of $Z$; by definition it is the mixture, in the sense of , of the Gaussian density $ p(s,x)=(2\pi s)^{-n/2} e^{-\frac{x \cdot Q^{-1} x}{2s}}$, $Q\in{\mathds R}^{n\times n}$ is positive definite, and the probability measure $\rho(ds)$. If the law $p^\rho(dx)$ is absolutely continuous with respect to $n$-dimensional Lebesgue measure, the density $p^\rho(x)$ is given by $$\label{dist-e06}
p^\rho(x)=\int_0^\infty (2\pi s)^{-n/2} \exp\left[ -\frac{1}{2s}\, x\cdot Q^{-1}x\right] \rho(ds), \quad x\in{{{\mathds R}^n}}.$$ Set $\phi(\lambda):=\mathcal{L} \rho(\lambda)$. Then we can calculate the characteristic function of $Z$ as a composition of the characteristic functions of a symmetric Gaussian distribution and $\phi$, $$\label{dist-e08}
{\mathcal{F}}^{-1}[p^\rho](\xi)= \phi\left(-\frac{1}{2} \xi \cdot Q \xi\right).$$ In general, we do not assume that the mixing probability measure $\rho$ is infinitely divisible.
The following theorem from [@BKS] gives sufficient conditions when normal variance-mean mixture is Fourier self-reciprocal. Its proof follows directly from the representation .
\[dist-09\] Let $p^\rho(x)$ be a normal variance-mean mixture given by and assume that the mixing probability measure $\rho(ds)=\rho(s)\,ds$ is absolutely continuous with respect to Lebesgue measure on $[0,\infty)$. If the density $\rho(s)$, $s{\geqslant}0$, satisfies $$\label{dist-e09}
\rho(s)=s^{(n-4)/2}\, \rho\left(\frac{1}{s}\right),$$ then $p^\rho=c \, {\mathcal{F}}^{-1}[p^\rho]$ with $c = p^\rho(0)$.
For our purposes the following simple corollary is important.
\[dist-11\] If the probability density $p^\rho(x)$, $x\in{{{\mathds R}^n}}$, is infinitely divisible and Fourier self-reciprocal, then $p^\rho \in {\mathsf{N}}$.
We have seen in Example \[dist-05\]c) that the one-dimensional symmetric Meixner and the one-dimensional generalized hyperbolic distributions are (extended) Fourier self-reciprocal, hence of class ${\mathsf{N}}$. Using the mixing result from Corollary \[dist-11\] we can extend this result to the $n$-dimensional generalized hyperbolic distribution with parameters $\lambda=n/4$, $\beta=\mu=0$ and $\kappa=\eta$. Let us remark that for $\lambda=n/4$ and $\kappa\neq \eta$, although Theorem \[dist-09\] does not cover this case, we are still in the generalized self-reciprocal setting. Indeed, $$\rho_{\eta,\kappa,\lambda} (s)
=\left(\frac{\kappa}{\eta}\right)^{2\lambda} s^{\frac{n-4}{2}} \rho_{\kappa,\eta,\lambda} (1/s).$$
Let $Y\sim\rho$ be a mixing random variable where $\rho(ds):=\rho_{\eta,\kappa,\lambda}(s)\,ds$ is the generalized inverse Gaussian distribution $$\label{dist-e14}
\rho_{\eta,\kappa,\lambda}(s)
=\frac{(\kappa/\eta)^\lambda}{2K_\lambda (\eta \kappa)}\, s^{\lambda-1} \exp\left[-\frac{1}{2}\,\big(\eta^2 s^{-1} + \kappa^2 s\big)\right],
\quad s>0,$$ with parameters $\eta, \kappa$ and $\lambda$ satisfying $$\label{dist-e15}
\begin{cases}
\eta{\geqslant}0, \quad \kappa>0 &\text{if\ \ } \lambda>0;\\
\eta>0, \quad \kappa>0 &\text{if\ \ } \lambda=0;\\
\eta>0, \quad \kappa{\geqslant}0 &\text{if\ \ } \lambda<0.
\end{cases}$$ From [@BNH] we know that $\rho_{\eta,\kappa,\lambda}(s)$ is an infinitely divisible probability density on $(0,\infty)$.
Consider the probability density $p^\rho$ as in with mixing probability $\rho = \rho_{\eta,\kappa,\lambda}$ as in . The formulae for $p^\rho$ and ${\mathcal{F}}^{-1}{p^\rho}$ can be explicitly calculated, see e.g. [@BNH] and [@BKS], $$\begin{gathered}
p^\rho(x)
=\label{dist-e16}
\frac{(\kappa/\eta)^{\lambda}}{(2\pi)^{n/2}K_\lambda(\eta \kappa)}\,
\frac{K_{\lambda-n/2} \big(\kappa\sqrt{\eta^2+x\cdot Q^{-1}x}\big)}{\big(\kappa^{-1}\sqrt{\eta^2+x\cdot Q^{-1} x} \big)^{n/2-\lambda}},\quad x\in{{{\mathds R}^n}},
\\
{\mathcal{F}}^{-1}[p^\rho](\xi)
=\label{dist-e18}
\left(\frac{\kappa^2}{\kappa^2+\xi\cdot Q \xi}\right)^{\lambda/2}\,
\frac{K_\lambda\big(\eta \sqrt{\kappa^2+\xi\cdot Q \xi}\big)}{K_\lambda(\eta \kappa)}, \quad \xi\in{{{\mathds R}^n}}.\end{gathered}$$ The probability distributions with the densities $p^\rho$ are called *generalized hyperbolic distributions* with parameters $\eta$, $\kappa$ and $\lambda$ (satisfying ). Since $\rho = \rho_{\eta,\kappa,\lambda}$ is infinitely divisible, so is $p^\rho$; as $K_\lambda = K_{-\lambda}$, it is easy to see that $p^\rho$ is extended (Fourier) self-reciprocal for $\lambda = n/4$, hence $p^\rho\in{\mathsf{N}}$ by Corollary \[dist-11\].
For general $\lambda{\geqslant}0$ and parameters $\eta,\kappa$ satisfying we can use the fact that and have the same structure to conclude that $-\ln \bigl(p^\rho(x)\big/p^\rho(0)\bigr)$ is a continuous negative definite function. In the table in Section \[table\] below we give several one-dimensional examples of such $p^\rho$. Our considerations show
\[dist-13\] Let $Q\in{\mathds R}^{n\times n}$ be a positive definite matrix, let $\eta,\kappa,\lambda$ satisfy the conditions and denote by $\rho = \rho_{\eta,\kappa,\lambda}$ the generalized inverse Gaussian distribution . Then the generalized hyperbolic distribution $p^\rho$ from has the property that $$\psi_{\eta,\kappa,\lambda}(x)
:=
-\ln \frac{p^\rho(x)}{p^\rho(0)}
=
-\ln\left(\frac{\eta^{n/2-\lambda}\, K_{\lambda-n/2} \big(\kappa \sqrt{\eta^2 + x\cdot Q^{-1}x} \big)}{K_{\lambda-n/2}(\kappa\eta) \big(\eta^2 + x\cdot Q^{-1}x \big)^{n/4 - \lambda/2}}\right),
\quad x\in{{{\mathds R}^n}},$$ is a continuous negative definite function which induces a metric $\delta_{\eta,\kappa,\lambda}(x,y) = \sqrt{\psi_{\eta,\kappa,\lambda}(x-y)}$ of class ${\mathcal{MCN}}({{{\mathds R}^n}})$. In particular, $p^\rho\in{\mathsf{N}}$.
We only have to show that $\delta_{\eta,\kappa,\lambda}\in{\mathcal{MCN}}({{{\mathds R}^n}})$. Since $$K_\nu(x) \approx \sqrt{\frac{\pi}{2|x|}} \, e^{-c|x|}
\quad\text{as}\quad
|x|\to \infty,$$ cf. [@erd-et-al vol. 2, §7.4.1], we get $\delta^2_{\eta,\kappa,\lambda}(x,0)= |x|+o(|x|)$ as $|x|\to\infty$. This implies that $\delta_{\eta,\kappa,\lambda}\in{\mathcal{MCN}}({{{\mathds R}^n}})$ by Lemma \[cndf-05\].
Let us add one more example of class ${\mathsf{N}}$ distributions obtained by mixing. Consider the probability density $p_2^S$ obtained by mixing with the probability measure $\rho^S_2(ds)$ whose Laplace transform is $$\label{dist-e20}
\mathcal{L}[\rho^S_2](\lambda)
=\left( \frac{\sqrt{2\lambda}}{\sinh \sqrt{2\lambda}}\right)^2.$$ The density $p_2^S$ and its characteristic function ${\mathcal{F}}^{-1}[p_2^S]$ can be calculated explicitly, $$ p_2^S(x)=\frac{\frac{\pi}{2} \left(\frac{\pi x}{2} \coth(\frac{\pi x}{2})-1\right)}{\sinh^2 (\frac{\pi x}{2})}
\quad\text{and}\quad
{\mathcal{F}}^{-1}[p_2^S](\xi)=\left(\frac{\xi}{\sinh\xi}\right)^2,$$ see [@PY03 p. 312, Table 6]. Note that $p_2^S$ is infinitely divisible because the mixing measure $\rho_2^S$ is infinitely divisible.
\[dist-15\] The transition probability density $p_2^S$ belongs to the class ${\mathsf{N}}$.
The proof of Lemma \[dist-15\] relies on the following proposition from [@SSV Corollary 9.16].
\[dist-17\] Let $g$ and $h$ be two entire functions of orders $0<\rho_1,\rho_2<2$ such that $g(0)=h(0)=0$; let $(a_n)_{n{\geqslant}1}$ and $(b_n)_{n{\geqslant}1}$ be two strictly increasing sequences of positive numbers, and let $(\pm a_n)_{n{\geqslant}1}$ and $(\pm b_n)_{n{\geqslant}1}$ be simple roots of $g$ and $h$, respectively. Moreover, assume that $a_n<b_n$ for all $n{\geqslant}1$. Then the function $\phi(t)=h(it) / g(it)$ is the characteristic function of an infinitely divisible probability measure on ${\mathds R}$.
We have to show that $\delta^2(x,0):=-\ln \bigl(p_2^S(x)\big/ p_2^S(0)\bigr)$ is negative definite. Write $\delta^2(x,0)$ in the form $\delta^2(x,0)=h(ix)\big/ g^3(ix)$, where $$h(z):=\frac{\pi}{2} \Big(\frac{\pi z}{2} \cos (\frac{\pi z}{2})-\sin (\frac{\pi z}{2}) \Big)
\quad\text{and}\quad
g(z):=\sin (\frac{\pi z}{2}).$$ We want to use Proposition \[dist-17\]. Clearly, both $g$ and $h$ are entire functions, so we need to check only the assumption about their zeroes. The zeroes of $g(z)$ are $(\pm 2k)_{k{\geqslant}0}$. Observe that $h(z)=(\frac{\pi z}{2})^2\, \frac{d}{dz} \left( \sin \bigl(\frac{\pi z}{2}\bigr) \big/ \frac{\pi z}{2} \right)$. Since $\sin \bigl(\frac{\pi z}{2}\bigr)\big/\frac{\pi z}{2}$ is an entire function of order $1$ with exclusively real roots, Laguerre’s theorem, see [@T39 Theorem 8.5.2], shows that the roots of $\frac d{dz} \left( \sin \bigl(\frac{\pi z}{2}\bigr)\big/\frac{\pi z}{2}\right)$ are also real and that they are located between the zeroes of $\sin \bigl(\frac{\pi z}{2}\bigr)\big/\frac{\pi z}{2}$, i.e. between $(\pm 2k)_{k>0}$. Moreover, $z=0$ is a root of $\frac d{dz} \left( \sin\bigl(\frac{\pi z}{2}\bigr)\big/\frac{\pi z}{2}\right)$ of multiplicity $1$ because it is located between $-2$ and $2$. Hence, the function $h$ has a root $z=0$ of multiplicity $3$, and otherwise simple roots $b_k, k>0$, located between $(2k)_{k>0}$ on the right half-axis, and $-b_k, k>0$, located between $(-2k)_{k>0}$ on the left half-axis. Thus we can rewrite $h(z)$ using the Hadamard representation theorem in the form $$h(z)=\Big(\frac{\pi z}{2}\Big)^3 \prod_{k=1}^\infty \Big(1-\frac{z^2}{b_k^2}\Big).$$ Therefore, $$\delta^2(z,0)
=\frac{ \prod_{k=1}^\infty \big(1-\frac{z^2}{b_k^2}\big)}{\sin\bigl(\frac{\pi z}{2}\bigr)\big/\frac{\pi z}{2}} \left(\frac{(\frac{\pi z}{2})}{\sin (\frac{\pi z}{2})}\right)^2
=\phi_1(z)\phi_2(z),$$ where $$\phi_1(z):=\frac{ \prod_{k=1}^\infty \big(1-\frac{z^2}{b_k^2}\big)}{\sin\bigl(\frac{\pi z}{2}\bigr)\big/\frac{\pi z}{2}}
\quad\text{and}\quad
\phi_2(z):= \left(\frac{(\frac{\pi z}{2})}{\sin (\frac{\pi z}{2})}\right)^2.$$ For $\phi_1$ we can apply Proposition \[dist-17\], since both numerator and denominator are entire functions of order $1$, with simple zeroes satisfying the conditions of the proposition. Hence, $\phi_1(i x )$ is the characteristic function of an infinitely divisible distribution. As a power of infinitely divisible characteristic functions, $\phi_2(z)$ is again an infinitely divisible characteristic function.
Let us close this section with an interesting negative result on the class ${\mathsf{N}}$.
\[dist-21\] Let $p:{{{\mathds R}^n}}\to[0,\infty)$ be a rotationally symmetric probability density which is of the form $p(x) = c \,e^{-f(|x|)}$ for some even and increasing function. If $p$ is not the normal distribution and if $$\lim_{r\to\infty} \frac{f(r)}{r\ln r} = \infty$$ then $p$ cannot be infinitely divisible.
Let $X$ be a (non-degenerate) $n$-dimensional random variable with probability density $p(x)$. If $X$ is infinitely divisible and not normally distributed then, by a straightforward modification of [@Sato Theorem 26.1] (see [@SH04 Proposition IV.9.8] for the exact formula in the one-dimensional setting), we have $$ \limsup_{r\to\infty} \frac{-\ln {\mathds P}(|X|>r)}{r\ln r}=\frac{1}{S}$$ where $S = \inf\{R>0\::\: {\operatorname{\mathrm{supp}}}\nu \subset B(0,R)\}$ and $\nu$ is the Lévy measure in the Lévy-Khintchine representation of the characteristic exponent of $X$. If $\nu=0$, we set $S=0$. Since $X$ is neither degenerate nor Gaussian, we have $S>0$.
On the other hand, since $f$ is increasing on $(0,\infty)$, we find for all $r>0$ $$\int_{|x|>r} e^{-f(|x|)}\,dx
{\leqslant}e^{-\frac{f(r)}{2}} \int_{|x|>r} e^{-\frac{f(|x|)}{2}}\,dx
= C \, e^{-\frac{f(r)}{2}}.$$ Thus, ${\mathds P}(|X|>r){\leqslant}2C\, e^{-\frac{f(r)}{2}}$, and $$\limsup_{r\to\infty} \frac{-\ln {\mathds P}(|X|>r)}{r\ln r}
{\geqslant}\limsup_{r\to\infty} \frac{-\ln\big(e^{-\frac{f(r)}{2}}\big)}{r\ln r}
=\frac 12 \limsup_{r\to\infty } \frac{f(r)}{r\ln r}
= \infty.$$ We conclude that $S=0$ and ${\operatorname{\mathrm{supp}}}\nu =\emptyset$ contradicting our assumptions.
\[dist-23\] Let $(p_t(x))_{t{\geqslant}0}$ be the transition densities of an $n$-dimensional symmetric Lévy process. If the characteristic exponent $\psi(\xi)$ satisfies $\psi(\xi){\geqslant}f(|\xi|)$ where $f:(0,\infty)\to(0,\infty)$ is increasing and satisfies $\lim_{r\to\infty} \frac{f(r)}{r\ln r}=\infty$, then none of the densities $p_t(x)$ is of class ${\mathsf{N}}$.
In particular,
1. the transition densities of a rotationally symmetric $\alpha$-stable process with $1<\alpha<2$ are not of class ${\mathsf{N}}$;
2. the convolution of a density of class ${\mathsf{N}}$ with the normal density is not of class ${\mathsf{N}}$.
Fix $t>0$ and apply Lemma \[dist-21\] to the density $p(\xi) := e^{-t\psi(\xi)}/p_t(0)$. Part b) is already contained in [@Ru70], where it is shown that any probability density decreasing faster than $e^{-c|x|^{1+\alpha}}$ as $|x|\to \infty$, cannot be infinitely divisible.
In Section \[table\] below we have compiled a list of some known examples of distributions of class ${\mathsf{N}}$.
The transition function of certain processes obtained by subordination {#sub}
======================================================================
Subordination in the sense of Bochner provides a good tool to get examples and insights. Our first general result for to hold, Theorem \[sub-15\], uses subordination. Since the composition of two Bernstein functions is again a Bernstein function, we can understand this theorem as a result for a certain class of subordinate Brownian motions.
\[sub-15\] Let $(p_t(x))_{t>0}$ be the transition densities of a Lévy process in ${\mathds R}$ such that the characteristic functions are of the form ${\mathcal{F}}^{-1} p_t(\xi) = e^{-tf(|\xi|)}$ with some Bernstein function $f$ satisfying $f(0)=0$ and $c_t := \int_0^\infty e^{-tf(r)}\,dr < \infty$ for all $t>0$. Then there exists, for each $t>0$, a complete Bernstein function $g_t$ such that $$\frac{p_t(x)}{p_t(0)} = e^{-g_t(|x|^2)}.$$
For every $t>0$ we know by our assumptions that $\rho_t(x):=e^{-tf(x)}$, $x>0$, is completely monotone, hence $\mathcal L\eta_t(x) = e^{-t f(x)}$ for some convolution semigroup $(\eta_t)_{t{\geqslant}0}$ of probability measures on ${\mathds R}$ with supports in $[0,\infty)$, i.e. a subordinator. Since $f$ is a Bernstein function and $|\xi|$ is the characteristic exponent of a Cauchy process, $f(|\xi|)$ is the characteristic exponent of a subordinate Cauchy process. Therefore, $$\begin{aligned}
p_t(x)
= \frac {1}{\pi} \int_0^\infty \frac s{s^2+ x^2}\,\eta_t(ds)
\end{aligned}$$ where we used that $x\mapsto \frac s{\pi\,(s^2+x^2)}$, $s>0$, is the transition density of the Cauchy process.
Denote by $\eta^*_t$ the pull-back of the measure $s\,\eta_t(ds)$ under the map $J:s\mapsto s^2$. Then $$p_t(x) = \frac 1{\pi} \int_{[0,\infty)} \frac 1{s+ u}\,\eta^*_t(ds)\bigg|_{u=|x|^2}
= h_t(u)\big|_{u=|x|^2}$$ where $h_t\in\mathcal S$ is a Stieltjes function. In particular, $h_t$ is an infinitely divisible completely monotone function, cf. [@SSV Proposition 7.11 and Definition 5.6]. Since $h_t(0+) = p_t(0) < \infty$, we can use [@SSV Lemma 5.7] to see that there is some Bernstein function $g_t$ such that $$\frac{h_t}{h_t(0+)} = e^{-g_t}
\qquad\text{or}\qquad
\frac{p_t(x)}{p_t(0)}=e^{-g_t(u)}\bigg|_{u=|x|^2}.$$
Fix $t>0$. Since $h_t\in\mathcal S$ and $h_t(0+) < \infty$, we see that $e^{g_t} = \frac 1{h_t} \in {\mathcal{CBF}}$ and $\frac 1{h_t(0+)} > 0$. Since we know already that $g_t\in{\mathcal{BF}}$, we can use [@SSV Remark 6.11] to conclude that $g_t\in{\mathcal{CBF}}$ with the representation $$g_t(s) = \beta_t + \int_0^\infty \frac{s}{s+r}\,\frac{\eta_t(r)}{r}\,dr$$ where $\eta_t : (0,\infty)\to[0,1]$ is measurable.
**a)** Theorem 9.5 in [@SSV] shows that the measure $\pi_t$ satisfying $\mathcal L\pi_t = h_t$ is a mixture of exponential distributions.
**b)** The converse of Theorem \[sub-15\] is, in general, not true. Consider, for example, the function $r\mapsto r^\alpha$, $r>0$, with $\frac{1}{2}<\alpha<1$; this is a complete Bernstein function, but the probability density given by $p(x)= e^{-|x|^{2\alpha}} \big/ \int_{\mathds R}e^{-|y|^{2\alpha}}\,dy$ is not infinitely divisible, cf. Lemma \[dist-21\].
**c)** Theorem \[sub-15\] admits a generalization to dimensions $n {\leqslant}3$. Let $f$ and $\eta_t$ be as in Theorem \[sub-15\] and assume that $(p_t(x))_{t>0}$ are the transition densities of an $n$-dimensional Lévy process such that $\mathcal F^{-1}p_t(\xi) = e^{-tf(|\xi|)}$.
As in the proof of Theorem \[sub-15\] we see that $p_t(x)$ can be written as a mixture of $n$-dimensional Cauchy distributions: $$p_t(x) = \frac{\Gamma\big(\frac{n+1}2\big)}{\pi^{\frac{n+1}2}} \int_{[0,\infty)} \frac{\lambda}{(\lambda^2+|x|^2)^{\frac{n+1}2}}\,\eta_t(d\lambda).$$ In particular, $$\label{sec7-eq1}
\frac{p_t(x)}{p_t(0)}
= \int_{[0,\infty)} \left(\frac{\lambda^2}{\lambda^2 + |x|^2}\right)^{\frac{n+1}2} \frac{\lambda^{-n}\,\eta_t(d\lambda)}{\int_{[0,\infty)}\lambda^{-n}\,\eta_t(d\lambda)}
=: \frac{h_t(|x|^2)}{h_t(0+)}.$$ If $\frac{n+1}{2}{\leqslant}2$, i.e. if $n{\leqslant}3$, we know from [@SH04 Theorem VI.4.9] that $h_t(s)/h_t(0+)$, $t>0$, is an infinitely divisible completely monotone function. Now we can follow the argument of the proof of Theorem \[sub-15\]. Note that Theorem \[sub-15\] tells us more: if $n=1$ the function $g_t(x)$ is a complete Bernstein function, which does not follow from .
Further examples of processes related to mixtures {#proc}
=================================================
In this section we give a few classes of transition probabilities $(p_t(x))_{t>0}$ where, for each $t>0$, $p_t(x)$ belongs to the class ${\mathsf{N}}$. These densities are obtained as variance-mean mixtures of the type with certain probability measures.
Let ${\nu}_t(x)$ be the standard $n$-dimensional Gaussian density $${\nu}_t(x)=\frac{1}{(2\pi t)^\frac{n}{2}}\, e^{-\frac{|x|^2}{2t}},\quad t>0,\; x\in{{{\mathds R}^n}}$$ and observe that $$\label{proc-e42}
\frac{{\nu}_t(x)}{{\nu}_t(0)}
= e^{-\frac{|x|^2}{2t}}
=\left(\frac{{\nu}_1(x)}{{\nu}_1(0)}\right)^{\frac{1}{2t}}
={\mathcal{F}}\bigl[{\nu}_{\frac{1}{2t}}\bigr](x).$$ The following proposition should be compared with Theorem \[dist-09\].
\[proc-41\] Assume that, for each $t>0$, $m_t(ds) $ is an infinitely divisible probability measure on $[0,\infty)$. Denote by $n_t(ds):=\big(\frac{s}{\pi}\big)^{\frac{n}{2}}\,G_\sharp m_t(ds)$ where $G_\sharp m_t$ denotes the pull-back with respect to the map $G: s\mapsto (2s)^{-1}$. If the probability measure $n^*_t(ds):=n_t(ds)/\mathcal{L}[n_t](0)$ is infinitely divisible, i.e. if $$ \mathcal{L}[n^*_t](\lambda)=e^{-f_t(\lambda)}$$ for some Bernstein function $f_t$, then the variance-mean mixture $$ p_t(x):=\int_0^\infty {\nu}_s(x)\,m_t(ds)$$ is infinitely divisible, and it is of the form $$ p_t(x) = p_t(0)\,e^{-f_t(|x|^2)}.$$
Using we see $$\begin{aligned}
p_t(x)
&=\int_{[0,\infty)} {\nu}_s(x)\,m_t(ds)\\
&= \int_{[0,\infty)} {\mathcal{F}}\bigl[{\nu}_{\frac{1}{2s}}\bigr](x)\, {\nu}_s(0)\, m_t(ds)\\
&= \int_{[0,\infty)} {\mathcal{F}}\bigl[{\nu}_\tau\bigr](x)\,{\nu}_{G^{-1}(\tau)}(0)\, G_\sharp m_t(d\tau)\\
&= \int_{[0,\infty)} e^{-\tau |x|^2} \left(\frac{\tau}{\pi}\right)^{\frac{n}{2}}\, G_\sharp m_t(d\tau)\\
&= \frac{\mathcal{L}[n_t](0)}{\pi^{\frac{n}{2}}} \, \mathcal{L}\bigl[n^*_t\bigr]\bigl(|x|^2\bigr)\\
&= p_t(0) e^{-f_t(|x|^2)}.
\qedhere\end{aligned}$$
\[proc-43\] **a)** Let $(\phi_t)_{t>0}$ be a family of completely monotone functions and set $n_t(s):=s^{-3/2} \phi_t(s)$ and which is normalized to become a probability density $n^*_t(s):= n_t(s)/\mathcal{L}[n_t](0)$. If each $n^*_t$ is an infinitely divisible probability density, then the probability density obtained by mixing $$\label{proc-e52}
p_t(x)=\int_{\mathbb{R}} {\nu}_s(x)\,n^*_t(s)\,ds$$ is infinitely divisible, and it is of the form $$ p_t(x)=p_t(0) e^{-f_t(|x|^2)},$$ where $f_t\in{\mathcal{CBF}}$ for each $t>0$. Moreover, $$\label{proc-e56}
f_t(s) = \int_0^\infty \frac{s}{s+r}\,\frac{\eta_t(r)}{r}\,dr$$ where $\eta_t : (0,\infty)\to [0,1]$ is a measurable function.
*Indeed:* The fact that $p_t(x)$ is infinitely divisible follows from and the infinite divisibility of $n^*_t$ is infinitely divisible. We can rewrite $p_t(x)$ in the following way: $$\begin{aligned}
p_t(x)
&=\frac{1}{\mathcal{L}[n_t](0)} \int_0^\infty {\nu}_s(x)\,s^{-\frac{3}{2}}\, \phi_t\left(\frac{1}{2s}\right) ds\\
&= \frac{\sqrt{2}}{\sqrt{\pi}\mathcal{L}[n_t](0)} \int_0^\infty e^{-x^2 r} \phi_t(r)\,dr\\
&= \frac{\sqrt{2}}{\sqrt{\pi}\mathcal{L}[n_t](0)}\,\mathcal{L}[\phi_t](|x|^2).
\end{aligned}$$ Since $\phi_t$ is completely monotone, its Laplace transform $\mathcal{L}[\phi_t]$ is a Stieltjes function and our calculation shows that $\infty > p_t(0) = \mathcal{L}[\phi_t](0+)$. Therefore, $1/\mathcal{L}[\phi_t]$ is a complete Bernstein function satisfying $1/\mathcal{L}[\phi_t](0+)>0$. By [@SSV Remark 6.11] we find that $$\frac 1{\mathcal{L}[\phi_t](s)} = \text{const}\cdot e^{-f_t(s)}$$ where $f_t(s)$ is a complete Bernstein function of the form .
**b)** Let $p_t(x)$ be the mixture of a Laplace density with a completely monotone density, i.e. $$p_t(x)
= \int_0^\infty \frac{s}{2} \, e^{-s|x|}\phi_t(s)\,ds, \quad t>0,$$ where $\phi_t$ is, for each $t>0$, a completely monotone probability density. Then $$\label{proc-e57}
p_t(x) = c_t \, e^{-g_t(|x|)},$$ where $c_t>0$ is some constant, depending on $t$, and $g_t$ is for each $t>0$ a Bernstein function.
*To see this*, recall that the mixture of Laplace densities is an infinitely divisible probability density, see [@SH04 Theorem IV.10.1]. Since $\phi_t(s)$ is completely monotone, $\frac s2\, \phi_t(s)/\int_0^\infty \frac s2\, \phi_t(s)\,ds$ is an infinitely divisible probability density, see [@SH04 Corollary VI.4.6]. Thus, $p_t(x)$ is of the form , since it is (up to a constant) a Laplace transform of an infinitely divisible density.
Towards a geometric understanding of transition functions of Feller processes {#fel}
=============================================================================
In this short section we propose a geometrical approach to understand transition functions of more general processes. This is more of a programme for further studies which should have geometric interests in its own right.
Recall that transition functions for diffusions generated by a second order elliptic differential operator are best understood when using the Riemannian metric associated with the principal part of the generator. Moreover, diffusions generated by subelliptic second order differential operators should be studied in the associated sub-Riemannian geometry. A similar remark applies to diffusions defined on (or defining a) metric measure space. We refer to [@Gri1], [@Gri2] and [@St] and the references therein.
Let $(X_t)_{t{\geqslant}0}$ be a Feller process, i.e. a Markov process such that the semigroup $$ T_t u(x) = {\mathds E}^x u(X_t) = \int_{{{\mathds R}^n}}u(y)\,p_t(x,dy),\quad u\in C_\infty({{{\mathds R}^n}})$$ has the Feller property: it preserves the space $C_\infty({{{\mathds R}^n}})$ of continuous functions vanishing at infinity. We assume that the kernel $p_t(x,dy)$ has a density which we denote, by some abuse of notation, again by $p_t(x,y)$. If the domain $D(A)$ of the generator $A$ of the Feller semigroup $(T_t)_{t{\geqslant}0}$ contains the test functions $C_c^\infty({{{\mathds R}^n}})$, then $A$ is a pseudo-differential operator with negative definite symbol, i.e. $$ Au(x)
= -q(x,D)u(x)
= -\int_{{{\mathds R}^n}}e^{ix\xi} q(x,\xi)\,{\mathcal{F}}u(\xi)\,d\xi,\quad u\in\mathcal S({{{\mathds R}^n}}),$$ where $q:{{{\mathds R}^n}}\times{{{\mathds R}^n}}\to{\mathds C}$ is a locally bounded function such that $q(x,\cdot)$ is, for every $x\in{{{\mathds R}^n}}$, a continuous negative definite function. Throughout we assume that $q(x,\xi)$ is real-valued. We refer to [@J2; @J3] where many examples of this kind are studied. Moreover, we refer to [@J] and, in particular, to [@Sch; @Sch-Sch] where $q(x,\xi)$ was calculated as $$ q(x,\xi) = - \lim_{t\to 0} \frac{{\mathds E}^x\left[e^{i\xi(X_t-x)} - 1\right]}{t}.$$ In [@BB1; @BB2] B. Böttcher has proved that for a large class of operators $-q(x,D)$ the symbol of $T_t$ $$ \lambda_t(x,\xi) = e^{-ix\xi}\,{\mathds E}^x\left[e^{i\xi X_t}\right],\quad t>0,\; x,\xi\in{{{\mathds R}^n}},$$ is asymptotically given by $$ \lambda_t(x,\xi) e^{-tq(x,\xi)} + r_0(t;x,\xi)
\quad\text{as}\quad t\to 0,$$ where $r_0(t;x,\xi)$ tends for $t\to 0$ weakly to zero in the topology of a certain symbol class.
The techniques employed in [@J-Sch] will also yield for certain elliptic differential operators $-L(x,D)$ generating the semigroup $(T_t)_{t{\geqslant}0}$ that for the subordinate semigroup $(T_t^f)_{t{\geqslant}0}$ it holds $$ \lambda_t(x,\xi)
=e^{-t f(L(x,\xi))} + \tilde r_0(t;x,\xi)
\quad\text{as $t\to 0$},$$ and $f(L(x,\xi))$ is, in a certain sense, an approximation of the symbol of $f(L(x,D))$.
Thus we have in several non-trivial cases $$ T_t u(x) = \int_{{{\mathds R}^n}}e^{ix\xi} e^{-tq(x,\xi)}\,{\mathcal{F}}u(\xi)\,d\xi + \cdots$$ and if $T_t u(x) = \int_{{{\mathds R}^n}}p_t(x,y) u(y)\,dy$ we might try as an approximation for $p_t(x,y)$ $$ p_t(x,y) = \int_{{{\mathds R}^n}}e^{i(x-y)\xi} e^{-tq(x,\xi)}\,d\xi + \cdots$$ Now let us assume that for every fixed $x\in{{{\mathds R}^n}}$ the continuous negative definite function $q(x,\cdot)$ belongs to ${\mathcal{MCN}}({{{\mathds R}^n}})$. In this case we would find $$\begin{gathered}
p_t(x,y) = \int_{{{\mathds R}^n}}e^{i(x-y)\xi} e^{-t d^2_{q(x,\cdot)}(\xi,0)} \,d\xi + \cdots\end{gathered}$$ and, in particular, $$\begin{aligned}
p_t(x,x)
&= \int_{{{\mathds R}^n}}e^{-t q(x,\xi)}\,d\xi + \cdots\\
&= \int_{{{\mathds R}^n}}e^{-t d^2_{q(x,\cdot)}(\xi,0)} \,d\xi + \cdots
\end{aligned}$$ where $d^2_{q(x,\cdot)}(\xi,\eta) = \sqrt{q(x,\xi-\eta)}$. Moreover, we might think to search for $p_t(x,y)$ an expression of the form $$\label{fel-e22}
p_t(x,y)
= p_t(x,x)\, e^{-\delta^2(t,x_0;x,y)}+\cdots,\quad x,y\in B^{d_{q(x_0,\cdot)}}(x_0,\epsilon),$$ with a suitable metric $\delta(t,x_0;\cdot,\cdot)$ associated with $q(x_0,\xi)$ in the sense of Section \[ptx\].
Thus we propose to switch from $({{{\mathds R}^n}},d_\psi)$ and $({{{\mathds R}^n}},\delta_{\psi,t}(\cdot,\cdot)$ to generalizations of Riemannian manifolds: Take ${{{\mathds R}^n}}$ with its standard differentiable structure and identify the tangent space $T_x{{{\mathds R}^n}}\simeq {{{\mathds R}^n}}$. Consider now the families of metrics $$ \begin{aligned}
d_{q(x,\cdot,\cdot)}& : {{{\mathds R}^n}}\times{{{\mathds R}^n}}\to {\mathds R}\\
&(\xi,\eta)\mapsto \sqrt{q(x,\xi-\eta)}
\end{aligned}
\qquad\text{and}\qquad
\begin{aligned}
\delta(t,x;\cdot,\cdot)& : {{{\mathds R}^n}}\times{{{\mathds R}^n}}\to {\mathds R}\\
&(y,z)\mapsto \delta(t,x;y,z)
\end{aligned}$$ ($\delta(t,x;\cdot,\cdot)$ as proposed in ) and start to study the corresponding geometric structures. Note that in case that $d_{q(x,\cdot,\cdot)}$ or $\delta(t,x;\cdot,\cdot)$ are related to a continuous negative definite function, our objects to study are manifolds ${{{\mathds R}^n}}$ with tangent spaces equipped with a metric such that they allow an isometric embedding into a Hilbert space which is, in general, infinite dimensional.
A list of probability distributions of class ${\mathsf{N}}$ {#table}
===========================================================
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$X_t$ ${\mathds E}e^{i\xi \cdot X_t} $ $p_t(x) $ $\psi(\xi)$ $-\ln \bigl[p_t(x)\big/p_t(0) \bigr]\vphantom{\Bigg]}$
--------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[Generalized hyperbolic, $n=1, t=1$. [@BKS]]{} [$\Bigl[\frac{\kappa^2}{\kappa^2+\xi^2}\Bigr]^{\frac{\lambda\vphantom{\int\limits^f}}{2}} [$\frac{(\frac\kappa\delta)^{\lambda}}{(2\pi)^{\frac {n\vphantom\int}2}K_\lambda(\delta \kappa)} [$-\ln \left[\Bigl[\frac{\kappa^2}{\kappa^2+\xi^2}\Bigr]^{\frac{\lambda}{2}} [$-\ln \Biggl[\frac{(\frac\kappa\delta)^{\frac{n}{2}+\lambda}K_{\lambda-n/2} \bigl(\kappa \sqrt{\delta^2+x^2}\bigr)}{\bigl(\sqrt{\delta^2+x^2}\,\kappa^{-1} \bigr)^{\frac {n\vphantom\int}2-\lambda}} \Biggr]$]{}
\frac{K_\lambda\bigl(\delta\sqrt{\kappa^2+\xi^2}\bigr)}{K_\lambda(\delta \kappa)}$]{} \frac{K_{\lambda-n/2} \bigl(\kappa\sqrt{\delta^2+x^2}\bigr)} \frac{K_\lambda\bigl(\delta\sqrt{\kappa^2+\xi^2}\bigr)}{K_\lambda(\delta \kappa)}\right]$]{}
{\bigl(\sqrt{\delta^2+x^2}\,\kappa^{-1}\bigr)^{\frac{n\vphantom\int}2-\lambda}}$]{}
Normal $N(0,1)$,$F(du)=\delta_t(u)$ $e^{-t|\xi|^2}$ $(2\pi t)^{-n/2} e^{-\frac{|x|^2}{2t}}$ $|\xi|^2$ $\frac{|x|^2}{2t}$
Cauchy in $\mathbb{R}$, $n=1$,$\lambda=-1, \alpha=0, \delta=t$.[@KPST83 p. 116] $e^{-t|\xi|}$ $\frac{1}{\pi} \frac{t}{|x|^2 +t^2}$ $|\xi|$ $\ln\Big(\frac{\xi^2+t^2}{t^2}\Big)$
Laplace in $\mathbb{R}$, $n=1$,$\lambda=1, \alpha=t, \delta=0$.[@KPST83 p. 116] $\frac{t^2}{\xi^2+t^2}$ $\frac{t}{2} e^{-t|x|}$ $\ln\Big(\frac{\xi^2+t^2}{t^2}\Big)$ $t|x|$
Hyperbolic, $t=n=\lambda=1$, [@J3 pp. 415-6] $ \frac{\alpha}{K_1(\alpha \delta )} \frac{K_1\big(\delta \sqrt{\alpha^2+\xi^2}\big)}{\sqrt{\alpha^2+\xi^2}}$ $\frac{1}{2\delta K_1(\alpha \delta )} e^{-\alpha \sqrt{\delta^2+x^2}}$ $-\ln \left[ \frac{\alpha}{K_1(\alpha \delta )} \frac{K_1\big(\delta \sqrt{\alpha^2+\xi^2}\big)}{\sqrt{\alpha^2+\xi^2}} \right] $ $ \alpha(\sqrt{\delta^2+x^2}-\delta)$
Relativistic Hamiltonian, $n=1, \lambda=-\frac{1}{2}, \delta=t$. [@J1 p. 182] $e^{-t \big( \sqrt{\alpha^2+\xi^2}-\alpha\big)}$ $\frac{\alpha \delta e^{a \delta} }{\pi} \frac{K_1\big(\alpha \sqrt{\delta^2 +x^2}\big)}{\sqrt{\delta^2 +x^2}}$ $\sqrt{\alpha^2+\xi^2}-\alpha$ $-\ln \left[ \frac{\delta K_1\big(\alpha \sqrt{\delta^2+x^2}\big)}{K_1(\alpha \delta)} \right]$
Meixner process $C_t$ in $\mathbb{R}$ [@PY03 p. 312] $\big(\frac{1}{\cosh \xi}\big)^t$ $ \frac{2^{t-1}}{\pi\Gamma(t)} |\Gamma(\frac{t+i\xi}{2})|^2$ $\ln \cosh \xi$ $ -\ln \Big| \frac{\Gamma(\frac{t+i\xi}{2})}{\Gamma(\frac{t}{2})}\Big|^2$
\[\] $t=1$ $\frac{1}{\cosh \xi}$ $\frac{1}{2 \cosh (\frac{\pi x}{2})}$ $\ln \cosh \xi$ $ \ln ( \cosh (\frac{\pi x}{2}))$
\[\] $t=2$ $\big(\frac{1}{\cosh \xi}\big)^2$ $\frac{x}{2 \sinh (\frac{\pi x}{2})}$ $\ln \cosh \xi$ $\ln \frac{ \sinh (\frac{\pi x}{2})}{\frac{\pi x}{2}} $
[@PY03 p. 312], $t=1$ $\frac{\xi}{\sinh \xi}$ $\frac{\pi}{4 \cosh^2 (\frac{\pi x}{2})}$ $\ln \frac{\sinh \xi}{\xi}$ $2 \ln \cosh (\frac{\pi x}{2})$
\[\] $t=2$ $\big(\frac{\xi}{\sinh \xi}\big)^2$ $\frac{\frac{\pi}{2} (\frac{\pi x}{2} \coth(\frac{\pi x}{2})-1)}{\sinh^2 (\frac{\pi x}{2})}$ $\ln \big(\frac{\sinh \xi}{\xi}\big)$ $-\ln \left[\frac{\frac{3\pi x}{2} \coth(\frac{\pi x}{2})-3}{\sinh^2 (\frac{\pi x}{2})}\right]$
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[99]{}
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[^1]: *N. Jacob*: Mathematics Department, Swansea University, Singleton Park, Swansea SA2 8PP, UK, <n.jacob@swansea.ac.uk>
[^2]: *V. Knopova*: V.M.Glushkov Institute of Cybernetics NAS of Ukraine, 03187, Kiev, Ukraine, <vicknopova@googlemail.com>
[^3]: *S. Landwehr*: Heinrich Heine University Düsseldorf, German Diabetes Center at the Heinrich Heine University Düsseldorf, Leibniz Center for Diabetes Research, Institute of Biometrics and Epidemiology, Auf’m Hennekamp 65, 40225 Düsseldorf, Germany, <sandra.landwehr@ddz.uni-duesseldorf.de>
[^4]: *R.L. Schilling*: Institut für Mathematische Stochastik, Technische Universität Dresden, 01062 Dresden, Germany, <rene.schilling@tu-dresden.de>
|
---
abstract: 'In many contexts it is extremely costly to perform enough high quality experimental measurements to accurately parameterize a predictive quantitative model. However, it is often much easier to carry out large numbers of experiments that indicate whether each sample is above or below a given threshold. Can many such categorical or “coarse” measurements be combined with a much smaller number of high resolution or “fine” measurements to yield accurate models? Here, we demonstrate an intuitive strategy, inspired by statistical physics, wherein the coarse measurements are used to identify the salient features of the data, while the fine measurements determine the relative importance of these features. A linear model is inferred from the fine measurements, augmented by a quadratic term that captures the correlation structure of the coarse data. We illustrate our strategy by considering the problems of predicting the antimalarial potency and aqueous solubility of small organic molecules from their 2D molecular structure.'
author:
- 'Alpha A. Lee'
- 'Michael P. Brenner'
- 'Lucy J. Colwell'
bibliography:
- 'hopfield\_refs.bib'
title: Optimal design of experiments by combining coarse and fine measurements
---
A large class of scientific questions asks whether dependent variables can be accurately predicted by using training data to learn the parameters of quantitative models. Classical statistics shows that this is possible if sufficiently many high resolution measurements are available, though the cost of performing these experiments can be prohibitive. On the other hand, in many settings, it can be straightforward to evaluate whether a measurement is above or below a certain threshold, raising the question of how such measurements can be incorporated into the modelling framework.
Examples abound in disparate fields. For instance, predicting the solubility of organic molecules is a fundamental challenge in physical chemistry [@llinas2008solubility]. Although accurate measurements are extremely difficult to obtain [@palmer2014experimental], determining whether a molecule is soluble at a particular concentration is comparatively simple. Similarly, in drug discovery, biochemical assays that determine whether a molecule binds to a given receptor are much simpler than measuring protein-ligand binding affinity [@malo2006statistical]. In protein biophysics, a key challenge is to predict the effect of amino acid changes on protein phenotype. Here, threshold measurements are naturally provided by homologous sequences from the same protein family [@morcos2014coevolutionary; @figliuzzi2015coevolutionary; @hopf2015quantification; @barrat2016improving; @levy2017potts]. In contrast, experimentally measuring the phenotypic change is much more difficult. A related problem is to predict the viral fitness landscape given HIV sequences obtained from patients; again collecting patient samples is much easier than measuring fitness directly [@shekhar2013spin; @ferguson2013translating]. In single-cell RNA sequencing, decomposition methods that extract the correlation structure of shallow gene expression measurements is an ongoing challenge [@dixit2016perturb; @buettner2016scalable].
Despite the ubiquity of this problem, to our knowledge there is no principled method for combining numerous binary/categorical (“coarse”) measurements with fewer quantitative (“fine”) measurements to produce a predictive model. Although regression approaches can account for a prior estimate of sample error [@bishop2006pattern], this is not the same as combining two qualitatively distinct forms of data to build a more accurate model.
In this Letter, we introduce an intuitive method that combines coarse and fine measurements. The coarse measurements provide sets of labelled samples – those data above and below some threshold – and the proposed method extracts features from the correlations of the variables in each set. Their contribution to the dependent variable is then determined by using the fine measurements to build a regression model for these features. Our model augments a quantitative linear model with a quadratic term which captures the correlation structure extracted from the coarse data. We illustrate our approach by applying it to solubility prediction, and interpret the approach in the context of the Ising model.
To fix ideas, we assume each sample is characterized by a vector of $p$ properties $\mathbf{f}_i \in \mathbb{R}^p$. The binary data indicates that $N_+$ ($N_-$) samples are above (below) some threshold. In addition, we are given $\mathbf{y} \in \mathbb{R}^{M}$, quantitative measurements for $M$ additional samples. These measurements could be binding affinity, solubility etc. We construct matrices $R_{\pm} \in \mathbb{R}^{N_\pm \times p}$ for samples above/below the threshold, with columns of $R_{+}$ and $R_{-}$ normalized separately to have zero mean and unit variance. Intuitively, if there are combinations of the $p$ properties that are always present in either sample set, then these properties should be good predictors of the measurement. Such persistent correlations can be identified from the eigendecomposition of each sample covariance matrix $C_\pm$ $$\begin{aligned}
C_{\pm} &= \frac{1}{N_\pm} R_\pm^{T} R_\pm \nonumber\\
&=\sum_{i=1}^{N_\pm} \lambda^{\pm}_i \mathbf{u}^{\pm}_i \otimes \mathbf{u}^{\pm}_i,
\label{sample_eigendecomposition}\end{aligned}$$ where $\{ \lambda^{\pm}_i\}$, $\{ \mathbf{u}^{\pm}_i \}$ are the eigenvalues and eigenvectors (note we perform separate eigendecompositions for the two matrices $C_\pm$). Each $\mathbf{u}^{\pm}_i$ identifies a particular combination of the $p$ properties, explaining a fraction $\lambda^{\pm}_i/\sum_i \lambda^{\pm}_i$ of the variance [@bishop2006pattern]. Each matrix $C_{\pm}$ is an unbiased estimator of the corresponding true covariance matrix. The quality of this estimator depends on data sampling. For example, one may inadvertently assay certain samples (easy to obtain, measure etc.), which could distort the estimator by causing an eigenvector with large eigenvalue to be localized on features common to these samples, even though they do not predict the output variable. For protein sequences, a natural source of such spurious correlations is phylogeny [@dutheil2012detecting; @obermayer2014inverse]. Here we propose that whereas the eigenvectors $\mathbf{u}^{\pm}_i$ reliably identify data features, their significance as estimated by the corresponding eigenvalues $ \lambda^{\pm}_i $ can be severely corrupted by imperfect sampling. Later we justify this ansatz with ideas from statistical physics, and show that this characterization applies to a large class of problems. This ansatz suggests a strategy to mitigate the corruption by using the additional quantitative measurements to determine the significance of each feature. We posit a general quadratic model $$y_i = \mathbf{h}^T \mathbf{f}_i + \mathbf{f}_i^{T} J \mathbf{f}_i + \epsilon_i.
\label{regression}$$ Here $\mathbf{h}$ is the variable-specific effect, $J$ captures the coupling between variables, and $\epsilon_i \sim N(0,\sigma)$ models random error. There are $p$ parameters in $\mathbf{h}$ and $p(p-1)/2$ parameters in $J$. If one had $M\gg p(p+1)/2$ quantitative measurements, these parameters could be estimated using linear least squares regression. However, it is costly to perform many detailed measurements, so we turn instead to the matrices $C_{\pm}$. We pose the ansatz $$J = \sum_{k=1}^{\hat{p}_+} c^{+}_{k} \mathbf{u}^{+}_{k} \otimes \mathbf{u}^{+}_{k} + \sum_{k =1}^{\hat{p}_-} c^{-}_{k} \mathbf{u}^{-}_{k} \otimes \mathbf{u}^{-}_{k} .
\label{ansatz}$$ Here $\hat{p}_\pm \le p$, since some eigenvectors will reflect noise due to finite sampling [@laloux1999noise; @plerou1999universal; @bai2010spectral]. Our ansatz reflects the hypothesis that the eigendecomposition of $C_{\pm}$ captures variable-variable correlations. If the number of samples is much smaller than the number of variables, random matrix theory provides a rigorous way to determine $\hat{p}_\pm $ [@laloux1999noise; @plerou1999universal; @bai2010spectral; @bun2016; @lee2016predicting; @bun2017cleaning]; this case will be discussed in detail later. Relaxing this assumption, we include all eigenvectors and determine the parameters $\mathbf{h}$, ${c}^{+}$ and ${c}^{-}$ by regressing against the the few quantitative measurements available. We note that the ansatz (\[ansatz\]) reduces the number of variables to $p+\hat{p}_+ + \hat{p}_- $. In the case where the coarse measurements yield multiple categories (or a single category), our method generalizes by forming separate correlation matrices for each category, and positing that $J$ is a sum of the outer product of all eigenvectors with coefficients determined by regressing against the quantitative data. Our method generalises to Generalised Linear Models with a link function on the right hand side of Equation (\[regression\]).
To illustrate our approach, we consider two examples: predicting the potency of chemicals against malaria and the equilibrium aqueous solubility of molecules.
*Antimalarials* – Developing accurate models that can rank the potency of a library of compounds against a target is an important unsolved challenge in drug discovery. We consider a published antimalarial screening campaign [@guiguemde2010chemical]: binary but high throughput assays reported 1528 ÒactiveÓ compounds against malaria, lower throughput but quantitative assays measured the potency ($\mathrm{pIC}_{50}$) of only 1189 compounds [@guiguemde2010chemical; @Riniker2017]. Figure \[results\]A shows that by combining binary and quantitative measurements, an hitherto unattempted strategy, an order of magnitude less quantitative measurements could have been performed to yield a model with similar predictive accuracy (c.f. Supplementary Information showing the same result for the Pearson correlation coefficient). Moreover, the model with coarse measurements clearly outperforms the linear model without the coarse measurements and the “null” quadratic model where the vectors $\mathbf{u}^{\pm}_k$ are random orthogonal vectors (i.e. random $R_{\pm}$). In the Supplemental Information we show that our model also outperforms direct quadratic regression. The compounds are described using the 1024-bit Morgan6 Fingerprint [@rogers2010extended] generated with `rdKit` [@rdkit].
*Solubility* – Predicting the aqueous solubility of molecules is a fundamental problem in physical chemistry important to a plethora of chemical industries. However, accurate solubility assays are low throughput ($\sim$ 1 hour/compound [@box2005]). Figure \[results\]B shows that one could obtain an accurate solubility model ($r^2 = 0.85$, MAE = 0.61) if one were to combine the outcome of a coarse solubility assay that could only tell whether a compound is soluble ($< 10^{-4}~\mathrm{mol/L}$) or not ($> 10^{-2}~\mathrm{mol/L}$) with much fewer quantitive solubility data. We use a standard dataset of the solubility of 1144 organic molecules [@delaney2004esol], and describe the molecule by concatenating the Avalon Fingerprint [@gedeck2006qsar], the MACCS Fingerprint [@durant2002reoptimization], and the 1024-bit Morgan6 Fingerprint [@rogers2010extended]. Our result compares favourably with other models that also use binary molecular fingerprints, e.g. kernel partial least squares regression achieves $r^2 = 0.83$ [@zhou2008scores].
![Combining coarse and fine measurements accurately predicts antimalarial activity and solubility. The predictive accuracy of (A) $\mathrm{pIC}_{50}$ against malaria and (B) solubility as a function of the number of quantitative measurements given to the model with coarse measurements (blue line), and without (red line). Including random quadratic terms (orange line) is not effective; error bars obtained over 30 random partitions of data into training (90%) and verification (10%) sets. (Inset) Out-of-sample solubility prediction with 90% of the full dataset has a mean absolute error of 0.61 ($r^2$ = 0.85). The estimate is arrived at by analysing 10 random partitions of the data into training and verification sets.[]{data-label="results"}](solubilityv5)
To understand why our heuristic strategy is successful, we consider a model problem where data is generated according to Eqn. (\[regression\]), which is the maximum entropy model [@schneidman2006weak; @cocco2009neuronal; @lee2015statistical], analogous to the Ising model. We thus interpret the dependent variable as an “energy”, noting that the logarithm of the solubility is proportional to the solvation energy. The interaction matrix $J$ can be decomposed into a sum of outer products of eigenvectors $\boldsymbol{\zeta}_i $ (Hopfield patterns [@hopfield1982neural]), and eigenvalues $E_i$ (Hopfield energies) as $$J = \sum_{i=1}^{m} E_i \boldsymbol{\zeta}_i \otimes \boldsymbol{\zeta}_i.$$ Furthermore, to model the binary features used in solubility prediction, we make the assumption that the independent variable is a vector of $\pm 1$.
To simulate binary measurements we randomly draw samples from the uniform distribution, evaluate Eqn. (\[regression\]) to determine the energy of each sample, and retain those samples that fall below a certain energy. Consider an interaction matrix $J$ with $p=100$, and $m=3$ randomly generated patterns. To fix ideas, henceforth let all patterns be attractive with $(E_1,E_2,E_3) = (- 30, - 25, - 20)$, $\mathbf{h}=0$ and $\epsilon_i=0$. Using this model, we generate 5000 random vectors, consider the $N = 500$ samples with lowest energy as above threshold samples, and compute the eigendecomposition of the resulting correlation matrix.
Figure \[pattern\]A shows that the eigenvalue distribution of the sample correlation matrix $C_{+}$ follows the Marčenko-Pastur distribution expected for a random matrix [@marvcenko1967distribution], $$\rho(\lambda) = \frac{\sqrt{ \left[ \left(1+\sqrt{\gamma} \right)^2 - \lambda \right]_+ \left[ \lambda - \left(1-\sqrt{\gamma} \right)^2 ) \right]_+}}{2 \pi \gamma \lambda}
\label{MP}$$ where $(\cdot)_+ =\mathrm{max}(\cdot,0)$, $\gamma = p/N$, with the exception of three distinct eigenvalues. Figure \[pattern\]B shows that their corresponding eigenvectors are indeed the Hopfield patterns that we put in. Therefore, the large eigenvectors of $C_{+}$ correspond to eigenvectors of $J$. Note that the random matrix theory framework applies because $m\ll p$, i.e. the signal is low rank compared to the noise. If the signal was high rank, all eigenvectors should be included and their significance determined by regression against fine measurements, as in the examples discussed above.
![Hopfield patterns can be recovered from threshold sampling. (A) Histogram of eigenvalues agrees with the Marčenko-Pastur distribution (red curve) save for three significant eigenvalues. (B) The top eigenvector is the lowest energy Hopfield pattern; the other eigenvectors are shown in Supplemental Information. Random matrix cleaning allows us to successfully (C) recover the coupling matrix $J$ and (D) predict Hopfield energies. []{data-label="pattern"}](thres_hopfield3)
Turning to eigenvalues, in this model, which features *uniform* sampling, we find that the eigenvalues are proportional to the Hopfield energy $E_i$. This allows us to “clean” the correlation matrix, by using the $q$ eigenvectors above the Marčenko-Pastur threshold to construct a rank $q$ approximation $J_{\mathrm{MP}}$ of the correlation matrix (here $q = 3$). Figure \[pattern\]C shows that $J_{\mathrm{MP}}$ accurately reconstructs $J$, and allows accurate prediction of the energy of particular states (Figure \[pattern\]D). Analogously, the eigendecomposition of $C_-$ allows one to recover *repulsive* patterns with positive Hopfield energies (see Supplemental Information).
Since the Hopfield patterns are energy minima, taken together Figure \[pattern\]A-D imply that the probability of the system visiting a particular basin under uniform sampling is proportional to the energy of that minima. Therefore, the hypervolume of each energy basin is proportional to the basin depth. This fact can be derived by noting that Eqn. (\[regression\]) is a quadratic form, so the Hessian matrix is a constant. Therefore, all local minima have the same mean curvature. Given that low lying energy minima are wide, we can extract the position of energy minima in the space of input variables by studying the correlation structure of the binary dataset. We note that the correlation between basin hypervolume and basin depth appears in many complex physical systems beyond the Ising model [@doye1998thermodynamics; @doye2005characterizing; @pickard2011ab].
A lingering question is whether our inference procedure is robust to the choice of threshold. To test this, we consider $m$ Hopfield patterns, chosen as eigenvectors of a symmetrized $p\times p$ Gaussian random matrix, with the Hopfield energy chosen to be Gaussian distributed with mean $10$ and unit variance. We draw 10000 samples randomly and compute the correlation matrix with the lowest energy $N$ samples. Figure \[sample\] shows that our method is robust: the correlation coefficient between $J$ and $J_{\mathrm{MP}}$ is large and constant for a wide range of thresholds and number of Hopfield patterns. The question of how many energy minima can be recovered from the binary data and a thermodynamic interpretation is discussed in the Supplemental Information.
![Hopfield inference with random matrix cleaning is robust to the energy threshold. Here $\rho$ is the Pearson correlation coefficient between the entries in $J$ and $J_{\mathrm{MP}}$. The results are computed by averaging over $50$ realizations. []{data-label="sample"}](n_samples)
We now turn to consider two common scenarios that break the assumptions made so far, stratified sampling and more complex energy landscapes.
*Stratified sampling*: Thus far we assumed that the sampling is uniform before thresholding. However, in many settings, the sampling is biased. To model this effect, we draw 5000 random samples, but freeze the first 5 variables to $+1$ for the first 2500 samples and the last 5 variables to $-1$ for the remaining 2500 samples. We then evaluate the energy, and take again the lowest $10^{\mathrm{th}}$ percentile. Figure \[freeze\_spin\]A-D shows that the frozen variables introduce sample-sample correlations, and now there are 4 significant eigenvectors with the first Hopfield pattern demoted to the second largest eigenvector (Figure \[freeze\_spin\]C). As such, the informative eigenvectors are still present in $J_{\mathrm{MP}}$, but the eigenvalues are misplaced.
![Few quantitative measurements enable $J$ to be inferred accurately for stratified datasets. (A) There are now four significant eigenvectors but still only three Hopfield patterns in the model. (B)-(D) The top eigenvector is uncorrelated with all Hopfield patterns, and the Hopfield patterns are demoted to the second to fourth significant eigenvectors. (E) Random matrix cleaning does not recover the coupling matrix. (F) Blue data points: The elements of the coupling matrix recovered by incorporating quantitative data and using Eqn. (\[ansatz\]); Orange data points: the elements of the coupling matrix recovered using the quantitative data only and ridge regression (with $J_{ij}$ being the coefficients). []{data-label="freeze_spin"}](phylogenyv2)
In this case, naïve random matrix cleaning does not recover $J$ (Figure \[freeze\_spin\]E), since there is no *a priori* reason to discard the first eigenvector unless we know the Hopfield patterns beforehand. We need additional information – for which we turn to the quantitative measurements – to accurately recover the Hopfield energies. Figure \[freeze\_spin\]F shows that an additional 500 quantitative measurement allow us to recover the coupling matrix (MAE = 0.12) using ridge regression and the ansatz Eqn. (\[ansatz\]). The error is significantly larger (MAE = 0.31) if only the quantitative measurements are used.
*Complex energy landscapes*: The geometric property that the depth of an energy minima is related to its hypervolume is not universal to all energy landscapes [@wales1998archetypal; @wales2003energy]. A natural question is whether the significant eigenvectors and eigenvalues of the correlation matrices of samples below/above an energy threshold allow us to infer features of a complex energy landscapes. We consider a landscape that comprises a sum of Gaussians $$H(\mathbf{f}) = \sum_{i} E_i \exp\left( - E_i^2 (\mathbf{f}\cdot \boldsymbol{\zeta}_i)^2\right).
\label{gaussian_landscape}$$ This landscape has the property that the depth of each energy minima, $E_i$ (located at $\boldsymbol{\zeta}_i$), is *inversely* proportional to its width $1/E_i$. As above, we let $(E_1,E_2,E_3) = (- 30,- 25, - 20)$ and generate Hopfield patterns by diagonalising a symmetrized Gaussian random matrix. We draw $5000$ samples and threshold to find the $500$ lowest energy samples. Figure \[gaussian\_landscape\] shows that there are again three significant eigenvectors above the Marčenko-Pastur threshold, but the lowest energy Hopfield pattern is demoted to the third eigenvector, while the highest energy Hopfield pattern is promoted to the top eigenvector. This is expected: the eigenvalue corresponding to each minimum is proportional to the number of samples near that minimum, i.e. the basin volume, which in this case is not proportional to basin depth. However, the eigenvectors still indicate the locations of the energy minima, motivating the approach described in Eqn. , where we use these eigenvectors to identify features.
![For energy landscapes where basin depth is not proportional to basin width, the eigenvectors indicate the locations of energy minima but the eigenvalues are awry. (A) There are three significant eigenvectors above the Marčenko-Pastur threshold. (B) - (D) The top eigenvector is correlated with the highest energy minimum, and the last significant eigenvector is correlated with the lowest energy minimum.[]{data-label="gaussian_landscape"}](gaussian_landscape)
In conclusion, we develop a general strategy, grounded in statistical physics, which integrates coarse and fine measurements to yield a predictive model. Since coarse measurements are often significantly less costly to obtain, our strategy provides a new avenue for experiment design. Although our Letter only considered an Ising-type model, the fact that the eigenvectors of the correlation matrix of coarse measurements point toward energy minima suggests a natural way to integrate our result into more complex non-linear models, for example by using $\mathbf{f}\cdot \mathbf{u}_i$, the overlaps between the sample vector and each eigenvector, as inputs to a general nonlinear function such as an artificial neural network.
The authors thank R Monasson for insightful discussions. AAL acknowledges the support of the George F. Carrier Fellowship and the Winton Advanced Research Fellowship. LJC acknowledges a Next Generation fellowship, and a Marie Curie CIG \[Evo-Couplings, Grant 631609\]. MPB is an investigator of the Simons Foundation, and acknowledges support from the National Science Foundation through DMS-1411694.
|
---
abstract: 'An operator satisfies the Global Comparison Property if anytime a function touches another from above at some point, then the operator preserves the ordering at the point of contact. This is characteristic of degenerate elliptic operators, including nonlocal and nonlinear ones. In previous work, the authors considered such operators in Riemannian manifolds and proved they can be represented by a min-max formula in terms of Lévy operators. In this note we revisit this theory in the context of Euclidean space. With the intricacies of the general Riemannian setting gone, the ideas behind the original proof of the min-max representation become clearer. Moreover, we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity.'
address:
- |
Department of Mathematics\
University of Massachusetts, Amherst\
Amherst, MA 01003-9305
- |
Department of Mathematics\
Michigan State University\
619 Red Cedar Road\
East Lansing, MI 48824
author:
- Nestor Guillen
- 'Russell W. Schwab'
bibliography:
- '../refs.bib'
date: 'Monday 18th February, 2019 (This is the revised version per referee suggestions.)'
title: 'Min-max formulas for nonlocal elliptic operators on Euclidean Space'
---
[^1]
Introduction {#sec:Introduction}
============
A map $I:C^2_b(\mathbb{R}^d)\to C^0_b(\mathbb{R}^d)$ is said to satisfy the **Global Comparison Property** (GCP) if $$\begin{aligned}
\label{equation:GCP}
u\leq v \textnormal{ in } \mathbb{R}^d \textnormal{ and } u(x)=v(x) \Rightarrow I(u,x)\leq I(v,x).\end{aligned}$$ The Laplacian operator, as well as its fractional powers $-(-\Delta)^{\alpha/2}$ ($\alpha \in (0,2)$) all satisfy this property. More generally, given a Lévy measure $\nu(dy)$ (a measure on $\mathbb{R}^d\setminus \{0\}$ such that $\min\{1,|y|^2\}$ is integrable with respect to $\nu$) the operator $$\begin{aligned}
I(u,x) = \int_{\mathbb{R}^d} u(x+y)-u(x)-\chi_{B_1}(y) \nabla u(x)\cdot y\;\nu(dy),\end{aligned}$$ will have the GCP. The GCP is also satisfied by Dirichlet-to-Neumann maps for elliptic equations, generators of Markov processes, Bellman-Isaacs operators in control and differential games, among many examples. When the operator is known a priori to be local, then nonlinear examples of maps with the GCP are of the form, $$\begin{aligned}
I(u,x) = F(D^2u(x),\nabla u(x),u(x)),\end{aligned}$$ where $F:\mathbb{S}_d \times \mathbb{R}^d\times \mathbb{R}\to\mathbb{R}$ is monotone in its first argument, and Lipschitz continuous in all arguments.
The main contribution of this article is to address when certain operators acting on $C^2_b(\real^d)$ must necessarily enjoy a structure similar to those examples above. The canonical object used to address this question will be a linear operator we choose to say is “of Lévy type”: those operators for which there exist functions, $A(x)\in\mathbb{S}_d$, $B(x)\in\real^d$, $C(x)\in\real$, and measures $\mu(x,dy)$ so that $$\begin{aligned}
\label{eqIN:LevyTypeLinear}
L(u,x) & = \tr(A(x)D^2u(x))+B(x)\cdot \nabla u(x)+C(x)u(x)\\
&+ \int_{\mathbb{R}^d} u(x+y)-u(x)-\ind_{B_1(0)}(y)\nabla u(x)\cdot y\;\mu(x,dy),\nonumber\\
&\text{with}\ A(x)\geq 0,\ \text{and}\ \sup_x\int_{\real^d}\min({\left| y \right|}^2,1)\mu(x,dy)<\infty.
\nonumber \end{aligned}$$ We will review some recent results that show for $I:C^2_b(\real^d)\to C_b(\real^d)$ that enjoys the GCP, is Lipschitz, and has a natural structural constraint, there exists a family of functions, $f_{ab}$ and linear operators of Lévy type, $L_{ab}$, so that $$\begin{aligned}
\label{eqIN:MinMaxMeta}
I(u,x) = \min\limits_{a}\max\limits_{b} \{ f_{ab}(x)+L_{ab}(u,x) \}.\end{aligned}$$ For linear operators, in the 1960’s Courrège [@Courrege-1965formePrincipeMaximum] showed that all of those that satisfy the GCP must have the form given in (\[eqIN:LevyTypeLinear\]). All of our results here should be considered an extension of Courrège’s result to the nonlinear setting.
In our previous work, [@GuSc-2016MinMaxNonlocalarXiv], we showed such a min-max representation in (\[eqIN:MinMaxMeta\]). The result in [@GuSc-2016MinMaxNonlocalarXiv] in fact dealt with a more general situation where $I:C^2_b(M)\to C^0_b(M)$ where $M$ is a complete Riemannian manifold. We will review the proof of this result in the context of Euclidean space, where many of the arguments simplify greatly. Moreover, we prove two refinements of the main result from [@GuSc-2016MinMaxNonlocalarXiv] relevant to the Euclidean case, one involving translation invariant operators and one for operators that behave continuously with respect to translation operators. Stated informally, our results are the following:
\[theoremA\] An operator $I(u,x)$ that is Lipschitz and satisfies the GCP admits a min-max formula in terms of Lévy type operators.
\[theoremB\] In the previous theorem, assume further that $I(u,x)$ commutes with translations. Then the Lévy operators appearing in the min-max formula all commute with translations.
\[theoremC\] Instead of translation invariance assume that the finite differences of $I(u,x)$ commute with translations up to a certain error depending on a modulus of continuity $\omega(\cdot)$. Then the Lévy operators appearing in the min-max formula have continuous coefficients with common modulus of continuity of the form $C\omega(2(\cdot))$.
Theorem \[theoremA\] above is a special case of the main result in [@GuSc-2016MinMaxNonlocalarXiv], and Theorems \[theoremB\] and \[theoremC\] are new.
Assumptions and main results {#subsection:assumptions and main results}
----------------------------
Here are our main assumptions.
\[assumption:GCP\] The map $I:C^2_b(\mathbb{R}^d)\to C^0_b(\mathbb{R}^d)$ is Lipschitz continuous and has the Global Comparison Property .
\[assumption:translation invariance\] The map $I:C^2_b(\mathbb{R}^d)\to C^0_b(\mathbb{R}^d)$ is translation invariant. Namely, for any $x,z\in\mathbb{R}^d$ and $u\in C^2_b(\mathbb{R}^d)$ we have $$\begin{aligned}
\label{eqn:translation invariance}
I(\tau_z u,x) = I(u,x+z),\;\textnormal{where } \tau_z u(x):=u(x+z).
\end{aligned}$$
\[assumption:tightness bound\] There is a non-increasing function $\rho:(0,\infty)\to \mathbb{R}$ with $\rho(R)\to 0$ as $R\to \infty$ such that if $u,v\in C^2_b(\mathbb{R}^d)$ are such that $u\equiv v$ in $B_{2R}(x_0)$, then $$\begin{aligned}
\| I(u)-I(v) \|_{L^\infty(B_R(x_0))} \leq \rho(R)\|u-v\|_{L^\infty(\mathbb{R}^d)}.
\end{aligned}$$
\[assumption:coefficient regularity\] There exists a modulus, $\om$, for all $v,u \in C^2_b(\mathbb{R}^d)$, $x, z \in\mathbb{R}^d$, $r>0$, we have $$\begin{aligned}
& | I(v+\tau_{-z}u,x+z)-I(v,x+z) -\left ( I(v+u,x)-I(v,x)\right ) | \\
& \leq \omega(|z|)C(r)\left ( \|u\|_{C^2(B_{2r}(x))} + \|u\|_{L^\infty(\mathcal{C}B_r(x))}\right ).
\end{aligned}$$ It is allowed that $C(r)\to\infty$ as $r\to0$; in some examples $C(r)$ may be bounded and in some it may be unbounded.
The meaning of Assumption \[assumption:GCP\] and Assumption \[assumption:translation invariance\] is self-evident. Assumption \[assumption:tightness bound\] seems rather technical, but it will be necessary to obtain compactness for a family of measures arising in the proof (and this assumption is satisfied by a broad family of examples). Note however that this assumption is not needed for the translation invariant case as well as the setting of Theorem \[theorem:MinMax Euclidean\] as these two theorems are obtained with different methods.
Last but not least, Assumption \[assumption:coefficient regularity\] can be thought of as a “coefficient regularity” assumption. For instance, in the linear and local case, in which $I$ is a Lévy operator without integral part, Assumption \[assumption:coefficient regularity\] is equivalent to the coefficients of the operator having modulus of continuity $C\omega(\cdot)$ for some constant $C>0$. In fact, Assumption \[assumption:coefficient regularity\] is stated so that it indeed linearizes to this usual assumption that one expects in the linear case.
As mentioned above, one can check that for linear operators, Assumption \[assumption:coefficient regularity\] is equivalent to the coefficients of the local part being uniformly continuous and the Lévy measures being uniformly continuous in the TV norm along shifts in the base point, i.e. $$\begin{aligned}
{\lVert\mu(x+x,\cdot)-\mu(x,\cdot)\rVert}_{TV(\mathcal{C}B_r)}\leq C\om({\left| z \right|}).
\end{aligned}$$ By its design, Assumption \[assumption:coefficient regularity\] is a technical artifact of our proof, and as such, it is unlikely to be sharp or even the most natural assumption. There is most likely room for improvement here. In fact, one indication of the possibility to make a more natural assumption lies in the fact that even when the original operator, $I$, is translation invariant (so the most regular dependence on $x$), it does not necessarily follow that $I$ also satisfies Assumption \[assumption:coefficient regularity\]. This also reflects the fact that we have taken a two completely different methods of proof for the results that concern translation invariant operators, and ones that have a modulus with respect to translations.
In Section \[section:examples\], we give a short list of some operators that fall within the scope of Assumptions \[assumption:GCP\]–\[assumption:coefficient regularity\] and Theorems \[theorem:MinMax Euclidean\]–\[theorem:minmax with beta less than 2\]. At the end of Section \[section:examples\], we give a list of which assumptions each example satisfies.
We note that one subtle improvement of the current work upon our previous one in [@GuSc-2016MinMaxNonlocalarXiv] is that because of a more streamlined proof for the translation invariant case, we were able to establish the non-translation invariant case, Theorem \[theorem:MinMax Euclidean\] (below), without the technical Assumption \[assumption:tightness bound\]. This is purely an artifact of using an approximation scheme in [@GuSc-2016MinMaxNonlocalarXiv] to treat all operators by the same method, and this turns out to have been not essential when one does not want the extra information provided by Theorems \[theorem:MinMax Euclidean ver2\] and \[theorem:minmax with beta less than 2\].
The first theorem uses the notion of “pointwise” $C^2$ or $C^1$, and so we will define that property here.
\[def:PointwiseC1C2\] For a fixed $x$ we say that $u\in C^2(x)$ (“pointwise $C^2$ at $x$”) if there exists a vector, $\grad u(x)$, and a symmetric matrix, $D^2u(x)$, such that $$\begin{aligned}
\text{as}\ y\to x,\ \ {\left| u(y)-u(x)-\grad u(x)\cdot(y-x)-\frac{1}{2}(y-x)\cdot \left( D^2u(x)(y-x)\right) \right|}\leq o({\left| y-x \right|}^2).
\end{aligned}$$ Similarly if $u$ only enjoys the existence of $\grad u(x)$ and $$\begin{aligned}
\text{as}\ y\to x,\ \ {\left| u(y)-u(x)-\grad u(x)\cdot(y-x) \right|}\leq o({\left| y-x \right|}),
\end{aligned}$$ we say that $u\in C^1(x)$ (“pointwise $C^1$ at $x$”).
Now we can restate Theorems 1–3 above, in more precise terms.
\[theorem:MinMax Euclidean\] If $I:C^2_b(\mathbb{R}^d)\to C^0_b(\mathbb{R}^d)$ satisfies Assumption \[assumption:GCP\], then, for each $x$, there exists a family of linear functionals on $C^2(x)$ that depend on $I$ and $x$, called $\mathcal{K}(I)_x$, so that for all $u\in C^2(x)$ $$\begin{aligned}
I(u,x) = \min\limits_{v\in C^2_b(\mathbb{R}^d)}\max\limits_{L \in \mathcal{K}(I)_x} \{ I(v,x)+L(u-v) \}.
\end{aligned}$$ Here, each $L\in\mathcal{K(I)}_x$, has the form $$\begin{aligned}
L(u)= \tr(A_xD^2u(x)) + B_x\cdot\nabla u(x) + C_x u(x)+
\int_{\mathbb{R}^d} u(x+y)-u(x)-\ind_{B_1(0)}(y)\nabla u(x)\cdot y\;\mu_x(dy),\end{aligned}$$ and for some universal $C$, the terms also satisfy the bound for all $x$: $$\begin{aligned}
{\left| A_x \right|} +{\left| B_x \right|} + {\left| C_x \right|} + \int_{\mathbb{R}^d} \min\{1,|y|^2\}\;\mu_x(dy) \leq C{\lVertI\rVert}_{\text{Lip},C^2_b\to C^0_b}.
\end{aligned}$$
The proof of Theorem \[theorem:MinMax Euclidean\] appears in Section \[sec:TheoremsWithoutWhitney\], which is at the end of Section \[section:Functionals with the GCP\].
We want to point out to the reader that the notation in Theorem \[theorem:MinMax Euclidean\] is intentional in its use of subscripts for e.g. $A_x$, etc. This is because our construction does not actually produce $L$ as a linear mapping $C^2_b\to C^0_b$, and so it is not correct to think of having a family of $L$ whose coefficients are actually *functions* of $x$. Rather, it just says that at each $x$ there is a family functionals that have the desired structure, but it is not clear that they can be put together across all $x$ to make a family of $x$-dependent operators.
This situation changes under other assumptions, and in the next two theorems, our method produces a family of linear operators mapping $C^2_b(\real^d)\to C^0_b(\real^d)$, all of the form (\[eqIN:LevyTypeLinear\]).
\[theorem:MinMax Translation Invariant\] If $I:C^2_b(\mathbb{R}^d)\to C^0_b(\mathbb{R}^d)$ satisfies Assumption \[assumption:GCP\] and Assumption \[assumption:translation invariance\] then there exists a family, $
\displaystyle
\{f_{ab}, L_{ab}\}_{a,b\in\mathcal{K}(I)},
$ that depends only on $I$, where for all $a,b$, $f_{ab}$ are constants, and $L_{ab}$ are linear translation invariant operators mapping $C^2_b(\real^d)\to C^0_b(\real^d)$ of the form (\[eqIN:LevyTypeLinear\]) (i.e. constant coefficients), and for all $u\in C^2_b(\mathbb{R}^d)$ and $x\in \mathbb{R}^d$ we have $$\begin{aligned}
I(u,x) = \min\limits_{a}\max\limits_{b} \{ f_{ab}+L_{ab}(u,x) \}.
\end{aligned}$$ Furthermore, for a universal $C$, for all $f_{ab}$ and $L_{ab}$, $$\begin{aligned}
{\left| f_{ab} \right|} + {\left| A_{ab} \right|} +{\left| B_{ab} \right|} + {\left| C_{ab} \right|} + \int_{\mathbb{R}^d} \min\{1,|y|^2\}\;\mu_{ab}(dy) \leq C{\lVertI\rVert}_{\text{Lip},C^2_b\to C^0_b}.
\end{aligned}$$
The proof of Theorem \[theorem:MinMax Translation Invariant\] appears in Section \[sec:TheoremsWithoutWhitney\], which is at the end of Section \[section:Functionals with the GCP\].
\[theorem:MinMax Euclidean ver2\] If $I:C^2_b(\mathbb{R}^d)\to C_b^0(\mathbb{R}^d)$ satisfies Assumption \[assumption:GCP\], Assumption \[assumption:tightness bound\], and Assumption \[assumption:coefficient regularity\], then, there exists a family, $
\displaystyle
\{f_{ab}, L_{ab}\}_{a,b\in\mathcal{K}(I)},
$ that depends only on $I$, where for all $a,b$, $f_{ab}\in C^0_b(\real^d)$ are functions, and $L_{ab}$ are linear operators mapping $C^2_b(\real^d)\to C^0_b(\real^d)$ of the form (\[eqIN:LevyTypeLinear\]), and for all $u \in C^2_b(\mathbb{R}^d)$, we have $$\begin{aligned}
I(u,x) = \min\limits_{a}\max\limits_{b} \{ f_{ab}(x)+L_{ab}(u,x) \},
\end{aligned}$$ and for a universal $C$, for all $f_{ab}$ and $L_{ab}$, $$\begin{aligned}
{\lVertf_{ab}\rVert}_{L^\infty} + {\lVertA_{ab}\rVert}_{L^\infty} +{\lVertB_{ab}\rVert}_{L^\infty} + {\lVertC_{ab}\rVert}_{L^\infty} + \sup_x\int_{\mathbb{R}^d} \min\{1,|y|^2\}\;\mu_{ab}(x,dy) \leq C{\lVertI\rVert}_{\text{Lip},C^2_b\to C^0_b}.
\end{aligned}$$
Furthermore, if $\om$ is as in Assumption \[assumption:coefficient regularity\], then the functions $f_{ab},A_{ab},B_{ab},C_{ab},$ all have a modulus of continuity $C\omega(2\cdot)$, while for each $r>0$ we have the estimate, $$\begin{aligned}
\label{eqIN:ThmModulusInTVNorm}
\|\mu_{ab}(x_1)-\mu_{ab}(x_2)\|_{\textnormal{TV}(\mathcal{C}B_r)} \leq C(r)\omega(2|x_1-x_2|),
\end{aligned}$$ where as above, $C(r)>0$, is a constant that may possibly (but not necessarily) have the property that $C(r)\to\infty$ as $r\to0$.
The proof of Theorem \[theorem:MinMax Euclidean ver2\] appears in Section \[sec:TheoremsThatUseWhitney\], which is at the end of Section \[section:Analysis of finite dimensional approximations\].
Finally, we give a theorem that reduces the possible terms in the min-max over (\[eqIN:LevyTypeLinear\]). Namely, there are instances in which there may be no second order terms or first order terms. To state this, we abuse notation slightly, and we give a shorthand as $C^{\beta}_b(\real^d)$ to mean the following: $$\begin{aligned}
\label{equation:Cbeta definition}
\begin{array}{rl}
&\text{if}\ \beta = 2 + \gamma,\ \textnormal{for}\ \gam\in (0,1),\ \text{then, we mean}\ C^\beta_b(\real^d) = C^{2,\gamma}_b(\real^d);\\
&\text{if}\ \beta = 2^+,\ \text{then, we mean}\ C^\beta_b(\real^d) = C^{2}_b(\real^d);\\
&\text{if}\ \beta = 2,\ \text{then, we mean}\ C^\beta_b(\real^d) = C^{1,1}_b(\real^d);\\
&\text{if}\ \beta = 1+\gam,\ \text{for}\ \gam\in(0,1), \text{then, we mean}\ C^\beta_b(\real^d) = C^{1,\gam}_b(\real^d);\\
&\text{if}\ \beta = 1^+,\ \text{then, we mean}\ C^\beta_b(\real^d) = C^1_b(\real^d);\\
&\text{if}\ \beta = 1,\ \text{then, we mean}\ C^\beta_b(\real^d) = C^{0,1}_b(\real^d);\\
&\text{if}\ \beta = \gam,\ \text{for}\ \gam\in(0,1), \text{then, we mean}\ C^\beta_b(\real^d) = C^{0,\gam}_b(\real^d).
\end{array}\end{aligned}$$
\[def:PointwiseCBeta\] For a fixed $x$, we say that $u\in C^{\beta}(x)$ (“pointwise $C^\beta(x)$”) if the same requirements of Definition \[def:PointwiseC1C2\] hold, but the estimate on the right hand side takes into account the different decay as follows:
- if, $\beta=2+\gam$, then $u$ has a second order Taylor expansion and the right hand side is $O({\left| y-x \right|}^{2+\gam})$;
- if, $\beta=2^+$, then $u$ has a second order Taylor expansion and the right hand side is $o({\left| y-x \right|}^{2})$;
- if, $\beta=2$, then we include this in the previous case whenever $u$ has a second order taylor expansion at $x$;
- if, $\beta=1+\gam$, then $u$ has a first order Taylor expansion and the right hand side is $O({\left| y-x \right|}^{1+\gam})$;
- if, $\beta=1^+$, then $u$ has a first order Taylor expansion and the right hand side is $o({\left| y-x \right|})$;
- if, $\beta=1$, then we include this in the previous case whenever $u$ has a first order taylor expansion at $x$;
- if, $\beta=\gam\in(0,1)$, then ${\left| u(y)-u(x) \right|}\leq C{\left| y-x \right|}^\gam$.
\[assumption:CBeta\] All of Assumptions \[assumption:GCP\] – \[assumption:coefficient regularity\] hold, but with all instances of $C^2_b(\real^d)$ replaced by $C^\beta_b(\real^d)$.
\[theorem:minmax with beta less than 2\] For each of Theorems \[theorem:MinMax Euclidean\], \[theorem:MinMax Translation Invariant\], \[theorem:MinMax Euclidean ver2\], we have the following variation: in each case assume that $I$ satisfies Assumption \[assumption:CBeta\], for some $\beta \in [0,2^+]$ (as enumerated above). Then, taking into account Definition \[def:PointwiseCBeta\] for Theorem \[theorem:MinMax Euclidean\], the min-max formula holds in each of the previous results with the following additions: if $\beta<2$ then $A_{ab} = 0$ for all $a,b$, while if $\beta <1$ then $B_{ab} = 0$ for all $a,b$ and the operators $L_{ab}$ take the form $$\begin{aligned}
L_{ab}(u,x) & = C_{ab}(x)u(x)+\int_{\mathbb{R}^d} u(x+y)-u(x)\;\mu_{ab}(x,dy).
\end{aligned}$$ Moreover, the smaller $\beta$, the more regular the Lévy measures $\mu_{ab}$ are at $y=0$, namely, we have $$\begin{aligned}
\sup \limits_{a,b,x}\int_{\mathbb{R}^d}\min\{1,|y|^\beta\}\mu_{ab}(x,dy) <\infty.
\end{aligned}$$
The proof of Theorem \[theorem:minmax with beta less than 2\] appears in Section \[sec:TheoremsThatUseWhitney\], which is at the end of Section \[section:Analysis of finite dimensional approximations\].
In Sections \[section:Finite Dimensional Approximations\] and \[section:Analysis of finite dimensional approximations\], one can see that at its heart, the fact that the modulus for $I$ is passed onto the coefficient functions in (\[eqIN:LevyTypeLinear\]) is a consequence of our choice to use a Whitney extension in an approximation to $I$, and the Whitney extension is well known to preserve a modulus of continuity. The actual details are a bit more involved, but that is the main reason. We note the presence of the factor of $2$ in the new modulus is a consequence of the Whitney Extension method; the interested reader can see [@Stei-71 Chapter VI].
A further comment regarding the assumptions is in order. Suppose that $I$ satisfies Assumption \[assumption:coefficient regularity\] with $\omega \equiv 0$. In this case, taking $v\equiv 0$ the assumption says that $$\begin{aligned}
I(\tau_{-h}u,x+h)-I(0,x+h) = I(u,x)-I(0,x), \end{aligned}$$ and if we further assume that $I(0,x)$ is constant (i.e. $I$ applied to the zero function returns a constant), then we have $$\begin{aligned}
I(\tau_{-h}u,x+h) = I(u,x),\end{aligned}$$ that is, $I$ is translation invariant. However, at first sight it is not clear what happens in the reverse direction. That is, we do not know how to show that a translation-invariant operator automatically satisfies Assumption \[assumption:coefficient regularity\] with $\omega \equiv 0$, and in fact we expect that this assumption can be modified so that it seamlessly includes the translation invariant operators as well.
Notation {#section_sub:Notation}
--------
For the readers’ convenience, a summary of symbols used in the paper is presented below.
[lll]{} Notation & Definition\
\
$d$ & space dimension\
$C^2_b$ & twice differentiable functions $f$ with bounded $f,\nabla f,$ and $D^2f$\
$C^\beta_b$ & bounded functions of class $C^\beta$, see for definition\
$\mathbb{S}_d$ & symmetric matrices of size $d\times d$\
$\|\cdot\|_{\textnormal{TV}}$ & total variation norm for a measure\
$L(X,Y)$ & space of bounded linear operators from $X$ to $Y$\
$\textnormal{c.h.}(E)$ & the convex hull of a set $E$\
$\mathcal{C}E$ & complement of a subset of $\mathbb{R}^d$\
$F^0(x,v)$ & upper gradient of a Lipschitz function (Definition \[definition:upper gradient\])\
$\partial F(x)$ & generalized gradient of $F$ at $x$ (Definition \[definition:generalized gradient\])\
$G_n$ & grid with step size $2^{-n}$\
$C(G_n)$ & space of real valued functions defined in $G_n$ (Definition \[definition:discrete function spaces\])\
$C_*(G_n)$ & subset of $C(G_n)$ of functions vanishing outside $[-2^n,2^n]\cap G_n$ (Definition \[definition:discrete function spaces\])\
$(\nabla_n)^1u(x)$ & discrete gradient for step size $2^{-n}$ (Definition \[definition:finite difference operators 1\])\
$(\nabla_n)^2u(x)$ & discrete Hessian for step size $2^{-n}$ (Definition \[definition:finite difference operators 2\])
Background {#section_sub:Background}
----------
There were roughly two reasons that motivated the results we present in this paper. First of all, the link between elliptic equations and a min-max formula for operators has a long history, and it has been exploited extensively in the case of *local* operators. Until [@GuSc-2016MinMaxNonlocalarXiv], the connection was not known for nonlocal, nonlinear operators. Even so, the link between the two was natural enough that there are at least a few results that assumed a structure like (\[eqIN:MinMaxMeta\]), including [@BaIm-07], [@JakobsenKarlsen-2006maxpple], [@KoikeSwiech-2013RepFormulaIntegroPDE-IUMJ], [@Schw-10Per], [@Schw-12StochCPDE], [@Silv-2011DifferentiabilityCriticalHJ], among many others. Thus the theorems here and in [@GuSc-2016MinMaxNonlocalarXiv] give a sort of a posteriori justification to min-max assumptions that appeared in earlier works. Secondly, a formula such as (\[eqIN:MinMaxMeta\]) can be very useful in connecting results about the integro-differential theory (of which, there has been a large volume recently) with some other pursuits that may not obviously relate to operators such as (\[eqIN:LevyTypeLinear\]). Two recent projects that exploit or were motivated by the min-max formulas are on some Hele-Shaw type free boundary evolutions in [@ChangLaraGuillenSchwab2018FBasNonlocal] and some Neumann homogenization problems [@GuSc-2014NeumannHomogPart1DCDS-A] [@GuSc-2018NeumannHomogPart2SIAM]. Both of these relate to linear and nonlinear Dirichlet-to-Neumann maps, studied in [@GuillenKitagawaSchwab2017estimatesDtoN], and there is plenty more to learn about the integro-differential structure in the nonlinear setting. The choice to pursue continuity properties such as the dependence given in (\[eqIN:ThmModulusInTVNorm\]), although a posteriori seems straightforward, was not initially obvious, and it was motivated by recent results about comparison theorems for viscosity solutions of integro-differential equations in [@GuillenMouSwiech2018coupling].
As mentioned earlier, for linear operators, the representation of (\[eqIN:LevyTypeLinear\]) goes back to Courrège [@Courrege-1965formePrincipeMaximum]. This was naturally connected with generators of Markov processes and boundary excursion processes for reflected diffusions. Hsu [@Hsu-1986ExcursionsReflectingBM] provides a similar representation for the Dirichlet to Neumann map for the Laplacian in a smooth domain $\Omega$, and this corresponds to studying the boundary process for a reflected Brownian motion. If $I$ is not necessarily linear but happens to satisfy the stronger *local* comparison principle, there are min-max results by many authors, e.g. Evans [@Evans-1984MinMaxRepresentations], Souganidis [@Souganidis-1985MaxMinRep], Evans-Souganidis [@EvansSoug-84DiffGameRepresentation] and Katsoulakis [@Katsou-1995RepresentationDegParEq]. In this case, the operator takes the form, $$\begin{aligned}
I(u,x) = F(x,u(x),\nabla u(x),D^2u(x)), \end{aligned}$$ which can be expressed as in Theorem \[theorem:MinMax Euclidean\], but with $\mu(x,dh)\equiv 0$. This was extended to even include the possibility of weak solutions acting as a *local* semi-group on $BUC(\real^d)$, related to image processing, in Alvarez-Guichard-Lions-Morel [@AlvarezLionsGuichardMorel-1993AxiomsImageProARMA], and to weak solutions of sets satisfying an order preserving set flow by Barles-Souganidis in [@BarlesSouganidis-1998NewApproachFrontsARMA]. In [@AlvarezLionsGuichardMorel-1993AxiomsImageProARMA] it was shown under quite general assumptions that certain nonlinear semigroups must be represented as the unique viscosity solution to a degenerate parabolic equation.
Although it is still too early to tell, one hopes that theorems like those presented here can create a bridge between some nonlocal equations for which regularity questions arise and the known results about such equations when a min-max structured is known to hold. In the *local* setting, there are a number of results that leverage the min-max to shed new light on certain issues, and it would be interesting to see if similar things can be done for the nonlocal theory (see the discussion in [@GuSc-2016MinMaxNonlocalarXiv Section 1] for an incomplete list of such results). The types of regularity results that could find new applications via the min-max theorems here fall into roughly three categories: Krylov-Safonov type results; regularity for translation invariant equations; and Schauder type regularity results. For Krylov-Safonov, this means that solutions of fully nonlinear equations can be shown to enjoy Hölder estimates depending only on the $L^\infty$ norm of the solution; some examples are: [@CaSi-09RegularityIntegroDiff], [@Chan-2012NonlocalDriftArxiv], [@ChDa-2012NonsymKernels], [@KassRangSchwa-2013RegularityDirectionalINDIANA], and [@SchwabSilvestre-2014RegularityIntDiffVeryIrregKernelsAPDE], among many others. For translation invariant equations, these are the results that show solutions to translation invariant equations very often enjoy $C^{1,\alpha}$ regularity under mild assumptions; some examples are: [@CaSi-09RegularityIntegroDiff], [@ChangLaraKriventsov2017], [@Kriventsov-2013RegRoughKernelsCPDE], [@Ros-OtonSerra-2016RegularityStableOpsJDE], [@Serra-2015RegFullyNonlinIntDiffRoughKernelsCalcVar], among others. Finally, for Schauder regularity, we mean results that show that for $x$-dependent operators, under certain regularity for the coefficients (such as Dini), solutions will have as much regularity as those equations with “constant coefficients”; some examples are: [@DongJinZhang2018DiniSchauderNonlocal], [@JinXiong-2015SchauderEstLinearParabolicIntDiffDCDS-A], [@MouZhang2018preprint], among others. On top of questions of the type of Krylov-Safonov regularity mentioned above, there is another family of regularity results that accompanies existence and uniqueness techniques for viscosity solutions of elliptic partial-differential / integro-differential equations, and it is typically referred to as the Ishii-Lions method, going back to [@IshiiLions-1990ViscositySolutions2ndOrder]. Both this Ishii-Lions regularity and comparison results could connect well with the operators treated in this paper, as many of the existing works on nonlocal equations assume a min-max. The types of results that could be applicable are like those in [@BaChIm-08Dirichlet], [@BaChIm-11Holder], [@BaChCiIm-2012LipschitzMixedEq], [@BaIm-07], and [@JakobsenKarlsen-2006maxpple], among others.
There is some more discussion of related works and background inside of the examples that we list in Section \[section:examples\].
Another description of operators satisfying the GCP
---------------------------------------------------
Let us describe an elementary but useful way to view operators satisfying the GCP, which is also related to the min-max representation. First, we introduce a family of functional spaces.
\[definition:L\_beta\^infinity spaces\] For $\beta \in [0,2^+]$ (using the abuse of notation in (\[equation:Cbeta definition\])) we define the space $L^\infty_{\beta}$ as follows. First, if $\beta \neq 1^+$, $$\begin{aligned}
L^\infty_\beta := \{ h \in L^\infty(\mathbb{R}^d) \;\mid\; |h(y)| = O(|y|^\beta) \textnormal{ as } |y|\to 0 \},
\end{aligned}$$ while for $\beta=1^+$, $$\begin{aligned}
L^\infty_\beta & := \{ h \in L^\infty(\mathbb{R}^d) \;\mid\; |h(y)| = o(|y|^\beta) \textnormal{ as } y\to 0\}.
\end{aligned}$$ (We note the first space requires “Big-O”, while the second space requires “little-o”.) The spaces $L^\infty_\beta$ are Banach spaces, with norms given by $$\begin{aligned}
\sup \limits_{y}|h(y)| \min\{1,|y|^\beta\}^{-1}.
\end{aligned}$$
Now, suppose we are given a continuous function $$\begin{aligned}
F: L^\infty_\beta(\mathbb{R}^d) \times \mathbb{S}_d \times \mathbb{R}^d\times \mathbb{R} \times \mathbb{R}^d\to\mathbb{R}.\end{aligned}$$ Assume that this function is monotone (non-decreasing) with respect to the first two variables. Then, given $u \in C^\beta_b(\mathbb{R}^d)$ define $$\begin{aligned}
I(u,x) := F(\delta_x u,D^2u(x),\nabla u(x),u(x),x)\end{aligned}$$ where we are using the notation $\delta_x u(y):= u(x+y)-u(x)-\nabla u(x)\cdot y \chi_{B_1(0)}(y)$ for $\beta\geq 1$, and $\delta_x u(y):=u(x+y)-u(x)$ for $\beta<1$. It is clear the operator $I$ thus defined has the GCP.
Do all operators with the GCP arise in this form? It is easy to see that the answer is positive, at least when $\beta<2$. Given $I:C^\beta(\mathbb{R}^d)\to C^0(\mathbb{R})$, with $\beta<2$, we define a function $$\begin{aligned}
F: L^\infty_\beta(\mathbb{R}^d) \times \mathbb{R}^d\times \mathbb{R} \times \mathbb{R}^d\to\mathbb{R}, \end{aligned}$$ by the formula $F(h,p,u,x) := I( \tau_{-x}h+\tau_{-x}p\cdot(\cdot)\chi_{B_1}+u,x)$. It is straightforward to see that for $F$ so defined and $u \in C^\beta_b(\mathbb{R}^d)$ we have $$\begin{aligned}
I(u,x) = F(\delta_x u,\nabla u(x),u(x),x).\end{aligned}$$
Real valued Lipschitz functions on Banach Spaces {#section:Real Valued Lipschitz Functions}
================================================
In this section we review various well known facts about Lipschitz functions on Banach spaces, following Clarke’s book [@Cla1990optimization Chapter 2]. We will refer most of the proofs to the relevant section in [@Cla1990optimization]. The section ends with Theorem \[theorem:MinMax for scalar functionals\] which yields a min-max formula for any real valued, Lipschitz $F$, such a result is neither new nor surprising, but we present it here in complete detail for the sake of completeness.
We fix a Banach Space, denoted by $X$, an open convex subset $\mathcal{K}\subset X$, and a function $$\begin{aligned}
F:\mathcal{K}\subset X\to \mathbb{R},\end{aligned}$$ which is assumed Lipschitz with constant $L>0$, that is $$\begin{aligned}
\label{eqnLipschitzFunctionals:Lipschitz constant}
|F(x)-F(y)|\leq L\|x-y\| \;\;\forall\;x,y\in \mathcal{K}.\end{aligned}$$
\[definition:upper gradient\] The upper gradient of $F$ at $x\in \mathcal{K}$ in the direction of $v\in X$, is defined as $$\begin{aligned}
F^0(x,v) := \limsup\limits_{t\searrow 0} \frac{F(x+tv)-F(x)}{t}.
\end{aligned}$$ This can be seen as a function $F^0:\mathcal{K}\times X\to \mathbb{R}$.
\[proposition:upper gradient properties\] The function $F^0(x,v)$ has the following properties
1. For any $x\in \mathcal{K},v\in X$, and $\lambda>0$ we have $F^0(x,\lambda v) =\lambda F^0(x,v)$.
2. For any $x\in \mathcal{K}$, and $v,w\in X$ we have $|F^0(x,v)-F^0(x,w)|\leq L\|v-w\|$.
3. If $(x_k,v_k)\to(x,v)$ then $\limsup F^0(x_k,v_k) \leq F^0(x,v)$.
4. $F^0(x,-v)=(-F)^0(x,v)$.
We refer the reader to [@Cla1990optimization Proposition 2.1.1].
\[definition:generalized gradient\] The generalized gradient of $F$ at $x\in \mathcal{K}$ is the subset of $X^*$ given by $$\begin{aligned}
\partial F(x) := \{ \ell \in X^* \mid F^0(x,v)\geq \langle \ell,v\rangle \;\;\forall\;v\in X\}.
\end{aligned}$$ We will denote by $\partial F$ the convex hull of the union of $\partial F(x)$, $$\begin{aligned}
\partial F := \hull \left ( \bigcup \limits_{x\in \mathcal{K}} \partial F(x)\right).
\end{aligned}$$
\[proposition:generalized gradient properties\] The set $\partial F(x)$, $x\in \mathcal{K}$, has the following properties
1. $\partial F(x)$ is a non-empty, convex, $\textnormal{weak}^*$-compact subset of $X^*$.
2. $\|\ell\| \leq L$ for every $\ell\in \partial F(x)$.
3. For any $v\in X$, we have that $$\begin{aligned}
F^0(x,v) = \max \limits_{\ell\in \partial F(x)} \langle \ell,v\rangle.
\end{aligned}$$
We refer the reader to [@Cla1990optimization Proposition 2.1.2].
The following theorem, due to Lebourg, is a generalization of the mean value theorem for differentiable functions.
\[theorem:Lebourg\] Let $x,y$ be points in $\mathcal{K}$. Then there exist $z$ of the form $z= tx+(1-t)y$ for some $t\in[0,1]$, such that for some $\ell \in \partial F(z)$ $$\begin{aligned}
F(x)-F(y) = \langle \ell,x-y\rangle.
\end{aligned}$$
We refer the reader to [@Cla1990optimization Theorem 2.3.7].
Using the generalized gradient and Lebourg’s theorem we can easily prove a min-max formula for Lipschitz functionals. Observe this is a general result for Lipschitz functionals in general Banach spaces, and it does not involve anything like GCP (functionals with the GCP on $C^\beta_b(\real^d)$ are considered in the next section).
\[theorem:MinMax for scalar functionals\] Let $F:\mathcal{K}\subset X\to\mathbb{R}$ be a Lipschitz function, with $\mathcal{K}$ convex, then for all $x\in \mathcal{K}$, $$\begin{aligned}
F(x) = \min\limits_{y\in \mathcal{K}}\max\limits_{\ell \in \partial F} \{F(y)+\langle \ell,y-x\rangle \}.
\end{aligned}$$
According to Theorem \[theorem:Lebourg\], given $x,y\in \mathcal{K}$ there is some $\ell\in\partial F$ such that $$\begin{aligned}
F(x)-F(y) = \langle \ell,x-y\rangle.
\end{aligned}$$ In other words, for any $x$ and $y$ in $\mathcal{K}$ we have the inequality $$\begin{aligned}
F(x) \leq \max \limits_{\ell \in \partial F}\left \{ F(y)+\langle \ell,x-y\rangle \right \}.
\end{aligned}$$ This also yields an equality for $y=x$, thus $F(x) = \min \limits_{y\in \mathcal{K}}\max \limits_{\ell \in \partial F}\left \{ F(y)+\langle \ell,x-y\rangle \right \}$.
Functionals with the GCP, revisited {#section:Functionals with the GCP}
===================================
Throughout this section $\mathcal{K}$ denotes an open convex set of $C^\beta_b(\mathbb{R}^d)$ (see ). Moreover, for $\rho>0$, we shall write $$\begin{aligned}
\mathcal{K}_{\rho} = \big \{ u \in C^\beta_b(\real^d) \;\mid \; \|v-u\|_{C^\beta}<\rho \Rightarrow v \in \mathcal{K} \big \}.\end{aligned}$$
\[definition:GCP with respect to a point\] Let $F$ be a map $F:\mathcal{K}\subset C^\beta_b(\mathbb{R}^d)\to\mathbb{R}$ and $x\in \mathbb{R}^d$. Such a functional is said to have the Global Comparison Property with respect to $x$ if $F(u)\leq F(v)$ for any pair of functions $u,v\in \mathcal{K}$ such that $u(y)\leq v(y)$ for all $y$ and $u(x)=v(x)$ –we will say in such a case that $v$ touches $u$ from above at $x$.
The following two auxiliary functions will be useful throughout the section: Fix $\phi_0:\mathbb{R}\to\mathbb{R}$, a nondecreasing $C^\infty$ function such that $0\leq \phi_0 \leq 1$, $\phi_0(x)=0$ for $x\leq 0$, $\phi_0(x)=1$ for $x\geq 1$. Then, given $r,R>0$ we define the functions $$\begin{aligned}
\phi_{r,R}(y) & := \phi_0\left ( \frac{|y|-R}{r} \right ) \label{equation:phi sub r R}\\
\psi_{r,R}(y) & := 1-\phi_{r,R}(y) \label{equation:psi sub r R}\end{aligned}$$ The following Proposition was first proved in [@GuSc-2016MinMaxNonlocalarXiv Lemma 4.15, Corollary 4.16], we review the proof here for the reader’s convenience.
\[proposition:GCP implies weak localization\] Suppose that $F:\mathcal{K}\subset C^\beta_b(\mathbb{R}^d)\to \mathbb{R}$ is a Lipschitz functional which has the $GCP$ with respect to $x$. Fix $\rho>0$. There is a constant $C(F,\rho)$ such that given $R>0$, $r\in (0,1)$, and $u,v \in \mathcal{K}_\rho$, then $$\begin{aligned}
|F(u)-F(v)| \leq C(F,\rho)r^{-\beta}\left ( \|u-v\|_{C^\beta(B_{R+r}(x))}+\|u-v\|_{L^\infty(\mathbb{R}^d\setminus B_{R}(x))} \right ).
\end{aligned}$$
\[remark:weak localization assumption versus proposition\] It is worth comparing Proposition \[proposition:GCP implies weak localization\] with Assumption \[assumption:tightness bound\]. In the latter, one is interested in how $I(u,x)$ depends very little on the values of $u$ far away from $x$ (so, as $r\to \infty$), whereas the former deals with a weak version of this property that holds only for $r\in (0,1)$ but which follows alone from the GCP without the need for further assumptions on $F$.
Take $\phi \in C^2_b(\mathbb{R}^d)$, such that $0\leq \phi \leq 1$ and $\phi(x)=0$. Then, for any $y$ we have $$\begin{aligned}
u(y) \leq w(y):= u(y)+ \phi(y) \left ( \|u-v\|_{L^\infty(\spt(\phi))}-(u(y)-v(y))\right ),
\end{aligned}$$ with the above being an equality for $y=x$. Now, let $\rho_0$ be chosen so that $$\begin{aligned}
2\|\phi\|_{C^2(\mathbb{R}^d)}\rho_0 \leq \rho.
\end{aligned}$$ Then, let us suppose that $u,v\in \mathcal{K}_{\rho}$ are such that $\|u-v\|_{C^\beta_b(\mathbb{R}^d)} \leq \rho_0$. In this case, we have $w\in \mathcal{K}$ since $u \in \mathcal{K}_\rho$ and in this case the GCP says that $$\begin{aligned}
F(u) \leq F(w).
\end{aligned}$$ Moreover, $F(w)\leq F(v)+L\|w-v\|_{C^\beta}$ and $w-v = (1-\phi) (u-v)+ \phi \|u-v\|_{L^\infty(\spt(\phi))}$, thus $$\begin{aligned}
F(u)-F(v) \leq L\|(1-\phi) (u-v)\|_{C^\beta}+L\|u-v\|_{L^\infty(\spt(\phi))}\|\phi\|_{C^\beta}.
\end{aligned}$$ Consider the function $\phi(y) = \phi_{r,R}(y-x)$. Thanks to $r\in (0,1)$, the following estimates hold $$\begin{aligned}
\|\phi\|_{C^\beta} & \leq Cr^{-\beta},\\
\|(1-\phi)(u-v)\|_{C^\beta} & \leq Cr^{-\beta} \|u-v\|_{C^\beta(B_{R+r})}.
\end{aligned}$$ Substituting these in the inequality for $F(u)-F(v)$, the desired inequality follows when $\|u-v\|_{C^\beta}$ is no larger than $\rho_0$. Otherwise, $\|u-v\|_{C^\beta}\geq \rho_0$ and iterating the inequality in the previous case one obtains that $$\begin{aligned}
|F(u)-F(v)| \leq C(F,\rho)r^{-\beta}\left ( \|u-v\|_{C^\beta(B_{R+r}(x))}+\|u-v\|_{L^\infty(\mathbb{R}^d\setminus B_{R}(x))} \right ).
\end{aligned}$$
\[lemma:generalized gradients inherit GCP and weak localization\] Let $F:\mathcal{K}\subset C^\beta_b(\mathbb{R}^d)\to\mathbb{R}$ be a Lipschitz functional which has the GCP with respect to $x$. Then, for every $\ell\in\partial F$ we have $$\begin{aligned}
\langle \ell,v\rangle\leq 0 \textnormal{ if } v\leq 0 \textnormal{ everywhere and } v(x)=0.
\end{aligned}$$ In other words, if $F$ has the GCP with respect to $x$, then any $\ell$ arising as a generalized gradient of $F$ also has the GCP with respect to $x$. Furthermore, for any such $\ell$ and $r\in (0,1)$ we have $$\begin{aligned}
|\langle \ell,v\rangle | \leq C r^{-\beta}\left ( \|v\|_{C^\beta(B_r)} + \|v\|_{L^\infty(\mathbb{R}^d)} \right ).
\end{aligned}$$
Let $u\in \mathcal{K}$, and let $v\in C^\beta_b(\mathbb{R}^d)$ be such that $$\begin{aligned}
v\leq 0\textnormal{ in } \mathbb{R}^d, v(x)=0.
\end{aligned}$$ Then, $u_t = u+tv$ touches $u$ from below at $x$ for each small $t$, therefore $F(u_t)\leq F(u)$ for every $t$, and $$\begin{aligned}
F^0(u,v) = \limsup\limits_{t\to 0} \frac{F(u+tv)-F(u)}{t} \leq 0.
\end{aligned}$$ Since, $$\begin{aligned}
\max \limits_{\ell\in \partial F(u)} \langle \ell,v\rangle = F^0(u,v),
\end{aligned}$$ it follows that $\langle \ell,v\rangle\leq 0$ for any $\ell\in \partial F(u)$, and the first part of the Lemma is proved. For the second part, one argues similarly, except that instead of invoking the GCP, one applies Proposition \[proposition:GCP implies weak localization\] in order to pass the same estimate for any $\ell \in \partial F$.
Fix a functional $\ell$ having the GCP with respect to $x$. Then, define $C_{\ell}$ by $$\begin{aligned}
\label{eqn:C sub ell definition}
C_{\ell} := \langle \ell,1\rangle.\end{aligned}$$ This associates a constant $C_{\ell}$ to any $\ell$ having the GCP. Likewise, we shall associate a vector $B_{\ell}$ and positive semi-definite matrix $A_{\ell}$. First, let us introduce some notation, $$\begin{aligned}
\label{eqFunctionalsGCP:DefOfSet-S}
\mathcal{S} := \{ \phi \in C^2_c(B_2(0))\;\mid\; \phi \equiv 1 \textnormal{ in a neighborhood of } 0,\; 0 \leq \phi \leq 1 \textnormal{ in all of } \mathbb{R}^d \}.\end{aligned}$$ Given $\phi,\eta\in\mathcal{S}$, define the function $$\begin{aligned}
\label{eqFunctionalsGCP:DefOfTalyorP}
P_{\phi,\eta,u,x}(\cdot) = \left \{ \begin{array}{rl}
u(x)+\phi(\cdot-x)(\nabla u(x),\cdot-x)+\tfrac{1}{2}\eta(\cdot-x)(D^2u(x)(\cdot-x),\cdot-x) & \textnormal{ if } \beta \in [2,3) ,\\
u(x)+\phi(\cdot-x)(\nabla u(x),\cdot-x) & \textnormal{ if } \beta \in [1,2) ,\\
u(x) & \textnormal{ if } \beta \in (0,1). \end{array}\right.\end{aligned}$$ For $x=0$ we will simply write $P_{\phi,\beta,u}$. Observe that, for example, if $\beta=2$ then $P_{\phi,\eta,u,x}$ is a smooth function which, in a neighborhood of $x$, coincides with the second order Taylor polynomial of the function $u$ at the point $x$.
\[definition:auxiliary A\_ell,eta and B\_ell,phi\] Given any $\phi \in \mathcal{S}$ let $B_{\ell,\phi}$ be the vector defined by $$\begin{aligned}
(B_{\ell,\phi},e) = \langle \ell, \phi(\cdot)(\cdot,e)\rangle,\;\;\forall\; \textnormal{ vectors } e.
\end{aligned}$$ At the same time, given $\eta \in \mathcal{S}$ let $A_{\ell,\eta}$ be the symmetric matrix defined by $$\begin{aligned}
\textnormal{tr}(A_{\ell,\eta}M) = \langle \ell, \eta(\cdot)\tfrac{1}{2}(M(\cdot),\cdot)\rangle,\;\;\forall\; \textnormal{ symmetric matrices } M.
\end{aligned}$$
The following lemmas will characterize all of functionals having the GCP with respect $0$ (compare with Courrege’s original proof [@Courrege-1965formePrincipeMaximum], see also [@GuSc-2016MinMaxNonlocalarXiv]).
\[lemma:preliminary Courrege theorem for a linear functional\] Let $\ell:C^\beta_b(\mathbb{R}^d)\to \mathbb{R}$ be a bounded linear functional which has the GCP with respect to $0$, and $\phi,\eta \in \mathcal{S}$ (defined in (\[eqFunctionalsGCP:DefOfSet-S\])). There is a positive measure $\mu_{\ell}$ on $\mathbb{R}^d\setminus \{0\}$ with $$\begin{aligned}
\int_{\mathbb{R}^d\setminus\{0\} } \min\{1,|y|^{\beta}\}\;\mu_{\ell}(dy) \leq C\|\ell\|,
\end{aligned}$$ such that for any $u \in C^\beta_b(\mathbb{R}^d)$ we have the following representation, $$\begin{aligned}
&\text{for}\ \beta\geq 2,\ \text{and}\ u\in C^\beta_b(\real^d)\intersect C^2(0),\\
&\ \ \ \ \ \langle \ell,u\rangle = C_{\ell}u(0)+(B_{\ell,\phi},\nabla u(0))+\tr(A_{\ell,\eta}D^2u(0))+\int_{\mathbb{R}^d}u(y)-P_{\phi,\eta,u}(y)\;\mu_\ell(dy),\\
&\text{for}\ \beta\in[1,2),\ \text{and}\ u\in C^\beta_b(\real^d)\intersect C^1(0),\\
&\ \ \ \ \ \langle \ell,u\rangle = C_{\ell}u(0)+(B_{\ell,\phi},\nabla u(0))+\int_{\mathbb{R}^d}u(y)-P_{\phi,\eta,u}(y)\;\mu_\ell(dy),\\
&\text{for}\ \beta\in(0,1),\ \text{and}\ u\in C^\beta_b(\real^d),\\
&\ \ \ \ \ \langle \ell,u\rangle = C_{\ell}u(0)+\int_{\mathbb{R}^d}u(y)-u(0)\;\mu_\ell(dy).
\end{aligned}$$ (The notation, $C^2(0)$ and $C^1(0)$, appears in Definition \[def:PointwiseC1C2\].)
We want to note that the dependence of $\mu$ only on $\ell$ is not a typo. Even though the vector $B_{\ell,\phi}$ and matrix $A_{\ell,\eta}$ clearly depend on the functions $\phi$ and $\eta$, the reader can see in the proof in (\[eqFunctionalsGCP:DefOfMuEll\]) that $\mu_\ell$ does not depend on $\phi$ or $\eta$.
It suffices to prove the representation formula for $u \in C^2_b(\mathbb{R}^d)$ (even if $\beta \neq 2$), as it trivially extends to all of $C^\beta_b(\mathbb{R}^d)$ by approximation. We fix $u \in C^2_b(\mathbb{R}^d)\intersect C^2(0)$. We recall $P_{\phi,\eta,u}$ is defined in (\[eqFunctionalsGCP:DefOfTalyorP\]). Since $P_{\phi,\eta,u} \in C^\beta_b(\mathbb{R}^d)$ for each fixed $\phi,\eta$, we may write $$\begin{aligned}
u & = u-P_{\phi,\eta,u}+P_{\phi,\eta,u},
\end{aligned}$$ and linearity gives $$\begin{aligned}
\langle \ell,u\rangle = \langle \ell, P_{\phi,\eta,u}\rangle+\langle \ell,u-P_{\phi,\eta,u}\rangle
\end{aligned}$$ Let us study each of these two terms. Using the definition of $C_\ell,B_{\ell,\phi},$ and $A_{\ell,\eta}$, we have for $\beta\geq2$ $$\begin{aligned}
\langle \ell,P_{\phi,\eta,u}\rangle & = u(0) \langle \ell,1\rangle + \sum \limits_{i=1}^d\partial_i u(0)\langle \ell, x_i \phi(x)\rangle +\frac{1}{2}\sum \limits_{i,j=1}^d \partial_{ij}^2u(0) \langle \ell, \eta(x)x_ix_j\rangle\\
& = C_{\ell}u(0)+(B_{\ell,\phi},\nabla u(0))+\tfrac{1}{2}\textnormal{tr}(A_{\ell,\eta}D^2u(0)),
\end{aligned}$$ as well as the corresponding expressions in the other cases when $\beta<2$. Next, we analyze the second term in the expression for $\langle \ell,u\rangle$ above, that is $$\begin{aligned}
\langle \ell, u-P_{\phi,\eta,u}\rangle.
\end{aligned}$$ First take the case $\beta \neq 1$. Given $w \in C^\beta_b(\mathbb{R}^d)$, define $\tilde w$ by $$\begin{aligned}
\tilde w(x) & := w(x) \frac{|x|^\beta}{1+|x|^\beta}.
\end{aligned}$$ Observe that since $\beta \neq 1$, the function $\tilde 1 = |x|^\beta(1+|x|^\beta)^{-1}$ belongs to $C^\beta_b(\mathbb{R}^d)$. The linear transformation $w \mapsto \tilde w$ defines a linear functional $\tilde \ell$ via the relation $$\begin{aligned}
\langle \tilde \ell, w\rangle := \langle \ell, \tilde w \rangle.
\end{aligned}$$ This clearly defines a bounded functional on $C^\beta_b(\mathbb{R}^d)$. In fact, however, this functional extends uniquely to a bounded functional in $C^0_b(\mathbb{R}^d)$: since $\tilde w$ is touched from above at $0$ by the function $\|w\|_{L^\infty}\tilde 1$, the GCP guarantees that $$\begin{aligned}
|\langle \tilde \ell, w\rangle| \leq \|w\|_{L^\infty}\langle \ell, \tfrac{|x|^\beta}{1+|x|^\beta} \rangle.
\end{aligned}$$ This shows $\tilde \ell$ is a uniquely defined continuous functional on $C_b^0(\mathbb{R}^d)$ whose norm as a functional on $C_b^0(\mathbb{R}^d)$ is no larger than $\|\ell\| \| \tfrac{|x|^\beta}{1+|x|^\beta}\|_{C^\beta}$. It follows there is a measure $\tilde \mu$ such that $$\begin{aligned}
\label{eqFunctionalsGCP:DefOfMuEll}
\langle \tilde \ell, w\rangle = \int_{\mathbb{R}^d} w(y)\;\tilde \mu(dy).
\end{aligned}$$ Moreover, since $\langle \tilde \ell,w\rangle \geq 0$ whenever $w\geq 0$, $\tilde \mu(dy)$ is a non-negative measure. Now, since $u \in C^2_b(\mathbb{R}^d)$, we have that the function $$\begin{aligned}
w(x) := (u(x)-P_{\phi,\eta,u}(x)) \frac{1+|x|^\beta}{|x|^\beta},
\end{aligned}$$ remains continuous as $x \to 0$, so $w \in C^0_b(\mathbb{R}^d)$ and thus $\langle \tilde \ell,w\rangle$ is well defined. In this case, we have $$\begin{aligned}
\langle \ell, u-P_{\phi,\eta,u} \rangle = \langle \tilde \ell,w \rangle,
\end{aligned}$$ and we obtain the formula $$\begin{aligned}
\langle \ell, u-P_{\phi,\eta,u}\rangle = \int_{\mathbb{R}^d}\left ( u(y)-P_{\phi,\eta,u}(y)\right ) \frac{1+|y|^\beta}{|y|^\beta}\;\tilde \mu(dy).
\end{aligned}$$ In particular, taking $\mu(dy) :=\tfrac{1+|y|^\beta}{|y|^\beta} \tilde \mu(dy)$, it follows that $$\begin{aligned}
\int_{\mathbb{R}^d\setminus\{0\}}\min\{1,|y|^\beta\}\mu(dy) \lesssim \|\ell\| \| \tfrac{|x|^\beta}{1+|x|^\beta}\|_{C^\beta}<\infty,
\end{aligned}$$ and $$\begin{aligned}
\langle \ell, u-P_{\phi,\eta,u}\rangle = \int_{\mathbb{R}^d\setminus\{0\} }u(y)-P_{\phi,\eta,u}(y)\;\mu(dy).
\end{aligned}$$ Revisiting the expression of $\ell$, we have when $\beta\geq2$ $$\begin{aligned}
\langle \ell,u\rangle = C_{\ell}u(0)+(B_{\ell,\phi},\nabla u(0))+\tfrac{1}{2}\textnormal{tr}(A_{\ell,\eta}D^2u(0)) + \int_{\mathbb{R}^d\setminus\{0\} }u(y)-P_{\phi,\eta,u}(y)\;\mu(dy),
\end{aligned}$$ and the analogous formulas follow for the other cases where $\beta \neq 1$, per the change in definition of the function $P_{\phi,\eta,u}$ in (\[eqFunctionalsGCP:DefOfTalyorP\]). It remains to consider the case $\beta = 1$.
Since $|x|$ is not a $C^1$ function, we are going to approximate it by a more regular function. For every small $\varepsilon>0$ we repeat the argument above with $\beta = 1+\varepsilon$ and conclude that for some $\mu_\varepsilon$ we have the formula $$\begin{aligned}
\langle \ell,u\rangle = C_{\ell}u(0)+(B_{\ell,\phi},\nabla u(0))+ \int_{\mathbb{R}^d\setminus\{0\} }u(y)-P_{\phi,\eta,u}(y)\;\mu_{\varepsilon}(dy),
\end{aligned}$$ and this measure $\mu_\varepsilon$ is positive and satisfies the bound $$\begin{aligned}
\int_{\mathbb{R}^d\setminus\{0\}}\min\{1,|y|^\beta\}\mu(dy) \lesssim \|\ell\| \| \tfrac{|x|^{1+\varepsilon}}{1+|x|^{1+\varepsilon}}\|_{C^1}.
\end{aligned}$$ Since $$\begin{aligned}
\sup \limits_{\varepsilon \in (0,1)} \| \tfrac{|x|^{1+\varepsilon}}{1+|x|^{1+\varepsilon}}\|_{C^1} < \infty,
\end{aligned}$$ it follows that the respective finite measures $\{\tilde \mu_{\varepsilon} \}_{\varepsilon \in (0,1)}$ have uniformly bounded mass. Therefore, it is not difficult to show (using $\ell$ to get tightness for the $\tilde \mu_\varepsilon$) that along a subsequence $\varepsilon \to 0$ we can find a limit $\tilde \mu$, and if we let $\mu:= (1+|y|)|y|^{-1}\tilde \mu$ then $$\begin{aligned}
\int_{\mathbb{R}^d\setminus \{0\}}\min\{1,|y|\}\mu(dy)<\infty,
\end{aligned}$$ and again, for any $u\in C^2_b(\mathbb{R}^d)$, $$\begin{aligned}
\langle \ell,u\rangle = C_{\ell}u(0)+(B_{\ell,\phi},\nabla u(0))+ \int_{\mathbb{R}^d\setminus\{0\} }u(y)-P_{\phi,\eta,u}(y)\;\mu(dy),
\end{aligned}$$
We consider the following special functions. For $\delta>0$, define (see for definition of $\psi_{r,R}$) $$\begin{aligned}
\phi_\delta(x) & := \psi_{\delta,1-2\delta}, \label{equation:auxiliary phi_delta}\\
\eta_\delta(x) & := \psi_{\delta,\delta}(x).\label{equation:auxiliary eta_delta}\end{aligned}$$ Note that $\phi_\delta \equiv 1$ inside $B_{1-2\delta}$ and $\phi_\delta \equiv 0$ outside $B_{1-\delta}$, while $\eta_\delta \equiv 1$ inside $B_\delta$ and $\eta_\delta \equiv 0$ outside $B_{2\delta}$. Furthermore, we note that $\delta \leq \delta'$ implies that $\eta_{\delta}\leq \eta_{\delta'}$.
\[lemma:A\_ell and B\_ell existence\] Assume that $\beta\in[0,3)$, $l:C^{\beta}_b(\real^d)\to\real$ is a bounded linear functional with the GCP with respect to $0$, and that $A_{\ell,\eta}$, $B_{\ell,\phi}$ are as in Definition \[definition:auxiliary A\_ell,eta and B\_ell,phi\]. Taking $\eta_\delta$ as in , the limit $$\begin{aligned}
A_{\ell} & := \lim\limits_{\delta \searrow 0} A_{\ell,\eta_\delta},
\end{aligned}$$ exists for all $\beta \in [0,3)$, and $A_\ell\equiv0$ if $\beta<2$. Moreover, if $\phi_\delta$ is as in , there is a sequence $\delta_k\searrow 0$ such that the following limit exists $$\begin{aligned}
B_{\ell} := \lim\limits_{k\to \infty} B_{\ell,\phi_{\delta_k}}.
\end{aligned}$$
Let $\eta_1,\eta_2 \in \mathcal{S}$ and such that $\eta_1\leq \eta_2$. Then for any positive semi-definite $M$ we have $$\begin{aligned}
\tfrac{1}{2}\eta_1(x)(Mx,x)\leq \tfrac{1}{2}\eta_2(x)(Mx,x),\;\textnormal{ with equality at } x=0.
\end{aligned}$$ Since $\ell$ has the GCP with respect to $0$, it follows that $$\begin{aligned}
\langle \ell, \tfrac{1}{2}\eta_1(x)(Mx,x)\rangle \leq \langle \ell,\tfrac{1}{2}\eta_2(x)(Mx,x)\rangle.
\end{aligned}$$ From this monotonicity and the elementary inequality $|\langle \ell, \tfrac{1}{2}\eta(x)(Mx,x)\rangle| \leq C|M|\max_{ij}\|\eta x_ix_j \|_{C^\beta}$ we conclude that the following limit exists for every positive semi-definite $M$ $$\begin{aligned}
\lim \limits_{\delta \searrow 0} \langle \ell, \tfrac{1}{2}\eta_{\delta}(x)(Mx,x)\rangle.
\end{aligned}$$ At the same time, when $\beta<2$ we have $\|\eta_{\delta}x_ix_j\|_{C^\beta} \to 0$ as $\delta \searrow 0$ for all $i,j$, so in this case the limit is zero. Now, given a symmetric matrix $M$, write $M = M^+-M^-$, where both $M^+$ and $M^-$ are positive semi-definite. Then, we also have that the limit $$\begin{aligned}
\lim \limits_{\phi\in\mathcal{S},\eta\searrow 0} \langle \ell, \tfrac{1}{2}\eta(x)(Mx,x)\rangle
\end{aligned}$$ exists for any symmetric matrix $M$. It is clear then that this limit is linear as a function of $M$, and therefore, there is a unique symmetric matrix $A_{\ell}$ such that $$\begin{aligned}
\label{eqn:A sub ell definition}
\tr(A_{\ell} M) = \lim\limits_{\eta \searrow 0}\langle \ell, \frac{1}{2}\eta(x)(Mx,x)\rangle.
\end{aligned}$$ Moreover, this matrix $A_{\ell}$ is positive semi-definite and $A_{\ell,\eta_\delta} \to A_{\ell}$ as $\delta \searrow 0$, and $A_{\ell} = 0$ when $\beta<2$. It remains to analyze the limit of $B_{\ell,\phi_{\delta}}$ along a subsequence. For every $\delta \in (0,1)$ $$\begin{aligned}
(B_{\phi_\delta})_i = \langle \ell,\phi_\delta x_i\rangle.
\end{aligned}$$ Now, recall the estimate from Lemma \[lemma:generalized gradients inherit GCP and weak localization\], which implies $$\begin{aligned}
|\langle \ell,\phi_\delta x_i\rangle| & \leq C(\|\phi_\delta x_i\|_{C^\beta(B_{1/2})}+\|\phi_\delta x_i\|_{L^\infty(\mathbb{R}^d)}).
\end{aligned}$$ A direct computation shows that $$\begin{aligned}
\sup\limits_{0<\delta<1}\|\phi_\delta x_i\|_{C^\beta(B_{1/2})}<\infty.
\end{aligned}$$ It follows that $$\begin{aligned}
\sup \limits_{0<\delta<1} |B_{\phi_\delta}| < \infty,
\end{aligned}$$ and by compactness, there must be a subsequence $\delta_k \to 0$ for which $\{B_{\ell,\phi_{\delta_k}}\}_k$ converges.
\[lemma:Courrege theorem for a linear functional\] Assume that $\beta\in[0,3)$. Let $\ell:C^\beta_b(\mathbb{R}^d)\to \mathbb{R}$ be a bounded linear functional which has the GCP with respect to $0$. For $\beta\geq 2$ and any $u \in C^\beta_b(\mathbb{R}^d)\intersect C^2(0)$, we have the representation $$\begin{aligned}
\langle \ell,u\rangle & = C_{\ell}u(0)+(B_{\ell},\nabla u(0))+\tr(A_{\ell}D^2u(0))+\int_{\mathbb{R}^d}u(y)-u(0)-\chi_{B_1(0)}(\nabla u(0),y)\;\mu_\ell(dy).
\end{aligned}$$ This representation is unique. This means that if there were $\tilde C$, $\tilde B$, $\tilde A$ and $\tilde \mu$ a measure in $\mathbb{R}^d\setminus \{0\}$ all such that $$\begin{aligned}
\langle \ell,u\rangle & = \tilde Cu(0)+(\tilde B,\nabla u(0))+\tr(\tilde AD^2u(0))+\int_{\mathbb{R}^d}u(y)-u(0)-\chi_{B_1(0)}(\nabla u(0),y)\;\tilde \mu(dy).
\end{aligned}$$ for all $u$, then $\tilde C = C_\ell$, $\tilde B = B_\ell$, $\tilde A = A_\ell$, and $\tilde \mu = \mu_\ell$. Furthermore, if $\beta<2$ and $u\in C^\beta(\real^d)\intersect C^1(0)$, then $A_{\ell}=0$, and if $\beta<1$, then $B_{\ell} = 0$ and the integrand on the right can be replaced with just $u(y)-u(0)$.
Let $\delta,\delta'\in(0,1)$. Applying Lemma \[lemma:preliminary Courrege theorem for a linear functional\] with the functions $\phi_{\delta}$ and $\eta_{\delta'}$, $$\begin{aligned}
\langle \ell,u\rangle & = C_{\ell}u(0)+(B_{\ell,\phi_{\delta}},\nabla u(0))+\tr(A_{\ell,\eta_{\delta'}}D^2u(0))+\int_{\mathbb{R}^d}u(y)-P_{\phi_{\delta},\eta_{\delta'},u}(y)\;\mu_\ell(dy).
\end{aligned}$$ Since $\min\{1,|y|^\beta\}$ is integrable against $\mu_\ell$, it follows that $$\begin{aligned}
\lim \limits_{\delta' \searrow 0}\int_{\mathbb{R}^d\setminus\{0\}} \eta_{\delta'}(y)(D^2u(0)y,y)\;\mu_\ell(dy) = 0.
\end{aligned}$$ Therefore, $$\begin{aligned}
\lim \limits_{\delta' \searrow 0}\int_{\mathbb{R}^d\setminus\{0\}} u(y)-P_{\phi_{\delta},\eta_{\delta'},u}(y)\;\mu_\ell(dy) = \int_{\mathbb{R}^d\setminus\{0\}} u(y)-u(0)-\phi_{\delta}(y)(\nabla u(0),y)\;\mu_\ell(dy).
\end{aligned}$$ Then, thanks to Lemma \[lemma:A\_ell and B\_ell existence\], the formula for $\langle \ell,u\rangle$ becomes (for every fixed $\delta\in (0,1)$) $$\begin{aligned}
\langle \ell,u\rangle = C_{\ell}u(0)+(B_{\ell,\phi_\delta},\nabla u(0))+\tr(A_{\ell}D^2u(0))+\int_{\mathbb{R}^d\setminus \{0\}} u(y)-u(0)-\phi(y)(\nabla u(0),y)\;\mu_\ell(dy).
\end{aligned}$$ Now, let $\delta_k\searrow 0$ be chosen so that $B_{\ell \phi_{\delta_k}}\to B_{\ell}$ (which can be done thanks to Lemma \[lemma:A\_ell and B\_ell existence\]). From the definition of $\phi_{\delta}$, we have that $$\begin{aligned}
u(y)-u(0)-\phi_{\delta_{k}}(y)(\nabla u(0),y) \textnormal{ is monotone in } k.
\end{aligned}$$ At the same time, for every $y\in \mathbb{R}^d$ we have $$\begin{aligned}
\lim \limits_{k\to\infty} \phi_{\delta_k}(y) = \chi_{B_1(0)}.
\end{aligned}$$ Therefore, by monotone convergence we have $$\begin{aligned}
\lim\limits_{k\to \infty} \int_{\mathbb{R}^d\setminus \{0\} } u(y)-u(0)-\phi_{\delta_k}(y)(\nabla u(0),y) \;\mu_\ell(dy) = \int_{\mathbb{R}^d\setminus \{0\} } u-u(0)-\chi_{B_1}(y)(\nabla u(0),y) \;\mu_\ell(dy).
\end{aligned}$$ From where it follows that $$\begin{aligned}
\langle \ell,u\rangle = C_{\ell}u(0)+(B_{\ell},\nabla u(0))+\tr(A_{\ell}D^2u(0))+\int_{\mathbb{R}^d\setminus \{0\}} u(y)-u(0)-\chi_{B_1(0)}(y)(\nabla u(0),y)\;\mu_\ell(dy),
\end{aligned}$$ as claimed. It remains to prove the uniqueness part. For this, it is enough to show that if for all $u$ we have $\langle \ell,u \rangle = 0$ and $$\begin{aligned}
\langle \ell,u\rangle & = C_{\ell}u(0)+(B_{\ell},\nabla u(0))+\tr(A_{\ell}D^2u(0))+\int_{\mathbb{R}^d}u(y)-u(0)-\chi_{B_1(0)}(\nabla u(0),y)\;\mu_\ell(dy),
\end{aligned}$$ then $C_\ell = 0, B_\ell = 0, A_\ell = 0$ $\mu_\ell = 0$. First, consider any $u$ with compact support which is disjoint from $\{0\}$, for such a $u$ we have $$\begin{aligned}
\langle \ell,u\rangle & = \int_{\mathbb{R}^d}u(y)\;\mu_\ell(dy),
\end{aligned}$$ Since $u$ can be any function with compact support in $\mathbb{R}^d\setminus \{0\}$, it follows that $\mu_\ell = 0$. Evaluating $\ell$ at the function $u(x)\equiv 1$ we obtain $C_\ell = 0$. Lastly, evaluating $\ell$ at all of the functions of the form $(x,e)$, $e\in\mathbb{R}^d$ and $(Mx,x)$, $M$ symmetric matrix, we see that $B_\ell \cdot e= 0$ for any vector $e$ and $\tr(AM) = 0$ for any symmetric matrix $M$, so that $B_\ell = 0$ and $A_\ell = 0$.
By a simple change of variables, Lemma \[lemma:Courrege theorem for a linear functional\] implies the following.
\[corollary:Courrege theorem for a linear functional, general base point\] Assume that $x$ is fixed, $\beta\in[0,3)$, and let $\ell:C^\beta_b(\mathbb{R}^d)\to \mathbb{R}$ be a bounded linear functional which has the GCP with respect to $x$. For $\beta\geq 2$ any $u \in C^\beta_b(\mathbb{R}^d)\intersect C^2(x)$ we have the representation $$\begin{aligned}
\langle \ell,u\rangle & = C_{\ell}u(x)+(B_{\ell},\nabla u(x))+\tr(A_{\ell}D^2u(x))+\int_{\mathbb{R}^d}u(x+y)-u(x)-\chi_{B_1(0)}(\nabla u(x),y)\;\mu_\ell(dy).
\end{aligned}$$ As before, this representation is unique, and when $\beta<2$ and $u\in C^\beta_b(\real^d)\intersect C^1(x)$, we have $A_{\ell} = 0$, while for $\beta<1$ we have $B_{\ell} = 0$ and the integrand can be replaced with just $u(x+y)-u(x)$.
Proofs of Theorems \[theorem:MinMax Euclidean\] and \[theorem:MinMax Translation Invariant\] {#sec:TheoremsWithoutWhitney}
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With Lemmas \[lemma:generalized gradients inherit GCP and weak localization\] and \[lemma:Courrege theorem for a linear functional\] and Corollary \[corollary:Courrege theorem for a linear functional, general base point\] in hand, we can now prove Theorems \[theorem:MinMax Translation Invariant\] and \[theorem:MinMax Euclidean\].
Consider the functional, $$\begin{aligned}
F(u) := I(u,0).
\end{aligned}$$ Now, by Theorem \[theorem:MinMax for scalar functionals\], we have that $$\begin{aligned}
F(u) = \min_a\max_b \{ f_{ab}+\langle \ell_{ab},u\rangle \}.
\end{aligned}$$ By Lemma \[lemma:generalized gradients inherit GCP and weak localization\], each $\ell_{ab}$ is a linear operator having the GCP with respect to $0$, in which case Lemma \[lemma:Courrege theorem for a linear functional\] says that for $u\in C^\beta_b(\real^d)\intersect C^2(0)$, $$\begin{aligned}
\langle \ell_{ab},u\rangle & = \tr(A_{ab}D^2u(0))+B_{ab}\cdot \nabla u(0)+C_{ab}u(0)+\int_{\mathbb{R}^d}u(y)-u(0)-\chi_{B_1}(0)(\nabla u(0),y)\;\mu_{ab}(dy).
\end{aligned}$$ The translation invariance of $I$ boils down to the identity $$\begin{aligned}
I(u,x) = F(\tau_x u).
\end{aligned}$$ Therefore, $$\begin{aligned}
I(u,x) = \min\limits_{a} \max\limits_{b} \{ f_{ab} + \langle \ell_{ab},\tau_x u\rangle\}
\end{aligned}$$ However, $\langle \ell_{ab},\tau_x u\rangle$ has a simple expression, namely $$\begin{aligned}
\tr(A_{ab}D^2u(x))+B_{ab}\cdot \nabla u(x)+C_{ab}u(x)+\int_{\mathbb{R}^d}u(x+y)-u(x)-\ind_{B_1(0)}\nabla u(x)\cdot y\;\mu_{ab}(dy),
\end{aligned}$$ and this proves the theorem.
The beginning of the proof is similar to that of the previous one. For each $x\in\mathbb{R}^d$, define a functional $$\begin{aligned}
F_x(u) := I(u,x),\;\;\forall\; u\in C^\beta_b(\mathbb{R}^d).
\end{aligned}$$ Applying Theorem \[theorem:MinMax for scalar functionals\], it follows that $$\begin{aligned}
F_x(u) := \min\limits_{v\in C^\beta_b(\mathbb{R}^d)} \max \limits_{ \ell \in \partial F_x} \{ F_x(v)+\langle \ell,u-v\rangle \}.
\end{aligned}$$ Applying Lemma \[lemma:generalized gradients inherit GCP and weak localization\], it follows that for any $\ell \in \partial F_x $ $$\begin{aligned}
& \langle \ell,u\rangle = Cu(x)+(B,\nabla u(x))+\tr(AD^2u(x))+\int_{\mathbb{R}^d} u(x+y)-u(x)-\chi_{B_1(0)}(\nabla u(x),y)\;\mu(dy).
\end{aligned}$$ Since $F_x(v) = I(v,x)$ this proves the Theorem, with $\mathcal{K}(I)_x = \{ L \mid L(u) = \langle \ell,u\rangle \textnormal{ for } \ell \in \partial F_x\}$ .
It is worthwhile to compare the proof of Theorem \[theorem:MinMax Euclidean\] above to the much longer and complicated one given in [@GuSc-2016MinMaxNonlocalarXiv]. The simplicity here is made possible by the use of a mean value theorem for Lipschitz functionals (Theorem \[theorem:Lebourg\]) in the infinite dimensional setting, which suffices to prove Theorem \[theorem:MinMax Euclidean\] as it involves a min-max formula in terms of linear functionals in $C^2_b$ and not linear operators from $C^2_b(\mathbb{R}^d)$ to $C^0_b(\mathbb{R}^d)$. The more complicated method from [@GuSc-2016MinMaxNonlocalarXiv] is however still of value, specially if one is interested in obtaining a min-max representation in terms of a family of linear operators from $C^2_b$ to $C_b^0$. Moreover, it is by adapting the method from [@GuSc-2016MinMaxNonlocalarXiv] that we are able to prove Theorem \[theorem:MinMax Euclidean ver2\], after analyzing the spatial properties of the finite dimensional approximations (see in Section \[section:Analysis of finite dimensional approximations\]).
Finite Dimensional Approximations to $C^\beta_b(\mathbb{R}^d)$ {#section:Finite Dimensional Approximations}
==============================================================
Graph approximations
--------------------
The following nested family of sets will be important in what follows $$\begin{aligned}
G_n & := 2^{-n} \mathbb{Z}^d. \end{aligned}$$ It will be convenient to write $h_n := 2^{-n}$. Then, $h_n$ represents the maximum possible distance between $x\in \mathbb{R}^d$ and $G_n$, and in particular $\textnormal{dist}(x,G_n)\leq h_n$ for all $x \in \mathbb{R}^d$. Observe that $$\begin{aligned}
G_1 \subset G_2 \subset G_3 \ldots, \end{aligned}$$ and note also the union of the sets $G_n$ is dense in $\mathbb{R}^d$.
\[definition:discrete function spaces\] We consider the following function spaces $$\begin{aligned}
C(G_n) & := \{ u:G_n \to \mathbb{R}^d\},\\
C_*(G_n) & := \{ u \in C(G_n) \mid u(x) = 0 \textnormal{ if } x\not\in [-2^n,2^n]^d\}.
\end{aligned}$$ These spaces will be related to $C^\beta_b(\mathbb{R}^d)$ by restriction, which we think of as a map denoted by $T_n$ and given by $$\begin{aligned}
T_n:C^\beta_b(\mathbb{R}^d)\to C(G_n),\;\; T_nu := u_{\mid G_n}.
\end{aligned}$$
\[remark:C star G\_n is finite dimensional\] The space $C_*(G_n)$ is a finite dimensional vector space.
Cube decomposition and partition of unity
-----------------------------------------
In this section we shall apply the Whitney theory to extend functions in a grid $r \mathbb{Z}^d$ to all of $\mathbb{R}^d$. Since it is in our interest for the Whitney construction to be compatible with the grid structure, we shall do the usual cube decomposition making sure the resulting family of cubes is invariant under translations by vectors in $r \mathbb{Z}^d$, the resulting construction is illustrated in Figure \[figure:periodic cube decomposition\].
\[lemma:cube decomposition of the complement of the lattice\] For every $r>0$, there exists a collection of cubes $\{Q_k\}_k$ such that
1. The cubes $\{Q_k\}_k$ have pairwise disjoint interiors.
2. The cubes $\{Q_k\}_k$ cover $\mathbb{R}^d \setminus r\mathbb{Z}^d$
3. $c_1 \textnormal{diam}(Q_k)\leq \textnormal{dist}(Q_k,\mathbb{Z}^d) \leq c_2 \textnormal{diam}(Q_k).$
4. For every $h \in r\mathbb{Z}^d$, there is a bijection $\sigma_h:\mathbb{N}\to\mathbb{N}$ such that $Q_k+h = Q_{\sigma_h k}$ for every $k\in\mathbb{N}$.
 \[figure:periodic cube decomposition\]
We consider the case $r=1$, once the collection of cubes is $\{Q_k\}_k$ obtained in this case, the general case follows via scaling by taking the family $\{rQ_k\}_k$ .
Consider the cube $Q_0 = [-1/2,1/2]^d$, let $\mathcal{M}_0$ denote the family of $2^d$ equal size cubes obtained from $Q_0$ by bisecting each of its sides. Let $\mathcal{M}_k$ denote the family of cubes obtained from applying this same procedure to each of the cubes in $\mathcal{M}_{k-1}$. Note that the side length of each cube in $\mathcal{M}_k$ is just $2^{-k}$. Now, we construct a family $\mathcal{F}_0$ as follows, with $R_k := \{ 2\sqrt{d}2^{-k}\leq |x|\leq 2\sqrt{d}2^{-(k-1)}\}$ for each $k\in \mathbb{N}$, then $$\begin{aligned}
\mathcal{F}_0 := \bigcup \limits_k \{ Q \in \mathcal{M}_k \; : \; Q\cap R_k \neq \emptyset \}.
\end{aligned}$$ Observe that if $Q \in \mathcal{F}_0$ then $Q\in\mathcal{M}_k$ for some $k$ and there is some $x\in Q$ such that $2\sqrt{d}2^{-k}\leq |x|$ and $|x|\leq 2\sqrt{d} 2^{-(k-1)}$. This means, $$\begin{aligned}
\sqrt{d}2^{-k}= 2\sqrt{d}2^{-k}-\textnormal{diam}(Q) \leq \textnormal{dist}(Q,0) \leq 2\sqrt{d}2^{-k},
\end{aligned}$$ and since $\textnormal{diam}(Q) = \sqrt{d}2^{-k}$, we conclude that $$\begin{aligned}
\textnormal{diam}(Q) \leq \textnormal{dist}(Q,0) \leq 4\textnormal{diam}(Q) \;\;\forall\;Q\in\mathcal{F}_0.
\end{aligned}$$ On the other hand, we have that $$\begin{aligned}
\bigcup \limits_{Q \in \mathcal{F}_0} Q = [-1/2,1/2]^d \setminus \{0\}.
\end{aligned}$$
If $\mathcal{F}$ denotes the subfamily of maximal cubes in $\mathcal{F}_0$, it follows that: the union of these cubes is still $[-1/2,1/2]^d \setminus \{0\}$, the inequality $\textnormal{diam}(Q) \leq \textnormal{dist}(Q,0) \leq 4\textnormal{diam}(Q)$ holds for each $Q\in\mathcal{F}$, and the cubes have pairwise disjoint interiors.
Denote by $\{Q_k\}_k$ an enumeration of the family of cubes of the form $Q+z$, where $Q\in \mathcal{F}$ and $z \in \mathbb{Z}^d$. It is clear that $\{Q_k\}_k$ covers all of $\mathbb{R}^d\setminus \mathbb{Z}^d$ and that these cubes have pairwise disjoint interiors. Furthermore, for any $h\in \mathbb{Z}^d$ the map $Q \to Q+h$ gives a bijection of the set $\{Q_k\}_k$ onto itself, therefore one can represent it via a bijection $\sigma_h:\mathbb{N}\to\mathbb{N}$ so that $Q_k +h = Q_{\sigma_h k}$. Last but not least, as each cube of the form $Q+z$ is closest to $z$ than to any other point in $\mathbb{Z}^d$, property (3) follows from the respectively inequality for the family $\mathcal{F}$.
\[remark:cube centers and hat ynk\] We apply Lemma \[lemma:cube decomposition of the complement of the lattice\] with $r=2^{-n}$, for some $n\in\mathbb{N}$, and for the rest of the section shall refer to the resulting cubes as $\{Q_{n,k}\}_k$.
Furthermore, for every $n$ and $k$, we will denote the center of $Q_{n,k}$ by $y_{n,k}$, and for each $n$ and $k$ we will denote by $\hat y_{n,k}$ the unique point in $G_n$ such that $$\begin{aligned}
\textnormal{dist}(y_{n,k},G_n) = |y_{n,k}-\hat y_{n,k}|,
\end{aligned}$$ (note that there is only one since by construction not a single center $y_{n,k}$ lies at equidistance to two different lattice points).
In particular, for each of the bijections $\sigma_h : \mathbb{N}\to \mathbb{N}$ from Lemma \[lemma:cube decomposition of the complement of the lattice\] we have $$\begin{aligned}
y_{n,k}+h = y_{n,\sigma_h k}, \; \hat y_{n,k} +h = \hat y_{n,\sigma_h k},\;\forall\;n,k.
\end{aligned}$$
\[remark:maximum number of overlapping cubes\] In all what follows, given a cube $Q$, we shall denote by $Q^*$ the cube with same center as $Q$ but whose sides are increased by a factor of $9/8$. Observe that for every $n$ and $k$, we have $Q_{n,k}^* \subset \mathbb{R}^d\setminus 2^{2-n}\mathbb{Z}^d$, and that any given $x$ lies in at most some number $C(d)$ of the cubes $Q_k^*$.
\[proposition:partition of unity properties\] For every $n$, there is a family of functions $\phi_{n,k}(x)$ such that
1. $0\leq \phi_{n,k}(x)\leq 1$ for every $k$ and $\phi_{n,k} \equiv 0$ outside $Q_{n,k}^*$ (using the notation in Remark \[remark:maximum number of overlapping cubes\])
2. $\sum_k \phi_{n,k}(x) =1 $ for every $x \in \mathbb{R}^d\setminus G_n$.
3. There is a constant $C$, independent of $n$ and $k$, such that $$\begin{aligned}
|\nabla^{i}\phi_{n,k}(x)| \leq \frac{C}{\diam(Q_{n,k})^i}.
\end{aligned}$$
4. For every $z \in G_n$, we have $$\begin{aligned}
\phi_{n,k}(x-z) = \phi_{n,\sigma_zk}(x),\;\;\forall\;k,\;x,
\end{aligned}$$ where $\sigma_z$ are the bijections introduced above.
Fix a $C^\infty$ function $\phi$ such that $$\begin{aligned}
& 0\leq \phi \leq 1,\\
& \phi \equiv 1 \textnormal{ in } Q_0 = [-1/2,1/2]^d,\\
& \phi \equiv 0 \textnormal{ outside } Q_0^*.
\end{aligned}$$ Let $\ell(Q)$ denote the common length for the sides of $Q_{n,k}$, and with $y_{n,k}$ as given in Remark \[remark:cube centers and hat ynk\] we define $$\begin{aligned}
\tilde \phi_{n,k} := \phi \left ( \frac{x-y_{n,k}}{\ell_{n,k}}\right ).
\end{aligned}$$ Consider the function $$\begin{aligned}
\Phi(x) = \sum \limits_{k} \tilde \phi_{n,k}(x).
\end{aligned}$$ It follows from Remark \[remark:maximum number of overlapping cubes\] that given any $x$ ,at most $C(d)$ of the terms appearing in the sum are non-zero in a neighborhood of $x$, and therefore $\Phi$ is a smooth function. Then, define $$\begin{aligned}
\phi_{n,k}(x) := \tilde \phi_{n,k}(x) \Phi(x)^{-1}.
\end{aligned}$$ It is clear that the functions $\{\phi_{n,k}\}_{k}$ satisfy properties (1) and (2). Property (3) follows easily from the chain rule, using the differentiability of the function $\phi$. It remains to check property (4), let $z \in G_n$, then $$\begin{aligned}
\phi_{n,k}(x-z) & = \phi \left ( \frac{x-(y_{n,k}+z)}{\ell(Q_{n,k})}\right )\Phi(x-z)^{-1}\\
& = \phi \left ( \frac{x-y_{n,\sigma_z k}}{\ell(Q_{n,\sigma_z k})}\right ) \Phi(x)^{-1} = \phi_{n,\sigma_z k}(x),
\end{aligned}$$ where we used that $\ell(Q_{n,k}) = \ell(Q_{n,\sigma_z k})$, which follows clearly from the definition of $\sigma_z$.
Discrete derivatives
--------------------
In what follows, it will be in our interest to approximate the first and second derivatives of a function $u \in C^\beta_b(\mathbb{R}^d)$ (see for our convention regarding the meaning of $C^\beta_b$) at a point $x\in G_n$ using only information about the values of $u$ on $G_n$. This motivates the following two definitions (we recall that $h_n = 2^{-n}$).
\[definition:finite difference operators 1\] The vector $(\nabla_n)^1u(x)$ is defined via the system of equations ($k=1,\ldots,d$) $$\begin{aligned}
(\nabla_n)^1u(x),e_k) := (2h_n)^{-1}[u(x+h_ne_k)-u(x-h_ne_k)]
\end{aligned}$$
\[definition:finite difference operators 2\] The matrix $(\nabla_n)^2u(x)$ is defined via the system of equations ($k,\ell=1,\ldots,d$), $$\begin{aligned}
( (\nabla_n)^2u(x) e_k,e_\ell) := h_n^{-2 }\left [ u(x+h_n e_k+h_n e_\ell) - u(x+h_n e_k) - u(x+h_n e_\ell) + u(x) \right ]
\end{aligned}$$
\[remark:finite difference translation invariance\] From the definition it is clear that these discrete derivatives commute with translations with respect to a vector $z\in G_n$. That is, given a function $u$ and $z\in G_n$ then for every $x\in G_n$ we have $$\begin{aligned}
((\nabla_n)^1 \tau_z u)(x) = ((\nabla_n)^1 u)(x+z)
\end{aligned}$$
Depending on how regular the function $u$ is, these discrete derivative operators enjoy quantitative “continuity estimates” as functions on $G_n$. An important point being that these estimates are uniform in $n$ once $u$ is fixed.
\[proposition:discrete derivatives converge to real derivatives\] There is a universal constant $C$ such that for $u \in C^{\beta}_b(\mathbb{R}^d)$ and $x \in G_n$, $$\begin{aligned}
& | (\nabla_n)^1u(x)-\nabla u(x)| \leq C\|u\|_{C^{\beta}}h_n^{\beta-1},\;\textnormal{ if } \beta \in [1,2],\\
& | (\nabla_n)^2u(x)-D^2 u(x)| \leq C\|u\|_{C^{\beta}}h_n^{\beta-2},\;\textnormal{ if } \beta \in [2,3).
\end{aligned}$$
See appendix.
\[proposition:discrete derivatives regularity\] Fix $u \in C^{\beta}_b(\mathbb{R}^d)$. Then, given $x_1,x_2 \in G_n$, we have $$\begin{aligned}
| u(x_1)-u(x_2)| & \leq C\|u\|_{C^{\beta}}|x_1-x_2|^{\beta},\;\textnormal{ if } \beta \in [0,1],\\
| (\nabla_n)^1u(x_1)-(\nabla_n)^1u(x_2)| & \leq C\|u\|_{C^{\beta}}|x_1-x_2|^{\beta-1},\;\textnormal{ if } \beta \in [1,2],\\
| (\nabla_n)^2u(x_1)-(\nabla_n)^2 u(x_2)| & \leq C\|u\|_{C^{\beta}}|x_1-x_2|^{\beta-2},\;\textnormal{ if } \beta \in [2,3].
\end{aligned}$$
See appendix.
The Whitney Extension and Projection operators.
-----------------------------------------------
\[definition:interpolating polynomials\]
$$\begin{aligned}
p^\beta_{u,k}(x) := \left \{ \begin{array}{ll}
u (\hat y_{n,k}) & \textnormal{ if } \beta\in [0,1)\\
u (\hat y_{n,k})+(\nabla^1_n u(\hat y_{n,k}),x-\hat y_{n,k}) & \textnormal{ if } \beta \in [1,2)\\
u (\hat y_{n,k})+(\nabla^1_n u(\hat y_{n,k}),x-\hat y_{n,k})+\tfrac{1}{2}\left ( \nabla^2_n u(\hat y_{n,k})(x-\hat y_{n,k}),(x-\hat y_{n,k})\right ) & \textnormal{ if } \beta \in [2,3)
\end{array}\right.\end{aligned}$$
We are now ready to define the Whitney extension operator. $$\begin{aligned}
E^\beta_n(u,x) := \left \{ \begin{array}{ll}
u(x) & \textnormal{ if } x\in G_n,\\
\sum \limits_{k} p^\beta_{u,k}(x)\phi_{n,k}(x) & \textnormal{ if } x\not\in G_n.
\end{array}\right.\end{aligned}$$
The projector operator $\pi_n^\beta:C^\beta_b(\mathbb{R}^d)\to C^\beta_b(\mathbb{R}^d)$ is given by $$\begin{aligned}
\label{equation:pi_n^beta definition}
\pi_n^\beta := E_n^\beta \circ T_n,\end{aligned}$$ where we recall that $T_nu = u_{\mid G_n}$ (Definition \[definition:discrete function spaces\]).
\[theorem:Whitney Extension Is Bounded\] There is a constant $C$ such that for any $n$ and any $u\in C^\beta_b(\mathbb{R}^d)$ we have $$\begin{aligned}
\|\pi_n^\beta u \|_{C^\beta(\mathbb{R}^d)} \leq C\|u\|_{C^\beta(\mathbb{R}^d)}.
\end{aligned}$$
This follows arguing exactly as in [@Stei-71 Chapter VI, Theorem 3 and 4], making use of the regularity estimates in Proposition \[proposition:discrete derivatives regularity\]. Since this is a standard argument, we omit the details.
\[proposition:Whitney extension translation invariance\] Let $z \in G_n$ and $u \in C^\beta_b$, then. $$\begin{aligned}
\pi_n^\beta(\tau_{z}u) = \tau_{z} \pi_n^\beta (u).
\end{aligned}$$
Let us show that $\pi_n^\beta(\tau_{z}u)(x) = \tau_{z} \pi_n^\beta (u)(x)$ for every $x \in \mathbb{R}^d$ and $z\in G_n$. Note that if $x\in G_n$ then the equality is trivial, so let us take $x\in \mathbb{R}^d\setminus G_n$ and $z \in G_n$, then we have $$\begin{aligned}
\pi_n^\beta(\tau_{z}u)(x) = \sum \limits_{k} p^\beta_{\tau_{z}u,k}(x)\phi_{n,k}(x).
\end{aligned}$$ Furthermore, it is not difficult to check that (see Remark \[remark:finite difference translation invariance\]) $$\begin{aligned}
p^\beta_{\tau_{z}u,k}(x) = p^\beta_{u,\sigma_z k}(x+z),
\end{aligned}$$ while part (4) of Proposition \[proposition:partition of unity properties\] implies that $$\begin{aligned}
\phi_{n,k}(x) = \phi_{n,\sigma_z k}(x+z).
\end{aligned}$$ From these two identities we conclude that $$\begin{aligned}
\pi_n^\beta(\tau_{z}u)(x) = \sum \limits_{k} p^\beta_{u,\sigma_z k}(x+z)\phi_{n,\sigma_z k}(x+z) = \sum \limits_{k} p^\beta_{u,k}(x+z)\phi_{n,k}(x+z) = \tau_{z} \pi_n^\beta (u)(x),
\end{aligned}$$ where we used that $\sigma_z$ is bijective, this proves the proposition.
\[remark:approximation of directions\] Given $\varepsilon \in (0,1)$ there is a $C>1$ such that for every $n\in \mathbb{N}$, $x_0 \in G_n$, and unit vector $x_* \in \mathbb{R}^d$ there is some $x_1 \in G_n$ and $s>0$ such that $$\begin{aligned}
|s x_* -(x_1-x_0)| \leq h_n,\; C^{-1}h_n^{\varepsilon}\leq |x_1-x_0| \leq C h_n^{\varepsilon}.
\end{aligned}$$ Indeed, this follows from the fact that $h_n^{\varepsilon} x_* \in [-h_n^{\varepsilon},h_n^\varepsilon]^d$ and that $[-h_n^{\varepsilon},h_n^\varepsilon]^d \cap (G_n-x_0)$ is a $h_n$-net in $[-h_n^{\varepsilon},h_n^\varepsilon]^d$, so there is $x_1 \in [-h_n^{\varepsilon},h_n^\varepsilon]^d \cap (G_n-x_0)$ such that $|h_n^\varepsilon x_*-(x_1-x_0)| \leq h_n$. Then, the inequalities for $|x_1-x_0|$ follow from two applications of the triangle inequality and the fact that $\varepsilon<1$ and $h_n \leq 1/2$ for all $n\geq 1$.
\[proposition:discrete derivative estimate at a local minimum\] Let $w \in C^{\beta}_b(\mathbb{R}^d)$ be such that $w(x)\geq 0$ for every $x\in G_n$ and such that $w(x_0) = 0$ at some $x_0 \in G_n$. Then, there is a universal $C$ such that $$\begin{aligned}
|\nabla \pi^\beta_n w(x_0)| & \leq C\|w\|_{C^{\beta}} h_n^{\min\{2,\beta\}-1},\;\textnormal{ if } \beta \geq 1,\\
|(\nabla^2 \pi^\beta_n w(x_0))_-| & \leq C\|w\|_{C^{\beta}} h_n^{(\min\{3,\beta\}-2)/2},\;\textnormal{ if } \beta \geq 2.
\end{aligned}$$ Here, for a given symmetric matrix $D$, $D_{-}$ denotes it’s negative part.
Fix any $x\in G_n$. Thanks to Proposition \[proposition:discrete derivatives converge to real derivatives\] and the fact that $|x-x_0|\geq h_n$ we have $$\begin{aligned}
|w(x)-w(x_0)- (\nabla \pi^\beta_n w(x_0),x-x_0)| \leq C\|w\|_{C^{\beta}}|x-x_0|^{\min\{2,\beta\}}.
\end{aligned}$$ Since $w(x_0)=0$, and $w(x)\geq 0$ by assumption, $$\begin{aligned}
0 \leq (\nabla \pi^\beta_n w(x_0),x-x_0) + C\|w\|_{C^{\beta}}|x-x_0|^{\min\{2,\beta\}}.
\end{aligned}$$ It is easy to see there is some $x_1 \in G_n$ such that $|x_0-x_1| = h_n$ and $$\begin{aligned}
(\nabla \pi^\beta_n w(x_0),x_1-x_0) = -|\nabla \pi^\beta_n w(x_0)|_{\ell^\infty}|x_1-x_0|,
\end{aligned}$$ and therefore, $$\begin{aligned}
(\nabla \pi^\beta_n w(x_0),x_1-x_0) \leq -C_d^{-1}|\nabla \pi^\beta_n w(x_0)||x_1-x_0|.
\end{aligned}$$ Combining these inequalities and recalling Theorem \[theorem:Whitney Extension Is Bounded\] it follows that $$\begin{aligned}
|\nabla \pi^\beta_n w(x_0)|\leq C\|w\|_{C^{\beta}}h_n^{\min\{2,\beta\}-1}.
\end{aligned}$$ This proves the estimate for the gradient when $\beta\geq 1$. Now assume $\beta \geq 2$, the beginning of the argument in this case goes along similar lines. For any $x\in G_n$ we have that $$\begin{aligned}
|w(x)-w(x_0)- (\nabla \pi^\beta_n w(x_0),x-x_0)-\tfrac{1}{2}(\nabla^2 \pi^\beta_n w(x_0)(x-x_0),x-x_0 )| \leq C\|w\|_{C^{\beta}}|x-x_0|^{\min\{3,\beta\}},
\end{aligned}$$ where we have once again used Theorem \[theorem:Whitney Extension Is Bounded\]. Thus, since $w(x_0) = 0$ and $w(x)\geq 0$ for $x \in G_n$, $$\begin{aligned}
(\nabla \pi^\beta_n w(x_0),x-x_0)+\tfrac{1}{2}(\nabla^2 \pi^\beta_n w(x_0)(x-x_0),x-x_0 ) + C\|w\|_{C^{\beta}}|x-x_0|^{\min\{3,\beta\}} \geq 0.
\end{aligned}$$ Now, since we are on a lattice, it is obvious that for any $x\in G_n$ we have that $x' := 2x_0-x \in G_n$. In this case we can add up the inequalities for $x$ and $x'$, and conclude that $$\begin{aligned}
& (\nabla \pi^\beta_n w(x_0),x-x_0)+\tfrac{1}{2}(\nabla^2 \pi^\beta_n w(x_0)(x-x_0),x-x_0 )\\
& +(\nabla \pi^\beta_n w(x_0),x'-x_0)+\tfrac{1}{2}(\nabla^2 \pi^\beta_n w(x_0)(x'-x_0),x'-x_0 ) + 2C\|w\|_{C^{\beta}}|x-x_0|^{\min\{3,\beta\}} \geq 0.
\end{aligned}$$ Since $x'-x_0 = -(x-x_0)$, we conclude that $$\begin{aligned}
& (\nabla^2 \pi^\beta_n w(x_0)(x-x_0),x-x_0 )+ 2C\|w\|_{C^{\beta}}|x-x_0|^{\min\{3,\beta\}} \geq 0,\;\forall\;x\in G_n.
\end{aligned}$$ Let $x_* \in \mathbb{R}^d$ be a unit vector such that $$\begin{aligned}
-(\nabla^2 \pi^\beta_n w(x_0)x_*,x_* ) = |(\nabla^2 \pi^\beta_n w(x_0))_-|
\end{aligned}$$ According to Remark \[remark:approximation of directions\], there is $x_1 \in G_n$ and $s>0$ such that $$\begin{aligned}
|sx_*-(x_1-x_0)| \leq h_n,\;\; C^{-1} h_n^{\varepsilon}\leq |x_1-x_0| \leq Ch_n^{\varepsilon}.
\end{aligned}$$ For this $x_1$ we have $$\begin{aligned}
|(\nabla^2 \pi^\beta_n w(x_0))_-|s^2 & = -(\nabla^2 \pi^\beta_n w(x_0)x_*,x_*)s^2 \\
& \leq -(\nabla^2 \pi^\beta_n w(x_0)(x_1-x_0),x_1-x_0 ) + C\|w\|_{C^\beta} |sx_*-(x_1-x_0)|.
\end{aligned}$$ This, together with the previous step, shows that $$\begin{aligned}
C^{-2}|(\nabla^2 \pi^\beta_n w(x_0))_-|(h_n^{\varepsilon})^2 \leq 2C\|w\|_{C^\beta}h_n^{\min\{3,\beta\}\varepsilon} + C\|w\|_{C^\beta} h_n,
\end{aligned}$$ again having used Theorem \[theorem:Whitney Extension Is Bounded\]. Simplifying, this becomes $$\begin{aligned}
|(\nabla^2 \pi^\beta_n w(x_0))_-| \leq C\|w\|_{C^{\beta}}(h_n^{(\min\{3,\beta\}-2)\varepsilon}+h_n^{1-\varepsilon}).
\end{aligned}$$ Choosing $\varepsilon = 1/2$, and noting $\min\{3,\beta\}-2)\leq 1$, we conclude that $$\begin{aligned}
|(\nabla^2 \pi^\beta_n w(x_0))_-| \leq C\|w\|_{C^{\beta}}h_n^{(\min\{3,\beta\}-2)/2}.
\end{aligned}$$
We fix an auxiliary function $\eta_0:[0,\infty) \to\mathbb{R}_+$, with $\eta_0 \in C^\infty(\mathbb{R}_+)$, and $$\begin{aligned}
\label{equation:eta_0 approximation to min of 1 and y squared}
\;0\leq \eta_0 \leq 1,\;\eta_0'(t) \geq 0 \textnormal{ for all }t, \eta_0(t) = t \textnormal{ for } t\leq 1/2,\;\eta_0(t) = 1 \textnormal{ for } t \geq 1.\end{aligned}$$ The function $\eta_0$, as well as the following two estimates, will be useful in the next section. Essentially, $\eta_0(t)$ should be thought of as a smooth replacement for $\min\{1,t\}$.
\[lemma:Whitney Extension Is Almost Order Preserving\] Let $1\leq \beta<\beta_0<3$, and consider $w\in C^{\beta_0}_b(\mathbb{R}^d)$ and $x_0\in G_n$ such that $$\begin{aligned}
w \geq 0 \textnormal{ in } G_n \textnormal{ and } w(x_0) = 0.
\end{aligned}$$ Then, there is a function $R_{\beta_0,n,w,x_0}$ such that $R(x_0) = 0$, and $$\begin{aligned}
& \pi_n^\beta w(x)+R_{\beta_0,n,w,x_0}(x) \geq 0,\;\;\forall\;x\in\mathbb{R}^d,\\
& \|R_{\beta_0,n,w,x_0}\|_{C^\beta(\mathbb{R}^d)} \leq Ch_n^{\gamma} \|w\|_{C^{\beta_0}(\mathbb{R}^d)},
\end{aligned}$$ for some constant $\gamma = \gamma(\beta,\beta_0) \in (0,1)$.
\[remark:Whitney extension is order preserving for beta<1\] For $\beta\in(0,1)$, it is straightforward that $w\geq 0$ in $G_n$ guarantees that $\pi_n^\beta w\geq 0$ everywhere, that is, the Whitney extension for $\beta \in (0,1)$ is order preserving. Accordingly, Lemma \[lemma:Whitney Extension Is Almost Order Preserving\] is only needed for $\beta>1$.
We consider the cases $1\leq \beta<2$ and $\beta\geq 2$ separately. First suppose $\beta \in [1,2)$. Let $\phi_0(t)$ be a smooth function such that $0\leq \phi_0(t)\leq 1$ for all $t$, $\phi_0(t)=1$ for $t\leq 1/4$ and $\phi_0(t)=0$ for $t\geq 1$. Then set $$\begin{aligned}
\tilde w(x) = \pi^\beta_n w(x) - ( \nabla \pi^\beta_nw(x_0), x-x_0) \phi_0(x-x_0).
\end{aligned}$$ For each $x\in \mathbb{R}^d$, let $\hat x$ denote a point in $G_n$ such that $|x-\hat x| = \textnormal{dist}(x,G_n)\leq h_n$. Then, since $w(\hat x)\geq 0$ for any $\hat x$ (from the assumption), we have $$\begin{aligned}
\tilde w(x) & = \tilde w(\hat x) + (\tilde w(x)-\tilde w(\hat x))\\
& \geq - ( \nabla \pi^\beta_n w (x_0),x-x_0) \phi_0(x-x_0) - C\|\tilde w\|_{C^{\beta_0}} |\hat x-x|\\
& \geq - ( \nabla \pi^\beta_nw (x_0),x-x_0) \phi_0(x-x_0) - C\|\tilde w\|_{C^{\beta_0}} h_n.
\end{aligned}$$ By Proposition \[proposition:discrete derivative estimate at a local minimum\], we have $|\nabla \pi^\beta_nw (x_0)| \leq C\|w\|_{C^{\beta_0}}h_n$ when $\beta_0>1$, therefore, $$\begin{aligned}
\tilde w(x) \geq -C\|w\|_{C^{\beta_0}}h_n,\;\;\forall\;x\in\mathbb{R}^d,
\end{aligned}$$ where we have used Theorem \[theorem:Whitney Extension Is Bounded\] to bound $\|\pi_n^\beta w\|_{C^\beta_0}$. On the other hand, since $\beta_0 >1 $ and $\nabla \tilde w(x_0)=0$, we have $$\begin{aligned}
\tilde w(x) & \geq -\|\tilde w\|_{C^{\beta_0}}|x-x_0|^{\beta_0},\\
& \geq -C\|w\|_{C^{\beta_0}}|x-x_0|^{\beta_0}\;\;\forall\;x\in\mathbb{R}^d,
\end{aligned}$$ Now, we take $\eta_0$ as in and define the function $$\begin{aligned}
\tilde R(x) := 2C\|w\|_{C^{\beta_0}}h_n \eta_0 \left ( \frac{|x-x_0|^{\beta_0}}{h_n} \right ).
\end{aligned}$$ If $|x-x_0|^{\beta_0} \geq h_n/2$, then $$\begin{aligned}
\tilde w(x)+\tilde R(x) & = \tilde w(x) + C\|w\|_{C^{\beta_0}}h_n \geq 0.
\end{aligned}$$ If on the contrary, $|x-x_0|^{\beta_0} \leq h_n/2$, then $$\begin{aligned}
\tilde w(x)+\tilde R(x) & = \tilde w(x) + C\|w\|_{C^{\beta_0}}|x-x_0|^{\beta_0} \geq 0.
\end{aligned}$$ We conclude that $$\begin{aligned}
\tilde w(x)+\tilde R(x) \geq 0,\;\forall\;x\in\mathbb{R}^d.
\end{aligned}$$ On the other hand, an elementary computation (see the Appendix) shows that $$\begin{aligned}
\|\tilde R\|_{C^{\beta}} \leq C h_n^{\gamma} \|w\|_{C^{\beta_0}}.
\end{aligned}$$ Finally, let $$\begin{aligned}
R_{\beta_0,n,w,x_0}(x) := \tilde R(x) - ( \nabla \pi^\beta_n(x_0), x-x_0) \phi_0(x-x_0).
\end{aligned}$$ We conclude that $\|R_{\beta_0,n,w,x_0}\|_{C^\beta} \leq C h_n^{\gamma}\|w\|_{C^{\beta_0}}$ and $$\begin{aligned}
\pi^\beta_n w(x)+R_{\beta_0,n,w,x_0}(x) \geq 0,\;\forall\;x\in\mathbb{R}^d.
\end{aligned}$$ This proves the Proposition when $\beta \in [1,2)$. The argument for $\beta\geq 2$ is similar, we only highlight the main differences. This time, we subtract not just the first order part of $w$ near $x_0$, but also the second order part, namely we consider the function $$\begin{aligned}
\tilde{\tilde w} := \pi_n^\beta w(x) - ( \nabla \pi^\beta_n(x_0), x-x_0) \phi_0(x-x_0) - \tfrac{1}{2}( (\nabla^2 \pi^\beta_n(x_0))_-(x-x_0), x-x_0) \phi_0(x-x_0).
\end{aligned}$$ Then, one applies again Proposition \[proposition:discrete derivative estimate at a local minimum\] and use the regularity of $w$ to obtain (in analogy to the previous case) $$\begin{aligned}
\tilde{\tilde w}(x) \geq -C\|w\|_{C^{\beta_0}} \max \{ h_n, |x-x_0|^{\beta_0}\}
\end{aligned}$$ The respective function $\tilde{\tilde R}$ is defined exactly as $\tilde R$ and one argues as in the previous case.
\[remark:Whitney extension is almost order preserving L infinity version\] The argument in the proof provides -after small modifications- a closely related result: if instead of $w \in C^{\beta}_b(\mathbb{R}^d)$ we assume that $w \in C^0_b(\mathbb{R}^d)$ and that for some $M>0$ and $\beta_0>\beta$ we have $$\begin{aligned}
|w(x)| \leq M|x-x_0|^{\beta_0},\;\forall\;x\in \mathbb{R}^d,
\end{aligned}$$ then there is as before a function $\hat R_{\beta_0,n,w,x_0}$ such that $\hat R_{\beta_0,n,w,x_0}(x_0)=0$ and $ \pi_n^\beta w(x)+R_{\beta_0,n,w,x_0}(x)\geq 0$ for all $x$, but this time the $C^{\beta}$ estimate for $\hat R_{\beta_0,n,w,x_0}$ is $$\begin{aligned}
\|\hat R_{\beta_0,n,w,x_0}\|_{C^{\beta}} \leq Ch_n^{\gamma}(\|w\|_{L^\infty}+M).
\end{aligned}$$
The following proposition will be useful later in the proof of Proposition \[proposition:touching from above function with decay at point operator estimate\].
\[proposition:touching from above function with decay at a point\] Let $1\leq \beta < \beta_0<3$ or $\beta \in (0,1)$ and $\beta_0 = \beta$. Fix $f \in C^\infty_c(\mathbb{R}^d)$, and let $\eta_0$ be as in . Let $x_0 \in G_n$ and $w(x) = f(x-x_0) \eta_0 (|x-x_0|^{\beta_0})$, then $$\begin{aligned}
\pi^\beta_n(w,x) & \leq C\|f\|_{L^\infty} \eta_0(|x-x_0|^{\beta_0}),\;\forall\;x\in\mathbb{R}^d, \textnormal{ if } \beta\in (0,1),\\
\pi^\beta_n(w,x) & \leq C\|f\|_{L^\infty} \eta_0(|x-x_0|^{\beta_0}) + \hat R_{\beta_0,n,w,x_0}(x) ,\;\forall\;x\in\mathbb{R}^d, \textnormal{ if } \beta \in [1,2],
\end{aligned}$$ for some function $\hat R_{\beta_0,n,w,x_0}$ such that $\hat R_{\beta_0,n,w,x_0}(x_0) = 0$ and $$\begin{aligned}
\|\hat R_{\beta_0,n,w,x_0}\|_{C^{\beta}} \leq C\|f\|_{L^\infty}h_n^{\gamma},
\end{aligned}$$ where $\gamma$ is as in Lemma \[lemma:Whitney Extension Is Almost Order Preserving\].
Define the function $\tilde w(x) := (\|f\|_{L^\infty}-f(x-x_0))\eta_0(|x-x_0|^{\beta_0})$. Then $\tilde w(x_0) = 0$ and $$\begin{aligned}
|\tilde w(x)| \leq 2\|f\|_{L^\infty}\eta_0(|x-x_0|^{\beta_0}),\;\forall\;x\in\mathbb{R}^d,
\end{aligned}$$ while, since $\eta_0\geq 0$, we also have $\tilde w(x)\geq 0$ for every $x\in G_n$. If $\beta \in [1,2]$, using Lemma \[lemma:Whitney Extension Is Almost Order Preserving\] and the function $\hat R_{\beta_0,n,w,x_0}$ from Remark \[remark:Whitney extension is almost order preserving L infinity version\], we have $$\begin{aligned}
\pi^\beta_n(\tilde w,x) + \hat R_{\beta_0,n,w,x_0}(x) \geq 0,\;\;\forall\;x,
\end{aligned}$$ This inequality, after some rearranging, yields (for $\beta\in[1,2]$) $$\begin{aligned}
\pi^\beta_n(w,x)\leq \|f\|_{L^\infty} \pi^\beta_n(\eta_0(|\cdot-x_0|^{\beta_0}),x) + \hat R_{\beta_0,n,w,x_0}(x) ,\;\;\forall\;x\in\mathbb{R}^d.
\end{aligned}$$ Since we also have $\|\tilde w\|_{L^\infty} \leq C\|f\|_{L^\infty}$, we have again by Remark \[remark:Whitney extension is almost order preserving L infinity version\] $$\begin{aligned}
\|\hat R_{\beta_0,n,w,x_0}\|_{C^{\beta}} \leq C\|f\|_{L^\infty}h_n^{\gamma},
\end{aligned}$$ and the Proposition is proved in this case. For $\beta \in (0,1)$ we argue along similar lines, using Remark \[remark:Whitney extension is order preserving for beta<1\] instead of Lemma \[lemma:Whitney Extension Is Almost Order Preserving\].
Convergence of the projection operators
---------------------------------------
\[lemma:projection operators convergence\] Let $0<\beta<\beta_0<3$, there is a constant $C$ such that if $u\in C^{\beta_0}_b(\mathbb{R}^d)$, then $$\begin{aligned}
\|\pi_n^\beta u - u \|_{C^{\beta}} \leq Ch_n^{\gamma}\|u\|_{C^{\beta_0}}.
\end{aligned}$$ Here, $\gamma= \gamma(\beta_0,\beta) \in (0,1)$.
For notational simplicity let us write $f(x) = \pi_n^\beta u(x)$ throughout the proof.
Since $u=f$ throughout $G_n$, for an arbitrary $x\in G_n$ we have (with $\hat x$ denoting a point in $G_n$ such that $\textnormal{dist}(x,G_n) = |x-\hat x|$), with $\alpha := \min\{1,\beta_0\}$ $$\begin{aligned}
|u(x)-f(x)| & \leq |f(x)-f(\hat x)| + |u(\hat x)-u(x)|\\
& \leq |x-\hat x|^\alpha [f]_{C^{\alpha}} + |x-\hat x|^{\alpha} [u]_{C^{\alpha}}\\
& \leq C\|u\|_{C^{\beta_0}}h_n^\alpha \leq C\|u\|_{C^{\beta_0}}h_n^\alpha,
\end{aligned}$$ where we made use of Theorem \[theorem:Whitney Extension Is Bounded\] to obtain $[f]_{C^\alpha} \leq C\|u\|_{C^\beta}$. This shows that $\|u-f\|_{L^\infty}$ goes to zero at some rate determined by $\beta_0$ and the size of $\|u\|_{C^{\beta_0}}$. To prove the lemma we need to also bound the Hölder seminorm of $u-f$ and its derivatives, according to $\beta_0$.
**The case $\beta, \beta_0 \in [0,1)$**. Fix $x_1,x_2 \in \mathbb{R}^d$. First, suppose that $|x_1-x_2| \leq \max\{|x_1-\hat x_1|,|x_2-\hat x_2|\}$, then $$\begin{aligned}
|f(x_1)-u(x_1)-(f(x_2)-u(x_2))| \leq [f-u]_{C^{\beta_0}}|x_1-x_2|^{\beta_0} \leq C\|u\|_{C^{\beta_0}}|x_1-x_2|^{\beta_0}.
\end{aligned}$$ In this case, and since $0\leq \beta <\beta_0<1$, we have that $|x_1-x_2|^{\beta_0-\beta} \leq \max \{|x_1-\hat x_1|^{\beta_0-\beta},|x_2-\hat x_2|^{\beta_0-\beta}\} \leq h_n^{\beta_0-\beta}$. Then, using Theorem \[theorem:Whitney Extension Is Bounded\] $$\begin{aligned}
|f(x_1)-u(x_1)-(f(x_2)-u(x_2))| \leq [f-u]_{C^\beta}|x_1-x_2|^{\beta} \leq C\|u\|_{C^{\beta_0}}h_n^{\beta_0-\beta}|x_1-x_2|^{\beta}.
\end{aligned}$$ Next, suppose that $|x_1-x_2| > \max\{|x_1-\hat x_1|,|x_2-\hat x_2|\}$. In this case $$\begin{aligned}
|f(x_1)-u(x_1)-(f(x_2)-u(x_2))| & \leq \|f\|_{C^{\beta_0}}|x_1-\hat x_1|^{\beta_0}+\|u\|_{C^{\beta_0}}|x_2-\hat x_2|^{\beta_0}\\
& \leq C\|u\|_{C^{\beta_0}}h_n^{\beta_0-\beta}|x_1-x_2|^{\beta},
\end{aligned}$$ where once again Theorem \[theorem:Whitney Extension Is Bounded\] was used. Combining these two estimates, we conclude that $$\begin{aligned}
[f-u]_{C^\beta} = \sup \limits_{x_1 \neq x_2} \frac{|f(x_1)-u(x_1)-(f(x_2)-u(x_2))|}{|x_1-x_2|^{\beta}} \leq C\|u\|_{C^{\beta_0}}h_n^{\beta_0-\beta}.
\end{aligned}$$ Then, using that $h_n \leq 1$ for all $n\geq 1$, we have $$\begin{aligned}
\|f-u\|_{C^\beta} \leq Ch_n^{\gamma}\|u\|_{C^{\beta_0}}.
\end{aligned}$$
**The case $\beta, \beta_0 \in [1,2)$**. In this case we trivially have the same estimates from the previous case, and only need the bounds for first derivative. This is done as follows, first $$\begin{aligned}
|\nabla f(x)-\nabla u(x)| \leq |\nabla f(x)-\nabla f(\hat x)| + |\nabla f(\hat x)-\nabla u(\hat x)| + |\nabla u(x)-\nabla u(\hat x)|.
\end{aligned}$$ Then, using Theorem \[theorem:Whitney Extension Is Bounded\], we have $$\begin{aligned}
|\nabla f(x)-\nabla u(x)| & \leq [\nabla f]_{C^{\beta_0-1}}h_n^{\beta_0-1} + |\nabla f(\hat x)-\nabla u(\hat x)| + [\nabla u]_{C^{\beta_0-1}}h_n^{\beta_0-1}\\
& \leq C\|u\|_{C^{\beta_0}}h_n^{\beta_0-1}+ |\nabla f(\hat x)-\nabla u(\hat x)|.
\end{aligned}$$ Recall that $\nabla f(\hat x) = (\nabla_n)^1 u(\hat x)$, and use Proposition \[proposition:discrete derivatives converge to real derivatives\] to conclude that $$\begin{aligned}
|\nabla f(x)-\nabla u(x)| & \leq C\|u\|_{C^{\beta_0}}h_n^{\beta_0-1}+ C\|u\|_{C^{\beta_0}}h_n^{\beta_0-1}.
\end{aligned}$$ The Hölder seminorm $[\nabla f-\nabla u]_{C^\beta}$ is bounded with the same argument used to bound $[f-u]_{C^{\beta}}$ in the previous case, we omit the details.
**The case $\beta = 2, \beta_0 \in (2,3)$**. Right as before, we note that $$\begin{aligned}
|D^2 f(x)-D^2 u(x)| \leq |D^2f(x)-D^2f(\hat x)| + |D^2 f(\hat x)-D^2 u(\hat x)| + |D^2 u(x)-D^2 u(\hat x)|.
\end{aligned}$$ Then, applying Theorem \[theorem:Whitney Extension Is Bounded\] and Proposition \[proposition:discrete derivatives converge to real derivatives\] as in the previous case, we have $$\begin{aligned}
|D^2 f(x)-D^2 u(x)| & \leq [D^2 f]_{C^{\beta_0-2}}h_n^{\beta_0-2} + |D^2 f(\hat x)-D^2 u(\hat x)| + [D^2 u]_{C^{\beta_0-2}}h_n^{\beta_0-2}\\
& \leq 2C\|u\|_{C^{\beta_0}}h_n^{\beta_0-2} + |\nabla f(\hat x)-\nabla u(\hat x)|\\
& \leq 3C\|u\|_{C^{\beta_0}}h_n^{\beta_0-2}.
\end{aligned}$$ For the Hölder seminorm, we repeat the argument used in the case $\beta \in (0,1)$, again we leave the details to the reader.
\[remark:projection operators C0 convergence\] If $u\in C^0_b(\mathbb{R}^d)$, then the same argument from Lemma \[lemma:projection operators convergence\] can be used to show $$\begin{aligned}
\lim \limits_{n\to \infty} \|u-\pi_n^0(u)\|_{L^\infty(\mathbb{R}^d)} = 0,
\end{aligned}$$ the rate of convergence being determined by the modulus of continuity of $u$.
Analysis of $I(u,x)$ via the finite dimensional approximations {#section:Analysis of finite dimensional approximations}
==============================================================
In this section we introduce a sequence of operators $I_n$ which approximate $I$. The operators $I_n$ behave like operators in a finite dimensional vector space in the sense that they arise from a composition between linear maps with a Lipschitz map from a finite dimensional space onto itself. This allows us to prove a min-max formula for $I_n(u,x)$ at least when $x\in G_n$ by using Clarke’s idea of a generalized gradient [@Cla1990optimization]. More precisely, we use the fact that $I_n$ factorizes via a map between finite dimensional vector spaces (which is what the spaces $C_*(G_n)$ were introduced for), where the generalized gradient can be used, and then lift this to corresponding maps from $C^\beta_b(\mathbb{R}^d)$ to $C_b^0(\mathbb{R}^d)$ using the Whitney extension. The majority of the section is concerned with deriving estimates and regularity properties for the linear operators arising in the min-max formula for $I_n$, and ultimately concluding such linear operators are pre-compact, which leads to a min-max formula for the original operator.
The operators $I_n$ and their min-max representation
----------------------------------------------------
We are going to approximate the operator $I(\cdot,x)$ via “finite dimensional approximations”, this referring to maps $I_n:C^\beta_b\to C^0_b$, which factorize through a finite dimensional space (see below).
We introduce a modification of the projection operator $\pi_n^0$ defined in . First, we define $$\begin{aligned}
\textnormal{Pr}_n:C(G_n) \to C_*(G_n),\;\; \textnormal{Pr}_n(u)(x) := u(x) \chi_{[-2^{n},2^{n}]^d}(x).\end{aligned}$$ That is, given $u\in C(G_n)$, we define $\textnormal{Pr}_n(u)$ as the function obtained by restricting $u$ to $G_n \cap [-2^{n},2^{n}]^d$ and then extending it to the rest of $G_n$ by zero. Then, we define the modified Whitney extension, $$\begin{aligned}
\hat E^\beta_n := E_n^\beta \circ \textnormal{Pr}_n,\end{aligned}$$ and the modified projection operator $$\begin{aligned}
\hat \pi_n^\beta := \hat E^\beta_n \circ T_n.\end{aligned}$$ These are, respectively, bounded linear maps from $C(G_n)$ to $C^\beta_b(\mathbb{R}^d)$ and from $C^0_b(\mathbb{R}^d)$ to $C^\beta_b(\mathbb{R}^d)$. Now we are ready to introduce the finite dimensional approximations to the operator $I$, define $$\begin{aligned}
\label{equation:I_n definition}
I_n &= \hat \pi_n^0 \circ I \circ \hat \pi_n^\beta,\;\;I_n:C^\beta_b(\mathbb{R}^d) \to C_b^0(\mathbb{R}^d).\end{aligned}$$ That is, to compute $I_n(u,x)$, we first compute the modified projection $\hat \pi_n^\beta u$, and compute $I(\hat \pi_n^\beta u)$, to which we later apply the modified projection $\hat \pi_n^0$. In particular, $I_n$ only depends on the values of $u$ on $G_n \cap [-2^n,2^n]^d$. Associated to this, we introduce a map, $i_n$, defined as follows $$\begin{aligned}
\label{equation:little i n via I n}
i_n:C_*(G_n)\to C_*(G_n),\ \ i_n & = \textnormal{Pr}_n \circ T_n \circ I \circ E_n^\beta.\end{aligned}$$ From the definition of $I_n$, we have $I_n = E_n^\beta \circ \textnormal{Pr}_n \circ T_n \circ I \circ E_n^\beta \circ \textnormal{Pr}_n \circ T_n$, thus we see $I_n$ and $i_n$ are themselves related by $$\begin{aligned}
\label{equation:I n via little i n}
I_n & = E_n^0 \circ i_n \circ \textnormal{Pr}_n \circ T_n.\end{aligned}$$ The situation for both $I_n$ and $i_n$ is represented in the following two diagrams, $$\begin{tikzcd}
C^\beta_b(\mathbb{R}^d) \arrow{r}{I_n} \arrow[swap]{d}{\hat \pi_n^\beta} & C^0_b(\mathbb{R}^d)\\
C^\beta_b(\mathbb{R}^d) \arrow{r}{I} & C^0_b(\mathbb{R}^d) \arrow{u}{\hat \pi_n^0}
\end{tikzcd} \quad\quad\quad \begin{tikzcd}
C_*(G_n) \arrow{r}{i_n} \arrow[swap]{d}{E_n^\beta} & C_*(G_n) \\
C^\beta_b(\mathbb{R}^d) \arrow{r}{I} & C^0_b(\mathbb{R}^d) \arrow{u}{\textnormal{Pr}_n \circ T_n}
\end{tikzcd}$$ Now, the space $C_*(G_n)$ is finite dimensional (Remark \[remark:C star G\_n is finite dimensional\]), and the map $i_n:C_*(G_n)\to C_*(G_n)$ is Lipschitz continuous. Therefore, tools available for Lipschitz functions in the finite dimensional setting can be applied to $i_n$ and then related to $I_n$ via .
We recall the generalized derivative of $i_n$ in the sense of Clarke [@Cla1990optimization Section 2.6].
\[definition:Clarke differential\] Let $V$ be a Banach space, and $T:V\to V$ a Lipschitz continuous function. We define the set of generalized derivatives of $T$, by $$\begin{aligned}
\mathcal{D}T := \textnormal{c.h.}\{ L:V\to V \mid L = \lim \limits_{k} L_k \textnormal{ where } L_k= DT(x_k),\; T \textnormal{ is differentiable at } x_k \;\forall\; k \}.
\end{aligned}$$
By Rademacher’s theorem, the set $\mathcal{D}T$ is not empty when $V$ is finite dimensional. Applying this to $i_n:C_*(G_n)\to C_*(G_n)$, we have, first, that $\mathcal{D}i_n$ is non-empty, and secondly that $\mathcal{D}I_n$ is non-empty as well, this is proved in Lemma \[lemma:characterization of D I sub n\], where we describe the relationship between $\mathcal{D}i_n$ to $\mathcal{D}I_n$. The following Lemma is the mean value theorem for nonsmooth Lipschitz functions between finite dimensional spaces (note the similarity with Theorem \[theorem:Lebourg\]).
\[lemma:min max finite dimensional map\] Assume that $I: C^\beta_b(\real^d)\to C^0_b(\real^d)$ is Lipschitz. For any $u,v \in C_*(G_n)$, there is a $L \in \mathcal{D}i_n$ such that $$\begin{aligned}
i_n(u,x) - i_n(v,x) = L(u-v,x).
\end{aligned}$$
We refer the reader to [@Cla1990optimization Proposition 2.6.5] for a proof of the lemma.
The second lemma is basically the chain rule.
\[lemma:characterization of D I sub n\] Assume that $I: C^\beta_b(\real^d)\to C^0_b(\real^d)$ is Lipschitz. The set $\mathcal{D}I_n$ is non-empty, and for any $L \in \mathcal{D}I_n$ there is a $\tilde L \in \mathcal{D}i_n$ such that $$\begin{aligned}
L =E_n^0 \circ \tilde L \circ T_n,
\end{aligned}$$ conversely, any $L$ defined in this way for some $\tilde L \in \mathcal{D}i_n$ belongs to $\mathcal{D}I_n$.
Note that $I_n$ is differentiable at a point $u$ if and only if $i_n$ is differentiable at $\tilde u = T_nu$, a fact which follows applying the chain rule to the identities and . Furthermore, at such $u$’s we have $$\begin{aligned}
DI_n(u) = E_n^* \circ Di_n(\tilde u) \circ T_n.
\end{aligned}$$ If $u_k$ is a sequence along which $I_n$ is differentiable, and $L_k:=DI_n(u_k)$ converges to some $L$, then the sequence $\tilde L_k := Di_n(\tilde u_k)$ has a limit $\tilde L$, and $L = E_n^* \circ \tilde L \circ T_n$, taking the convex hull and by the linearity of $E_n^*$ and $T_n$, the lemma follows.
The following remark will not be of any relevance until the proof of Theorem \[theorem:MinMax Euclidean ver2\] at the end of this section, but we include it here to illustrate how Lemmas \[lemma:min max finite dimensional map\] and \[lemma:characterization of D I sub n\] immediately yield a min-max formula for $I_n(u,x)$ (for $x\in G_n$).
\[remark:MinMax for In\] Fix $n$ and let $x\in G_n$. Then for any $u \in C^\beta_b(\mathbb{R}^d)$ we have $$\begin{aligned}
\label{equation:minmax for I_n}
I_n(u,x) \leq \max \limits_{L\in \mathcal{D}I_n} \{ I_n(v,x)+L(u-v,x)\},\;\;\forall\;x\in G_n, u,v\in C^\beta_b(\mathbb{R}^d).
\end{aligned}$$ Indeed, according to Lemma \[lemma:min max finite dimensional map\] given $u$ and $v$ says there is some $\tilde L \in\mathcal{D}i_n$ such that $$\begin{aligned}
i_n(u)-i_n(v) = \tilde L(u-v).
\end{aligned}$$ In this case, we have $E_n^0(i_n(u))-E_n^0(i_n(v)) = E_n^0(\tilde L(u-v))$, and thus setting $L := E_n^0 \circ \tilde L \circ T_n \in \mathcal{D}I_n$, we have $$\begin{aligned}
I_n(u) = I_n(v) + L(u-v),
\end{aligned}$$ and immediately follows.
Next we make an elementary observation regarding the nature of the operators $L \in \mathcal{D}I_n$. This observation is merely a consequence of the factorization of $I_n$ through the space $C(G_n)$.
\[remark:structure of D little i n kernel\] For each $L \in \mathcal{D}I_n$ there is a function $K = K_L$, $K: G_n \times G_n \to \mathbb{R}$ such that $$\begin{aligned}
\label{equation:L in DI_n formula}
Lu(x) = \sum \limits_{y\in G_n} K(x,y)u(x+y),\;\;\forall\;u\in C^\beta_b(\mathbb{R}^d).
\end{aligned}$$ Indeed, simply let us use the basis functions $\{e_y\}_{y\in G_n} \subset C(G_n)$ given by $$\begin{aligned}
e_y(x) = \left \{ \begin{array}{rl}
1 & \textnormal{ if } x =y,\\
0 & \textnormal{ if } x\neq y.
\end{array}\right.
\end{aligned}$$ Observe that for any $u \in C^{\beta}_b(\mathbb{R}^d)$ the function $T_nu$ has finite support, and in particular $T_nu = \sum_{y\in G_n}u(y)e_y$ as the sum on the right has at most a finite number of non-zero terms. Thanks to Lemma \[lemma:characterization of D I sub n\], there is some $\tilde L \in \mathcal{D}i_n$ such that $L = E^0_n \circ \tilde L \circ T_n$ and therefore, $$\begin{aligned}
Lu(x) = \sum \limits_{y\in G_n} (\tilde Le_y)(x)u(y) = \sum \limits_{y\in G_n-x} (\tilde Le_{x+y})(x)u(x+y),\;\forall\;x\in G_n.
\end{aligned}$$ Then, defining $K_L(x,y) = (\tilde Le_{x+y})(x)$ for $x,y\in G_n$ the identity follows.
For the rest of this section we analyze the operators $I_n$ and the sets $\mathcal{D}I_n$ and obtain in the limit a min-max formula for $I_n$. We shall focus on operators satisfying Assumption \[assumption:coefficient regularity\]. As we see below this property is inherited –to some extent– by the operators $I_n$, and by any operator $L \in \mathcal{D}I_n$, this fact is covered in the next two propositions. In the subsections that follow, we will use the spatial regularity afforded by Assumption \[assumption:coefficient regularity\] to show that the operators in the family $\mathcal{D}I_n$ have coefficients enjoying some regularity, which in the limit yields regular coefficients.
\[proposition:coefficient regularity inherited by In\] Let $I$ be Lipschitz and satisfy Assumption \[assumption:coefficient regularity\]. Let $x_1,x_2 \in G_n$ and $h = x_1 - x_2$, and $r\geq 2^{4-n}$. Then, for any $u,v \in C^\beta_b(\mathbb{R}^d)$ we have $$\begin{aligned}
& | I_n(v+\tau_{-h}u,x_1)-I_n(v,x_1) -\left ( I_n(v+u,x_2)-I_n(v,x_2)\right ) |\\
& \leq \omega(|h|)C(2r)\left ( \|u\|_{C^\beta(B_{4r}(x_2))} + \|u\|_{L^\infty(\mathcal{C}B_r(x_2))}\right ).
\end{aligned}$$ where $\omega(\cdot)$ is the modulus of continuity and $C(\cdot)$ the function given by Assumption \[assumption:coefficient regularity\].
Observe that $$\begin{aligned}
I_n(v+\tau_{-h}u,x_1)-I_n(v,x_1) = I(\pi_n^\beta v + \pi_n^\beta (\tau_{-h}u),x_1) - I_n(\pi_n^\beta,x_1),
\end{aligned}$$ and recall that Proposition \[proposition:Whitney extension translation invariance\] says that $\pi_n^\beta (\tau_{-h}u) = \tau_{-h}\pi_n^\beta (u)$ when $G_n+h = G_n$.
Therefore, applying the bound in Assumption \[assumption:coefficient regularity\] with $\tfrac{3}{2}r$, $$\begin{aligned}
& | I_n(v+\tau_{-h}u,x_1)-I_n(v,x_1) -\left ( I_n(v+u,x_2)-I_n(v,x_2)\right ) |\\
& = | I(\pi_n^\beta v + \tau_{-h}(\pi_n^\beta u),x_1) - I_n(\pi_n^\beta,x_1)-\left ( I(\pi_n^\beta v+\pi_n^\beta u,x_2)-I(\pi_n^\beta v,x_2)\right ) |\\
& \leq \omega(|x_1-x_2|)C(3r/2)\left ( \|\pi_n^\beta u\|_{C^\beta(B_{3r}(x))} + \|\pi_n^\beta u\|_{L^\infty(\mathcal{C}B_{3r/2}(x))}\right ).
\end{aligned}$$ Now, provided $r \geq 2^{4-n}$, we have $$\begin{aligned}
\|\pi_n^\beta u\|_{C^\beta(B_{3r}(x))} & \leq C \|u\|_{C^\beta(B_{4r}(x))},\\
\|\pi_n^\beta u\|_{L^\infty(\mathcal{C}B_{3r/2}(x))} & \leq C\|u\|_{L^\infty(\mathcal{C}B_{r}(x))},
\end{aligned}$$ the proposition follows.
\[proposition:coefficient regularity for L\] Let $I$ be Lipschitz and satisfy Assumption \[assumption:coefficient regularity\]. Given $L\in \mathcal{D}I_n$, $x_1,x_2 \in G_n$, $r\geq 2^{4-n}$ and $u \in C^\beta_b(\mathbb{R}^d)$, we have the inequality $$\begin{aligned}
\label{eqn:coefficient regularity}
|L(\tau_{-h}u ,x_1)-L(u,x_2)| \leq \omega(|h|) C(2r) \left (\|u\|_{C^\beta(B_{4r}(x_2))}+ \|u\|_{L^\infty(\mathcal{C}B_{r}(x_2))} \right ).
\end{aligned}$$ Here, $h = x_1-x_2$ and $\omega(\cdot)$ and $C(\cdot)$ are given by Assumption \[assumption:coefficient regularity\].
Consider any $v \in C^\beta_b(\mathbb{R}^d)$ such that $I_n$ is differentiable at $v$ with derivative $L$. Then, $$\begin{aligned}
L(\tau_{-h}u ,x_1) & = \lim \limits_{s\to 0} \frac{1}{s} \left ( I_n(v+s \tau_{-h}u,x_1)-I_n(v,x_1)\right ),\\
L(u ,x_2) & = \lim \limits_{s\to 0} \frac{1}{s} \left ( I_n(v+s u,x_2)-I_n(v,x_2)\right ).
\end{aligned}$$ By Proposition \[proposition:coefficient regularity inherited by In\], we have $$\begin{aligned}
& | L(\tau_{-h}u ,x_1) - L(u ,x_2) | \\
& = \limsup \limits_{s\to 0} \frac{1}{s} \left | I_n(v+s \tau_{-h}u,x_1)-I_n(v,x_1) - ( I_n(v+s u,x_2)-I_n(v,x_2))\right |,\\
& \leq \omega(|h|)C(2r) \limsup \limits_{s\to 0} \frac{1}{s} \left ( \|s u\|_{C^\beta(B_{2r}(x))} + \|s u\|_{L^\infty(\mathcal{C} B_{r}(x))} \right ),\\
& = \omega(|h|)C(2r) \left ( \| u\|_{C^\beta(B_{2r}(x))} + \| u\|_{L^\infty(\mathcal{C} B_{r}(x))} \right ).
\end{aligned}$$ This proves the desired inequality for those $L \in \mathcal{D}I_n$ which happen to be the derivative of $I_n$ at a point of differentiability. This property is clearly preserved under limits and convex combinations, so it follows any $L \in \mathcal{D}I_n$ has the desired property.
The following proposition is directly related to Proposition \[proposition:touching from above function with decay at a point\].
\[proposition:touching from above function with decay at point operator estimate\] Assume that $I$ is Lipschitz and satisfies Assumption \[assumption:GCP\]. For $f\in C^\infty_c(\mathbb{R}^d)$ let $w(x) = f(x-x_0)\eta_0(|x-x_0|^\beta)$ with $\eta_0$ as in , then $$\begin{aligned}
I(\pi_n^\beta u + \pi_n^\beta w,x)-I(\pi_n^\beta u,x) \leq C\|f\|_{L^\infty}.
\end{aligned}$$ If instead we have $w(x) = f(x-x_0)\eta_0(|x-x_0|^{\beta_0})$ with $f$ non-negative and some $\beta_0>\beta$, then $$\begin{aligned}
I(\pi_n^\beta u + \pi_n^\beta w,x)-I(\pi_n^\beta u,x) \geq -C\|f\|_{L^\infty} h_n^{\gamma},
\end{aligned}$$ for some constant $\gamma= \gamma(\beta_0,\beta) \in (0,1)$.
We apply Proposition \[proposition:touching from above function with decay at a point\], and we have with $\hat R_{\beta, n, w, x_0}$ from the same proposition, we have $$\begin{aligned}
\pi_n^\beta w(x) \leq \hat w(x):= C\|f\|_{L^\infty}\left ( \eta_0(|x-x_0|^\beta) + \hat R_{\beta, n,w,x_0}(x) \right ),\;\forall\;x\in\mathbb{R}^d,
\end{aligned}$$ with equality holding for $x=x_0$. It follows that $\pi_n^\beta u + \pi_n^\beta w$ is touched from above at $x_0$ by $\pi^\beta_n u+ \hat w$. Then, since $I(\cdot,x)$ has the GCP, $$\begin{aligned}
I(\pi_n^\beta u + \pi_n^\beta w,x) \leq I(\pi_n^\beta u + \hat w,x)
\end{aligned}$$ This means that $$\begin{aligned}
I(\pi_n^\beta u + \pi_n^\beta w,x_0)-I(\pi_n^\beta u,x_0) \leq I(\pi_n^\beta u + \hat w,x_0)-I(\pi_n^\beta u,x_0) \leq C\|\hat w\|_{C^\beta}.
\end{aligned}$$ Since $\|\hat w\|_{C^\beta} = \|f\|_{L^\infty} \| \eta_0(|\cdot-x_0|^\beta) + \hat R_{\beta, n,w,x_0}\|_{C^\beta} \leq C\|f\|_{L^\infty}$ the first inequality is proved. For the second inequality, we apply Remark \[remark:Whitney extension is almost order preserving L infinity version\] directly, and use that $I$ has the GCP to conclude that $$\begin{aligned}
I(\pi_n^\beta u + \pi_n^\beta w + \hat R_{\beta_0,n,w,x_0},x_0) \geq I(\pi_n^\beta u,x_0).
\end{aligned}$$ Then, using the Lipschitz property of $I$ we conclude that $$\begin{aligned}
I(\pi_n^\beta u + \pi_n^\beta w,x_0) - I(\pi_n^\beta u,x_0) \geq -C\| \hat R_{\beta_0,n,w,x_0}\|_{C^{\beta}} \geq -Ch_n^{\gamma} \|f\|_{L^\infty},
\end{aligned}$$ where we used that $|w(x)| \leq C\|f\|_{L^\infty}\min\{1,|x-x_0|^{\beta_0}\}$ and Remark \[remark:Whitney extension is almost order preserving L infinity version\] to obtain the last inequality.
\[proposition:tightness bound for I\_n\] Let $I$ be Lipschitz and satisfy Assumption \[assumption:tightness bound\]. Let $R\geq 1$ and $w\in C^\beta_b(\real^d)$ with $w\equiv 0$ in $B_{3R}(x_0)$, then for any $x \in \cap B_{R}(x_0)$ we have $$\begin{aligned}
|I(\pi_n^\beta u + \pi_n^\beta w,x)-I(\pi_n^\beta u,x)| \leq \rho(R)\|w\|_{L^\infty(\mathbb{R}^d)},
\end{aligned}$$ where $\rho$ is the rate coming from Assumption \[assumption:tightness bound\].
If $w\equiv 0$ in $B_{3R}(x_0)$, then $\pi^\beta_n \equiv 0$ in $B_{2R}(x_0)$. In other words, $\pi_n^\beta u$ and $\pi_n^\beta u + \pi_n^\beta w$ are identically equal in $B_{2R}(x_0)$. Therefore, Assumption \[assumption:tightness bound\] says that $$\begin{aligned}
|I(\pi_n^\beta u + \pi_n^\beta w,x)-I(\pi_n^\beta u,x)| \leq \rho(R)\|\pi_n^\beta w\|_{L^\infty(\mathbb{R}^d)},\;\;\forall x \in B_{R}(x_0).
\end{aligned}$$ By Proposition \[proposition:touching from above function with decay at a point\], $\|\pi_n^\beta w\|_{L^\infty(\mathbb{R}^d)} \leq \|w\|_{L^\infty(\mathbb{R}^d)}$, the proposition is proved.
Properties of $\mathcal{D}I_n$
------------------------------
For each $L\in \mathcal{D}I_n$ and $x\in G_n$ we define a Borel measure $\mu_L(x,dy)$ (which is possibly signed) as follows $$\begin{aligned}
\label{equation:definition of mu sub L}
\mu_L(x,dy) := \sum \limits_{y \in G_n\setminus \{0\}} K_L(x,y) \delta_{x+y}. \end{aligned}$$ where $K_L(x,y)$ is as in Remark \[remark:structure of D little i n kernel\]. From its definition, it is immediate that given $\phi \in C^{\beta}$ and $x\in G_n$ then $$\begin{aligned}
L(\phi,x) = \int_{\mathbb{R}^d} \phi(x+y)\;d\mu_L(x,dy).\end{aligned}$$
\[proposition:DI\_n Borel measures integrability\] Assume that $I$ is Lipschitz and satisfies Assumption \[assumption:GCP\]. For each $L\in \mathcal{D}I_n$ and $x\in G_n$, and $\eta_0(t)$ the function in , $$\begin{aligned}
\sup \limits_{n}\sup \limits_{x\in G_n} \int_{\mathbb{R}^d} f(y)\eta_0(|y|^{\beta})\;\mu_L(x,dy) & \leq C\|f\|_{L^\infty},\;\;\forall\;f\in C^\infty_c(\mathbb{R}^d).
\end{aligned}$$
Fix $x_0 \in G_n$. Let us assume first that $\beta \neq 1$. Let $w(x) = f(x-x_0)\eta_0(|x-x_0|^\beta)$, then $$\begin{aligned}
L(w,x_0) = \int_{\mathbb{R}^d} \phi(y) \eta_0(|y|^\beta) \;\mu_L(x_0,dy).
\end{aligned}$$ Therefore it suffices to show there is a universal constant such that $$\begin{aligned}
L(w,x_0) \leq C\|f\|_{L^\infty},\;\;\forall\;L\in\mathcal{D}I_n.
\end{aligned}$$ Let us prove this when $L$ arises as the derivative of $I_n$ at some $v\in C^\beta_b$, namely, that $$\begin{aligned}
L(\phi,x_0) = \lim \limits_{s\to 0} (I_n(v+s\phi,x_0) -I_n(v,x_0))/s.
\end{aligned}$$ In this case, we can apply Proposition \[proposition:touching from above function with decay at point operator estimate\] to the expression on the right and conclude that $$\begin{aligned}
\lim \limits_{s\to 0} (I_n(v+sw,x_0) -I_n(v,x_0))/s \leq C\|f\|_{L^\infty},
\end{aligned}$$ where we used that when $\beta \neq 1$ the function $\eta_0(|\cdot-x_0|^\beta)$ belongs to $C^\beta_b(\mathbb{R}^d)$ and the norm $\|\eta_0(|\cdot-x_0|^\beta)\|_{C^\beta}$ is bounded in terms of $\beta,d,$ and the function $\eta_0$. This the desired estimate for such $L$. Since this property is clearly preserved under limits and convex combinations, it follows that the property holds for all elements of $\mathcal{D}I_n$.
The case $\beta = 1$ proceeds similarly, except one first fixes $\varepsilon \in (0,1)$ and considers the function $\eta_0(|x-x_0|^{\beta+\varepsilon})$ instead. After proceeding as in the previous case, we obtain the estimate $$\begin{aligned}
\int_{\mathbb{R}^d} f(y)\eta_0(|y|^{\beta+\varepsilon})\;\mu_L(x_0,dy) & \leq C\|f\|_{L^\infty},
\end{aligned}$$ for every $L \in \mathcal{D}I_n$ and $x_0 \in G_n$. The constant $C$ is independent of $\varepsilon \in (0,1)$, since $\|\eta_0(|\cdot-x_0|^\beta)\|_{C^1}$ is independent of $\varepsilon$ when $\varepsilon>0$. Letting $\varepsilon \searrow 0$ for the integral on the left (and using the special form of $\mu_{L}(x_0,dy)$) one obtains the estimate in the case $\beta =1$.
\[proposition:DI\_n Borel measures negative part\] Assume that $I$ is Lipschitz and satisfies Assumption \[assumption:GCP\]. Let $f\in C^\infty_c(\mathbb{R}^d)$ be a non-negative function. There is a constant $C=C(I,d,\beta,\beta_0)$ such that given $\beta_0>\beta$ then for each $L\in \mathcal{D}I_n$ and $x\in G_n$, $$\begin{aligned}
\inf \limits_{n} \inf \limits_{x \in G_n} \int_{\mathbb{R}^d} f(y)\eta_0(|y|^{\beta_0})\;\mu_L(x,dy) & \geq -Ch_n^{\gamma}\|f\|_{L^\infty}.
\end{aligned}$$ As before, $\eta_0$ is the function in , and $\gamma = \gamma(\beta,\beta_0)$.
As in the proof of the previous proposition, we note that if $x_0 \in G_n$, $w(x) := f(x-x_0)\eta_0(|x-x_0|^{\beta_0})$, and $L \in \mathcal{D}I_n$, then $$\begin{aligned}
L(w,x_0) = \int_{\mathbb{R}^d} f(y) \eta_0(|y|^\beta) \;\mu_L(x_0,dy).
\end{aligned}$$ As in the previous Proposition, it suffices to show that $L(w,x_0) \geq -C\|f\|_{L^\infty} h_n^{\gamma}$, and from $\mathcal{D}I_n$’s definition, it suffices to show this for those $L's$ in $\mathcal{D}I_n$ which are the derivative of $I_n$ at some $u \in C^{\beta}_b(\mathbb{R}^d)$. In this case, given that $f\geq 0$, we may apply the second part of Proposition \[proposition:touching from above function with decay at point operator estimate\] to obtain $$\begin{aligned}
L(w,x_0) = \lim\limits_{s\to 0} \frac{I(\pi_n^\beta u + \pi_n^\beta(sw),x_0)-I(\pi_n^\beta u,x_0)}{s} \geq \lim\limits_{s\to 0} -\frac{C\|s f\|_{L^\infty }h_n^\gamma}{s} = -Ch_n^\gamma\|f\|_{L^\infty},
\end{aligned}$$ and the proposition is proved.
Let us recall the function $$\begin{aligned}
P_{\phi,\eta,u,x}(\cdot) = u(x)+\phi(\cdot-x)(\nabla u(x),\cdot-x)+\tfrac{1}{2}\eta(\cdot-x)(D^2u(x)(\cdot-x),(\cdot-x)).\end{aligned}$$ In this section we introduce a variation on this function. This modification takes into account the geometry of the grid $G_n$ as well as the regularity exponent $\beta$, and will be used in a way analogous to the previous section. $$\begin{aligned}
P_{\phi,\eta,u,x}^{(n)}(\cdot) = \left \{ \begin{array}{lr}
u(x) & \textnormal{ if } \beta \in(0,1),\\
u(x) + \phi(\cdot-x)((\nabla_n)^1 u(x),\cdot-x) & \textnormal{ if } \beta \in [1,2),\\
u(x) + \phi(\cdot-x)((\nabla_n)^1 u(x),\cdot-x)+ \tfrac{1}{2} \eta(\cdot-x)((\nabla_n)^2u(x)(\cdot-x),\cdot-x) & \textnormal{ if } \beta \in [2,3).
\end{array}\right.\end{aligned}$$ Associated with this, we introduce functions in $G_n$ taking (respectively) scalar, vector, and matrix values.
First, some notation. To functions $\eta,\phi \in \mathcal{S}$ we associate the following family of functions $$\begin{aligned}
\phi_{i}(y) = \phi(y)y_i,\;i=1,\ldots,d,\;\;\eta_{ij}(y) = \eta(y)y_iy_j,\;i,j=1,\ldots,d.\end{aligned}$$
Then, for $L\in\mathcal{D}I_n$ and $\eta,\phi \in \mathcal{S}$ we define a symmetric matrix $A_{L,\eta}$, a vector $B_{L,\phi}$, and a scalar $C_{L}$. These are functions in $G_n$ defined by the formulas, $$\begin{aligned}
{(A_{L,\eta}(x))}_{ij} & = L(\tau_{-x}\eta_{ij},x), \;i,j=1,\ldots,d, \label{eqn:AsubL definition}\\
{(B_{L,\phi}(x))}_i & = L(\tau_{-x}\phi_i,x), \;i=1,\ldots,d, \label{eqn:BsubL definition}\\
C_L(x) & = L(1,x). \label{eqn:CsubL definition}.\end{aligned}$$ The functions $A_{L,\eta},B_{L,\phi},C_{L},$ and $\mu_L$ give us a representation for $L(u,x)$ for $x\in G_n$.
\[proposition:L in DIn first representation formula\] Assume that $I$ is Lipschitz. Let $L\in \mathcal{D}I_n$, then for $\beta\in[2,3)$ and $u\in C^\beta_b(\real^d)$ we may write it as $$\begin{aligned}
L(u,x) & = C_L(x)u(x)+B_{L,\phi}(x)\cdot (\nabla_n)^1 u(x)+\tr(A_{L,\eta}(x)(\nabla_n)^2u(x))\\
& \;\;\;\;+\int_{\mathbb{R}^d} u(x+y)-P_{\phi,\eta,u,x}^{(n)}(x+y) \;\mu_L(x,dy).
\end{aligned}$$ For $\beta \in [1,2)$ $$\begin{aligned}
L(u,x) & = C_L(x)u(x)+B_{L,\phi}(x)\cdot (\nabla_n)^1 u(x)+\int_{\mathbb{R}^d} u(x+y)-P_{\phi,\eta,u,x}^{(n)}(x+y) \;\mu_L(x,dy),
\end{aligned}$$ and for $\beta \in [0,1)$ $$\begin{aligned}
L(u,x) & = C_L(x)u(x)+\int_{\mathbb{R}^d} u(x+y)-u(x)\;\mu_L(x,dy).
\end{aligned}$$
We do the case $\beta \geq 2$ explicitly, as the others are identical. Let us compute $L(u,x)$ by adding and subtracting $L(P_{\phi,\eta,u,x}^{(n)},x)$, $$\begin{aligned}
L(u,x) & = L(u-P_{\phi,\eta,u,x}^{(n)},x)+L(P^{(n)}_{\phi,\eta,u,x},x).
\end{aligned}$$ From Remark \[remark:structure of D little i n kernel\], , we have that $$\begin{aligned}
L(u-P_{\phi,\eta,u,x}^{(n)},x) = \int_{\mathbb{R}^d} u(x+y)-P_{\phi,\eta,u,x}^{(n)}(x+y)\;\mu_L(x,dy)
\end{aligned}$$ As for the other term, we observe that $$\begin{aligned}
L(P_{\phi,\eta,u,x}^{(n)},x) & = u(x)L(1,x) + \sum \limits_{i=1}^d (\nabla_1u)^n_i(x) L(\tau_{-x}\phi_i,x)+ \tfrac{1}{2}\sum \limits_{i,j=1}^d (\nabla_n)^2_{ij}u(x)L(\tau_{-x}\eta_{ij},x).
\end{aligned}$$ Rewriting the terms on the right and gathering the terms, we conclude that $$\begin{aligned}
L(P_{\phi,\eta,u,x}^{(n)},x) & = C_L(x)u(x) + (B_{L,\phi}(x),(\nabla_n)^1u(x)) + \tr(A_{L,\eta}(x)(\nabla_n)^2u(x)).
\end{aligned}$$ The remaining cases of $\beta$ follow from the corresponding definition of $P^{(n)}_{\phi,\eta,u}$ in those cases.
The next two propositions say that the terms appearing Proposition \[proposition:L in DIn first representation formula\] satisfy a uniform continuity in $G_n$. The first refers to the measure $\mu_L$.
\[proposition:TV norm continuity and tightness estimate for discrete Levy measures\] Assume $I$ satisfies Assumptions \[assumption:GCP\], \[assumption:tightness bound\], and \[assumption:coefficient regularity\], as stated for $C^\beta_b(\real^d)$. Let $L \in D I_n$, $x_1,x_2 \in G_n$, and $r\geq 2^{4-n}$. There is a constant $C(r)$ such that for any $\zeta \in C_c(\mathbb{R}^d)$ such that $\zeta \equiv 0$ in $B_r$, $$\begin{aligned}
\left |\int_{\mathcal{C}B_r} \zeta(y)\;\mu_L(x_1,dy)-\int_{\mathcal{C} B_r} \zeta(y)\;\mu_L(x_2,dy) \right | \leq C(r)\|\zeta\|_{L^\infty}\omega(|x_1-x_2|),
\end{aligned}$$ where $\omega$ is the modulus from Assumption \[assumption:coefficient regularity\]. In particular, $$\begin{aligned}
\left \| \mu_L(x_1,dy)-\mu_L(x_2,dy) \right \|_{\textnormal{TV}(\mathcal{C}B_r )} \leq C(r)\omega(|x_1-x_2|).
\end{aligned}$$ On the other hand, if $\zeta \in C^0(\mathbb{R}^d)$ is such that $\zeta\equiv 0$ in $B_{3R}(0)$ for some $R>1$, then for any $x_0 \in G_n$ we have $$\begin{aligned}
\int_{\mathbb{R}^d} \zeta(y) \;\mu_L(x_0,dy) \leq \rho(R)\|\zeta\|_{L^\infty(\mathbb{R}^d)},
\end{aligned}$$ where $\rho(\cdot)$ is the function from Assumption \[assumption:tightness bound\].
From the fact that $\tau_{-x_1}\zeta$ and $\tau_{-x_2}\zeta$ vanish in, respectively, $B_r(x_1)$ and $B_r(x_2)$, we have $$\begin{aligned}
L( \tau_{-x_1}\zeta,x_1) - L(\tau_{-x_2}\zeta,x_2) & = \int_{\mathbb{R}^d} \zeta(y)\;d\mu(x_1,dy)-\int_{\mathbb{R}^d} \zeta(y)\;d\mu(x_2,dy) \\
&= \int_{\mathcal{C}B_r} \zeta(y)\;d\mu(x_1,dy)-\int_{\mathcal{C}B_r} \zeta(y)\;d\mu(x_2,dy).
\end{aligned}$$ Since $\zeta \equiv 0$ in $B_r$, Proposition \[proposition:coefficient regularity for L\] says that, as long as $r\geq 2^{4-n}$ $$\begin{aligned}
& \left | \int_{\mathcal{C}B_r} \zeta(y)\;d\mu(x_1,dy)-\int_{\mathcal{C}B_r} \zeta(y)\;d\mu(x_2,dy) \right | \leq \omega(|x_1-x_2|)C(r)\|\zeta\|_{L^\infty(\mathcal{C} B_{r})}.
\end{aligned}$$ This proves the first estimate, for the second one, fix $\zeta$ and $x_0 \in G_n$, and define $w(x) = \tau_{-x_0}\zeta$, then $$\begin{aligned}
L(w,x_0) = \int_{\mathbb{R}^d} \zeta(y) \;\mu_L(x_0,dy).
\end{aligned}$$ Therefore, as before, it suffices for us to bound $L(w,x_0)$ for every $L \in \mathcal{D}I_n$, and from the definition of $\mathcal{D}I_n$ it suffices to prove the bound for those $L$ such that $L = DI_n(v)$ at some $v$. In this case, Proposition \[proposition:tightness bound for I\_n\] says that $$\begin{aligned}
L(w,x_0) = \lim\limits_{s\to 0} \frac{1}{s} (I_n(v+sw,x_0)-I_n(v,x_0)) \leq \rho(R)\|w\|_{L^\infty(\mathbb{R}^d)} = \rho(R)\|\zeta\|_{L^\infty(\mathbb{R}^d)}
\end{aligned}$$
The following notation will be useful in what follows, $$\begin{aligned}
\alpha(r,\eta) & := C(2r)\left ( \max\limits_{1\leq i,j\leq d}\|\eta_{ij} \|_{C^\beta(B_{4r})} +\max\limits_{1\leq i,j\leq d}\|\eta_{ij} \|_{L^\infty(\mathcal{C}B_r)}\right ),\\
\beta(r,\phi) & := C(2r)\left ( \max\limits_{1\leq i\leq d}\|\phi_i \|_{C^\beta(B_{4r})} +\max\limits_{1\leq i\leq d}\|\phi_i \|_{L^\infty(\mathcal{C}B_r)}\right ), \end{aligned}$$ where $C(r)$ is as in Assumption \[assumption:coefficient regularity\] (see also Proposition \[proposition:coefficient regularity inherited by In\]).
\[proposition:discrete coefficient regularity for L in DI\_n\] Assume $I$ satisfies Assumptions \[assumption:GCP\], \[assumption:tightness bound\], and \[assumption:coefficient regularity\], as stated for $C^\beta_b(\real^d)$. Let $L \in \mathcal{D}I_n$, $r\geq 2^{4-n}$, and $x_1,x_2 \in G_n$, then $$\begin{aligned}
|A_{L,\eta}(x_1)-A_{L,\eta}(x_2)| & \leq \alpha(r,\eta)\omega(|x_1-x_2|),\\
|B_{L,\phi}(x_1)-B_{L,\phi}(x_2)| & \leq \beta(r,\phi)\omega(|x_1-x_2|),\\
|C_{L}(x_1)-C_{L}(x_2)| & \leq C(r)\omega(|x_1-x_2|).
\end{aligned}$$
Fix $x_1,x_2 \in G_n$ and let $h = x_2-x_1$. Applying Proposition \[proposition:coefficient regularity for L\] to $x=x_1$ and $h$, with the functions $1$, $\phi_i$, and $\eta_{ij}$, we see that for $r\geq 2^{4-n}$ $$\begin{aligned}
|L(\tau_{-x_2}\eta_{ij},x_2)-L(\tau_{-x_1}\eta_{ij},x_1)| & \leq \alpha(\eta,r)\omega(|x_1-x_2|),\\
|L(\tau_{-x_2}\phi_i,x_2)-L(\tau_{-x_1}\phi,x_1)| & \leq \beta(\phi,r)\omega(|x_1-x_2|),\\
|L(1,x_2)-L(1,x_1)| & \leq C\omega(|x_1-x_2|).
\end{aligned}$$ These inequalities respectively amount to the stated estimate for $A_{L,\eta}$, $B_{L,\phi}$, and $C_L$.
Properties of $\mathcal{D}_I$
------------------------------
Now, we define the set $\mathcal{D}_I$, which plays the role the Clarke differential played for $I_n$ (we recall that c.h. stands for “convex hull”). $$\begin{aligned}
\label{equation:DI definition}
\mathcal{D}_I := \textnormal{c.h.}\{ L \mid \exists \{L_{n_k}\}, n_k\to\infty,\; L_{n_k} \in \mathcal{D}I_{n_k} \textnormal{ s.t } L(u,\cdot) = \lim\limits_{k} L_{n_k}(u,\cdot)\;\forall\;u \}.\end{aligned}$$
\[remark:Clarke differential in the limit\] We would like to note a point about notation and definitions, namely why above we have $\mathcal{D}_I$ with $I$ as a subscript. This is to avoid confusion (or perhaps, to promote it) by distinguishing it from the generalized derivative in the sense of Clarke from Definition \[definition:Clarke differential\]. The objects are closely related, and in fact one would hope that $\mathcal{D}_I = \mathcal{D}I$, but we are not concerned with whether this is actually the case as the above definition works for our purposes.
The following is an important Lemma that says –among other things– that $\mathcal{D}_I$ is non-empty.
\[lemma:DI\_n sequences subconverge\] Assume $I$ satisfies Assumptions \[assumption:GCP\], \[assumption:tightness bound\], and \[assumption:coefficient regularity\], as stated for $C^\beta_b(\real^d)$. Given a sequence $n_k \to\infty$ and operators $L_{n_k}$ with $L_{n_k} \in \mathcal{D}I_{n_k}$ for every $k$, and $\phi,\eta \in \mathcal{S}$ we have the following
1. There is a subsequence $\bar n_k$ and functions $A(x),B(x),$ and $C(x)$ defined on $\mathbb{R}^d$ and taking values respectively in $\mathbb{S}(d)$, $\mathbb{R}^d$, and $\mathbb{R}$, such that if $x \in G_n$ for some $n$ then we have the convergence $$\begin{aligned}
A_{L_{\bar n_k},\eta}(x) \to A(x),\; B_{L_{\bar n_k},\phi}(x) \to B(x),\; C_{L_{\bar n_k}}(x) \to C(x).
\end{aligned}$$
2. There is a function $\mu(x)$ in $\mathbb{R}^d$, taking values on the space of Lévy measures in $\mathbb{R}^d$, such that for every $r>0$, and every $x$ as before we have the convergence $$\begin{aligned}
\lim \limits_{k\to \infty } \| \mu_{L{\bar n_k}}(x)-\mu(x)\|_{\textnormal{TV}(\mathcal{C}B_r)} = 0.
\end{aligned}$$
3. The functions $A,B,C,$ all have a modulus of continuity $C\omega(2(\cdot))$, while for each $r>0$ we have the estimate, $$\begin{aligned}
\label{equation:continuity estimate mu_L}
\|\mu(x_1)-\mu(x_2)\|_{\textnormal{TV}(\mathcal{C}B_r)} \leq C(r)\omega(2|x_1-x_2|).
\end{aligned}$$
4. If we define $L$ by $$\begin{aligned}
L(u,x) & := \textnormal{tr}(A(x)D^2u(x))+B(x)\cdot \nabla u(x) + C(x)u(x)\\
& \;\;\;\;+ \int_{\mathbb{R}^d} u(x+y)-P_{\phi,\eta,u,x}(x+y) \;\mu(x,dy)
\end{aligned}$$ Then, $L \in \mathcal{D}_I$.
5. Moreover, if $\beta<2$, then we have $A(x)\equiv0$. Furthermore, if $\beta<1$ then $B(x)\equiv 0$ and $L$ takes the form $$\begin{aligned}
L(u,x) = C(x)u(x) + \int_{\mathbb{R}^d} u(x+y)-u(x) \;\mu(x,dy).
\end{aligned}$$
Let us fixe $\eta$ and $\phi$. First of all, we invoke Proposition \[proposition:L in DIn first representation formula\] to obtain the collection of $A_{L_{n_k},\eta}$, $B_{L_{n_k},\phi}$, $C_{L_{n_k}}$, and $\mu_{L_{n_k}}$. Furthermore, already as a result of Proposition \[proposition:L in DIn first representation formula\], we have item (5) of the lemma.
*Step 1.* (Extension) We have a sequence of functions defined on varying, monotone increasing sets $G_n$. One way to show they converge (along a subsequence) to a function in $\mathbb{R}^d$ is by extending them to all of $\mathbb{R}^d$ and check whether the resulting sequences are pre-compact.
With this idea in mind, for each $n\in\mathbb{N}$ we apply the Whitney extension to $A_{L_n,\eta}$, $B_{L_n,\eta}$, $C_{L_n,\eta}$, $$\begin{aligned}
\hat A_{L_n,\eta}(x) := E_n^0 (A_{L_n,\eta})(x),\; \hat B_{L_n,\phi}(x) := E_n^0 (B_{L_n,\phi})(x),\;\hat C_{L_n}(x) := E_n^0 (C_{L_n})(x).
\end{aligned}$$ We repeat the same for $\mu_{L_{n}}$, resulting in a map $\hat \mu_{L_n}$ from $\mathbb{R}^d$ to the space of Lévy measures, given by the formula $$\begin{aligned}
\hat \mu_{L_n}(x,dy) = \sum \limits_{k=1}^\infty \phi_{n,k}(x)\mu(x_k,dy),
\end{aligned}$$ where $\{\phi_{k}\}_k$ is the partition of unity from Proposition \[proposition:partition of unity properties\]. The functions $\hat A_{L_{n},\eta}$, $\hat B_{L_{n},\phi}$, and $\hat C_{L_{n}}(x)$ all have modulus of continuity $C\omega(2(\cdot))$, thanks to Proposition \[proposition:discrete coefficient regularity for L in DI\_n\] and the properties of the Whitney extension operator, see [@Stei-71 Chapter VI, Theorem 3]. The same proof from reference [@Stei-71] can be applied with minor modifications to show that for every $r>0$ we have $$\begin{aligned}
\|\hat \mu_{L_n}(x_1)-\hat \mu_{L_n}(x_2)\|_{\textnormal{TV}(\mathcal{C}B_r)} \leq C(r) \omega(2|x_1-x_2|).
\end{aligned}$$ Furthermore, for every $x$, by Proposition \[proposition:TV norm continuity and tightness estimate for discrete Levy measures\], $$\begin{aligned}
|\hat \mu_{L_n}(x)|(\mathcal{C}B_R) \leq \rho(R),
\end{aligned}$$ where $\rho(R) \to 0$ as $R\to \infty$. This shows that for each $r>0$, the functions $\{ \hat \mu_{L_n}\mid_{\mathcal{C}B_r}\}_n$ are an equicontinuous family of functions taking values inside the space of measures $\nu$ which are supported in $\mathcal{C}B_r$ and such that $\nu(\mathcal{C}B_R) \leq \rho(R)$ for all $R\geq r$. This space, equipped with the total variation distance, is a compact metric space.
*Step 2.* (Cantor diagonalization) We now use a standard Cantor diagonalization argument to obtain locally uniform convergence along a subsequence. We construct a family nested sequences $\tilde n^m_k$ in the following recursive manner. First, $\tilde n^1_k$ is a subsequence of $n_k$ along which the functions converge uniformly in $B_1$ to functions $A^1(x),B^1(x)$, and $C^1(x)$) defined in $B_1$. Next, suppose that for $m \in \mathbb{N}$ we have build a nested family of sequences $\tilde n^1_k,\ldots,\tilde n^m_k$ such that the functions $A_{L_{\tilde n^m_k},\eta},\ldots$, etc converge uniformly in $B_m(0)$ to functions $A^m(x)\ldots$, etc. In this case, we choose $\tilde n^{m+1}_k$ to be a subsequence of $\tilde n^m_k$ along which $A_{L_{\tilde n^{m+1}_k},\eta},\ldots$ converge uniformly in $B_{m+1}$ to functions $A^{m+1}(x)\ldots$ and so on.
Having constructed these $\tilde n^m_k$, we define the sequence $\tilde n_k$ as $\tilde n_k := n^k_k$. The resulting sequences converge locally uniformly, respectively, to $A(x),B(x)$, and $C(x)$.
*Step 3.* (Cantor diagonalization continued)
As noted at the end of Step 1, for every $r>0$, the sequence $\{\hat \mu_{L_{\tilde n_k}}\}_k$ is an equicontinuous family of functions taking values in a compact metric space. Therefore, we can apply the Arzela-Ascoli type theorem found in [@GreenValentine1961arzela p. 202] to obtain a subsequence $\bar n^1_k$ of $\tilde n_k$ and a measure $\mu^1$ such that $$\begin{aligned}
\lim \limits_{k\to \infty}\sup \limits_{x\in B_1} \| \hat \mu_{L_{\bar n_k^1}}(x)-\mu^1( x)\|_{\textnormal{TV}(\mathcal{C}B_{1/2})} = 0.
\end{aligned}$$ Now, suppose we have repeated this $m$ times: we have $\bar n^m_k$ (a subsequence of $\bar n^{m-1}_k$), as well as a measure $\mu^m$ such that $$\begin{aligned}
\lim \limits_{k\to \infty} \sup \limits_{x\in B_m} \| \hat \mu_{L_{\bar n_k^m}}(x)-\mu^m( x)\|_{\textnormal{TV}(\mathcal{C}B_{1/2^m})} = 0.
\end{aligned}$$ Then, using again the compactness theorem in [@GreenValentine1961arzela p. 202] we pick a subsequence $\bar n^{m+1}_k$ of $\bar n^m_k$ and a measure $\mu^{m+1}$ such that $$\begin{aligned}
\lim \limits_{k\to \infty} \sup \limits_{x\in B_{m+1}} \| \hat \mu_{L_{\bar n_k^m}}(x)-\mu^{m+1}( x)\|_{\textnormal{TV}(\mathcal{C}B_{1/2^{m+1}})} = 0.
\end{aligned}$$ Observe that the measures $\{ \mu^{m}\}$ are such that $\mu^{m+1}_{\mid \mathcal{C}B_{1/2^m}}(x) = \mu^m(x)$ for all $x\in B_m$, which uniquely defines a direct limit measure $\mu(x)$ for each $x\in \mathbb{R}^d\setminus \{0\}$. Letting $\bar n_k := \bar n_k^k$ we see that for every $R>0$ and $r>0$ we have $$\begin{aligned}
\lim \limits_{k\to \infty} \sup \limits_{x\in B_{R}} \| \hat \mu_{L_{\hat n_k^k}}(x)-\mu( x)\|_{\textnormal{TV}(\mathcal{C}B_{r})} = 0.
\end{aligned}$$ Since $\bar n_k$ is a subsequence of $\tilde n_k$, we still have convergence of $A_{L_{\bar n_k},\eta},\ldots$ to $A(x),\ldots$. Moreover, the continuity estimates in the previous step all pass to the limit to give respective estimates for $A(x),B(x),C(x),$ and $\mu(x)$ in the respective metrics.
Last but not least, we note that while $\{\mu_{L_{\bar n_k}}\}_k$ are a sequence of signed measures, their limit $\mu$ will be a measure, which follows at once from Proposition \[proposition:DI\_n Borel measures negative part\].
*Step 4.* (Convergence)
First, note that for fixed $u$, we have that as $n\to \infty$, $$\begin{aligned}
u(x+\cdot)-P^{(n)}_{\phi,\eta,u,x}(x+\cdot) \to u(x+\cdot)-P_{\phi,\eta,u,x}(x+\cdot) \textnormal{ in } L^\infty(\mathbb{R}^d),
\end{aligned}$$ which in particular guarantees that, for every fixed $r>0$, $$\begin{aligned}
\lim \limits_{k\to \infty} \int_{\mathcal{C}B_r}u(x+y)-P^{(n_k)}_{\phi,\eta,u,x}(x+y)\;\mu_{L_{n_k}}(x,dy) = \int_{\mathcal{C}B_r}u(x+y)-P_{\phi,\eta,u,x}(x+y)\;\mu(x,dy).
\end{aligned}$$ Then, by the bound in Proposition \[proposition:DI\_n Borel measures integrability\], we conclude that $$\begin{aligned}
\lim \limits_{k\to \infty} \int_{\mathbb{R}^d}u(x+y)-P^{(n_k)}_{\phi,\eta,u,x}(x+y)\;\mu_{L_{n_k}}(x,dy) = \int_{\mathbb{R}^d}u(x+y)-P_{\phi,\eta,u,x}(x+y)\;\mu(x,dy).
\end{aligned}$$ Therefore, and taking into account the convergence of $\hat A_{L_{\tilde n_k},\eta}, \hat B_{L_{\tilde n_k},\phi},$ and $\hat C_{L_{\tilde n_k}}$, and with $L(u,x)$ defined as in the statement of the Lemma, $x\in G_n$, and $u\in C^{\beta}_b(\mathbb{R}^d)$, we have $$\begin{aligned}
\lim \limits_{k \to \infty} L_{\tilde n_k}(x) & = \lim\limits_{k\to \infty} \big \{ \tr(\hat A_{L_{\tilde n_k},\eta}D^2u(x)) + \hat B_{L_{\tilde n_k},\phi}\cdot \nabla u(x) + \hat C_{L_{\tilde n_k}}(x)u(x) \big \} \\
& \;\;\;+ \lim \limits_{k\to \infty}\int_{\mathbb{R}^d} u(x+y)-P^{(\tilde n_k)}_{\phi,\eta,u,x}(x+y) \;\hat \mu_{L_{n_k}}(x,dy)\\
& = \tr(AD^2u(x)) + B\cdot \nabla u(x) + C(x)u(x)\\
& \;\;\;+ \int_{\mathbb{R}^d}u(x+y)-P_{\phi,\eta,u,x}(x+y) \;\hat \mu(x,dy),
\end{aligned}$$ and we conclude that $L \in \mathcal{D}_I$.
It is to be expected that every $L \in \mathcal{D}_I$ satisfies the GCP, and thus, it has to be an operator of Lévy type. This is proved in the lemma below, and further, we show that the coefficients in the operator inherit a modulus of continuity from Assumption \[assumption:coefficient regularity\].
\[lemma:L in DI representation formula first form\] Assume $I$ satisfies Assumptions \[assumption:GCP\], \[assumption:tightness bound\], and \[assumption:coefficient regularity\], as stated for $C^\beta_b(\real^d)$. Given $L \in \mathcal{D}_I$, and any $\phi, \eta \in \mathcal{S}$, the operator $L$ can be represented as $$\begin{aligned}
L(u,x) & = C_L(x)u(x)+B_{L,\phi}(x)\cdot \nabla u(x)+\tr(A_{L,\eta}(x)D^2u(x))\\
& \;\;\;\;+ \int_{\mathbb{R}^d}u(x+y)-P_{\phi,\eta,u,x}(x+y)\;\mu(x,dy).
\end{aligned}$$ Here, $\mu_L(x,dy)$ is a Lévy measure satisfying the continuity estimate , and $$\begin{aligned}
(A_{L,\eta})_{ij}(x) & = L( \tau_{-x}\eta_{ij},x),\\
(B_{L,\phi})_i(x) & = L( \tau_{-x}\phi_{i},x),\\
C_{L}(x) & = L(1,x),
\end{aligned}$$ all have modulus of continuity $C\omega(2(\cdot))$.
Fix $\phi,\eta \in \mathcal{S}$. Assume first that $L$ is the limit of a sequence $L_{n_k}$ with $L_{n_k} \in \mathcal{D}I_{n_k}$. Then, by Lemma \[lemma:DI\_n sequences subconverge\] there is a subsequence $\tilde n_k$ as well as (matrix, vector, scalar, measure)-valued functions $A,B,C$, and $\mu$, all such that $$\begin{aligned}
C_{L_{\tilde n_k}}(x) \to C(x),\; B_{L_{\tilde n_k},\phi_k}(x) \to B(x),\; A_{L_{\tilde n_k},\eta_k}(x) \to A(x),\;\mu_{L_{\tilde n_k}}(x,dy) \to \mu(x,dy).
\end{aligned}$$ and, as a result, we have $$\begin{aligned}
L(u,x) & = \tr(A(x)D^2u(x))+B(x)\cdot \nabla u(x)+C(x)u(x)\\
& \;\;\;\;+ \int_{\mathbb{R}^d} u(x+y)-P_{\phi,\eta,u,x}(y)\;\mu(x,dy).
\end{aligned}$$ The estimate in Proposition \[proposition:DI\_n Borel measures integrability\] in the limit as $n\to \infty$ implies that $$\begin{aligned}
\int_{\mathbb{R}^d} \eta_0(|y|^{\beta})\;\mu(x,dy) \leq C,
\end{aligned}$$ for some constant $C$ independent of $x$ and $L$. Meanwhile, also the $n\to \infty$ limit of the estimate in Proposition \[proposition:DI\_n Borel measures negative part\] implies that $\mu(x,dy)$ is a non-negative measure in $\mathbb{R}^d\setminus \{0\}$. The positivity of $\mu$ means that the previous estimate is equivalent to $$\begin{aligned}
\int_{\mathbb{R}^d} \min\{1,|y|^{\beta}\}\;\mu(x,dy) \leq C.
\end{aligned}$$ Since $L_{\tilde n_k}(u,x) \to L(u,x)$, for every $u$, we have in particular, for $x \in \bigcup G_k$ $$\begin{aligned}
(A_{L_{\tilde n_k},\eta})_{ij}(x) = L_{\tilde n_k}(\tau_{-x} \eta_{ij},x) \to L(\tau_{-x}\eta_{ij},x).
\end{aligned}$$ From where it follows that $(A_{L,\eta})_{ij}(x) = L( \tau_{-x}\eta_{ij},x)$ (and thus for all $x$, by continuity), the exact same argument yields that $ (B_{L,\phi})_i(x) = L( \tau_{-x}\phi_{i},x)$, and $C_{L}(x) = L(1,x)$, and the lemma is proved.
Let us now simplify things by doing away with the auxiliary functions $\phi$ and $\eta$. To accomplish this, we shall make use of the auxiliary functions from Section \[section:Functionals with the GCP\]. $$\begin{aligned}
\label{equation:phi delta and psi delta recalled}
\phi_{\delta}(x) = \psi_{\delta,1-\delta},\;\eta_{\delta}(x) = \psi_{\delta,\delta}(x),\end{aligned}$$ where we recall the two-parameter of functions $\psi_{r,R}(x)$ was defined in . An important property of these one-parameter families is the bound $$\begin{aligned}
\label{equation:auxiliary phi and eta functions are bounded uniformly in delta}
\sup \limits_{\delta \in (0,1)} \{ \|\phi_{\delta}\|_{C^\beta(B_{1/2})} + \|\phi_\delta\|_{L^\infty(\mathbb{R}^d)} + \max \limits_{ij}\|\eta_{\delta}x_ix_j\|_{C^\beta(\mathbb{R}^d)} \} < \infty.\end{aligned}$$
\[corollary:L in DI representation formula final form\] Assume $I$ satisfies Assumptions \[assumption:GCP\], \[assumption:tightness bound\], and \[assumption:coefficient regularity\], as stated for $C^\beta_b(\real^d)$. Then, any $L \in \mathcal{D}_I$ has the form, $$\begin{aligned}
L(u,x) & = C(x)u(x)+B(x)\cdot \nabla u(x)+\tr(A(x)D^2u(x))\\
& \;\;\;\;+ \int_{\mathbb{R}^d}u(x+y)-u(x)-\chi_{B_1(0)}(y)\nabla u(x)\cdot y\;\mu(x,dy).
\end{aligned}$$ Moreover, $A,B,$ and $C$ each have modulus of continuity $C\omega(2(\cdot))$, and for every $r>0$ and any $x_1,x_2\in\mathbb{R}^d$ we have $$\begin{aligned}
\|\mu_L(x_1)-\mu_{L}(x)\|_{\textnormal{TV}(\mathcal{C} B_r)} \leq C(r) \omega(2|x_1-x_2|).
\end{aligned}$$ If $\beta<2$, then $A\equiv 0$, while if $\beta<1$ then $B \equiv 0$ and the integrand with respect to $\mu(x,dy)$ in the formula above is replaced with $u(x+y)-u(x)$.
Take a decreasing sequence $\delta_k$ such that $\delta_k \to 0$, and let us take the functions $\phi_{\delta_k}$ and $\eta_{\delta_k}$, as defined in . Then for each $k$, $L$ has the representation $$\begin{aligned}
L(u,x) & = C_L(x)u(x)+B_{L,\phi_{\delta_k}}(x)\cdot \nabla u(x)+\tr(A_{L,\eta_{\delta_k}}(x)D^2u(x))\\
& \;\;\;\;+ \int_{\mathbb{R}^d}u(x+y)-P_{\phi_{\delta_k},\eta_{\delta_k},u,x}(x+y)\;\mu(x,dy),
\end{aligned}$$ where $A_{L,\eta_{\delta_k}}$, $B_{L,\phi_{\delta_k}},$ and $C_{L}$ are as in Lemma \[lemma:L in DI representation formula first form\]. Now, $L$ satisfies the estimate $$\begin{aligned}
|L(\tau_{-x_1}(\eta_{\delta_k})_{ij},x_1)-L(\tau_{-x_2}(\eta_{\delta_k})_{ij},x_2)| \leq \alpha(1,\eta_{\delta_k})\omega(2|x_1-x_2|)
\end{aligned}$$ Thanks to , it follows that $\alpha(1,\eta_{\delta_k}) \leq C$ for all $k$. It follows that $\{A_{L,\eta_{\delta_k}}\}_k$ has a uniform modulus of continuity. The same argument yields a modulus of continuity for $\{B_{L,\phi_{\delta_k}}\}_k$ and for the function $C(x)$, all given by $C\omega(2|x_1-x_2|)$, with $C$ independent of $k$ and $\omega$ being the modulus from Assumption \[assumption:coefficient regularity\]. This equicontinuity means these sequences of functions are pre-compact at least when restricted to any compact subset of $\mathbb{R}^d$, by the Arzela-Ascoli theorem. Therefore, after a Cantor diagonalization argument we see that along some subsequence $m_k \to \infty$ these functions converge locally uniformly in $\mathbb{R}^d$ to functions $A(x)$, $B(x)$, respectively. Of course, the functions $A,B,$ and $C$ all inherit the modulus of continuity $C\omega(2(\cdot))$. The respective TV-norm continuity estimate for $\mu_L$ follows by applying Proposition \[proposition:TV norm continuity and tightness estimate for discrete Levy measures\] and passing to the limit (always recalling that, $\mathcal{D}_I$ is the convex hull of such limit points).
With the convergence established, we have $$\begin{aligned}
\lim\limits_{k\to \infty}\big ( B_{L,\phi_{\delta_{m_k}}}(x)\cdot\nabla u(x) + \tr(A_{L,\eta_{\delta_{m_k}}}(x)D^2u(x)) \big ) = B(x)\cdot\nabla u(x) + \tr(A(x)D^2u(x)),
\end{aligned}$$ and so, for every $u$ we have the formula $$\begin{aligned}
L(u,x) & = C(x)u(x)+B(x)\cdot \nabla u(x)+\tr(A(x)D^2u(x))\\
& \;\;\;\;+ \lim\limits_{k \to \infty}\int_{\mathbb{R}^d}u(x+y)-P_{\phi_{\delta_{k}},\eta_{\delta_k},u,x}(x+y)\;\mu(x,dy),
\end{aligned}$$ It remains to compute the limit of the integral, observe that $$\begin{aligned}
\int_{\mathbb{R}^d}\eta_{\delta_k}(y)(D^2u(x)y,y)\;\mu(x,dy) = \int_{B_{\delta_k}}\eta_{\delta_k}(y)(D^2u(x)y,y)\;\mu(x,dy),
\end{aligned}$$ which means that $$\begin{aligned}
\left | \int_{\mathbb{R}^d}\eta_{\delta_k}(y)(D^2u(x)y,y)\;\mu(x,dy) \right | \leq C|D^2u(x)| \int_{B_{\delta_k}} |y|^2 \;d\mu(x,dy).
\end{aligned}$$ Therefore, $$\begin{aligned}
\lim\limits_{k\to 0}\int_{\mathbb{R}^d}\eta_{\delta_k}(y)(D^2u(x)y,y)\;\mu(x,dy) = 0.
\end{aligned}$$ On the other hand, for every $y$ we have $$\begin{aligned}
\lim \limits_{k\to \infty} \Big ( u(x+y)-P_{\phi_{\delta_k},\eta_{\delta_k},u,x}(y)\Big) = u(x+y)-u(x)-\chi_{B_1}(y)\nabla u(x)\cdot y,
\end{aligned}$$ and the limit is monotone. Therefore, by monotone convergence we conclude that $$\begin{aligned}
\lim\limits_{k\to \infty} \int_{\mathbb{R}^d}u(x+y)-P_{\phi_{\delta_k},\eta_{\delta_k},u,x}(y)\;\mu(x,dy) = \int_{\mathbb{R}^d}u(x+y)-u(x)-\chi_{B_1}(y)\nabla u(x)\cdot y\;\mu(x,dy).
\end{aligned}$$ and with this the Corollary is proved.
Limits of $I_n$
---------------
\[lemma:I\_n converges to Iu for nice u\] Assume that $I:C^\beta_b(\real^d)\to C^0_b(\real^d)$ is Lipschitz. Let $K>0$ and $0<\beta<\beta_0<3$. If $u \in C^{\beta_0}_b(\mathbb{R}^d)$ is supported in $B_K$, and $2^{n-2}\geq K$, then $$\begin{aligned}
\|I_nu-Iu\|_{L^\infty(B_K \cap G_n)} \leq C2^{-n \gamma} \|u\|_{C^{\beta_0}(\mathbb{R}^d)},
\end{aligned}$$ for a universal constant $C$ and $\gamma= \gamma(\beta_0,\beta) \in (0,1)$. Furthermore, we have $$\begin{aligned}
\lim \limits_{n\to\infty} \|I(u)-I_n(u)\|_{L^\infty(B_K)} = 0.
\end{aligned}$$
Let $u$ be compactly supported in $B_K$, and be such that $\|u\|_{C^{\beta_0}} \leq M$. First, note that since $2^{n-2} \geq K$, then we have $$\begin{aligned}
\hat \pi_n^\beta u = \pi_n^\beta u,
\end{aligned}$$ thus, $I_n(u) = \hat \pi_n^0\circ I \circ \pi_n^\beta (u)$. Keeping this in mind, using the Lipschitz property of $I$, we have $$\begin{aligned}
\|I(u)-I(\hat \pi_n^\beta u)\|_{L^\infty(\mathbb{R}^d)} \leq C\|u-\pi_n^\beta u\|_{C^\beta(\mathbb{R}^d)}.
\end{aligned}$$ Since $2^{n-2}\geq K$ we have that $I(\hat \pi_n^\beta u) = \hat \pi_n^0 I(\hat \pi_n^\beta u) = I_n(u)$ when restricted to $B_K \cap G_n$, which thanks to Lemma \[lemma:projection operators convergence\] implies the first estimate. Next, Theorem \[theorem:Whitney Extension Is Bounded\] guarantees that $$\begin{aligned}
\|\hat \pi_n^0 I(u)-\hat \pi_n^0 I(\hat \pi_n^\beta u)\|_{L^\infty(K)} \leq C\|I(u)-I(\hat \pi_n^\beta u)\|_{L^\infty(\mathbb{R}^d)} \leq C\|u- \pi_n^\beta u\|_{L^\infty(\mathbb{R}^d)}.
\end{aligned}$$ Thus, $$\begin{aligned}
\|I_n(u)-I(u)\|_{L^\infty(K)} & \leq \|\hat \pi_n^0 I(u)-I_n(u)\|_{L^\infty(K)} + \|\hat \pi_n^0( I(u))-I(u)\|_{L^\infty(K)}\\
& \leq C \|u- \pi_n^\beta u\|_{C^\beta(\mathbb{R}^d)} + \|\hat \pi_n^0 (I(u))-I(u) \|_{L^\infty(K)}.
\end{aligned}$$ Applying Lemma \[lemma:projection operators convergence\] to the first term and Remark \[remark:projection operators C0 convergence\] to the second, we conclude that $$\begin{aligned}
\lim\limits_{n\to \infty}\|I_nu-Iu\|_{L^\infty(K)} =0.
\end{aligned}$$
\[corollary:I\_n pointwise convergence to I\] Assume $I$ satisfies Assumptions \[assumption:GCP\], \[assumption:tightness bound\], and \[assumption:coefficient regularity\], as stated for $C^\beta_b(\real^d)$. Then for every $u \in C^\beta_b(\mathbb{R}^d)$ and every $R>0$, $$\begin{aligned}
\lim \limits_{n\to \infty} \|I_nu-Iu\|_{L^\infty(B_R)} = 0.
\end{aligned}$$
Fix $u \in C^\beta_b(\mathbb{R}^d)$ and $R,\varepsilon>0$. For $K>0$ (to be determined later), we may decompose $u$ as $u = u_0+u_1$, where $u_0$ is compactly supported in $B_{2K+1}$ and $u_1 \equiv 0$ in $B_{2K}$, all such that $$\begin{aligned}
\|u_i \|_{C^\beta(\mathbb{R}^d)} \leq C\|u\|_{C^\beta(\mathbb{R}^d)},\;i=1,2.
\end{aligned}$$ The constant $C>1$ being independent of $K$. Now, by Assumption \[assumption:tightness bound\] and since $u \equiv u_0$ in $B_{2K}$, we have $$\begin{aligned}
|I(u_0)-I(u)| \leq \rho(K)\|u-u_0\|_{L^\infty(\mathbb{R}^d)} \leq 2C\rho(K)\|u\|_{C^\beta(\mathbb{R}^d)}.
\end{aligned}$$ Choose $K$ large enough so that $K\geq 2R$ and $2C\rho(R)\|u\|_{C^\beta(\mathbb{R}^d)} \leq \varepsilon/2$. Then, with this $K$, we apply Lemma \[lemma:I\_n converges to Iu for nice u\] two times, and conclude that there is some $n_0>0$ such that $$\begin{aligned}
|I_n(u_0)-I(u_0)| + |I_n(u_0)-I_n(u)| \leq \varepsilon/2 \textnormal{ whenever } n \geq n_0.
\end{aligned}$$ On the other hand, in all $\mathbb{R}^d$ we have the pointwise inequality, $$\begin{aligned}
|I_n(u)-I(u)| \leq |I_n(u_0)-I(u_0)| + |I_n(u_0)-I_n(u)| + |I(u_0)-I(u)|,
\end{aligned}$$ and it follows that, for $x \in B_{R}$ and $n\geq n_0$, that $$\begin{aligned}
|I_n(u,x)-I(u,x)| \leq \varepsilon,
\end{aligned}$$ and the corollary is proved.
Proofs of Theorems \[theorem:MinMax Euclidean ver2\] and \[theorem:minmax with beta less than 2\] {#sec:TheoremsThatUseWhitney}
-------------------------------------------------------------------------------------------------
We conclude this section with the proofs of the remaining theorems.
Consider the set $\mathcal{D}_I$. The proof will boil down to showing that for any $u,v\in C^{\beta_0}_c(\mathbb{R}^d)$ and any $x\in \mathbb{R}^d$ there is some $L\in \mathcal{D}_I$ such that $$\begin{aligned}
I(u,x) \leq I(v,x)+L(u-v,x). \end{aligned}$$ Fix $u,v$ and $x$. Then, by Remark \[remark:MinMax for In\], for every $n$ we have $$\begin{aligned}
I_n(u,x) \leq \max \limits_{L_n\in \mathcal{D}I_n} \{ I_n(v,x)+L_n(u-v,x)\}.\end{aligned}$$ In particular, for every $n$, there is some $L_n \in \mathcal{D}I_n$ such that (with this same $u,v$ and $x$) $$\begin{aligned}
I_n(u,x) \leq I_n(v,x)+L_n(u-v,x). \end{aligned}$$ Let us obtain an inequality as we let $n\to \infty$ along some subsequence. Thanks to Corollary \[corollary:I\_n pointwise convergence to I\], for every $x \in \mathbb{R}^d$ we have $$\begin{aligned}
\lim \limits_{n\to \infty} I_n(u,x) = I(u,x),\;\;\lim\limits_{n\to\infty} I_n(v,x) = I(v,x).\end{aligned}$$ On the other hand, Lemma \[lemma:DI\_n sequences subconverge\] says there is a subsequence $n_k$ and an operator $L$ such that $L_{n_k}(u-v,x)$ converges to $L(u-v,x)$, and moreover $L\in\mathcal{D}_I$, by the definition of $\mathcal{D}_I$. Then, we conclude that $$\begin{aligned}
I(u,x) & \leq I(v,x)+L(u-v,x) \leq \sup \limits_{L\in \mathcal{D}_I} \{ I(v,x)+L(u-v,x)\}.\end{aligned}$$ The above holds for any pair of functions $u$ and $v$ and any point $x\in\mathbb{R}^d$. Taking the minimum over all $v$, we obtain for any $u$ and $x$, $$\begin{aligned}
I(u,x) & = \min \limits_{v\in C^\beta_b(\mathbb{R}^d)} \max \limits_{L\in \mathcal{D}_I} \left \{ I(v,x)-L(v,x)+L(u,x) \right \}. \end{aligned}$$ Using $v\in C^\beta_b(\mathbb{R}^d)$ and $L \in \mathcal{D}_I$ as the set of labels, which we rename $ab$, and letting $f_{ab}(x)$ correspond to the functions $I(v,x)-L(v,x)$, we obtain the desired min-max representation.
The $L^\infty$ bounds for the coefficients follow from the construction of $A_{\eta_k}$, etc... in (\[eqn:AsubL definition\]), (\[eqn:BsubL definition\]), (\[eqn:CsubL definition\]). The continuity of the coefficients and the Lévy measures follows from Lemma \[lemma:DI\_n sequences subconverge\].
For the versions of Theorems \[theorem:MinMax Euclidean\] and \[theorem:MinMax Translation Invariant\] with $\beta<2$ we apply the last part of Lemma \[lemma:Courrege theorem for a linear functional\] to conclude the functionals (or translation invariant operators) appearing in the min-max all have the corresponding simpler form. As for Theorem \[theorem:MinMax Euclidean ver2\], we use instead the last part of Corollary \[corollary:L in DI representation formula final form\] to obtain the simpler expresion for the Lévy operators in the cases where $\beta<2$.
Some Examples {#section:examples}
=============
In this section we list some examples to which our results apply, yet the integro-differential structure given in either (\[eqIN:LevyTypeLinear\]) or (\[eqIN:MinMaxMeta\]) is not readily apparent from the definition of the operator itself. We emphasize that most cases of the *linear* examples that we list were already contained in the classic work of Courrège [@Courrege-1965formePrincipeMaximum], but we include them here for the sake of illustration. In all of these examples, the operators satisfy the GCP and the other technical requirements to apply the results presented above. We do not intend to give all details, but rather just make a list, with some appropriate references. At the end of the section, we list how these examples relate to Assumptions \[assumption:GCP\]–\[assumption:coefficient regularity\].
The statement of the examples.
------------------------------
\[EX:Generator\] The generator of a Markov process. Assume that $X_t$ is a Markov process taking values in $\real^d$, and that $\expct_x$ is the expectation of the process, having started from $x$ at $t=0$. The generator is defined as the operator $$\begin{aligned}
L(u,x) = \lim_{t\to0}\frac{\expct(u(X_t))-\expct(u(X_0))}{t},
\end{aligned}$$ over all $u$ for which the limit exists. (See Liggett [@Liggett-ContinuousTimeMarkovBookAMS Chapter 3].)
Thanks to the fact that $\expct$ preserves ordering, one can immediately see that $L$ enjoys the GCP. When $X_t$ is such that $L:C^2_b\to C^2_b$, this example is covered by Courrège [@Courrege-1965formePrincipeMaximum]; but if $X_t$ is such that $L:C^\beta_b\to C^0_b$ (in a Lipschitz fashion) for some $0<\beta<2$, then by Theorem \[theorem:minmax with beta less than 2\], there are fewer terms (see the list just above Theorem \[theorem:minmax with beta less than 2\] for our use of the notation $C^\beta_b(\real^d)$). In this context, the result of Courrège can be seen as a version of the Lévy-Khintchine formula for a process whose increments need not be stationary.
\[EX:D-to-N\] The Dirichlet-to-Neumann map for linear, elliptic operators on half-space. Assume that $L$ is an operator that admits unique bounded solutions on $\real^{d+1}_+$ and that has a comparison principle. What we mean by this is the following: we can take $u\in C^{1,\al}_b(\real^d)$ and associate to it the unique bounded solution, $U_u$ of $$\begin{aligned}
L(U_u,X) = 0\ \ \text{in}\ \real^{d+1}_+,\ \ \text{and}\ \ U_u=u\ \text{on}\ \real^d\times{0}.
\end{aligned}$$ A couple of reasonable examples would be $$\begin{aligned}
L(U,X) = \tr(A(X)D^2U(X))\ \ \text{or}\ \ L(U,X) = \dive(A(X)\nabla U),\end{aligned}$$ where $A$ is uniformly elliptic and Hölder continuous. The Dirichlet-to-Neumann map is then defined as $$\begin{aligned}
I(u,x) := \partial_n U_u(x). \end{aligned}$$
First of all, the assumptions on $A$ are such that for some $\al'$, $U_u\in C^{1,\al'}_b\left(\overline{\real^{d+1}_+}\right)$ and hence the normal derivative is well defined (see, e.g. [@GiTr-98 Chapters 8, 9]). It is not hard to check that this operator satisfies the GCP, and this fact comes entirely from the property that the solution operator, by the assumed comparison principle, preserves ordering of solutions whenever the boundary data are ordered (it has nothing to do with linearity of the solution operator). This is, again, within the context of Courrège’s result, but we can invoke Theorem \[theorem:minmax with beta less than 2\] to remove extra terms of order higher than $1$. Ellipticity and scaling show that this is always an operator of order $1$ (and will map $C^{1,\al}\to C^{\al'}$). We note that in this example, via linear equations with nice coefficients, one can derive lots of information about the operator $\partial_n U_u$ by directly using the Poisson kernel that represents the solution $U_u$.
In the context of periodic equations, one can use the results in Sections \[section:Finite Dimensional Approximations\] and \[section:Analysis of finite dimensional approximations\] to show that the coefficients in the resulting Lévy operators will share the same periodicity. In fact, this is very straightforward if $I$ is linear. If instead one looks at almost periodic coefficients, it seems reasonable to hope that the coefficients will also be almost periodic, but we have not checked this claim. If it is the case, there could be an application to some boundary homogenization problems with irrationally oriented half-spaces inside a periodic medium, related to [@GuSc-2018NeumannHomogPart2SIAM]. Operators related to the Dirichlet-to-Neumann mapping of this example are also of interest in conformal geometry, see Chang-Gonzalez [@ChGo-2011FracLaplaceGeometry]. It is also possible to consider an elliptic equation with weights in order to obtain some operators of order different than 1, e.g. Caffarelli-Silvestre [@CaffarelliSilvestre-2007ExtensionCPDE].
\[EX:BoundaryProcess\] The boundary process of a reflected diffusion. (See Hsu [@Hsu-1986ExcursionsReflectingBM], or [@IkedaWatanabe-1981SDE Chp. IV, Sec. 7] and/or [@SatoUeno-1965MultiDimDiffBoundryMarkov Sec. 8].)
In this context, one starts with a diffusion in $\real^{d+1}_+$, say $X_t$, so that $X_t$ reflects off of the bottom boundary whenever it reaches it. Under a time rescaling of $X_t$ (because it spends zero time on the boundary), the resulting process can be viewed at times only when it hits $\real^{d}\times\{0\}$, and induces a pure jump process on $\real^d\times\{0\}$. This process is generated by an operator of the form (\[eqIN:LevyTypeLinear\]) with $A\equiv 0$. It turns out that this generator for the boundary process is exactly the Dirichlet-to-Neumann mapping from the previous example. This process was studied in a smooth domains for Brownian motion by Hsu [@Hsu-1986ExcursionsReflectingBM].
\[EX:Subordination\] Subordinated diffusions and Bernstein functions. (See Schilling-Song-Vondraček [@SchillingSongVondracek-2012BookBernsteinFunctions].)
The time-rescaling of the reflected diffusion in the previous example is just one choice of a rescaling, and in general one can time-rescale diffusions on $\real^d$ (so no boundary space here) in a myriad of fashions to create new stochastic processes from one reference Brownian motion. This is a process known as subordination, and it can be used to create operators with generators in the class (\[eqIN:LevyTypeLinear\]), starting with one that may simply only contain the second order term. The generator for the subordinated process will enjoy the GCP because the generator of the original diffusion also enjoys the GCP. This technique has played a large and fundamental role in the study of Lévy processes, and one can see it in use in e.g., the book of Schilling-Song-Vondraček [@SchillingSongVondracek-2012BookBernsteinFunctions], especially [@SchillingSongVondracek-2012BookBernsteinFunctions Chapter 13]. The subordination formula is closely related to an extension into plus one space variables, and this extension was used to create operators of fractional order that enjoy the GCP in the work of Stinga-Torrea [@StinTor2010Extension] and also provide other properties of the fractional operators.
\[EX:MongeAmpere\] The Monge-Ampère operator, $\MA(u,x)=\det(D^2u)$.
When one restricts this operator to the subset of $C^2$ of convex functions, then $\MA$ is in fact (degenerate) elliptic and locally Lipschitz. Specifically for each $\delta>0$, $\MA$ is uniformly elliptic (depending upon $\delta$), Lipschitz, and translation invariant as a mapping, $$\begin{aligned}
\MA: \{u\in C^2_b(\real^d) : \frac{1}{\delta}>D^2u> \delta\}\to C^0_b(\real^d).
\end{aligned}$$ Thus, $\MA$, must enjoy a min-max structure. Experts have known and utilized this min-max propert of $\MA$ in the study of fully nonlinear elliptic equations for a long time, and one can show that $$\begin{aligned}
(\MA(u,x))^{1/d} =\frac{1}{d} \inf\{ \tr(AD^2u(x)) : A\geq 0,\ \text{and}\ \det(A)=1\}. \end{aligned}$$ In fact, this formula is intimately connected with various investigations into nonlocal operators that should be an analog of $\MA$ in the fractional setting (as of yet, there is not one that is considered better than others). Some works that address nonlocal analogs of $\MA$ are: [@CaffarelliCharro-2015FractionalMongeAmpereAPDE], [@CaffarelliSilvestre-2016NonlocalMongeAmpereCAG], and [@GuSc-12ABParma].
\[EX:CaSiExtremal\] General nonlocal operators as treated in Caffarelli-Silvestre [@CaSi-09RegularityIntegroDiff] [@CaSi-09RegularityByApproximation]. These are simply operators that are assumed to satisfy the GCP, are defined for all functions in $C^{1,1}(\real^d)$, map $C^2_b(\real^d)\to C^0_b(\real^d)$, and satisfy a form of uniform ellipticity that is given by the existence of concave respectively convex operators, $\M^-_\L$ and $\M^+_\L$ so that $$\begin{aligned}
\label{eqEX:ExtremalIneq}
\text{for all}\ u,v\in C^{1,1}(\real^d),\
\M^-_\L(u-v,x)\leq I(u,x)-I(v,x)\leq \M^+_\L(u-v,x).
\end{aligned}$$ Here, $\L$ is a class of linear operators that is usually a particular subset of those that satisfy the Lévy type condition (\[eqIN:LevyTypeLinear\]).
This context for nonlocal operators was given in [@CaSi-09RegularityIntegroDiff Definition 3.1], and it played an important role in many of the results– especially when $\L$ is chosen to contain certain classes of operators. These operators, in cases in which they are Lipschitz fall into the scope of our results, and furthermore, the role of the extremal operators gives extra information about the min-max formula. In particular, as shown in [@GuSc-2016MinMaxNonlocalarXiv Section 4.6], when ellipticity occurs with respect to $\M^\pm_\L$, then the min-max may be restricted to only utilize linear functionals (or linear operators) that also satisfy the extremal inequality in (\[eqEX:ExtremalIneq\]). This also appeared in a homogenization result by one of the authors in which they were unable to show that the limit operator had an explicit integro-differential formula, but rather was only integro-differential and uniformly elliptic in the sense of [@CaSi-09RegularityIntegroDiff Definition 3.1] ( see the homogenization in [@Schw-10Per]).
\[EX:DtoNFullyNonBananas\] The Dirichlet to Neumann map for fully nonlinear elliptic equations. In Example \[EX:D-to-N\], the linearity of $L$ is not necessary, and the function $U_u$ can also be taken to solve a fully nonlinear, uniformly elliptic equation in $\real^{d+1}_+$. These equations always possess a comparison principle (by definition), and under most reasonable assumptions, the solution $U_u$ will be globally $C^{1,\al'}$, allowing for the normal derivative to be defined classically (see [@SilvestreSirakov-2013boundary] for this regularity).
This was a main topic in the recent paper by the authors and Kitagawa [@GuillenKitagawaSchwab2017estimatesDtoN]. It turns out that the extremal operators (as in Example \[EX:CaSiExtremal\]) for the nonlinear D-to-N not only play a crucial role in investigating the Lévy measures in the min-max, but they also take a refreshingly simple form. The extremal operators in this case, $\M^\pm_\L$ of Example \[EX:CaSiExtremal\], are simply the Dirichlet-to-Neumann operators for the solutions of the corresponding extremal operators for the elliptic second order equation in $\real^{d+1}_+$. These are usually called the Pucci extremal operators (see [@CaCa-95]), and solutions to their equations are generally very well behaved. In [@GuillenKitagawaSchwab2017estimatesDtoN], the properties of the Lévy measures in the min-max are linked to the harmonic measures for linear equations with bounded measurable coefficients (e.g. [@Kenig-1993PotentialThoeryNonDiv]), but there is still more to learn about them before they can be connected with existing integro-differential theory.
\[EX:HeleShawFTW\] An operator that drives surface evolution in one and two phase free boundary problems related to a type of Hele-Shaw flow. Given $f\in C^{1,\al}(\real^d)$, such that $0<\inf f\leq \sup f<\infty$, we can define the unique solution, $U_f$, of the elliptic equation, $$\begin{aligned}
&\Delta U_f = 0\ \text{in}\ \{(x,x_{d+1}) : 0<x_{d+1}<f(x) \},\\
&U_f=1\ \text{on}\ \{x_{d+1}=0\},\
U_f=0\ \text{on}\ \{ (x,d_{d+1}) : x_{d+1}=f(x) \}.
\end{aligned}$$ This allows to define a (fully nonlinear) operator on $f$ as $$\begin{aligned}
I(f,x):= \partial_n U_f(x,f(x)),
\end{aligned}$$ that is, the normal derivative of the solution on the upper boundary given by the graph of $f$.
For Hele-Shaw flow in the simplified setting that the free boundary is parametrized by the graph of $f(\cdot,t)$, it can be shown that the free boundary evolves by a normal velocity that at each time is given by $I(f,x)$. The interpretation here is that fluid flows into the domain under a pressure at the bottom boundary, $x_{d+1}=0$, and the top edge of the fluid exists at $x_{d+1}=f(x)$, with $U_f$ representing the pressure of the fluid. This pressure induces a force on the fluid, which is given by $\partial_n U_f(x,f(x))$ at the top boundary. This operator, and its implications for rewriting a class of free boundary problems that are similar to Hele-Shaw was studied by the authors and Chang Lara in [@ChangLaraGuillenSchwab2018FBasNonlocal]. In particular, the min-max formula makes it straightforward to convert the free boundary flow into a nonlocal parabolic equation for $f$, and this parabolic equation is very similar to ones that have already been studied in the nonlocal literature (e.g. [@Silv-2011DifferentiabilityCriticalHJ]). When $U_f$ is defined to be harmonic in the domain determined by $f$, standard regularity theory immediately gives estimates that show there is some $\al'$ so that the mapping from $f$ to $I(f)$ is Lipschitz from $C^{1,\al}(\real^d)$ to $C^{\al'}(\real^d)$. In [@ChangLaraGuillenSchwab2018FBasNonlocal] it was also shown that the same Lipschitz property can be obtained when $U_f$ is defined as the solution of a nonlinear uniformly elliptic second order equation instead of just the Laplacian. This operator gives a good example of what can be said in the translation invariant case of the min-max, and its properties are studied initially in [@ChangLaraGuillenSchwab2018FBasNonlocal]. Even in the simplest case of defining $U_f$ to be harmonic, the resulting operator $I$ will always be inherently nonlinear and nonlocal.
Relationship to Assumptions \[assumption:GCP\]–\[assumption:coefficient regularity\]
------------------------------------------------------------------------------------
Here we list how each of the above examples fits within the context of Assumptions \[assumption:GCP\]–\[assumption:coefficient regularity\].
**(Example \[EX:Generator\]).** By construction, this $L$ is always linear. Thus, Assumption \[assumption:GCP\] follows from simply saying that $L$ is a bounded operator on $C^{\beta}$, which of course requires assumptions on the process, $X_t$, or more specifically the transition probability measure for $X_t$. Again, via linearity, Assumption \[assumption:translation invariance\] follows whenever the process, $X_t$, has stationary and independent increments. Assumptions \[assumption:tightness bound\] and \[assumption:coefficient regularity\] will be an extra requirement on the transition probability measure for $X_t$. In particular (although a bit circular), Assumption \[assumption:coefficient regularity\], in view of linearity, is equivalent to the martingale problem for $X_t$ having a solution and the generator having uniformly continuous coefficients.
**(Example \[EX:D-to-N\]).** (The interested reader can see [@GuillenKitagawaSchwab2017estimatesDtoN] for more details.) Assumption \[assumption:GCP\] holds for $C^{1,\al}\to C^{\al'}$ when $A$ is $\al$-Hölder continuous. Assumption \[assumption:translation invariance\] holds if $A$ is a constant. Assumption \[assumption:tightness bound\] holds in both of the above settings, by using a barrier argument (which is easier implemented for the non-divergence equation). Since $I$ is linear, Assumption \[assumption:coefficient regularity\] holds when $A$ is Hölder continuous. Indeed, by linearity, checking Assumption \[assumption:coefficient regularity\] is equivalent to estimating $$\begin{aligned}
I(\tau_{-z}u,x+z)-I(u,x).\end{aligned}$$ In the case of divergence equations, one can write down the equations satisfied for $V=\tau_{-z}U_u$, and then also the equation satisfied by $W:= U_{\tau_{-z}u}-V$. The desired estimate is then equivalent to estimating ${\left| \partial_n W(x+z) \right|}$, i.e. a global Lipschitz estimate for $W$. Since $W$ satisfies $$\begin{aligned}
\dive(A(X)\grad W(X))= -\dive((A(X)-A(x-z))\grad V),\end{aligned}$$ we see that by global Lipschitz estimates, $$\begin{aligned}
{\left| \grad W \right|}\leq C{\lVert(A-A(\cdot-z))\grad V\rVert}_{L^\infty}\leq C{\left| z \right|}^\al,\end{aligned}$$ by the original assumption that $A$ is Hölder continuous. (Note, the Lipschitz estimates here are a standard modification to, e.g. [@GruterWidman-1982GreenFunUnifEllipticManMath Lemma 3.2] to allow for a right hand side of the form $\dive(f)$ with $f\in L^\infty$.)
**(Example \[EX:BoundaryProcess\]).** In most reasonable situations in which the diffusion has regular coefficients, this is contained in the previous example.
**(Example \[EX:Subordination\]).** This, of course, depends heavily on the original Markov process and the choice of subordinator. However, one of the most classical situations starts with a Brownian motion and then uses a Lévy stable subordinator. In this case, the resulting operator is translation invariant, and Assumptions \[assumption:GCP\] and \[assumption:translation invariance\] follow more or less by construction.
**(Example \[EX:MongeAmpere\]).** This is a translation invariant operator, and as mentioned already satisfies the Lipschitz property on the specified convex subsets of $C^2$. So, Assumptions \[assumption:GCP\] and \[assumption:translation invariance\] hold.
**(Example \[EX:CaSiExtremal\]).** As this is a general example, the operators only satisfy the given assumptions when explicitly required to do so. However, the interesting part of this example arises from the fact that the knowledge of the extremal inequalities in (\[eqEX:ExtremalIneq\]) in fact gives more detailed information about the linear operators that will appear in the min-max of Theorems \[theorem:MinMax Euclidean\]–\[theorem:minmax with beta less than 2\]. This is discussed in [@GuSc-2016MinMaxNonlocalarXiv Section 4.6].
**(Example \[EX:DtoNFullyNonBananas\]).** This operator satisfies Assumption \[assumption:GCP\] as a mapping of $C^{1,\al}\to C^{\al'}$ (for some $0<\al'<\al$) under standard assumptions about $F$. The relevant regularity theory comes from Silvestre-Sirakov [@SilvestreSirakov-2013boundary]. It can also be checked by using the same type of barrier argument that works for Example \[EX:D-to-N\] will show Assumption \[assumption:tightness bound\] is also satisfied. Due to the nonlinear nature of the D-to-N in this setting, it is not obvious how to show that Assumption \[assumption:coefficient regularity\] is satisfied– we do not know if it satisfied or not. Thus, the best one can say about this operator when it is not translation invariant is the outcome of Theorem \[theorem:MinMax Euclidean\]. We simply note to the interested reader that because of the lack of exact cancelation from the fact that the mapping is not linear, one probably needs more detailed information about $F$. Indeed, using the extremal operators would not help because it would produce $$\begin{aligned}
I(v+\tau_{-z} u, x+z)-I(v,x+z) -(I(v+u,x)-I(v,x))
&\leq M^+(\tau_{-z}u,x+z) - M^-(u,x)\\
=M^+(u,x)-M^-(u,x).\end{aligned}$$ Here we use $M^\pm$ as the extremal operators for $I$, and also that these are translation invariant. This estimate completely neglects the influence of the shift, $\tau_z$, and so it would not be useful (furthermore, one expects that $M^+(u,x)>M^-(u,x)$).
**(Example \[EX:HeleShawFTW\]).** As it is stated above, this operator, $I$, is actually translation invariant, and so it is straightforward to check that Assumptions \[assumption:GCP\] and \[assumption:translation invariance\] hold. In the case that the equation for $U$ (i.e. $\Delta U=0$) is replaced by either a fully nonlinear operator and/or and operator that is not translation invariant, it is harder to check all of the applicable assumptions. Again, for fully nonlinear equations that define $U$, in [@ChangLaraGuillenSchwab2018FBasNonlocal] $I$ was checked to be Lipschitz as a map of $C^{1,\al}\to C^{\al'}$ (which took a reasonably non-trivial amount of work).
Additional proofs and computations
==================================
Fix $u \in C^{\beta}_b(\mathbb{R}^d)$, and let $x \in G_n$, then by the regularity of $u$, $$\begin{aligned}
|u(x \pm h_n e_k)- (u(x) \pm h_n \nabla u(x_0)\cdot e_k) | \leq C\|u\|_{C^\beta} h_n^{\min\{\beta-1,1\}}.
\end{aligned}$$ Therefore, $$\begin{aligned}
|u(x + h_ne_k) - u(x + h_n e_k) -2 h_n \nabla u(x_0)\cdot e_k | \leq C\|u\|_{C^\beta} h_n^{\min\{\beta-1,1\}}
\end{aligned}$$ For the second estimate, we shall make use of $$\begin{aligned}
|u(x + h_n e_k)- (u(x) + h_n \nabla u(x_0)\cdot e +h_n^2\tfrac{1}{2}(D^2u(x) e,e)) | \leq C\|u\|_{C^\beta} h_n^{\min\{\beta-2,1\}}.
\end{aligned}$$ Therefore, $$\begin{aligned}
& u(x+h_n e_k+h_n e_\ell) - u(x+h_n e_k) - u(x+h_n e_\ell) + u(x)\\
& ``='' u(x) + h_n \nabla u(x_0)\cdot (e_k+e_\ell) +h_n^2\tfrac{1}{2}(D^2u(x)(e_k+e_\ell,e_k+e_\ell)\\
& \;\;\;\; -(u(x) + h_n \nabla u(x_0)\cdot e_k +h_n^2\tfrac{1}{2}(D^2u(x) e_k,e_k))\\
& \;\;\;\; -(u(x) + h_n \nabla u(x_0)\cdot e_\ell +h_n^2\tfrac{1}{2}(D^2u(x) e_\ell,e_\ell)) + u(x)\\
& = h_n^2\tfrac{1}{2} \left ( (D^2u(x)(e_k+e_\ell,e_k+e_\ell)-(D^2u(x) e_k,e_k))-(D^2u(x) e_\ell,e_\ell)) \right )\\
& = h_n^2 (D^2u(x)e_k,e_\ell)
\end{aligned}$$ It follows that $$\begin{aligned}
|u(x+h_n e_k+h_n e_\ell) - u(x+h_n e_k) - u(x+h_n e_\ell) + u(x)- h_n^2 (D^2u(x)e_k,e_\ell)| \leq C\|u\|_{C^\beta} h_n^{\min\{\beta-2,1\}},
\end{aligned}$$ and the proposition is proved.
Fix $u \in C^{\beta}_b(\mathbb{R}^d)$.
*Step 1*. Let $x \in G_n$, then $$\begin{aligned}
& | (\nabla_n)^1u(x)-\nabla u(x)| \leq C\|u\|_{C^{\beta}}h_n^{\beta-1},\;\textnormal{ if } \beta \in [1,2],\\
& | (\nabla_n)^2u(x)-D^2 u(x)| \leq C\|u\|_{C^{\beta}}h_n^{\beta-2},\;\textnormal{ if } \beta \in [2,3].
\end{aligned}$$
*Proof of Step 1.* By the regularity of $u$, $$\begin{aligned}
|u(x \pm h_n e_k)- (u(x) \pm h_n \nabla u(x_0)\cdot e_k) | \leq C\|u\|_{C^\beta} h_n^{\min\{\beta-1,1\}}.
\end{aligned}$$ Therefore, $$\begin{aligned}
|u(x + h_ne_k) - u(x + h_n e_k) -2 h_n \nabla u(x_0)\cdot e_k | \leq C\|u\|_{C^\beta} h_n^{\min\{\beta-1,1\}}
\end{aligned}$$
*Step 2.* Given $x\in G_n$, we have $$\begin{aligned}
& | (\nabla_n)^1u(x)| \leq C\|u\|_{C^{1}},\;| (\nabla_n)^2u(x)| \leq C\|u\|_{C^{2}}.
\end{aligned}$$
*Step 3.*
$$\begin{aligned}
|(\nabla_n)^1u(\hat x)-(\nabla_n)^1u(\hat y)| \leq C\|u\|_{C^\beta} d(\hat x,\hat y)^{\beta-1}, \;\textnormal{ if } \beta \in [1,2],\\
|(\nabla_n)^2u(\hat x)-(\nabla_n)^2u(\hat y)| \leq C\|u\|_{C^\beta} d(\hat x,\hat y)^{\beta-2}, \;\textnormal{ if } \beta \in [2,3].
\end{aligned}$$
$$\begin{aligned}
\nabla \tilde R(x) = 2C\|w\|_{C^{\beta_0}} \eta'\left ( \frac{|x-x_0|^{\beta_0}}{h_n} \right ) \beta_0 |x-x_0|^{\beta_0-1}\frac{(x-x_0)}{|x-x_0|}
\end{aligned}$$
If $|x-x_0|^{\beta_0}\leq h_n$, then $$\begin{aligned}
\nabla \tilde R(x) = 2C\|w\|_{C^{\beta_0}}\beta_0 |x-x_0|^{\beta_0-1}\frac{(x-x_0)}{|x-x_0|}
\end{aligned}$$ This expression is zero except when $|x-x_0|\leq h_n^{1/\beta_0}$, so $$\begin{aligned}
|\nabla \tilde R(x)| \leq 2C\|w\|_{C^{\beta_0}}\beta_0 h_n^{1-1/\beta_0}.\\
\end{aligned}$$ Furthermore, for $x,x'$ such that $|x-x_0|^{\beta_0}\leq h_n$, we have $$\begin{aligned}
|\nabla \tilde R(x)-\nabla \tilde R(x')| & \leq 2C\beta_0 \|w\|_{C^\beta_0} \left | |x-x_0|^{\beta_0-1}\frac{(x-x_0)}{|x-x_0|}-|x'-x_0|^{\beta_0-1}\frac{(x'-x_0)}{|x'-x_0|} \right | \\
& \leq C\|w\|_{C^{\beta_0}}h_n^{\beta_0-\beta}|x-x'|^{\beta}.
\end{aligned}$$ In conclusion, $$\begin{aligned}
\|\tilde R\|_{L^\infty}+ \|\nabla \tilde R\|_{L^\infty} + [\nabla \tilde R]_{C^{\beta-1}} \leq C\|w\|_{C^{\beta_0}}(h_n+h_n^{1-1/\beta_0}+h_n^{\beta_0-\beta}) \leq C\|w\|_{C^{\beta_0}}h_n^\gamma.
\end{aligned}$$
[^1]: The authors gratefully acknowledge partial support from the National Science Foundation while this work was in progress: N. Guillen DMS-1700307 and R. Schwab DMS-1665285. The authors also thank the anonymous referee for some suggestions that we believe improved the presentation of our results.
|
---
abstract: 'The ordinal approach to evaluate time series due to innovative works of Bandt and Pompe has increasingly established itself among other techniques of nonlinear time series analysis. In this paper, we summarize and generalize the theory of determining the Kolmogorov-Sinai entropy of a measure-preserving dynamical system via increasing sequences of order generated partitions of the state space. Our main focus are measuring processes without information loss. Particularly, we consider the question of the minimal necessary number of measurements related to the properties of a given dynamical system.'
---
<span style="font-variant:small-caps;">Karsten Keller</span>
<span style="font-variant:small-caps;">Sergiy Maksymenko</span>
<span style="font-variant:small-caps;">Inga Stolz</span>
(Communicated by the associate editor name)
Introduction {#sec1}
============
Since the invention of permutation entropy by Bandt and Pompe [@bandt_pompe_2002] and the proof of its coincidence with Kolmogorov-Sinai entropy for piecewise monotone interval maps by Bandt et al. in [@bandt_et_al_2002], there is some increasing interest in considering time series and dynamical systems from the pure ordinal point of view (see Amig[ó]{}, [@amigo_2010]). The idea behind this viewpoint is that much information of a system is already contained in ordinal patterns describing the up and down of its orbits. This ordinal view can be particularly useful when having physical quantities for which the statement that a measuring value is larger than another one is well interpretable, but concrete purely given differences of measuring values are not. A prominent example is the (indirect) measurement of temperature as the mean kinetic energy of the particles of a system by a thermometer. One can make statements about what is warmer or colder, but, for example, the interpretation of an increase by $1^{\circ}\mathrm{C}$ with not knowing the baseline value is complicated.
This paper is generally discussing the Kolmogorov-Sinai entropy from the ordinal viewpoint. It reviews and particularly extends and generalizes former results given by Antoniouk et al. [@antoniouk_et_al_2013], Amig[ó]{} [@amigo_2012], Keller [@keller_2012], Keller and Sinn [@keller_sinn_2010; @keller_sinn_2009] and Amig[ó]{} et al. [@amigo_2005]. Aspects of entropy estimation are touched.
*The framework.* The basic model of our discussion is a *measure-preserving dynamical system* $(\Omega,\mathcal{A},\mu,T)$, i.e. $\Omega$ is a non-empty set whose elements are interpreted as the states of a system, $\mathcal{A}$ is a sigma-algebra on $\Omega$, $\mu: \mathcal{A}\to[0,1]$ is a probability measure, and $T:\Omega\hookleftarrow$ is a $\mathcal{A}$-$\mathcal{A}$-measurable $\mu$-preserving map describing the dynamics of the system. *$\mu$-preserving* means that $\mu(T^{-1}(A))= \mu(A)$ for all $A \in \mathcal{A}$; the measure $\mu$ is then called $T$-invariant.
We want to have some kind of regularity of $T$ by assuming at least one of the following conditions: $$\begin{aligned}
T\mbox{ is \emph{ergodic} with respect to }\mu\mbox{, i.e. }\hspace{6cm}\nonumber\\\mu(A)\in\{0,1\}\mbox{ for all }A\in \mathcal{A}\mbox{ with }T^{-1}(A)= A,\label{ergodic}\\
\Omega\mbox{ can be embedded into some compact metrizable space so that }\mathcal{A}=\mathcal{B}(\Omega). \label{nonergodic}\end{aligned}$$ Here and in the whole paper, $\mathcal{B}(\Omega)$ denotes the *Borel $\sigma$-algebra* in the case that $\Omega$ is a topological space. As usual, equivalent to $T$ is ergodic with respect to $\mu$, we say that $\mu$ *is ergodic for* $T$.
Often the states of a system, whatever they are, cannot be accessed directly, but information on them can be obtained by measurements. In this paper such measurements are assumed to be given via *observables* $X_1,X_2,X_3,\ldots $ defined as ${\mathbb R}$-valued random variables on the probability space $(\Omega ,\mathcal{A},\mu)$. So the measurements are provided by a stochastic process - we say *sequence of observables* ${\bf X}=(X_i)_{i\in {\mathbb N}}$ - whose realization has components $(X_i(T^{\circ t}(\omega)))_{t\in {\mathbb N}_0}$. Here $X_i(T^{\circ t}(\omega))$ is interpreted as the $i$-th measured value from the system at time $t$ when starting in state $\omega\in\Omega$.
A priori we have infinitely many observables providing more and more information, the finite case, however, is included by equality of all $X_i$; $i\geq n$ for some $n\in {\mathbb N}$. We will write $\mathbf{X}=(X_i)_{i=1}^n$ in the case of finitely many observables and $\mathbf{X}=X$ in the case of only one observable $X$.
Unless otherwise stated, in the following $(\Omega,\mathcal{A},\mu,T)$ is a measure-preserving dynamical system and $\mathbf{X}=(X_i)_{i\in {\mathbb N}}$ a sequence of observables.
*Kolmogorov-Sinai entropy.* In order to recall the Kolmogorov-Sinai entropy, let $q \in \mathbb{N}$ and $\mathcal{P} = \{P_1,P_2,\ldots,P_q\} \subset \mathcal{A}$ be a finite partition of $\Omega$, i.e. $\Omega=\bigcup_{l=1}^q P_l$, $P_l\neq\emptyset$ for $l=1,2,\ldots ,q$, $P_{l_1}\cap P_{l_2}=\emptyset$ for different $l_1,l_2\in\{1,2,\ldots ,q\}$, and let $A = \{1,2,\ldots,q\}$ be the corresponding alphabet. Each word $a_1a_2\ldots a_t$ of length $t \in \mathbb{N}$ defines a set $$P_{a_1 a_2\ldots a_t}
:=
\{\omega \in \Omega \mid (\omega,T(\omega),\ldots,T^{\circ t-1}(\omega)) \in P_{a_1} \times P_{a_2} \times \ldots \times P_{a_t} \},$$ and the collection of all non-empty sets obtained for such words of length $t$ provides a partition $\mathcal{P}_t \subset \mathcal{A}$ of $\Omega$. In particular, $\mathcal{P}_1=\mathcal{P}$.
The *entropy rate* of $T$ with respect to an initial partition $\mathcal{P}$ is given by $$h_\mu(T,\mathcal{P})
=
\lim \limits_{t\to\infty} \frac{1}{t} H_{\mu}(\mathcal{P}_t),$$ where $H_\mu(\mathcal{C})$ denotes the *(Shannon) entropy* of a finite partition $\mathcal{C}=\{C_1,C_2,\ldots,\linebreak C_q\}\subset\mathcal{A}$ of $\Omega$; $q \in \mathbb{N}$, i.e. $$H_\mu(\mathcal{C})
=
- \sum_{l=1}^q \mu(C_l) \ln(\mu(C_l))$$ (with $0\ln(0) := 0$), and the *Kolmogorov-Sinai entropy* is defined by $$h_{\mu}^{\mathrm{KS}}(T)
=
\sup_{\mathcal{P} \text{ finite partition }} h_{\mu}(T,\mathcal{P}).$$ Although the Kolmogorov-Sinai entropy is well-defined, its determination is not easy. In some special cases one can find finite partitions already determining it, usually called generating partitions (see Definition \[def:gen\_partition\]), however, do not exist or are not accessible. As a substitute, we want to consider special sequences of partitions only depending on the ordinal structure of a dynamical system.
*Ordinal partitioning.* For a single observable $X$ on $(\Omega,\mathcal{A},\mu,T)$ and $s,t\in \mathbb{N}_0$ with $s<t$, consider the bisection $$\label{bisection}
\begin{split}
\mathcal{P}^{X,T}_{s,t}
=
\{&\{\omega \in \Omega \mid X ( T^{\circ s}(\omega)) < X ( T^{\circ t}(\omega))\} ,
\\ &\{\omega \in \Omega \mid X ( T^{\circ s}(\omega)) \geq X ( T^{\circ t}(\omega))\}\}
\end{split}$$ of $\Omega$ and, for observables $X_1,X_2,\ldots ,X_n$ on $(\Omega,\mathcal{A},\mu,T)$ and $d,n\in {\mathbb N}$, the partition $$\label{ppart}
\mathcal{P}_d^{(X_i)_{i=1}^n,T}
=
\bigvee_{i=1}^n\ \bigvee_{0\leq s<t\leq d}\mathcal{P}_{s,t}^{X_i,T},$$ i.e. the coarsest partition refining all bisections $\mathcal{P}_{s,t}^{X_i,T}$; $i=1,2,\ldots n$, $0\leq s<t\leq d$. (If one of the sets of the right hand side of is empty, $\mathcal{P}^{X,T}_{s,t}$ is considered to consist of only one set.)
The partition $\mathcal{P}_d^{(X_i)_{i=1}^n,T}$ is called *ordinal partition* of *order* $d$ *associated to $(X_i)_{i=1}^n$*. By definition its parts contain all states with equal ordinal measurement structure for an initial orbit part.
*A central statement.* Clearly, in order to preserve information of the given system, the observables should separate orbits of the system in a certain sense. In order to give a precise description, let in the following $\sigma((\mathbf{X} \circ T^{\circ t})_{t\in {\mathbb N}_0})$ be the $\sigma$-algebra generated by all random variables $X_i\circ T^{\circ t}$; $i\in {\mathbb N}$, $t\in {\mathbb N}_0$ and write $\mathcal{F}\overset{\mu}{\supset}\mathcal{G}$ if for each $G\in \mathcal{G}$ there exists some $F\in \mathcal{F}$ with $\mu(F\,\Delta\,G)=0$.
The following generalization of a statement in Antoniouk et al. [@antoniouk_et_al_2013] says that if there is no information loss by measuring with observables, all information is preserved also by only considering measurements from the ordinal viewpoint.
\[main\] Let $(\Omega, \mathcal{A},\mu, T$) be a measure-preserving dynamical system and $\mathbf{X} = (X_i)_{i \in \mathbb{N}}$ be a sequence of observables such that $\sigma((\mathbf{X} \circ T^{\circ t})_{t\in\mathbb{N}_0}) \overset{\mu}{\supset} \mathcal{A}$. Assume that or holds. Then $$\label{equ:hKS_lim_perm}
h_\mu^{\mathrm{KS}}(T)
=
\lim\limits_{d,n\to\infty} h_\mu(T,\mathcal{P}_d^{(X_i)_{i=1}^n,T})
=
\sup_{d,n\in {\mathbb N}} h_\mu(T,\mathcal{P}_d^{(X_i)_{i=1}^n,T}).$$
When Bandt and Pompe [@bandt_pompe_2002] invented the permutation entropy, they considered one-dimensional systems with coincidence of states and measurements. This fits into the given general approach as follows: $\Omega$ is a Borel subset of ${\mathbb R}$ and only one observable is considered to be the *identity map* $\mathrm{id}$ from $\Omega$ into ${\mathbb R}$. In this situation the assumptions of Theorem \[main\] are satisfied and so it holds $$h_\mu^{\mathrm{KS}}(T)
=
\lim\limits_{d\to\infty} h_\mu(T,\mathcal{P}_d^{\mathrm{id},T})=\sup_{d\in {\mathbb N}} h_\mu(T,\mathcal{P}_d^{\mathrm{id},T})$$ (compare [@keller_sinn_2010; @keller_sinn_2009]).
*Structure of the paper.* The paper is organized as follows. In Section \[sec2\] we provide a proof of Theorem \[main\] on the basis of Antoniouk et al. [@antoniouk_et_al_2013]. We, moreover, discuss this statement from different perspectives in Section \[sec3\] by presenting its modifications and variants. Section \[sec4\] is devoted to the concept of permutation entropy, in particular to the two different approaches to it given by Bandt et al. in [@bandt_et_al_2002] and Amig[ó]{} et al. in [@amigo_2005], respectively, and to its relation to the Kolmogorov-Sinai entropy. The ordinal approach to dynamical systems opens new perspectives to the estimation of system complexity. Advantages and limitations of this approach are discussed in Section \[sec5\]. The natural question of how many observables are necessary for satisfying the assumptions of Theorem \[main\] is in the focus of Section \[sec6\]. The corresponding discussion is strongly related to Takens’ delay embedding and similar ideas (see Takens [@takens_81] and Sauer [@sauer_et_al_1991]).
Kolmogorov-Sinai entropy from the ordinal viewpoint {#sec2}
===================================================
This section is devoted to the proof of Theorem \[main\].
*Preliminaries.* In the following we write $\mathcal{F} \overset{\mu}{=} \mathcal{G} $ if $\mathcal{F}\overset{\mu}{\supset}\mathcal{G}$ and $\mathcal{F}\overset{\mu}{\subset}\mathcal{G}$, and denote by $\mathbf{1}_A$ the *indicator function* of a subset $A \subset \Omega$. Moreover $\sigma (\diamondsuit)$ denotes the $\sigma$-algebra generated by a set $\diamondsuit$ of subsets of $\Omega$, by a sequence or double sequence $\diamondsuit$ of sets of subsets of $\Omega$, or by a random variable $\diamondsuit$ on $\Omega$.
Given two finite partitions $\mathcal{C},\mathcal{D}\subset \mathcal{A}$ of $\Omega$, we write $\mathcal{C}\prec\mathcal{D}$ if $\mathcal{D}$ is *finer* than $\mathcal{C}$ or, equivalently, if $\mathcal{C}$ is *coarser* than $\mathcal{D}$, that is, each element $C \in \mathcal{C}$ is a finite union of some elements of $\mathcal{D}$. Note that $\prec$ on the set of finite partitions of $\Omega$ contained in $\mathcal{A}$ is a partial order.
The *join* $\bigvee_{r=1}^m \mathcal{C}_r$ of $m \in \mathbb{N}$ finite partitions $\mathcal{C}_r = \{C_r^{(1)},C_r^{(2)},\ldots,C_r^{(\vert \mathcal{C}_r \vert)}\} \subset \mathcal{A}$ of $\Omega$ with $r=1,2,\ldots, m$ is the coarsest partition refining all $\mathcal{C}_r$; $r=1,2,\ldots, m$, i.e. $$\bigvee_{r=1}^m \mathcal{C}_r
=
\{\bigcap_{r=1}^m C_r^{(l_r)}\neq\emptyset \mid l_r\in\{1,2,\ldots,\vert \mathcal{C}_r\vert\}\mbox{ for }r=1,2,\ldots ,m\}.$$ For an observable $Y$ on $(\Omega,\mathcal{A},\mu,T)$ we consider the finite partitions $$\mathcal{P}_d^{Y,T}:=\bigvee_{0\leq s<t\leq d}\mathcal{P}_{s,t}^{Y,T}\mbox{ and }\widetilde{\mathcal{P}}_d^{Y,T}:=\bigvee_{0<t\leq d}\mathcal{P}_{0,t}^{Y,T}$$ (compare ) for $d\in {\mathbb N}$ and the $\sigma$-algebras $\Sigma^{Y,T}$ and $\widetilde{\Sigma}^{Y,T}$ generated from all $\mathcal{P}_d^{Y,T}$ and $\widetilde{\mathcal{P}}_d^{Y,T}$; $d\in {\mathbb N}$, respectively.
Besides $\mathcal{P}_d^{(X_i)_{i=1}^n,T}=\bigvee_{i=1}^n \mathcal{P}_d^{X_i,T}$ (compare ), for $d,n\in {\mathbb N}$ we are interested in the finite partitions $$\label{less}
\widetilde{\mathcal{P}}_d^{(X_i)_{i=1}^n,T}
:=
\bigvee_{i = 1}^n\widetilde{\mathcal{P}}_d^{X_i,T}.$$ Furthermore, we need the following $\sigma$-algebras associated to these partitions:
$$\Sigma^{\mathbf{X},T}
:=
\sigma\left(\left(\mathcal{P}_d^{(X_i)_{i=1}^n,T}\right)_{d,n\in\mathbb{N}}\right)
=
\sigma\left(\left(\Sigma^{X_i,T}\right)_{i\in\mathbb{N}}\right)$$
and $$\widetilde{\Sigma}^{\mathbf{X},T}
:=
\sigma\left( \left(\widetilde{\mathcal{P}}_d^{(X_i)_{i=1}^n,T} \right)_{d,n \in \mathbb{N} } \right)
=
\sigma\left(\left(\widetilde{\Sigma}^{X_i,T}\right)_{i\in\mathbb{N}}\right).$$
*The proof.* Although we consider dynamical systems equipped with infinitely many observables, we can follow closely the argumentation in the paper Antoniouk et al. [@antoniouk_et_al_2013]. So let us first recall or modify those statements of that paper used in our proof.
[@antoniouk_et_al_2013 Lemma 3.2]\[distribution\] Let $F:\mathbb{R}\to[0,1]$ be the distribution function of an observable $X$, that is $F(a) = \mu(\{\omega\in \Omega \mid X(\omega)\leq a\})$ for all $a \in \mathbb{R}$. Then $$\sigma(F \circ X)
\overset{\mu}{=}
\sigma(X).$$
[@antoniouk_et_al_2013 Lemma 3.3]\[convergence\] Let $T: \Omega \hookleftarrow$ be an ergodic map and let $I_d:\Omega\to\mathbb{R}$ be defined by $I_d(\omega) := \sum_{t= 1}^{d} \mathbf{1}_{\{X ( T^{\circ t}(\omega)) \leq X(\omega)\}}$ for all $d \in \mathbb{N}$ and $\omega \in \Omega$. Then $$F(X(\omega))
=
\lim\limits_{d\to\infty} \frac{I_d(\omega)}{d} \text{ for a.e.~} \omega \in \Omega.$$
By very slight modifications we can extend [@antoniouk_et_al_2013 Corollary 3.4 and Corollary 3.5] to countably many observables:
\[inclusions\] Let $T: \Omega \hookleftarrow$ be an ergodic map. Then $$\sigma(\mathbf{X})
\overset{\mu}{\subset}
\widetilde{\Sigma}^{\mathbf{X},T}
\subset\Sigma^{\mathbf{X},T}.$$
Compare to [@antoniouk_et_al_2013 Corollary 3.4]. The $\sigma$-algebra $\widetilde{\Sigma}^{\mathbf{X},T}$ is generated by the $\sigma$-algebras $\widetilde{\Sigma}^{X_i,T}
:=
\sigma((\bigvee_{0<t\leq d}\mathcal{P}_t^{X_i,T})_{d \in \mathbb{N}})$; $i \in \mathbb{N}$. Therefore by $\sigma(X_i) \overset{\mu}{\subset} \widetilde{\Sigma}^{X_i,T}$ for all $i \in \mathbb{N}$ it follows the assumption. This is true since $\frac{I_d}{d}:\Omega\to[0,1]$ is $\widetilde{\Sigma}^{X,T}$-$\mathcal{B}([0,1])$-measurable for all $d \in \mathbb{N}$ and hence so is $F \circ X$ and $X$ by Lemma \[distribution\] and Lemma \[convergence\]. The inclusion $\widetilde{\Sigma}^{\mathbf{X},T}
\subset\Sigma^{\mathbf{X},T}$ is given by construction (compare and ).
\[inclusion\] Let $T: \Omega \hookleftarrow$ be an ergodic map. Then $$\sigma((\mathbf{X} \circ T^{\circ t})_{t \in \mathbb{N}_0})
\overset{\mu}{\subset}
\Sigma^{\mathbf{X},T}.$$
For fixed $n\in {\mathbb N}$, in [@antoniouk_et_al_2013 Proof of Corollary 3.5] it is shown that $$\label{puneq}
\mathcal{P}_d^{X_i \circ T,T} \prec \mathcal{P}_{d+1}^{X_i,T}\mbox{ for all }d \in \mathbb{N}\mbox{ and }i=1,2,\ldots,n$$ implying $$\label{sigmauneq}
\Sigma^{X_i\circ T^{\circ t},T}\subset \Sigma^{X_i,T}\mbox{ for all }i=1,2,\ldots,n\mbox{ and }t\in\mathbb{N}_0.$$ Moreover, Corollary \[inclusions\] gives $$\label{suneq}
\sigma(\mathbf{X}\circ T^{\circ t}) \overset{\mu}{\subset} \Sigma^{\mathbf{X}\circ T^{\circ t},T}
\mbox{ for all }t \in \mathbb{N}_0.$$ Consequently, $\sigma((\mathbf{X} \circ T^{\circ t})_{t \in \mathbb{N}_0}) \overset{\mu}{\subset} \Sigma^{\mathbf{X},T}$.
\[increasing\] $(\mathcal{P}_d^{(X_i)_{i=1}^n,T})_{d,n \in \mathbb{N}}$ is an increasing sequence in $n$ for fixed $d$, and for fixed $n$ it is an increasing sequence in $d$.
In particular, $(\mathcal{P}_{d_j}^{(X_i)_{i=1}^{n_j},T})_{d_j,n_j \in \mathbb{N}}$ is an increasing sequence in $j$ if $(d_j)_{j \in \mathbb{N}}$ and $(n_j)_{j \in \mathbb{N}}$ are increasing sequences in $\mathbb{N}$.
Given $d,n \in \mathbb{N}$, it holds $$\begin{split}
\mathcal{P}_d^{(X_i)_{i=1}^n,T}&=\bigvee_{i=1}^n\ \bigvee_{0\leq s<t\leq d}\mathcal{P}_{s,t}^{X_i,T},\\
\mathcal{P}_{d+1}^{(X_i)_{i=1}^n,T}&=\bigvee_{i=1}^n\ \bigvee_{0\leq s<t\leq d+1}\mathcal{P}_{s,t}^{X_i,T},\\
\mathcal{P}_d^{(X_i)_{i=1}^{n+1},T}&=\bigvee_{i=1}^{n+1}\ \bigvee_{0\leq s<t\leq d}\mathcal{P}_{s,t}^{X_i,T},
\end{split}$$ implying $\mathcal{P}_d^{(X_i)_{i=1}^n,T}\prec \mathcal{P}_{d+1}^{(X_i)_{i=1}^n,T},\mathcal{P}_d^{(X_i)_{i=1}^{n+1},T}$ and so the above statements.
For completing the proof of Theorem \[main\], we apply the following statement (see Walters [@walters_2000 Theorem 4.22]):
\[walters\] For a sequence $(\mathcal{C}_d)_{d \in \mathbb{N}}$ of finite partitions $\mathcal{C}_d\in \mathcal{A}$ of $\Omega$ increasing with respect to $\prec$ and satisfying $\sigma(( \mathcal{C}_d)_{d \in \mathbb{N}})\overset{\mu}{\supset}\mathcal{A}$, it holds $$h_{\mu}^{\mathrm{KS}}(T)
=\lim\limits_{d\to\infty} h_{\mu}(T,\mathcal{C}_d).$$
First suppose that $T$ is an ergodic map. Then under the assumptions of Theorem \[main\] and by Corollary \[inclusion\] it holds $\mathcal{A} \overset{\mu}{\subset} \sigma((\mathbf{X} \circ T^{\circ t})_{t \in \mathbb{N}_0}) \overset{\mu}{\subset} \Sigma^{\mathbf{X},T}$. Since by Lemma \[increasing\] $(\mathcal{P}_{d_j}^{(X_i)_{i=1}^{n_j},T})_{d_j,n_j \in \mathbb{N}}$ is an increasing sequence in $j$ with respect to $\prec$ for increasing sequences $(d_j)_{j \in \mathbb{N}}$ and $(n_j)_{j \in \mathbb{N}}$ in $\mathbb{N}$, the assertion of Theorem \[main\] follows from Lemma \[walters\].
In the non-ergodic case the ergodic decomposition theorem is consulted. For a thorough treatment we refer the reader to Einsiedler and Ward [@einsiedler_2010] and Einsiedler et al. [@einsiedler_et_al_2015]. In particular, the ergodic decomposition theorem claims that under certain conditions any $T$-invariant measure $\mu$ can be decomposed into ergodic components and subsequently the entropy rate as well as the Kolmogorov-Sinai entropy of $T$ with respect to $\mu$ can be written as the integral of the entropies with respect to the decomposition.
In order to complete the proof of Theorem \[main\], we apply the following statement (see Einsiedler and Ward [@einsiedler_2010 Theorem 6.2], Einsiedler et al. [@einsiedler_et_al_2015 Theorem 5.27] and Keller and Sinn [@keller_sinn_2010] for the case of a non-invertible $T$):
\[ergodicdecomposition\] Let $(\Omega,\mathcal{A},\mu,T)$ be a measure-preserving dynamical system satisfying . Then there exists a probability space $(\Omega^\ast,\mathcal{A}^\ast,\nu)$ and a map $\omega^\ast \mapsto \mu_{\omega^\ast}$ associating to each $\omega^\ast \in \Omega^\ast$ a probability measure $\mu_{\omega^\ast}$ on $(\Omega,\mathcal{A})$ such that the following is valid:
$\Omega^\ast$ can be embedded into some compact metrizable space so that $\mathcal{A}=\mathcal{B}(\Omega^\ast)$, the map $\omega^\ast \in \Omega^\ast \to \int_{\Omega^\ast} f \,\mathbf{d} \mu_{\omega^\ast}$ is $\mathcal{A^\ast}$-$\mathcal{B}(\mathbb{R})$-measurable for every essentially bounded measurable function $f:\Omega \to \mathbb{R}$, the measure $\mu_{\omega^\ast}$ is ergodic $T$-invariant for $\nu$-a.e. $\omega^\ast \in \Omega^\ast$, and $$\mu
=
\int_{\Omega^*} \mu_{\omega^\ast} \,\mathbf{d}\:\!\nu(\omega^\ast).$$ Moreover, it holds $$h^{\mathrm{KS}}_\mu(T)
=
\int_{\Omega^*} h^{\mathrm{KS}}_{\mu_{\omega^\ast}}(T)\,\mathbf{d}\:\!\nu(\omega^\ast)
\label{decomposition}$$ and $$h_\mu(T,\mathcal{P})
=
\int_{\Omega^*} h_{\mu_{\omega^\ast}}(T,\mathcal{P})\,\mathbf{d}\:\!\nu(\omega^\ast)
\text{ for each finite partition } \mathcal{P} \subset\mathcal{A} \text{ of } \Omega.
\label{entropy}$$
Altogether we obtain $$\begin{aligned}
h^{\mathrm{KS}}_{\mu}(T)
&\overset{\text{\eqref{decomposition}}}{=}&
\int_{\Omega^*} h^{\mathrm{KS}}_{\mu_{\omega^\ast}}(T)\,\mathbf{d}\:\!\nu(\omega^\ast)\\
&\overset{\text{Theorem \ref{main}}}{\underset{\text{ergodic case}}{=}}&
\int_{\Omega^*} \lim\limits_{j \to \infty}h_{\mu_{\omega^\ast}}(T,\mathcal{P}_{d_j}^{(X_i)_{i=1}^{n_j},T})\,\mathbf{d}\:\!\nu(\omega^\ast)\\
&\overset{\text{monotone }}{\underset{\text{convergence}}{=}}&
\lim\limits_{j \to \infty} \int_{\Omega^*} h_{\mu_{\omega^\ast}}(T,\mathcal{P}_{d_j}^{(X_i)_{i=1}^{n_j},T})\,\mathbf{d}\:\!\nu(\omega^\ast)\\
&\overset{\text{\eqref{entropy}}}{=}&
\lim\limits_{j \to \infty} h_{\mu}(T,\mathcal{P}_{d_j}^{(X_i)_{i=1}^{n_j},T}).\end{aligned}$$ Here $(n_j)_{j\in {\mathbb N}}$ and $(d_j)_{j\in {\mathbb N}}$ are strictly increasing sequences of natural numbers.
Modifications and conseqences of Theorem \[main\]. {#sec3}
==================================================
We want to have a closer look at Theorem \[main\]. For this recall that $\mathbf{X} \circ T^{\circ t}$ can be interpreted as a measurement of a system at time $t$. As discussed in Section \[sec1\], there is no information loss when taking a pure ordinal viewpoint in the case that these measurements have ‘separating properties’.
*Less comparisons.* The main Theorem \[main\] can be given in a relaxed version if the considered observables provide a ‘separation’ from the outset (compare also [@keller_sinn_2009; @keller_et_al_2007]). In order to determine the Kolmogorov-Sinai entropy, this means, in the case of ‘separating’ original observables, one does not need all comparisons between the elements of an orbit but only comparisons between points and their iterates.
\[main2\] Let $(\Omega, \mathcal{A},\mu, T$) be a measure-preserving dynamical system and $\mathbf{X} = (X_i)_{i \in \mathbb{N}}$ be a sequence of observables such that $\sigma(\mathbf{X}) \overset{\mu}{\supset} \mathcal{A}$. Assume that or holds. Then $$h_\mu^{\mathrm{KS}}(T)
=
\lim\limits_{d,n\to\infty} h_\mu(T,\widetilde{\mathcal{P}}_d^{(X_i)_{i=1}^n,T})
=
\sup_{d,n\in {\mathbb N}} h_\mu(T,\widetilde{\mathcal{P}}_d^{(X_i)_{i=1}^n,T}).$$
For an ergodic map $T$ we have that $ \mathcal{A} \overset{\mu}{\subset} \widetilde{\Sigma}^{\mathbf{X},T}$, which follows from Corollary \[inclusions\] and the assumption $\sigma(\mathbf{X}) \overset{\mu}{\supset} \mathcal{A}$. Moreover $(\widetilde{\mathcal{P}}_d^{(X_i)_{i=1}^n,T})_{d,n \in \mathbb{N}}$ is an increasing sequence in $d$ and $N$ with respect to $\prec$, as it can be shown analogical to the proof of Lemma \[increasing\]. Thus, for $T$ ergodic the assertion follows by Lemma \[walters\]. To show the non-ergodic case one can use the the ergodic decomposition theorem as in the proof of Theorem \[main\]. It seems that the assumption $\sigma(\mathbf{X}) \overset{\mu}{\supset} \mathcal{A}$ in Theorem \[main2\] cannot be replaced by the assumption $\sigma((\mathbf{X} \circ T^{\circ t})_{t\in\mathbb{N}_0}) \overset{\mu}{\supset} \mathcal{A}$ in Theorem \[main\]. At least, the argumentation of the proof of Corollary \[inclusion\] cannot be adapted. Whereas $$\sigma(\mathbf{X}\circ T^{\circ t}) \overset{\mu}{\subset} \widetilde{\Sigma}^{\mathbf{X}\circ T^{\circ t},T}
\mbox{ for all }t \in \mathbb{N}_0$$ is true as is, the analogue $$\widetilde{\mathcal{P}}_d^{X_i \circ T,T}\prec\widetilde{\mathcal{P}}_{d+1}^{X_i,T}\mbox{ for all }d \in \mathbb{N}\mbox{ and }i=1,2,\ldots,n$$ of is false. Therefore the analogue $$\widetilde{\Sigma}^{X_i\circ T^{\circ t},T} \subset \widetilde{\Sigma}^{X_i,T}\mbox{ for all }i=1,2,\ldots,n\mbox{ and }t\in\mathbb{N}_0$$ of is not guaranteed. Let us give an example.
\[example\] Let $\Omega=[0,1]$ and $T:\Omega\hookleftarrow$ be defined by $$T(\omega)
=
\left\{
\begin{array}{rl}
2\omega &\mbox{for }\omega\leq\frac{1}{2}\\
2-2\omega &\mbox{else}
\end{array}
\right. .$$ ($T$ is the tent map preserving the equidistribution on $[0,1]$.) Let $$Y=2\cdot {\bf 1}_{[0,\,1/3]}+3\cdot {\bf 1}_{]1/3,\,2/3]}+{\bf 1}_{]2/3,\,1]},$$ $\omega_1 = 1$ and $\omega_2 = \frac{5}{6}$. Then $$\begin{split}
(Y(T^{\circ t}(\omega_1))_{t\in {\mathbb N}_0}&=(1,2,2,2,2,2,\ldots),\\
(Y(T^{\circ t}(\omega_2))_{t\in {\mathbb N}_0}&=(1,2,3,3,3,3,\ldots).
\end{split}$$ It follows that $\omega_1$ and $\omega_2$ are separated by $\mathcal{P}^{Y \circ T,T}_{0,1}$ and hence for all $\widetilde{\mathcal{P}}^{Y \circ T, T}_d$; $d \in \mathbb{N}$, but are not separated by $\widetilde{\mathcal{P}}^{Y,T}_d$ for all $d \in \mathbb{N}$. Consequently, $\widetilde{\mathcal{P}}^{Y \circ T, T}_d\hspace{-2mm}\not{\!\!\prec}\ \widetilde{\mathcal{P}}^{Y,T}_{d+l}$ for all $d \in \mathbb{N}$ and $l \in \mathbb{N}_0$.
*Other partitions.* For a single observable $X$ on a measure-preserving dynamical system $(\Omega,\mathcal{A},\mu,T)$ and $s,t\in \mathbb{N}_0$ with $s<t$, let $$\begin{split}
\mathcal{Q}^{X,T}_{s,t}
=
\{&\{\omega \in \Omega \mid X ( T^{\circ s}(\omega)) > X ( T^{\circ t}(\omega))\} ,\\
&\{\omega \in \Omega \mid X ( T^{\circ s}(\omega)) \leq X ( T^{\circ t}(\omega))\}\}
\end{split}$$ and $$\begin{split}
\mathcal{R}^{X,T}_{s,t}
=
\{&\{\omega \in \Omega \mid X ( T^{\circ s}(\omega)) < X ( T^{\circ t}(\omega))\} ,\\
&\{\omega \in \Omega \mid X ( T^{\circ s}(\omega)) > X ( T^{\circ t}(\omega))\},\\
&\{\omega \in \Omega \mid X ( T^{\circ s}(\omega)) = X ( T^{\circ t}(\omega))\}\}.
\end{split}$$ Further, for observables $X_1,X_2,\ldots ,X_n$ on $(\Omega,\mathcal{A},\mu,T)$ and $d\in {\mathbb N}$, let $$\label{qpart}
\mathcal{Q}_d^{(X_i)_{i=1}^n,T}
=
\bigvee_{i=1}^n\ \bigvee_{0\leq s<t\leq d}\mathcal{Q}_{s,t}^{X_i,T}$$ and $$\label{rpart}
\mathcal{R}_d^{(X_i)_{i=1}^n,T}
=
\bigvee_{i=1}^n\ \bigvee_{0\leq s<t\leq d}\mathcal{R}_{s,t}^{X_i,T}.$$ (If one of the sets of the right hand side of or is empty, then it is not considered in order to have only nonempty sets.) Then the following is valid:
The statement of Theorem \[main\] remains true when substituting$\mathcal{P}_d^{(X_i)_{i=1}^n,T}$ by $\mathcal{Q}_d^{(X_i)_{i=1}^n,T}$ or $\mathcal{R}_d^{(X_i)_{i=1}^n,T}$.
Application of Theorem \[main\] to $-{\bf X}=(-X_i)_{i\in {\mathbb N}}$ provides $$h_\mu^{\mathrm{KS}}(T)
=
\lim\limits_{d,n\to\infty} h_\mu(T,\mathcal{P}_d^{(-X_i)_{i=1}^n,T})=h_\mu(T,\mathcal{Q}_d^{(X_i)_{i=1}^n,T}).$$ Moreover, each $\mathcal{R}_d^{(X_i)_{i=1}^n,T}$ is finer than $\mathcal{P}_d^{(X_i)_{i=1}^n,T}$ implying $$h_\mu(T,\mathcal{R}_d^{(X_i)_{i=1}^n,T})\geq h_\mu(T,\mathcal{P}_d^{(X_i)_{i=1}^n,T}).$$ Therefore $$h_\mu^{\mathrm{KS}}(T)
\geq
\lim\limits_{d,n\to\infty} h_\mu(T,\mathcal{R}_d^{(X_i)_{i=1}^n,T})
\geq
\lim\limits_{d,n\to\infty} h_\mu(T,\mathcal{P}_d^{(X_i)_{i=1}^n,T})
\overset{\text{Theorem \ref{main}}}{=} h_\mu^{\mathrm{KS}}(T).$$ The existence of the limit $$\lim\limits_{d,n\to\infty} h_\mu(T,\mathcal{R}_d^{(X_i)_{i=1}^n,T})$$ and its coincidence with the corresponding supremum is obvious (compare discussion for $\mathcal{P}_d^{(X_i)_{i=1}^n,T}$ in Section \[sec2\]).
Let us consider an order $\prec$ between observables $X,Y$ by $X\prec Y$ iff for all $\omega_1,\omega_2\in\Omega$ the following holds (compare [@amigo_2012]): $$Y(\omega_1)\leq Y(\omega_2)\mbox{ implies }X(\omega_1)\leq X(\omega_2).$$ One easily shows the following:
For $X\prec Y$ it holds $\mathcal{R}_d^{X,T}\prec \mathcal{R}_d^{Y,T}$.
Note that for $X\prec Y$ not generally $\mathcal{P}_d^{X,T}\prec \mathcal{P}_d^{Y,T}$ and $\mathcal{Q}_d^{X,T}\prec \mathcal{Q}_d^{Y,T}$. After the following corollary being an immediate consequence of Theorem \[main\], we will illustrate this point by an example.
\[mainordered\] Let $(\Omega, \mathcal{A},\mu, T)$ be a measure-preserving dynamical system and $\mathbf{X}=(X_i)_{i\in {\mathbb N}}$ be a sequence of observables with $X_1\prec X_2\prec X_3\prec\ldots$ and $\sigma(\{\mathbf{X} \circ T^{\circ t}\}_{t\in {\mathbb N}_0}) \overset{\mu}{\supset} \mathcal{A}$. Assume that or holds. Then $$h_\mu^{\mathrm{KS}}(T)
=
\lim\limits_{d,i\to\infty} h_\mu(T,\mathcal{R}_d^{X_i,T})=\sup_{d,i\in {\mathbb N}} h_\mu(T,\mathcal{R}_d^{X_i,T}).$$
See Example \[example\] and let $$X=2\cdot {\bf 1}_{[0,\,5/8]}+{\bf 1}_{]5/8,\,1]}$$ and $$Y=4\cdot {\bf 1}_{[0,\,1/8]\cup [3/8,5/8]}+3\cdot{\bf 1}_{]1/8,\,3/8[}+{\bf 1}_{]5/8,\,1]}.$$ Obviously, $X\prec Y$. Let $\omega_1=\frac{1}{4}$ and $\omega_2=\frac{3}{4}$. Then $$\begin{split}
(X(T^{\circ t}(\omega_1))_{t\in {\mathbb N}_0}&=(2,2,1,2,2,2,2,2,2,\ldots),\\
(X(T^{\circ t}(\omega_2))_{t\in {\mathbb N}_0}&=(1,2,1,2,2,2,2,2,2,\ldots),\\
(Y(T^{\circ t}(\omega_1))_{t\in {\mathbb N}_0}&=(3,4,1,4,4,4,4,4,4,\ldots),
\end{split}$$ and $$(Y(T^{\circ t}(\omega_2))_{t\in {\mathbb N}_0}=(1,4,1,4,4,4,4,4,4,\ldots).$$ From this, on one hand it follows that $\omega_1$ and $\omega_2$ are separated by $\mathcal{P}^{X,T}_{0,1}$, i.e. lie in different elements of $\mathcal{P}^{X,T}_{0,1}$, hence are separated by $\mathcal{P}^{X,T}_d$ for all $d\in {\mathbb N}$. On the other hand, this implies that $\omega_1$ and $\omega_2$ are not separated by $\mathcal{P}^{Y,T}_d$ for all $d\in {\mathbb N}$.
Therefore for no $d\in {\mathbb N}$ the partition $\mathcal{P}^{Y,T}_d$ is finer than $\mathcal{P}^{X,T}_d$. The similar is true for $\mathcal{Q}^{Y,T}_d$ and $\mathcal{Q}^{X,T}_d$, since $\mathcal{Q}^{Z,T}_d=\mathcal{P}^{-Z,T}_d$ for an observable $Z$ on $(\Omega,\mathcal{A},\mu,T)$.
\[refinement\] Each finite partition $\mathcal{C}=\{C_1,C_2,\ldots ,C_q\}\subset\mathcal{A}$; $q \in \mathbb{N}$ is generated by observables of the form $X=\sum_{l=1}^q\alpha_l\cdot {\bf 1}_{C_l}$ in the sense that $C_l=X^{-1}(\alpha_l)$ for all $l=1,2,\ldots, q$, where $\alpha_l$; $l=1,2,\ldots , q$ are different real numbers. If a partition $\mathcal{D}\subset\mathcal{A}$ is finer than $\mathcal{C}$, than it can be written as $$\mathcal{D}=\bigcup_{l=1}^q\{D_j^{(l)}\mid j=1,2,\ldots, m_l\}$$ with $m_1,m_2,\ldots ,m_q\in {\mathbb N}$ and $C_l=\bigcup_{j=1}^{m_l}D_j^{(l)}$.
If $X=\sum_{l=1}^q\alpha_l\cdot {\bf 1}_{C_l}$ for different $\alpha_l\in {\mathbb N}$ and if $m>m_l$ for all $l=1,2,\ldots ,q$, then for $$Y=\sum_{l=1}^q\sum_{j=1}^{m_l}(\alpha_l\cdot m+j)\,{\bf 1}_{D_j^{(l)}}$$ it holds $X\prec Y$. This shows that an increasing sequence $(\mathcal{C}_d)_{d\in {\mathbb N}}$ can be ‘generated’ by a sequence $(X_d)_{d\in {\mathbb N}}$ of observables with $X_1\prec X_2\prec X_3\prec\ldots$ .
Permutation entropy {#sec4}
===================
The idea of considering dynamical systems from the ordinal viewpoint is strongly related to the invention of the permutation entropy, which we want to discuss now. We first give a definition of it in our general framework:
Given a sequence $\mathbf{X}=(X_i)_{i \in \mathbb{N}}$ of observables on a measure-preserving dynamical system $(\Omega,\mathcal{A},\mu,T)$, we define the *permutation entropy* $h_\mu(T,{\bf X})$ with respect to $\mathbf{X}$ by $$\label{PE}
h^\ast_\mu(T,{\bf X})
= \lim_{n\to\infty}\limsup_{d\to\infty} \frac{1}{d}\,H_\mu(T,\mathcal{P}_d^{(X_i)_{i=1}^n,T}).$$
Originally, by Bandt et al. in [@bandt_et_al_2002] the definition of permutation entropy was given directly for one-dimensional systems. In our framework, this is $h^\ast_\mu(T,\mathrm{id})$ with $T$ being an interval map.
[*Permutation and Kolmogorov-Sinai entropy.*]{} One reason for investigating the permutation entropy is its close relationship to the well-established Kolmogorov-Sinai entropy first observed by Bandt et al. in [@bandt_et_al_2002]. In their seminal paper they have shown that both entropies are coinciding for piecewise monotone interval maps $T$, i.e. for selfmaps $T$ on intervals splitting into finitely many subintervals on which $T$ is continuous and monotone.
Moreover, in the case that $\sigma((\mathbf{X} \circ T^{\circ t})_{t\in\mathbb{N}_0}) \overset{\mu}{\supset} \mathcal{A}$ and that or holds, the Kolmogorov-Sinai entropy is not larger than permutation entropy. It holds for finitely many observables $$\lim\limits_{d\to\infty} h_\mu(T,\mathcal{P}_d^{(X_i)_{i=1}^n,T})
\leq \limsup_{d\to\infty} \frac{1}{d}\,H_\mu(T,\mathcal{P}_d^{(X_i)_{i=1}^n,T})
\text{ for all } n \in \mathbb{N}$$ (see Keller et al. [@keller_et_al_2012 Corollary 3]), hence the corresponding inequality for infinitely many ones follows by $n$ approaching to infinity. So let us summarize:
Let $(\Omega, \mathcal{A},\mu, T$) be a measure-preserving dynamical system and $\mathbf{X} = (X_i)_{i \in \mathbb{N}}$ be a sequence of observables such that $\sigma((\mathbf{X} \circ T^{\circ t})_{t\in\mathbb{N}_0}) \overset{\mu}{\supset} \mathcal{A}$. Assume that or holds. Then $$h_\mu^{\mathrm{KS}}(T)
\leq
h^\ast_\mu(T,{\bf X}).$$
[ *The approach of Amig[ó]{} et al. [@amigo_2012; @amigo_2005].*]{} This approach to permutation entropy different to the original is based on a refining sequence of finite partitions and is justified by the following statement due to Amig[ó]{} et al. [@amigo_2012; @amigo_2005]. We express the statement by finite-valued observables and refer here to Remark \[refinement\].
For a measure-preserving dynamical system $(\Omega, \mathcal{A},\mu, T$) the following is valid:
1. If $X$ is a finitely-valued observable, and $\mathcal{P}$ the finite partition generated by $X$, then $$h_\mu(T,\mathcal{P})=h^\ast_\mu(T, X).$$
2. If $(X_i)_{i\in {\mathbb N}}$ is a sequence of finitely-valued observables with $X_1\prec X_2\prec X_3\prec\ldots$ and the corresponding sequence of finite partitions generates $\mathcal{A}$, then $$\label{modified}
h_{\mu}^{\mathrm{KS}}(T)
=\lim\limits_{i\to\infty} h^\ast_\mu(T,X_i).$$
One immediately sees that by Lemma \[walters\] assertion (ii) follows directly from statement (i). Amig[ó]{} et al. took the right hand side of as their modified concept of permutation entropy before showing its equality to Kolmogorov-Sinai entropy.
We want to finish this section by stating the following general problem, which is interesting on the different levels from the original one-dimensional definition of permutation entropy to the generalization for finitely or infinitely many observables.
Are the Kolmogorov-Sinai entropy and the permutation entropy coinciding and, if not, under which assumptions?
Note that the pure combinatorial part of the problem is relatively well understood (see Unakafova et al. [@unakafova_et_al_2013], Keller et al. [@keller_et_al_2012]).
Ordinal time series analysis {#sec5}
============================
Ever since the idea of Bandt and Pompe [@bandt_pompe_2002] to consider the rank order of consecutive values of a time series instead of the values themselves, the ordinal approach attracts increasing attention and is applied in many scientific fields, for example in biomedical research, engineering and econophysics (see Amig[ó]{} et al. [@amigo_eta_al_2014; @amigo_eta_al_2013], Zanin et al. [@zanin_et_al_2012] and the references given there).
The reason is that the ordinal viewpoint brings with it many advantages especially for measuring complexity, such as robustness against small noise, simplicity of application and interpretation, and low computational costs. As mentioned, the determination of Kolmogorov-Sinai entropy is usually not easy, our discussion above, however, suggests that the ordinal approach can be used as a framework for estimating the Kolmogorov-Sinai entropy of dynamical systems and suchlike from real world data.
In the following we consider the theory developed in the previous sections in an applied context and discuss the pro and cons of using this approach in view of studying long and complex time series. A detailed exposition of this ordinal pattern approach is provided in Keller et al. [@keller_et_al_2007].
*Ordinal patterns.* The task of gaining information about an underlying system via measurements is a common everyday problem. As already mentioned, this issue is increasingly addressed by using information lying in the ordinal structure of a system or a time series obtained from it. This leads to considering the up and downs in a time series, which can be described via so-called ordinal patterns.
For $d\in \mathbb{N}$ denote the *set of permutations* of $\{0,1,\dots,d\}$ by $\Pi_d$. We say that a real vector $(x_s)_{s=0}^d$ has *ordinal pattern* $\boldsymbol{\pi} = (\pi_0,\pi_1,\dots,\pi_d) \in \Pi_d$ *of order $d$* if $$x_{\pi_0}
\geq
x_{\pi_1}
\geq
\dots
\geq
x_{\pi_{d-1}}
\geq
x_{\pi_{d}}$$ and $$\label{equality}
\pi_{u-1}>\pi_u\mbox{ if }x_{\pi_{u-1}} = x_{\pi_u}\mbox{ for any }u \in \{1,2,\dots,d\}.$$ Given a time series $(x_t)_{t\in {\mathbb N}_0}$, the *ordinal pattern of order $d$ at time $t$* is defined as that of $(x_{t+s})_{s=0}^d$ and denoted by $\boldsymbol{\pi}_t$.
In Figure \[figure\] we consider a time series of $50$ data points where exemplary the ordinal pattern $\boldsymbol{\pi}_{10}=(0,5,3,4,6,1,2) \in \Pi_6$ is emphasized, which corresponds to the order relation of the six successive values at $t=10$, that is $$x_{t} > x_{t+5} > x_{t+3} > x_{t+4} > x_{t+6} > x_{t+1} > x_{t+2};\ t= 10.$$
table [datap.data]{}; table [data1.data]{}; (axis cs:10,69) – node\[left\] (axis cs:10,170); (axis cs:11,55) – node\[left\] (axis cs:11,120); (axis cs:12,43) – node\[left\] (axis cs:12,110); (axis cs:13,66) – node\[left\] (axis cs:13,150); (axis cs:14,65) – node\[left\] (axis cs:14,140); (axis cs:15,68) – node\[left\] (axis cs:15,160); (axis cs:16,64) – node\[left\] (axis cs:16,130);
It is easily seen that, following the framework given in Section \[sec1\], two states $\omega_1 \in \Omega$ and $\omega_2\in\Omega$ belong to the same part of some ordinal partition $\mathcal{P}_d^{(X_i)_{i=1}^n,T}$ iff the ordinal patterns of the vectors $$(X_i(\omega_1),X_i(T(\omega_1)),\ldots ,X(T^{\circ d}(\omega_2))) \text{ and } (X_i(\omega_2),X_i(T(\omega_2)),\ldots ,\linebreak X(T^{\circ d}(\omega_2)))$$ coincide. Clearly, the other previous considered partitions (see Equations , and ), despite some adjustments in terms of equality, can be coherent assimilated to this ordinal approach by redefining ordinal patterns in terms of the equality of values. The setting is here in some sense arbitrary, however, the proposed definition of ordinal patterns has established itself. We will use it in the following to demonstrate how the previous covered theory provides interesting and promising tools for extracting the information saved in an ordinal pattern sequence or suchlike, for example, by estimating the permutation entropy (see Equation ) or by approximating the Kolmogorov-Sinai entropy.
In order to utilize ordinal patterns for the analysis of a system, sequential data $(x_t)_{t\in {\mathbb N}}$ obtained from a given measurement are transformed into a series $(\boldsymbol{\pi}_t)_{t\in {\mathbb N}_0}$ of ordinal patterns. Distributions of ordinal patterns obtained from this approach are the central objects of exploration.
Note that ordinal patterns do not provide a symbolic representation as it is usually considered, since partitions of the state space are not given a priori, but are created on the basis of the given dynamics. However, the ordinal patterns as ‘symbols’ are very simple objects being directly obtained from the orbits of the system and containing intrinsic causal information. For the relationship of symbolic dynamics and representations and ordinal time series analysis see Amig[ó]{} et al. [@amigo_eta_al_2014].
For simplicity, we now restrict our exposition to the one-dimensional case with only one measurement. What we have in mind is a measure preserving dynamical system $(\Omega,\mathcal{A},\mu,T)$, where $\Omega$ is a Borel subset of ${\mathbb R}$, acting as the model of a system, with a single observable $X$ being the identity map. The extension of the ideas to the general case is obvious.
*Estimation of ordinal quantities.* The naive and mainly used estimator of ordinal pattern probabilities, so of the probability of the ordinal partition parts, is the relative frequency of ordinal patterns in an orbit of some length. For some $t,d\in {\mathbb N}$, some ordinal pattern $\boldsymbol{\pi}$ of order $d$ and some $\omega\in\Omega$ the estimation is given by the number $$\begin{aligned}
\hat{p}_{\boldsymbol{\pi}}=\frac{1}{t-d+1}\#\{s\in\{0,1,\ldots ,t-d\}\mid (X(T^{\circ s}(\omega)),X(T^{\circ s+1}(\omega)),\\\ldots
,X(T^{\circ s+d}(\omega)))\mbox{ has ordinal pattern }\boldsymbol{\pi}\}.\end{aligned}$$ Here $t+1$ is the length of the considered orbit of $\omega$. Clearly, the estimation only makes sense in the ergodic case. Then, by Birkhoff’s ergodic theorem, the corresponding estimator is consistent.
If in the ergodic case all $\hat{p}_{\boldsymbol{\pi}}$; $\pi \in \Pi_d$ are determined, it follows immediately that in the simple case considered a reasonable estimator for is given by the *empirical permutation entropy of order* $d \in \mathbb{N}$: $$\hat{h}^\ast_\mu(T,X)= - \frac{1}{d} \sum_{\pi \in \Pi_d} \hat{p}_{\pi} \ln \hat{p}_{\pi}.$$ It gives furthermore also some information on the Kolmogorov-Sinai entropy.
*Assets and drawbacks.* Irrespective of the considered ordinal partition, the ordinal approach brings along some practical advantages and disadvantages. Note that most difficulties to overcome are common to any sort of time series analysis.
Considering the order relation between the values of a time series, small inaccuracies in measurements (e.g. errors between the state of a system and its observed value) are mostly negligible. Hence, the methods considered are relatively robust towards calibration differences of measuring instruments. Furthermore, the ordinal approach is easily interpretable and there already exist efficient methods to perform an ordinal time series analysis in real time. For a deeper discussion we refer to Riedl et al. [@riedl_et_al_2013] as well as Unakafova and Keller [@unakafova_keller_2013]. Last but not least, a foreknowledge of the data range when analyzing data is usually not necessary.
In contrast, the ordinal analysis of time series can be rather poor if the underlying system is so complex that such a large value $d$ is needed that the computational capacity is insufficient. If, for example, the permutation entropy of a dynamical system is very large, its estimation by the empirical permutation entropy is problematic. Note that generally also for simple systems the convergency of empirical permutation entropies of order $d$ to the permutation entropy can be rather slow, which is the reason for considering a conditional adaption of the permutation entropy (see Unakafov and Keller [@unakafov_keller_2014]).
In addition, the choice of a suitable order $d$ with respect to the length of the original time series is affected by common problems. Large values of $d$ are needed to evaluate encapsulated information as accurate as possible but a large $d$ grants $(d+1)!$ possible ordinal patterns which have to be considered if nothing is known about the original time series. If one chooses an overlarge $d$ relative to the length of a time series, it can happen that not all ordinal patterns which are substantial for describing the underlying dynamics are observed in the ordinal pattern distribution or suchlike. This is known as *undersampling*.
Moreover, ordinal time series analysis can lead to an arbitrary poor approximation of the Kolmogorov-Sinai entropy or poor representation of the underlying dynamics by the statistics, especially while working on wrong assumptions, e.g. a given system fails to be ergodic or the chosen observables cause information loss while measuring. The next section alludes to the latter problem.
Algebra reconstruction dimension {#sec6}
================================
Theorems \[main\] claims that the Kolmogorov-Sinai entropy of $T$ can be computed provided that we have sufficiently many observables “generating” $\mathcal{A}$ up to $\mu$-measure zero. Essential for applications, the natural question arises how we can decrease the number of observables as much as possible. In this section we briefly review the known results in this direction.
*Only one observable.* The following example shows that *theoretically* in most real cases we can find only one such observable.
\[exmp:std\_spaces\] Let $I=[0,1]$, $Z$ be a separable complete metric space (such spaces are called *Polish*), $\Omega\subset Z$ be its uncountable Borel subset, and $\mathcal{A} := \mathcal{B}(\Omega)$ be the Borel $\sigma$-algebra of $\Omega$. Then the pair $(\Omega,\mathcal{B}(\Omega))$ is called a *standard Borel space*. It is well known, e.g. see Kechris [@kechris_1995 Proposition 12.1], that then there exists a measurable isomorphism of $(\Omega,\mathcal{B}(\Omega))$ onto the space $\bigl(I, \mathcal{B}(I)\bigr)$, that is a bijection $\mathbf{X}:\Omega\to I$ such that $\mathbf{X}^{-1}(\mathcal{B}(I)) = \mathcal{B}(\Omega)$.
Let $\mu$ be a measure on $(\Omega,\mathcal{B}(\Omega)$ and $T:\Omega\to\Omega$ be any $\mu$-preserving map. Then $$%\label{equ:exmp:sigmaTX_BO}
\mathcal{B}(\Omega)
\supset
\sigma((\mathbf{X}\circ T^{\circ t})_{t\in\mathbb{N}_0})
\supset
\sigma(\mathbf{X})
=
\mathbf{X}^{-1}(\mathcal{B}(I))
=
\mathcal{B}(\Omega),$$ that is $\sigma((\mathbf{X}\circ T^{\circ t})_{t\in\mathbb{N}_0}) = \mathcal{B}(\Omega)$. Moreover, as every separable metric space $Z$ can be embedded into a Hilbert cube being a compact space, compare to Hurewicz and Wallmann [@hurewicz_wallman_1941 Chapter V, §5, Theorem V4], we see that condition holds for $\Omega \subset Z$ as well, and therefore by Theorem \[main\] the Kolmogorov-Sinai entropy $h^{\mathrm{KS}}(T)$ of $T$ can be computed via the formula .
Notice that the function $\mathbf{X}:\Omega\to[0,1] \subset \mathbb{R}$ from Example \[exmp:std\_spaces\] is not in general continuous and its explicit construction is very complicated. Therefore it is not useful for real applications. This leads to the following notion.
\[def:ard\] Let $(\Omega, \mathcal{B}(\Omega))$ be a standard Borel space with measure $\mu$ on $\mathcal{B}(\Omega)$, and $T:\Omega\to\Omega$ be a $\mathcal{B}(\Omega)$-$\mathcal{B}(\Omega)$-measurable map. By the *algebra reconstruction dimension* of $T$ with respect to $\mu$ we will mean the minimal integer number $n\geq1$ such that there exists a *continuous* map $\mathbf{X}:\Omega \to\mathbb{R}^n$ satisfying $$\label{equ:def:ard}
\sigma((\mathbf{X}\circ T^{\circ t})_{t\in\mathbb{N}_0}) \overset{\mu}{\supset} \mathcal{B}(\Omega).$$ This number will be denoted by $\mathrm{ard}_{\mu}(T)$. If such $n$ does not exist, then we will assume that $\mathrm{ard}_{\mu}(T)=\infty$.
Thus $\mathrm{ard}_\mu (T)$ is the minimal number of *continuous* observables needed to approximate the Kolmogorov-Sinai entropy via .
Given a map $T:\Omega\to \Omega$, a map $\mathbf{X}:\Omega\to\mathbb{R}^n$ and $t\in\mathbb{N}$ one can define the following *$t$-reconstruction* map $$\Lambda_{\mathbf{X}, T, t}
=
\bigl(\mathbf{X}, \mathbf{X}\circ T, \ldots, \mathbf{X}\circ T^{\circ t-1}\bigr) : \Omega\to\mathbb{R}^{nt}$$ and an *$\infty$-reconstruction* map $$\Lambda_{\mathbf{X}, T, \infty}
=
\bigl(\mathbf{X}, \mathbf{X}\circ T, \mathbf{X}\circ T^{\circ 2}, \ldots\bigr) : \Omega\to\mathbb{R}^{\infty}.$$ Evidently, $\Lambda_{\mathbf{X},T,1} = \mathbf{X}$, $$\sigma((\mathbf{X}\circ T^{\circ s})_{s=0}^{t-1}) = \sigma(\Lambda_{\mathbf{X}, T, t}),$$ and $$\sigma(\mathbf{X}) \ \subset \ \sigma(\Lambda_{\mathbf{X}, T, t}) \
\subset
\ \sigma(\Lambda_{\mathbf{X}, T, t+1}) \
\subset
\ \sigma(\Lambda_{\mathbf{X}, T, \infty}); \ t\in\mathbb{N}.$$ In particular, can be reformulated as follows: $$\label{equ:sigmaLambda_in_BOmega}
\sigma(\Lambda_{\mathbf{X}, T, \infty}) \ \overset{\mu}{\supset} \ \mathcal{B}(\Omega).$$
Before discussing $\mathrm{ard}_{\mu}(T)$ we will present an example for the existence of one separating observable, that is $\mathbf{X}:\Omega\to\mathbb{R}$ satisfying , and therefore allowing to approximate the Kolmogorov-Sinai entropy by formula , see Theorem \[th:gen\_partitions\] below. However, now this observable is “discrete”, i.e. it takes at most countable many values.
\[def:gen\_partition\] Let $(\Omega,\mathcal{A},\mu,T)$ be a measure-preserving dynamical system. An at most countable partition $\mathcal{C} = \{C_l\}_{l=1}^q \subset \mathcal{A}$ of $\Omega$ for some $q\in\mathbb{N}\cup\{\infty\}$, is called *generating* with respect to $T$, if $$\sigma ((T^{-t} \mathcal{C})_{t\in {\mathbb N}_0})\overset{\mu}{=} \mathcal{A},$$ where $T^{-t} \mathcal{C} = \{ (T^{\circ t})^{-1}C_l\}_{l=1}^q$.
The following lemma is evident.
\[lm:func\_for\_gen\_part\] Suppose a measure-preserving dynamical system $(\Omega,\mathcal{A},\mu,T)$ has a generating partition $\mathcal{C}=\{C_l\}_{l=1}^q$; $q\in\mathbb{N}\cup\{\infty\}$. Define a function $\mathbf{X}:\Omega\to\mathbb{R}$ by $\mathbf{X}=\sum_{l=1}^ql\cdot {\bf 1}_{C_l}$ (compare Remark \[refinement\]). Then $\sigma(\mathbf{X}) = \sigma(\mathcal{C})$, whence $$\sigma(\Lambda_{\mathbf{X}, T, \infty})
=
\sigma\bigl((\mathbf{X}\circ T^{\circ t})_{t\in\mathbb{N}_0}\bigr)
=
\sigma ((T^{-t} \mathcal{C})_{t\in {\mathbb N}_0})
\overset{\mu}{=}
\mathcal{A}.$$
In general, a $\mu$-preserving map does not have a generating partition. Nevertheless, for non-singular ergodic automorphisms of standard probability spaces such partitions do exist, what we discuss now. First we recall necessary definitions.
Let $(\Omega, \mathcal{A}, \mu)$ be a probability space. The measure $\mu$ is called *complete* if for any subset $A\in\mathcal{A}$ with $\mu(A)=0$ every its subset $B$ also belongs to $\mathcal{A}$.
A countable family of sets $\{A_l\}_{l\in\mathbb{N}} \subset \mathcal{A}$ is called a *complete basis* of $(\Omega, \mathcal{A}, \mu)$ if
1. for each $A\in\mathcal{A}$ there exists a $B \in \sigma(\{A_i\}_{l=1}^\infty)$ with $A \subset B$ and $\mu(B\setminus A) = 0$;
2. for any $\omega_1,\omega_2\in\Omega$ there exists an $l\in\mathbb{N}$ such that $\omega_1\in A_l$ and $\omega_2\in\Omega\setminus A_l$;
3. each intersection $\bigcap_{l\in\mathbb{N}} B_l$, where every $B_l$ is either $A_l$ or $\Omega\setminus A_l$, is non-empty.
A probability space $(\Omega, \mathcal{A}, \mu)$ is called *standard* if it has a complete basis and $\mu$ is complete.
It has been proved by Rohlin [@rohlin_1961] that every standard probability space with non-atomic measure is isomorphic with the probability space $(I, \mathcal{B}(I), \lambda)$, where $\lambda$ is the Lebesgue measure on $I$.
Recall also that a one-to-one transformation $T:\Omega\to\Omega$ is non-singular with respect to a measure $\mu$ if it is bi-measurable, i.e. $T^{-1}\mathcal{A}=\mathcal{A}$ and $T\mathcal{A}=\mathcal{A}$, and $\mu(A)=0$ if and only if $\mu(T(A))=0$ for all $A\in \mathcal{A}$.
The following theorem is a consequence of results by Rohlin [@rohlin_1961], Parry [@parry_1966] and Krieger [@krieger_1970] about the existence of countable and finite generating partitions of ergodic maps.
[@rohlin_1961; @parry_1966; @krieger_1970]\[th:gen\_partitions\] Let $(\Omega, \mathcal{B}(\Omega), \mu)$ be a standard probability space, and $T:\Omega\to\Omega$ be a non-singular ergodic $\mu$-preserving map. Then $\Omega$ has a countable generating partition with respect to $T$. Hence there is a discrete measurable function $\mathbf{X}:\Omega\to\mathbb{R}$ taking at most countable distinct values and satisfying .
Moreover, if $h^{\mathrm{KS}}(T)<\infty$, then $T$ admits a *finite* generating partition, and so $\mathbf{X}$ can be assumed to take only finitely many distinct values.
*The continuous case.* Notice that the function $\mathbf{X}$ from Theorem \[th:gen\_partitions\] is slightly better than the one from Example \[exmp:std\_spaces\], as it takes a discrete set of values mutually distinct for distinct elements of the generating partition $\mathcal{C}$. Nevertheless, it is hard to construct as it requires to know a generating partition for $T$, and so it is not useful for application as well.
Now we will consider the opposite situation when almost any continuous map $\mathbf{X}:\Omega\to\mathbb{R}^n$ satisfies .
\[lm:emb\_th\_ard\_2k1\] Let $\Omega$ be a Polish space admitting an embedding $\mathbf{X}:\Omega\to\mathbb{R}^{n}$. Then for any measure $\mu$ on $\mathcal{B}(\Omega)$ and any $\mu$-preserving map $T$, we have that $\mathrm{ard}_{\mu}(T)\leq n$. In particular, if $\dim\Omega=k$; $k \in \mathbb{N}$, then $\mathrm{ard}_{\mu}(T)\leq 2k+1$.
Since $\mathbf{X}$ is an embedding, we obtain that $\sigma(\mathbf{X}) = \mathbf{X}^{-1}(\mathcal{B}(\mathbb{R}^n)) = \mathcal{B}(\Omega)$, whence $\sigma(\Lambda_{\mathbf{X}, T, \infty}) = \mathcal{B}(\Omega)$ as well.
The second statement follows from the well known fact that every $k$-dimensional separable metric space $\Omega$ can be embedded into $\mathbb{R}^{2k+1}$, [@hurewicz_wallman_1941 Chapter V, §4, Theorem V3]. Moreover, by the same theorem the set of embeddings $\mathrm{Emb}(\Omega, \mathbb{R}^{2k+1})$ is residual (and, in particular, dense) in the space $C(\Omega,\mathbb{R}^{2k+1})$ of all continuous maps. Therefore *almost every* family of $2k+1$ *continuous* observables will allow to approximate the Kolmogorov-Sinai entropy of $T$.
The next statement is a slight generalization of Theorem 2.2 from Keller [@keller_2012].
\[th:keller\_diff\] Let $\Omega$ be a smooth manifold and $\mathcal{D}(\Omega)$ be the group of its $C^{\infty}$ diffeomorphisms. Then there exists a residual subset $\mathcal{W}$ of $\mathcal{D}(\Omega)$ such that $\mathrm{ard}_{\mu}(T)=1$ for each $T\in\mathcal{W}$ and any measure $\mu$ preserved by $T$.
Let $\dim\Omega=k$. For each $n\in\mathbb{N}$ let $$\mathcal{E}_{n}
=
\{ (\mathbf{X}, T) \in C^{\infty}(\Omega,\mathbb{R})\times\mathcal{D}(\Omega) \mid \Lambda_{\mathbf{X},T,n}:\Omega\to\mathbb{R}^{n} \ \text{is an embedding}\}.$$ Thus if $(\mathbf{X}, T)\in \mathcal{E}_{n}$, then $\mathrm{ard}_{\mu}(T) = 1$.
It is proved by Takens [@takens_81] that if $n\geq 2 k + 1$, then $\mathcal{E}_{n}$ is residual (and in particular non-empty and everywhere dense) in $C^{\infty}(\Omega,\mathbb{R})\times\mathcal{D}(\Omega)$. Thus we have that $\mathcal{E}_{2k+1} = \bigcap_{l=1}^{\infty} U_l$, where each $U_i$ is open and everywhere dense in the space $C^{\infty}(\Omega,\mathbb{R})\times\mathcal{D}(\Omega)$. Let $p:C^{\infty}(\Omega,\mathbb{R})\times\mathcal{D}(\Omega) \to \mathcal{D}(\Omega)$ be the natural projection, i.e. $p(\mathbf{X},T) = T$. It is a standard fact from general topology that $p$ is an open map, whence $$\mathcal{W} = p\bigl(\mathcal{E}_{2k+1}\bigr)
=
\bigcap_{l=1}^{\infty} p(U_l)$$ is a residual subset of $\mathcal{D}(\Omega)$. Then $\mathrm{ard}_{\mu}(T)=1$ for each $T\in \mathcal{W}$ and any measure $\mu$ preserved by $T$.
Notice that the latter result does not guarantee that for *any* measure $\mu$ on $\mathcal{B}(\Omega)$ preserved by some diffeomorphism $T$ there exists some other $\mu$-preserving diffeomorphism $T'$ with $\mathrm{ard}_{\mu}(T')=1$.
The following notion allows to decrease the dimension $2k+1$ in Lemma \[lm:emb\_th\_ard\_2k1\] by putting some restrictions on $\mu$.
Let $\mathbf{X}:\Omega\to R$ be a continuous map between topological spaces. Then the following subset of $\Omega$ $$N_{\mathbf{X}} = \{ \omega\in \Omega \mid \mathbf{X}^{-1}( \mathbf{X}(\omega)) \not= \{\omega\} \}$$ will be called the *set of non-injectivity* of $\mathbf{X}$.
[(Antoniouk et al. [@antoniouk_et_al_2013 Theorem 4.2])]{} Let $\mathbf{X}:\Omega\to R$ be a continuous map between Polish spaces and $\mu$ be a measure on $\mathcal{B}(\Omega)$. Suppose there exists a Borel subset $D$ such that $N_{\mathbf{X}} \subset D$ and $\mu(D)=0$. Then $\sigma(\mathbf{X}) \overset{\mu}{=} \mathcal{B}(\Omega)$.
Let $\Omega$ be a smooth manifold of dimension $k$. Say that a subset $Q \subset \Omega$ *has Lebesgue measure zero*, if for any local chart $\phi:\Omega \supset U \to \mathbb{R}^k$ in $\Omega$ the set $\phi(Q\cap U)$ has Lebesgue measure zero in $\mathbb{R}^k$. Notice that there is no natural definition of a set of *fixed positive Lebesgue measure*.
A measure $\mu$ on $\mathcal{B}(\Omega)$ will be said *Lebesgue absolutely continuous* if $\mu(Q)=0$ for each subset $Q\subset\Omega$ of measure zero.
\[th:noninj\_set\][(Antoniouk et al. [@antoniouk_et_al_2013 Theorem 2.13])]{} Let $\Omega$ be a smooth manifold of dimension $k$ and $\mu$ be a Lebesgue absolutely continuous measure on $\mathcal{B}(\Omega)$. For each $n\in\mathbb{N}$ let $$\mathcal{V}_{n} = \{ \mathbf{X} \in C^{\infty}(\Omega,\mathbb{R}^n) \mid N_{\mathbf{X}}\in\mathcal{B}(\Omega), \ \mu(N_{\mathbf{X}})=0\}.$$ If $n>k$, then $\mathcal{V}_{n}$ is residual in $C^{\infty}(\Omega,\mathbb{R}^n)$. Hence $\mathrm{ard}_{\mu}(T) \leq k+1$ for any (not necessarily continuous) $\mu$-preserving map $T:\Omega\to\Omega$.
Comparison of results
---------------------
It is convenient to compare these results in the following table, where it is assumed that $\Omega$ is a Polish space of dimension $k$.
$\Omega$ $\mu$ $T$ $\mathrm{ard}_{\mu}(T)$ Statement
----------------- -------------------------------- ------------------------------------- ------------------------- --------------------------------
Borel space any measure any $\mu$-preserving measurable map $\leq 2k+1$ Lemma \[lm:emb\_th\_ard\_2k1\]
Smooth manifold Lebesgue absolutely continuous any $\mu$-preserving measurable map $\leq k+1$ Theorem \[th:noninj\_set\]
Smooth manifold any measure preserved by $T$ generic diffeomorphism $1$ Theorem \[th:keller\_diff\]
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Received xxxx 20xx; revised xxxx 20xx.
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---
abstract: 'Let $\pi:X\to Y$ be a factor map, where $(X,T)$ and $(Y,S)$ are topological dynamical systems. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\geq 0$, and $f\in C(X)$. The $\ba$-weighted topological pressure of $f$, denoted by $P^\ba(X, f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We prove the following variational principle: $$P^\ba(X, f)=\sup\left\{a_1h_\mu(T)+a_2h_{\mu\circ\pi^{-1}}(S)+\int f \;d\mu\right\},$$ where the supremum is taken over the $T$-invariant measures on $X$. It not only generalizes the variational principle of classical topological pressure, but also provides a topological extension of dimension theory of invariant sets and measures on the torus under affine diagonal endomorphisms. A higher dimensional version of the result is also established.'
address:
- |
Department of Mathematics\
The Chinese University of Hong Kong\
Shatin, Hong Kong\
- |
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China\
[*and* ]{}\
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China\
author:
- 'De-Jun FENG'
- Wen Huang
title: Variational principle for weighted topological pressure
---
Introduction
============
Inspired by the theory of Gibbs states in statistical mechanics, Ruelle [@Rue73] introduced the notion of topological pressure to the theory of dynamical systems and established a variational principle for it. Ruelle only considered the case when the underlying dynamical systems satisfy expansiveness and specification. Later Walters [@Wal75] generalized these results to general topological dynamical systems. Topological pressure, and the associated variational principle and equilibrium measures constitute the main components of the thermodynamic formalism [@Rue78]. They play important roles in dimension theory of dynamical systems. Indeed they provide as a basic tool in studying dimension of invariant sets and measures for conformal dynamical systems (see e.g. [@Bow79; @Rue82; @Pes97]).
In this paper we aim to introduce a generalized notion of pressure for factor maps between general topological dynamical systems, and establish a variational principle for it. To be more precise, let us introduce some notation first. We say that $(X, T)$ is a [*topological dynamical system*]{} (TDS) if $X$ is a compact metric space and $T$ is a continuous map from $X$ to $X$. Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems. Suppose that $(Y, S)$ is a factor of $(X, T)$, in the sense that there exists a continuous surjective map $\pi: X\to Y$ such that $\pi\circ T=S\circ \pi$. The map $\pi$ is called a [*factor map*]{} from $X$ to $Y$. Let $f$ be a real-valued continuous function on $X$, and let $a_1>0$, $a_2\geq 0$. The main purpose of this paper is to consider the following.
\[que-1\] How can one define a meaningful term $P^{(a_1, a_2)}(T, f)$ such that the following variational principle holds? $$\label{e-var}
P^{(a_1, a_2)}(T, f)=\sup\left\{a_1h_\mu(T)+a_2h_{\mu\circ\pi^{-1}}(S)+\int f \;d\mu\right\},$$ where the supremum is taken over the set of all $T$-invariant Borel probability measures $\mu$ on $X$, and $h_\mu(T), h_{\mu\circ\pi^{-1}}(S)$ stand for the measure-theoretic entropies of $\mu$ and $\mu\circ \pi^{-1}$ with respect to $T$ and $S$, respectively (cf. [@Wal82]).
According to the variational principle of Ruelle and Walters, the left-hand side of equals $a_1P(T,\frac{1}{a_1}f)$ in the particular case when $a_2=0$, where $P(T,\cdot)$ stands for the classic topological pressure of continuous functions (cf. [@Wal82]). Our interest is on the general case that $a_2\neq 0$. This project is motivated from the study of dimension of invariant sets and measures on the tori under diagonal affine expanding maps.
Let $T$ be the endmorphism on the $2$-dimensional torus ${\Bbb T}^2=\R^2/\Z^2$ represented by an integral diagonal matrix $A={\rm diag}(m_1,m_2)$, where $2\leq m_1< m_2$. That is, $Tu=A u \;(\mbox{mod}\; 1)$ for $u\in {\Bbb T}^2$. In their seminal works, Bedford [@Bed84] and McMullen [@McM84] independently determined the Hausdorff dimension of the so-called [*self-affine Sierpinski gaskets*]{}, which are a particular class of $T$-invariant subsets of $\T^2$ defined as follows: $$K(T, \D):=\left\{\sum_{n=1}^\infty A^{-n}u_n:\; u_n\in \D \mbox{ for all }n\geq 1\right\},$$ where $\D$ runs over the non-empty subsets of $$\left\{\left(\begin{array}{l} i\\
j\end{array} \right):\; i=0,1,\ldots, m_1,\; j=0,1,\ldots, m_2-1\right\}.$$ Moreover, McMullen [@McM84] exhibited explicitly that for each $\D$, there exists an ergodic $T$-invariant measure $\mu$ supported on $ K(T, \D)$ with $\dim_H\mu=\dim_HK(T, \D)$, where $\dim_H$ denotes the Hausdorff dimension of a set or measure (cf. [@Fal03]). Later Kenyon and Peres [@KePe96] extended this result to any compact $T$-invariant set $K\subseteq {\Bbb T}^2$, that is, there is an ergodic $T$-invariant measure $\mu$ supported on $K$ so that $\dim_H\mu=\dim_HK$. Furthermore Kenyon and Peres [@KePe96] established the following variational principle for the Hausdorff dimension of $K$: $$\label{e-var-H}
\dim_HK=\sup\left\{ \frac{1}{\log m_2}h_\eta(T)+\left(\frac{1}{\log m_{1}}-\frac{1}{\log m_{2}}\right)h_{\eta\circ \pi^{-1}}(S) \right\},$$ where the supremum is taken over the collection of $T$-invariant Borel probability measures $\eta$ supported on $K$, $\pi: \T^2\to \T^1$ denotes the projection $(x,y)\mapsto x$, and $S: \T^1\to \T^1$ denotes the map $x\mapsto m_1 x(\mbox{mod}\; 1)$. It is easy to check that $(\T^1, S)$ is a factor of $(\T^2, T)$ with the factor map $\pi$. We emphasize that for any ergodic $T$-invariant measure $\eta$ on $\T^2$, the sum in the bracket of just equals $\dim_H\eta$ (cf. [@KePe96 Lemma 3.1]); i.e. $$\label{e-LY}
\dim_H\eta= \frac{1}{\log m_2}h_\eta(T)+\left(\frac{1}{\log m_{1}}-\frac{1}{\log m_{2}}\right)h_{\eta\circ \pi^{-1}}(S).$$ This is a version of Ledrappier-Young dimension formula for ergodic measures on $\T^2$. We remark that an extension of the variational relation to higher dimensional tori was also established by Kenyon and Peres [@KePe96].
Let us turn back to Question \[que-1\]. According to , if $\pi$ is the factor map $(x,y)\mapsto x$ between the toral dynamics $(K, T)$ and $(\pi(K), S)$ as in the above paragraph, and if $f\equiv 0$ on $K$, and $a_1=\frac{1}{\log m_2}$, $a_2=\frac{1}{\log m_{1}}-\frac{1}{\log m_{2}}$, then we can just define $P^{(a_1,a_2)}(f)$ to be the Hausdorff dimension of $K$. The problem arises how can we extend this to general factor maps between topological dynamical systems, as well as to general continuous functions $f$ and vectors $(a_1,a_2)$.
In [@BaFe12; @Fen11], Barral and the first author defined $P^{(a_1,a_2)}(f)$ (and called it [*weighted topological pressure*]{}) via relative thermodynamic formalism and subadditive thermodynamic formalism, in the particular case when the underlying dynamical systems $(X, T)$ and $(Y, S)$ are subshifts over finite alphabets. They also studied the dynamical properties of weighted equilibrium measures (i.e. the invariant measures $\mu$ which attain the supremum in ) and gave the applications to the multifractal analysis on Sirpinski gaskets/sponges [@BaFe12], and to the uniqueness of invariant measures of full dimension supported on affine-invariant subsets of tori [@Fen11]. Independently, in this subshift case Yayama [@Yay11] defined $P^{(a_1,a_2)}(f)$ for the particular case $f\equiv 0$, along the similar way.
However, the approach of [@BaFe12; @Fen11] in defining $P^{(a_1,a_2)}(f)$ relies on certain special property of subshifts and does not extend to general topological dynamical systems (see Section \[s-7.1\] for details). Moreover, the variational principle established therein does not give a new proof of Kenyon and Peres’ variational relation for the Hausdorff dimension.
In the paper, we define $P^{(a_1,a_2)}(f)$ in a new way, which is inspired from the dimension theory of affine invariant subsets of tori, and from the “dimension” approaches of Bowen [@Bow73] and Pesin-Pitskel’ [@PePi84] in defining the topological entropy and topological pressure for arbitrary subsets.
We will present our definition under a more general setting. Let $k\geq 2$. Assume that $(X_i, d_i)$, $i=1,\ldots, k$, are compact metric spaces, and $(X_i, T_i)$ are topological dynamical systems. Moreover, assume that for each $1\leq i\leq k-1$, $(X_{i+1}, T_{i+1})$ is a factor of $(X_i, T_i)$ with a factor map $\pi_i: X_i\to X_{i+1}$; in other words, $\pi_1,\ldots, \pi_{k-1}$ are continuous maps so that the following diagrams commute.
$$\begin{CD}
X_1 @ > \pi_1 >> X_2 @ >\pi_2>> \cdots @>\pi_{k-1}>> X_k\\
@V T_1VV @ VV T_2 V @. @VV T_k V\\
X_1 @ > \pi_1 >> X_2 @ >\pi_2>> \cdots @>\pi_{k-1}>> X_k
\end{CD}$$
For convenience, we use $\pi_0$ to denote the identity map on $X_1$. Define $\tau_i:\;X_1\to X_{i+1}$ by $\tau_i=\pi_i\circ\pi_{i-1}\circ
\cdots \circ \pi_0$ for $i=0,1,\ldots,k-1$.
Let $\M(X_i,T_i)$ denote the set of all $T_i$-invariant Borel probability measures on $X_i$, endowed with the weak-star topology. Fix $\ba=(a_1,a_2,\ldots,a_k)\in \R^k$ with $a_1>0$ and $a_i\geq 0$ for $i\geq 2$. For $\mu\in \M(X_1,T_1)$, we call $$h^{\ba}_\mu(T_1):=\sum_{i=1}^ka_ih_{\mu\circ \tau_{i-1}^{-1}}(T_i)$$ the [*$\ba$-weighted measure-theoretic entropy of $\mu$ with respect to $T_1$*]{}, or simply, the [*$\ba$-weighted entropy of $\mu$*]{}, where $h_{\mu\circ \tau_{i-1}^{-1}}(T_i)$ denotes the measure-theoretic entropy of $\mu\circ \tau_{i-1}^{-1}$ with respect to $T_i$.
\[de-1.1\]
For $x\in X_1$, $n\in \N$, $\epsilon>0$, denote $$\begin{split}
& B_n^\ba(x,\epsilon):=\left \{y\in X_1:\; d_i(T_i^j\tau_{i-1} x, T_i^j \tau_{i-1} y)< \epsilon \mbox{ for } 0\leq j\leq \lceil(a_1+\ldots +a_i)n\rceil-1, \right. \\
&\left. \mbox{}\qquad\quad \qquad \qquad i=1,\ldots, k\right\},
\end{split}$$ where $\lceil u\rceil$ denotes the least integer $\geq u$. We call $B^\ba_n(x, \epsilon)$ the $n$-th $\ba$-weighted Bowen ball of radius $\epsilon$ centered at $x$.
Following the approaches of Bowen [@Bow73] and Pesin-Pitskel’ [@PePi84] in defining topological entropies and topological pressures of non-compact subsets [@Bow73], and in which replacing Bowen balls by $\ba$-weighted Bowen balls, we can define the notions of $\ba$-weighted topological entropy and $\ba$-weighted topological pressure, respectively. To be concise, in this section we only give the definition of $\ba$-weighted topological entropy. The definition of $\ba$-weighted topological pressure will be given in Section \[s-3.1\].
Let $Z \subset X_1$ and $\epsilon> 0$. We say that an at most countable collection of $\ba$-weighted Bowen balls $\Gamma
=\{B^\ba_{n_j}(x_j,\epsilon)\}_j$ [*covers*]{} $Z$ if $Z \subset \bigcup_j
B^\ba_{n_j}(x_j,\epsilon)$. For $\Gamma=\{B^\ba_{n_j}(x_j,\epsilon)\}_j$, put $n(\Gamma) =\min_j
n_j$. Let $s \geq 0$ and define $$\Lambda^{\ba, s}_{N, \epsilon}(Z) =\inf\sum_j
\exp(-sn_j),$$ where the infinum is taken over all collections $\Gamma=\{B^\ba_{n_j}(x_j,\epsilon)\}$ covering $Z$, such that $n(\Gamma) \geq N$. The quantity $\Lambda^{\ba, s}_{N, \epsilon}(Z)$ does not decrease with $N$, hence the following limit exists: $$\Lambda^{\ba, s}_\epsilon(Z) = \lim_{N\to \infty} \Lambda^{\ba, s}_{N, \epsilon}(Z).$$ There exists a critical value of the parameter s, which we will denote by $\htop^\ba(T_1,Z,\epsilon)$, where $\Lambda^{\ba, s}_{\epsilon}(Z)$ jumps from $\infty$ to $0$, i.e. $$\Lambda^{\ba, s}_{\epsilon}(Z) = \left\{
\begin{array}{ll}
0, & s > \htop^\ba(T_1,Z,\epsilon),\\
\infty,& s < \htop^\ba (T_1,Z,\epsilon).
\end{array}
\right.$$ It is clear to see that $\htop^\ba(T_1,Z,\epsilon)$ does not decrease with $\epsilon$, and hence the following limit exists, $$\htop^\ba(T_1, Z) = \lim_{\epsilon\to 0}\htop^\ba(T_1,Z,\epsilon).$$
\[de-1.2\] We call $\htop^\ba(T_1, Z)$ the [*$\ba$-weighted topological entropy of $T_1$ restricted to $Z$*]{} or, simply, the [*$\ba$-weighted topological entropy of $Z$*]{}, when there is no confusion about $T_1$. In particular we write $\htop^\ba(T_1)$ for $\htop^\ba(T_1,X_1)$.
Similarly we will define the $\ba$-weighted topological pressure $P^\ba(T_1, f)$ of continuous functions $f$ on $X_1$ (see Section \[s-3.1\]). In the particular case when $f\equiv 0$, we have $P^\ba(T_1, 0)=\htop^\ba(T_1)$. The main result of this paper is the following variational principle for weighted topological pressure.
\[thm-1.1\] Let $f\in C(X_1)$. Then $$\label{e-main}
P^\ba(T_1, f)=\sup\left\{\int f d\mu+h_\mu^\ba(T_1):\; \mu\in \M(X_1, T_1)\right\}.$$
In Section 6, we will extend the above theorem to the case that $f$ is a sub-additive potential. As a corollary, taking $f\equiv 0$ in Theorem \[thm-1.1\], we obtain the following variational principle for weighted topological entropy.
\[cor-entropy\] $
\htop^\ba(T_1)=\sup\{h_\mu^\ba(T_1):\; \mu\in \M(X_1, T_1)\}.
$
Theorem \[thm-1.1\] and Corollary \[cor-entropy\] provide as weighted versions of Ruelle-Walters’ variational principle for topological pressure, and Goodwyn-Dinaburg-Goodman’s variational principle for topological entropy (cf. [@Wal82]). They are also topological extensions of Kenyon-Peres’ variational principle for Hausdorff dimension of toral affine invariant sets. Indeed, consider the aforementioned factor map $\pi$ between the toral dynamics $(K, T)$ and $(\pi(K), S)$ and let $a_1=\frac{1}{\log m_2}$, $a_2=\frac{1}{\log m_{1}}-\frac{1}{\log m_{2}}$. It is easy to see from our definition that $\htop^{(a_1, a_2)}(T, K)$ simply coincides with $\dim_HK$, and hence Corollary \[cor-entropy\] recovers and its higher dimensional versions given in [@KePe96]. Moreover, by Corollary \[cor-entropy\], we can generalize to a class of skew-product expanding maps on the $k$-torus (see Section \[s-7.2\] for details).
The proof of Theorem \[thm-1.1\] is quite sophisticated. Besides adopting some ideas from [@Wal75; @Mis76] and [@KePe96], we also introduce substantially new ideas in the proof. For the convenience of the readers, in the following we illustrate a rough outline of our proof.
To see the lower bound in , we first prove that for each ergodic measure $\mu\in \M(X_1, T_1)$, $$\label{e-BK}
\lim_{\epsilon \rightarrow 0} \liminf_{n\rightarrow +\infty} \frac{-\log \mu(B_n^{\bf a}(x,\epsilon))}{n}=
\lim_{\epsilon \rightarrow 0} \limsup_{n\rightarrow +\infty}
\frac{-\log \mu(B_n^{\bf a}(x,\epsilon))}{n}=h_\mu^{\bf a}(T_1)$$ for $\mu$-a.e. $x\in X_1$. The above formula is not only a weighted version of Brin-Katok’s Theorem [@BrKa83] on local entropy, but also a topological extension of the Ledrappier-Young dimension formula . The justification of is mainly adapted from Kenyon-Peres’ proof of in [@KePe96] and Brin-Katok’s argument in [@BrKa83]. Based on , the lower bound in follows from a simple covering argument.
The proof of the upper bound in is more complicated. First we apply the techniques in geometric measure theory to prove the following “dynamical” Frostman lemma: for any $0<s<P^{\ba}(T_1,f)$ and small enough $\epsilon>0$, there exists a Borel probability measure $\nu$ on $X_1$ and $N\in \N$ such that $$\label{e-dyn}
\nu(B_n^{\ba}(x,\epsilon))\leq
\exp\left(-sn + \frac{1}{a_1}S_{\lceil a_1n\rceil}f(x)\right), \quad \forall x\in X_1, \; n\geq N,$$ where $S_nf(x):=\sum_{i=0}^{n-1}f(T_1^i x)$. This is a key part in our proof. Notice that there exists a small $\tau\in (0,\epsilon)$ such that for any Borel partition $\alpha_i$ of $X_i$ with $\mbox{diam}(\alpha_i)<\tau$, $i=1,\ldots, k$, we have $$\bigvee_{i=1}^k\bigvee_{j=t_{i-1}(n)}^{t_i(n)-1}T^{-j}_1\pi^{-1}_{i-1} \alpha_i(x)\subseteq B^\ba_n(x,\epsilon), \quad \forall x\in X_1, \; n\geq N,$$ where $t_0(n)=0$, $t_i(n)=\lceil (a_1+\ldots +a_{i})n\rceil$, and $\vee$ stands for the join of partitions. Hence implies that $$\label{e-6.2''}
\sum_{i=1}^k H_{\nu}\Big(\bigvee_{j=t_{i-1}(n)}^{t_i(n)-1}T^{-j}_1\pi^{-1}_{i-1} \alpha_i\Big)\geq sn -\int \frac{1}{a_1}S_{\lceil a_1n\rceil}f(x) d\nu(x).$$ Then, as another key part, we use and entropy theory to show the existence of a $T_1$-invariant measure $\mu$ on $X_1$ such that $h^{\ba}_\mu(T_1)>s-\int f d\mu$, from which the upper bound follows. In the proof of this part, a combinatoric lemma (see Lemma \[lem-KP\]) established by Kenyon-Peres [@KePe96] plays an important role; besides this, we also use a delicate compactness argument based on the upper semi-continuity of certain entropy functions, and adopt some ideas from [@Wal75; @Mis76] as well. Reducing back to the aforementioned toral dynamics, our approach provides a new proof for the upper bound in Kenyon-Peres’ variational principle .
The paper is organized as follows. In Section \[s-2\], we prove the upper semi-continuity of certain entropy functions. In Section \[s-3\], we define weighted topological pressure for continuous functions and more generally for sub-additive potentials; we also establish a dynamical Frostman lemma for the weighted topological pressure. In Sections \[s-4\]-\[s-5\], we prove respectively the lower and upper bounds of Theorem \[thm-1.1\]. In Section \[s-6\], we extend Theorem \[thm-1.1\] to the sub-additive case. In Section 7, we give some remarks, examples and questions. In Appendix \[s-a\], we prove the formula .
Upper semi-continuity of certain entropy functions {#s-2}
==================================================
In this section, we prove the upper semi-continuity of certain entropy functions (see Lemma \[usc\]), which is needed in our proof of the upper bound part of Theorem \[thm-1.1\]. We begin with the following.
Let $Z$ be a compact metric space. A function $f:Z\rightarrow
[-\infty,+\infty)$ is called upper semi-continuous if one of the following equivalent conditions holds:
1. $\limsup\limits_{z_N\rightarrow z} f (z_N) \leq f (z)$ for each $z \in Z$;
2. for each $r\in \mathbb{R}$ the set $\{z \in Z : f (z) \ge r\}$ is closed.
By (C2), the infimum of any family of upper semi-continuous functions is again an upper semi-continuous function; both the sum and supremum of finitely many upper semi-continuous functions are upper semi-continuous functions.
\[appro\] Let $Z$ be a compact metric space and $f:Z\rightarrow
[-\infty,+\infty)$ be an upper semi-continuous function. Then for any $\mu\in \mathcal{M}(Z)$, $$\label{usc-1}\inf \limits_{g\in C(Z),g\ge f}\int_Z
g(z) d \mu(z)=\int_Z f(z) d \mu(z).$$
It is well known that the equality holds when $f$ is a real-valued upper semi-continuous function (see e.g. [@DS Appendix (A7)] for a proof). In the following we assume that $f$ is an upper semi-continuous function taking values in $[-\infty,+\infty)$.
By the upper semi-continuity of $f$, we have $\sup_{z\in Z}f(z)=\max_{z\in Z}f(z)<+\infty$. Thus $\int_Z f(z) d
\mu(z)$ is well defined and $\int_Z f(z) d \mu(z)\in
[-\infty,+\infty)$.
For $M\in \mathbb{N}$, let $f_M(z)=\max\{ f(z),-M\}$ for $z\in Z$. Then $f_M$ is an upper semi-continuous real-valued function, and thus $$\inf \limits_{g\in C(Z),g\ge f_M}\int_Z
g(z) d \mu(z)=\int_Z f_M(z) d \mu(z).$$ Since $$\sup \limits_{M\in
\mathbb{N}}\sup \limits_{z\in Z} f_M(z)\le \max\left\{\max \limits_{z\in
Z}f(z),0\right\}<+\infty$$ and $ f_M(z)\searrow f(z)$ as $M\rightarrow
+\infty$ for any $z\in Z$, one has $$\lim_{M\rightarrow +\infty} \int_Z f_M(z) d \mu(z)= \int_Z
\lim_{M\rightarrow +\infty} f_M(z) d \mu(z) = \int_Z f(z) d
\mu(z)$$ by Lebesgue’s monotone convergence theorem. Moreover $$\begin{aligned}
\inf \limits_{g\in C(Z),g\ge f}\int_Z g(z) d \mu(z)&=\inf_{M\in
\mathbb{N}} \left\{\inf\limits_{g\in C(Z),g\ge f_M} \int_Z g(z) d
\mu(z)\right\}\\
&=\inf_{M\in \mathbb{N}} \int_Z f_M(z) d
\mu(z) \\
&=\lim_{M\rightarrow +\infty} \int_Z f_M(z) d \mu(z)\\
&= \int_Z f(z) d
\mu(z).\end{aligned}$$ This completes the proof of the lemma.
Let $(X,T)$ be a TDS with a compatible metric $d$. For $\epsilon>0$ and $M\in \mathbb{N}$, we define $$\label{e-2014-6}
\mathcal{P}_X(\epsilon,M)=\{ \alpha: \alpha \text{ is a finite
Borel partition of }X \text{ with } \text{diam}(\alpha)<\epsilon,
\#(\alpha)\le M\},$$ where $\text{diam}(\alpha):=\max_{A\in \alpha}\text{diam}(A)$, and $\#(\alpha)$ stands for the cardinality of $\alpha$. Then we define $$\mathcal{P}_X(\epsilon)=\{\alpha: \alpha \text{ is a finite
Borel partition of }X \text{ with }
\text{diam}(\alpha)<\epsilon\}.$$ It is clear that for any $\epsilon>0$, there exists $N:=N(\epsilon)\in \mathbb{N}$ such that $\mathcal{P}_X(\epsilon,M)\neq\emptyset$ for any $M\ge N$. The main result of this section is the following.
\[usc\] Let $(X,T)$ be a TDS and $\epsilon>0$. Then
1. If $M\in \mathbb{N}$ with $\mathcal{P}_X(\epsilon,M)\neq \emptyset$, then the map $$\label{e-HEML}
\theta\in
\mathcal{M}(X)\mapsto H_\theta(\epsilon,M;\ell):=\inf_{\alpha\in
\mathcal{P}_X(\epsilon,M)}
\frac{1}{\ell}H_\theta\left(\bigvee_{i=0}^{\ell-1}T^{-i}\alpha\right)$$ is upper semi-continuous from $\mathcal{M}(X)$ to $[0,\log M]$ for each $\ell\in \mathbb{N}$.
2. The map $$\theta\in
\mathcal{M}(X)\mapsto H_\theta(\epsilon;\ell):=\inf_{\alpha\in
\mathcal{P}_X(\epsilon)}
\frac{1}{\ell}H_\theta\left(\bigvee_{i=0}^{\ell-1}T^{-i}\alpha\right)$$ is a bounded upper semi-continuous non-negative function for each $\ell\in \mathbb{N}$.
3. The map $$\mu\in \mathcal{M}(X,T)\mapsto
h_\mu(T,\epsilon):=\inf_{\alpha\in
\mathcal{P}_X(\epsilon)}h_\mu(T,\alpha)$$ is a bounded upper semi-continuous non-negative function.
We first prove (1). Let $M\in \mathbb{N}$ with $\mathcal{P}_X(\epsilon,M)\neq \emptyset$, and $\ell\in
\mathbb{N}$. Clearly, the map $H_{\bullet}(\epsilon,M;\ell)$ is defined from $\mathcal{M}(X)$ to $[0,\log M]$. Let $\theta_0\in \mathcal{M}(X)$. It is sufficient to show that the map $H_{\bullet}(\epsilon,M;\ell)$ is upper semi-continuous at $\theta_0$.
Let $\delta>0$. Then there exists $\alpha\in
\mathcal{P}_X(\epsilon,M)$ such that $$\label{e-diff}
\frac{1}{\ell}H_{\theta_0}\left(\bigvee_{i=0}^{\ell-1}T^{-i}\alpha\right)\le
H_{\theta_0}(\epsilon,M;\ell)+\delta.$$ Let $\alpha=\{
A_1,\ldots,A_u\}$. Then $u\le M$ and $\text{diam}(A_i)<\epsilon$ for $i=1,2,\ldots,u$. By Lemma 4.15 in [@Wal82], there exists $\delta_1= \delta_1(u,\delta)
> 0$ such that whenever $\gamma_1=\{E_1,\ldots, E_u\},\gamma_2=\{F_1,\ldots, F_u\}$ are two Borel partitions of $X$ with $\sum_{j=1}^u \sum_{i=0}^{\ell-1} \theta_0\circ
T^{-i}(E_j\Delta F_j)<\delta_1$, then $$\label{entropy}
\begin{split}
\frac{1}{\ell}& \left|H_{\theta_0}\left(\bigvee_{i=0}^{\ell-1}T^{-i}\gamma_1\right)-H_{\theta_0}\left(\bigvee_{i=0}^{\ell-1}T^{-i}\gamma_2\right)\right|
\\
&\le \frac{1}{\ell}\sum_{i=0}^{\ell-1} \left|H_{\theta_0\circ
T^{-i}}(\gamma_1|\gamma_2)+H_{\theta_0\circ
T^{-i}}(\gamma_2|\gamma_1)\right|<\delta.
\end{split}$$ Write $\eta=\sum_{i=0}^{\ell-1} \theta_0\circ T^{-i}$. Next, we are going to construct a Borel partition $\beta=\{ B_1,\ldots,B_u\}$ of $X$ so that $\text{diam}(\beta)<\epsilon$, $\sum_{j=1}^u \eta(A_j\Delta
B_j)<\delta_1$ and $\eta(\partial \beta)=0$.
In fact, note that $\eta(X)=\ell<\infty$, hence $\eta$ is regular on $X$. Thus there exist open subsets $V_j$ of $X$ such that $A_j\subseteq V_j$, $\text{diam}(V_j)<\epsilon$ and $\eta(V_j\setminus A_j)<\frac{\delta_1}{u^2}$ for $j=1,\ldots,u$. Clearly, $\mathcal{V}:=\{ V_1,\ldots,V_u\}$ is an open cover. Let $t>0$ be a Lebesgue number of $\mathcal{V}$. For any $x\in X$, there exists $0<t_x\le \frac{t}{3}$ such that $\eta(\partial
B(x,t_x))=0$. Thus $\{
B(x,t_x):x\in X\}$ forms an open cover of $X$. Take its finite subcover $\{ B(x_i,t_{x_i})\}_{i=1}^r$, that is, $\bigcup_{i=1}^r
B(x_i,t_{x_i})=X$. Obviously, each $B(x_i,t_{x_i})$ is a subset of some $V_{j(i)}$, $j(i)\in \{1,\ldots,u\}$ since $t_{x_i}\le
\frac{t}{3}$.
Let $I_j=\{ i\in \{1,\ldots,r\}: B(x_i,t_{x_i})\subset V_j\}$ for $j=1,\ldots,u$. Then $\bigcup_{j=1}^u I_j=\{1,\ldots,r\}$. Put $B_1=\bigcup_{i\in I_1} B(x_i,t_{x_i})$ and $B_j=\left(\bigcup_{i\in I_j}
B(x_i,t_{x_i})\right)\setminus \bigcup_{m=1}^{j-1}B_m$ inductively for $j=2,\ldots,u$. It is clear that $\beta=\{ B_1,\ldots,B_u\}$ is a Borel partition of $X$ with $B_j\subseteq V_j$ and $\eta(\partial
B_j)=0$ for $j=1,\ldots,u$. Now for each $j\in \{1,\ldots,u\}$, $$\begin{aligned}
A_j\Delta B_j&= (B_j\setminus A_j)\cup (A_j\cap (X\setminus
B_j))\subseteq (V_j\setminus A_j)\cup \bigcup_{k\neq j}(A_j \cap
B_k)\\
&\subseteq (V_j\setminus A_j)\cup \bigcup_{k\neq j}(A_j \cap
V_k)\subseteq (V_j\setminus A_j)\cup \bigcup_{k\neq j}(A_j \cap
(V_k\setminus A_k))\\
&\subseteq \bigcup_{k=1}^u (V_k\setminus A_k).\end{aligned}$$ Thus $\sum_{j=1}^u \eta(A_j\Delta
B_j)\le u\sum_{k=1}^u \eta (V_k\setminus A_k)<\delta_1$.
Summing up, we have constructed a Borel partition $\beta=\{B_1,\ldots, B_u\}\in \mathcal{P}_X(\epsilon,M)$ so that $\sum_{j=1}^u\eta(B_j\Delta A_j)<\delta_1$ and $\eta(\partial \beta)=0$. Now on the one hand, by and , we have $$\frac{1}{\ell}H_{\theta_0}\left(\bigvee_{i=0}^{\ell-1}T^{-i}\beta\right)\le
\frac{1}{\ell}H_{\theta_0}\left(\bigvee_{i=0}^{\ell-1}T^{-i}\alpha\right)+\delta\le
H_{\theta_0}(\epsilon,M;\ell)+2\delta.$$ On the other hand, since $\eta(\partial \beta)=0$, one has $\theta_0(T^{-i}\partial \beta)=0$ for $i=0,1,\cdots,\ell-1$. As $ \partial T^{-i}A\subseteq
T^{-i}\partial A$ for any $A\subseteq X$, one has $\theta_0(\partial
T^{-i}\beta)=0$ for $i=0,1,\cdots,\ell-1$. Moreover note that $\partial(A\cap B)\subseteq (\partial A)\cap (\partial B)$ for any $A,B\subseteq X$, we have $\theta_0(\partial(\bigvee_{i=0}^{\ell-1}T^{-i}\beta))=0$. Thus the map $\theta\in \mathcal{M}(X)\mapsto
\frac{1}{\ell}H_\theta(\bigvee_{i=0}^{\ell-1}T^{-i}\beta)$ is continuous at the point $\theta_0$. Therefore $$\begin{aligned}
\limsup_{\theta\rightarrow \theta_0} H_\theta(\epsilon,M;\ell)&\le
\limsup_{\theta\rightarrow \theta_0}
\frac{1}{\ell}H_\theta \left(\bigvee_{i=0}^{\ell-1}T^{-i}\beta\right)\\
&=\frac{1}{\ell}H_{\theta_0}\left(\bigvee_{i=0}^{\ell-1}T^{-i}\beta\right)\\
&\le H_{\theta_0}(\epsilon,M;\ell)+2\delta.\end{aligned}$$ Finally letting $\delta\searrow 0$, we see that the map $H_{\bullet}(\epsilon,M;\ell)$ is upper semi-continuous at $\theta_0$. This completes the proof of (1).
Now we turn to the proof of (2). Let $\ell\in \mathbb{N}$. Since $\mathcal{P}_X(\epsilon)=\bigcup_{M\in
\mathbb{N},\mathcal{P}_X(\epsilon,M)\neq
\emptyset}\mathcal{P}_X(\epsilon,M)$, we have $$H_\theta(\epsilon;\ell)=\inf_{M\in
\mathbb{N},\mathcal{P}_X(\epsilon,M)\neq
\emptyset}H_\theta(\epsilon,M;\ell)$$ for $\theta\in
\mathcal{M}(X)$. Moreover, by (1) and the fact that the infimum of any family of upper semi-continuous functions is again an upper semi-continuous one, we know that the map $$\theta\in \mathcal{M}(X)\mapsto
H_\theta(\epsilon;\ell):=\inf_{\alpha\in \mathcal{P}_X(\epsilon)}
\frac{1}{\ell}H_\theta\left(\bigvee_{i=0}^{\ell-1}T^{-i}\alpha\right)$$ is a bounded upper semi-continuous non-negative function. This proves (2).
In the end we prove (3). Note that $$\begin{aligned}
h_\mu(T,\epsilon)&=\inf \limits_{\alpha\in
\mathcal{P}_X(\epsilon)}h_\mu(T,\alpha)=\inf \limits_{\alpha\in
\mathcal{P}_X(\epsilon)} \inf_{\ell\ge
1}\frac{1}{\ell}H_\mu\left(\bigvee_{i=0}^{\ell-1}T^{-i}\alpha\right)\\
&=\inf_{\ell\ge 1}\inf \limits_{\alpha\in \mathcal{P}_X(\epsilon)}
\frac{1}{\ell}H_\mu\left(\bigvee_{i=0}^{\ell-1}T^{-i}\alpha\right)=\inf_{\ell\ge
1}H_\mu(\epsilon;\ell)\end{aligned}$$ for $\mu\in \mathcal{M}(X,T)$. Using (2) and the fact that the infimum of any family of upper semi-continuous functions is again an upper semi-continuous one, we know that the map $$\mu\in \mathcal{M}(X,T)\mapsto
h_\mu(T,\epsilon)$$ is a bounded upper semi-continuous non-negative function. This completes the proof of the lemma.
Weighted topological pressures and a dynamical Frostman lemma {#s-3}
=============================================================
In this section we introduce the definition of weighted topological pressure for (asymptotically) sub-additive potentials for general topological dynamical systems. Moreover, using some ideas from geometric measure theory, we establish a dynamical Frostman lemma (see Lemma \[lem-Frost\]) for weighted topological pressure, which plays a key role in our proof of Theorem \[thm-1.1\].
Weighted topological pressures for sub-additive potentials {#s-3.1}
----------------------------------------------------------
Assume that $(X, T)$ is a TDS. We say that a sequence $\Phi=\{\log \phi_n\}_{n=1}^\infty$ of functions on $X$ is a [*sub-additive potential*]{} if each $\phi_n$ is an upper semi-continuous nonnegative-valued function on $X$ such that $$\label{e-1.0}
0\leq \phi_{n+m}(x)\leq \phi_n(x)\phi_m(T^nx),\qquad \forall\;
x\in X, \; m,n\in \N.$$ In particular, $\Phi$ is called [*additive*]{} if each $\phi_n$ is a continuous positive-valued function so that $\phi_{n+m}(x)=\phi_n(x)\phi_m(T^nx)$ for all $x\in X$ and $m,n\in \N$; in this case, there is a continuous real function $g$ on $X$ such that $\phi_n(x)=\exp(\sum_{i=0}^{n-1}g(T^ix))$ for each $n$.
Let $k\geq 2$. Assume that $(X_i, d_i)$, $i=1,\ldots, k$, are compact metric spaces, and $(X_i, T_i)$ are TDS’s. Moreover, assume that for each $1\leq i\leq k-1$, $(X_{i+1}, T_{i+1})$ is a factor of $(X_i, T_i)$ with a factor map $\pi_i: X_i\to X_{i+1}$.
Let $\ba=(a_1,\ldots, a_k)\in \R^k$ with $a_1>0$ and $a_i\geq 0$ for $2\leq i\leq k$. For any $n\in \N$ and $\epsilon>0$, define $$\label{e-0.1}
{\mathcal T}^\ba_{n,\epsilon}:=\{A\subset X_1:\; A \mbox { is Borel subset of } B^\ba_n(x,\epsilon) \mbox{ for some }x\in X_1\},$$ where $B^\ba_n(x,\epsilon)$ is defined as in Definition \[de-1.1\].
Let $\Phi=\{\log \phi_n\}_{n=1}^\infty$ be a sub-additive potential on $X_1$. Let $Z\subseteq X_1$, $s \geq 0$ and $N\in \N$, define $$\Lambda^{\ba, s}_{\Phi, N, \epsilon}(Z) =\inf\sum_j
\exp\left(-sn_j+\frac{1}{a_1}\sup_{x\in A_j}\phi_{\lceil a_1n_j\rceil}(x)\right ),$$ where the infimum is taken over all countable collections $\Gamma=\{(n_j, A_j)\}_j$ with $n_j\geq N$, $A_j\in {\mathcal T}^\ba_{n_j,\epsilon}$ and $\bigcup_jA_j\supseteq Z$. The quantity $\Lambda^{\ba, s}_{\Phi, N, \epsilon}(Z) $ does not decrease with $N$, hence the following limit exists: $$\Lambda^{\ba, s}_{\Phi, \epsilon}(Z) = \lim_{N\to \infty} \Lambda^{\ba, s}_{\Phi, N, \epsilon}(Z) .$$ There exists a critical value of the parameter $s$, which we will denote by $P^\ba(T_1,\Phi, Z, \epsilon)$, where $\Lambda^{\ba, s}_{\Phi, \epsilon}(Z)$ jumps from $\infty$ to $0$, i.e. $$\Lambda^{\ba, s}_{\Phi, \epsilon}(Z) = \left\{
\begin{array}{ll}
0, & s > P^\ba (T_1,\Phi,Z,\epsilon),\\
\infty,& s < P^\ba (T_1,\Phi, Z, \epsilon).
\end{array}
\right.$$ Clearly $P^\ba(T_1,\Phi,Z, \epsilon)$ does not decrease with $\epsilon$, and hence the following limit exists, $$P^\ba (T_1,\Phi, Z) = \lim_{\epsilon\to 0}P^\ba (T_1,\Phi, Z, \epsilon).$$
We call $P^\ba (T_1,\Phi):=P^\ba (T_1,\Phi, X_1)$ the [*$\ba$-weighted topological pressure of $\Phi$ with respect to $T_1$*]{} or, simply, the [*$\ba$-weighted topological pressure of $\Phi$*]{}, when there is no confusion about $T_1$.
Let $f\in C(X_1)$. Define $\Phi=\{\log \phi_n\}_{n=1}^\infty$ by $\phi_n(x)=\exp(\sum_{j=0}^{n-1}f(T_1^jx))$. In this case, $\Phi$ is additive. We just define $P^\ba (T_1,f):=P^\ba (T_1,\Phi)$.
Taking $f\equiv 0$, one can see that $P^\ba (T_1,0)=\htop^\ba(T_1)$. Let $\Phi=\{\log \phi_n\}_{n=1}^\infty$ be a sub-additive potential on $X_1$. For any $\mu\in \M(X_1, T_1)$, define $$\label{e-1.2} \Phi_*(\mu):=\lim_{n\to
\infty}\int \frac{\log \phi_n(x)}{n}\; d\mu(x).$$ This limit always exists and takes values in $\R\cup \{-\infty\}$ (cf. [@Wal75 Theorem 10.1]).
In our proof of Theorem \[thm-1.1\], we need the following dynamical Frostman lemma.
\[lem-Frost\] Let $\Phi=\{\log \phi_n\}_{n=1}^\infty$ be a sub-additive potential on $X_1$. Suppose that $P^{\ba}(T_1,\Phi)>0$. Then for any $0<s<P^{\ba}(T_1,\Phi)$, there exist a Borel probability measure $\nu$ on $X_1$ and $\epsilon>0$, $N\in \N$ such that for any $x\in X_1$ and $n\geq N$ we have $$\label{e-dyn**}
\nu(B_n^{\ba}(x,\epsilon))\leq
\exp (-sn)\sup_{y\in B_n^{\ba}(x,\epsilon)} (\phi_{\lceil a_1n\rceil}(y))^{1/a_1}.$$
A non-weighted version of the above lemma was first proved by the authors in the particular case when $\phi_n\equiv 1$ (see [@FeHu12 Lemma 3.4]), using some ideas and techniques in geometric measure theory. In the remainder of this section, we will give the detailed proof of Lemma \[lem-Frost\], by adapting and elaborating the approach in [@FeHu12]. A key ingredient of our proof is the notion of average weighted topological pressure, which is an analogue of weight Hausdorff measure in geometric measure theory. The definition of this notion and some of its properties will be given in next subsection. In Subsection \[S-3\], we prove Lemma \[lem-Frost\].
Average weighted topological pressures {#s-average}
--------------------------------------
Let $\Phi=\{\log \phi_n\}_{n=1}^\infty$ be a sub-additive potential on $X_1$. For any function $f: X_1\to [0, \infty)$, for $s \geq 0$ and $N\in \N$, define $$\label{e-19}
\W_{\Phi, N, \epsilon}^{\ba, s} (f) =\inf\sum_j
c_j\exp\left(-sn_j+\frac{1}{a_1}\sup_{x\in A_j}\log \phi_{\lceil a_1n_j\rceil}(x)\right ),$$ where the infimum is taken over all countable collections $\Gamma=\{(n_j, A_j, c_j)\}_j$ with $n_j\geq N$, $A_j\in {\mathcal T}^\ba_{n_j,\epsilon}$, $0<c_j<\infty$, and $$\sum_{j} c_j\chi_{A_j}\geq f,$$ where $\chi_A$ denotes the characteristic function of $A$, i.e., $\chi_A(x)=1$ if $x\in A$ and $0$ if $x\in X_1\backslash A$.
For $Z\subseteq X_1$, we set $\W^{\ba,s}_{\Phi, N,\epsilon}(Z)=\W^{\ba,s}_{\Phi, N,\epsilon}(\chi_Z)$. The quantity $\W^{\ba,s}_{\Phi, N,\epsilon}(Z)$ does not decrease with $N$, hence the following limit exists: $$\W^{\ba,s}_{\Phi, \epsilon}(Z) = \lim_{N\to \infty} \W^{\ba,s}_{\Phi, N,\epsilon}(Z).$$ There exists a critical value of the parameter $s$, which we will denote by $P^\ba_{W}(T_1,\Phi, Z, \epsilon)$, where $\W^{\ba,s}_{\Phi, \epsilon}(Z)$ jumps from $\infty$ to $0$, i.e. $$\W^{\ba,s}_{\Phi, \epsilon}(Z) = \left\{
\begin{array}{ll}
0, & s > P^\ba_{W} (T_1,\Phi, Z, \epsilon),\\
\infty,& s < P^\ba_{W} (T_1,\Phi,Z, \epsilon).
\end{array}
\right.$$ Clearly $P^\ba_{W} (T_1,\Phi, Z, \epsilon)$ does not decrease with $\epsilon$, and hence the following limit exists, $$P^\ba_{W} (T_1,\Phi, Z) = \lim_{\epsilon\to 0}P^\ba_{W} (T_1,\Phi, Z, \epsilon).$$
We call $P^\ba_{W} (T_1,\Phi):= P^\ba_{W} (T_1,\Phi, X_1) $ the [*average $\ba$-weighted topological pressure of $\Phi$ with respect to $T_1$*]{} or, simply, the [*average $\ba$-weighted topological pressure of $\Phi$*]{}, when there is no confusion about $T_1$.
The main result of this subsection is the following.
\[pro-2.1\] Let $Z\subseteq X_1$. Then for any $s\geq 0$ and $\epsilon,\delta>0$, we have $$\label{e-ine}
\Lambda^{\ba, s+\delta}_{\Phi, N,6\epsilon}(Z)\leq \W^{\ba,s}_{\Phi, N,\epsilon}(Z)\leq \Lambda^{\ba, s}_{\Phi, N,\epsilon}(Z),$$ when $N$ is large enough. As a consequence, $P^\ba_{W} (T_1,\Phi)=P^\ba (T_1,\Phi)$.
Before giving the proof of Proposition \[pro-2.1\], we first state some lemmas.
\[lem-1.1\] For any $s\geq 0$, $N\in \N$ and $\epsilon>0$, both $\Lambda^{\ba,s}_{\Phi, N,\epsilon}$ and $\W^{\ba, s}_{\Phi, N,\epsilon}$ are outer measures on $X$.
It follows directly from the definitions $\Lambda^{\ba,s}_{\Phi, N,\epsilon}$ and $\W^{\ba, s}_{\Phi, N,\epsilon}$.
The following combinatoric lemma plays an important role in the proof of Proposition \[pro-2.1\].
\[lem-2.0\] Let $(X, d)$ be a compact metric space and $\epsilon>0$. Let $(E_i)_{i\in \I}$ be a finite or countable family of subsets of $X$ with diameter less than $\epsilon$, and $(c_i)_{i\in \I}$ a family of positive numbers. Let $t>0$. Assume that $F\subseteq X$ such that $$F\subseteq \left\{x\in X:\; \sum_i c_i\chi_{E_i} >t\right \}.$$ Then $F$ can be covered by no more than $\frac{1}{t}\sum_ic_i$ balls with centers in $\bigcup_{i\in \I} E_i$ and radius $6\epsilon$.
To prove Lemma \[lem-2.0\], we need the following well known covering lemma.
\[cf. Theorem 2.1 in [@Mat95]\] \[lem-2.1\]
Let $(X, d)$ be a compact metric space and ${\mathcal
B}=\{B(x_i,r_i)\}_{i\in \mathcal I}$ be a family of open balls in $X$. Then there exists a finite or countable subfamily ${\mathcal B'}=\{B(x_i,r_i)\}_{i\in {\mathcal I}'}$ of pairwise disjoint balls in ${\mathcal B}$ such that $$\bigcup_{B\in {\mathcal B}} B\subseteq \bigcup_{i\in {\mathcal I}'}B(x_i,5r_i).$$
Without loss of generality, assume that $\I\subseteq \N$. For any $i\in \I$, pick $x_i\in E_i$ and write $B_i=B(x_i, \epsilon)$ and $5B_i=B(x_i, 5\epsilon)$ for short. Clearly $E_i\subseteq B_i$. Define $$Z=\left\{x\in X:\; \sum_i c_i\chi_{B_i} >t\right \}.$$ We have $F\subset Z$. To prove the lemma, it suffices to show that $Z$ can be covered by no more than $\frac{1}{t}\sum_ic_i$ balls with centers in $\{x_i:i\in \I\}$ and radius $6\epsilon$. To avoid triviality, we assume that $\sum_ic_i<\infty$; otherwise there is nothing left to prove.
For $k\in \N$, define $$\I_{k}=\{i\in \I:\; i\leq k\} \quad\mbox{and}\quad
Z_{k}=\Big\{x\in Z:\; \sum_{i\in \I_{k}}c_i\chi_{B_i}(x)>t\Big\}.$$ We divide the remaining proof into two small steps.
[*Step 1. For each $k\in \N$, there exists a finite set $\J_{k}\subseteq \I_{k}$ such that the balls $B_i$ [($i\in \J_{k}$)]{} are pairwise disjoint, $Z_{k}\subseteq \bigcup_{i\in \J_{k}}5B_i$ and $$\#(\J_{k})\leq \frac{1}{t}\sum_{i\in \I_{k}}c_i.$$* ]{} To prove the above result, we adopt the argument from Federer [@Fed69 2.10.24] in the study of weighted Hausdorff measures (see also Mattila [@Mat95 Lemma 8.16]). Since $\I_{k}$ is finite, by approximating the $c_i$’s from above, we may assume that each $c_i$ is a positive rational, and then multiplying $c_i$ and $t$ with a common denominator we may assume that each $c_i$ is a positive integer. Let $m$ be the least integer with $m\geq t$. Denote $\B=\{B_i,\; i\in \I_{k}\}$ and define $u: \B\to \N$ by $u(B_i)=c_i$. We define by induction integer-valued functions $v_0,v_1,\ldots, v_m$ on $\B$ and sub-families $\B_1,\ldots, \B_m$ of $\B$ starting with $v_0=u$. Using Lemma \[lem-2.1\] we find a pairwise disjoint subfamily $\B_1$ of $\B$ such that $\bigcup_{B\in \B}B\subseteq \bigcup_{B\in \B_1} 5B$, and hence $Z_{k}\subseteq \bigcup_{B\in \B_1} 5B$. Then by repeatedly using Lemma \[lem-2.1\], we can define inductively for $j=1, \ldots, m$, disjoint subfamilies $\B_j$ of $\B$ such that $$\B_j\subseteq \{B\in \B:\; v_{j-1}(B)\geq 1\},\quad Z_{k}\subseteq \bigcup_{B\in \B_j} 5B$$ and the functions $v_j$ such that $$v_j(B)=\left\{\begin{array}{ll}
v_{j-1}(B)-1 & \mbox { for } B\in \B_j,\\
v_{j-1}(B) & \mbox { for } B\in \B\backslash \B_j.
\end{array}
\right.$$ This is possible since for $j<m$, $Z_{k}\subseteq \big\{x: \sum_{B\in \B:\; B\ni x}v_j(B)\geq m-j\big\}$, whence every $x\in Z_{k}$ belongs to some ball $B\in \B$ with $v_j(B)\geq 1$. Thus $$\begin{aligned}
\sum_{j=1}^m \#(\B_j) &=&\sum_{j=1}^m \sum_{B\in \B_j} (v_{j-1}(B)-v_j(B))=\sum_{B\in \B_j}\sum_{j=1}^m (v_{j-1}(B)-v_j(B)) \\
&\leq & \sum_{B\in \B}\sum_{j=1}^m (v_{j-1}(B)-v_j(B))\leq \sum_{B\in \B} u(B)=\sum_{i\in \I_{k}} c_i.\end{aligned}$$ Choose $j_0\in \{1,\ldots, m\}$ so that $\#(\B_{j_0})$ is the smallest. Then $$\#(\B_{j_0}) \leq \frac{1}{m}\sum_{i\in \I_{k}} c_i
\leq \frac{1}{t}\sum_{i\in \I_{k}} c_i.$$ Hence $\J_{k}:=\{i\in \I_k:\; B_i\in \B_{j_0}\}$ is desired.
[*Step 2. There exists $\I'\subset \I$ with $\#(\I')\leq \frac{1}{t}\sum_{i\in \I}c_i$ so that $Z\subseteq \bigcup_{i\in \I'} 6B_i$.* ]{}
Since $Z_{k}\uparrow Z$, $Z_{k}\neq \emptyset$ when $k$ is large enough. Let $\J_{k}$ be constructed as in Step 1. Then $\J_{k}\neq \emptyset$ when $k$ is large enough. Define $G_{k}=\{x_i:\; i\in \J_{k}\}$. Then $$\#(G_k)=\# (\J_k)\leq \frac{1}{t}\sum_{i\in \I_{k}}c_i\leq \frac{1}{t}\sum_{i\in \I} c_i.$$
Since the space of non-empty compact subsets of $X$ is compact with respect to the Hausdorff distance (cf. Federer [@Fed69 2.10.21]), there is a subsequence $(k_j)$ of natural numbers and a non-empty compact set $G\subset X$ such that $G_{k_j}$ converges to $G$ in the Hausdorff distance as $j\to \infty$. As any two different points in $G_{k}$ have a distance not less than $\epsilon$, so do the points in $G$. Thus $G$ is a finite set, moreover, $\#(G_{k_j})=\#(G)$ when $j$ is large enough. Hence $$\bigcup_{x\in G} B(x,5.5\epsilon) \supseteq \bigcup_{x\in G_{k_j}} B(x,5\epsilon)=\bigcup_{i \in \J_{k_j}} 5 B_i\supseteq Z_{k_j}$$ when $j$ is large enough, and thus $\bigcup_{x\in G} B(x,5.5\epsilon)\supseteq Z$. On the other hand, when $j$ is large enough, we have $$\bigcup_{x'\in G_{k_j}} B(x',6\epsilon) \supseteq \bigcup_{x\in G} B(x,5.5\epsilon),$$ hence we have $\bigcup_{x'\in G_{k_j}} B(x',6\epsilon)\supseteq Z$, with $\#(G_{k_j})\leq \frac{1}{t}\sum_{i\in \I}c_i.$
Return back to the metric spaces $(X_i, d_i)$ and TDS’s $(X_i, T_i)$, $i=1,\ldots, k$. For $n\in \N$, define a metric $d^\ba_n$ on $X_1$ by $$d^\ba_n(x,y)=\sup\left\{ d_i(T_i^j\tau_{i-1} x, T_i^j \tau_{i-1} y): \; 1\leq i\leq k, \;0\leq j\leq \lceil(a_1+\ldots +a_i)n\rceil-1\right\}.$$
\[lem-51\] Let $\epsilon>0$. Then there exist $\gamma>0$ such that for any $n\in \N$, $X_1$ can be covered by no more than $\exp(n\gamma)$ balls of radius $\epsilon$ in metric $d^\ba_n$.
By compactness, for each $1\leq i\leq k$, we can find a finite open cover $\alpha_i$ of $X_i$ with $\mbox{diam}(\alpha_i)<\epsilon$ (in metric $d_1$). Let $n>0$. Define $$\beta=\bigvee_{i=1}^k \left(
\bigvee_{j=0}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i\right).$$ Then $\beta$ is an open cover of $X_1$ with diameter less than $\epsilon$ (with respect to the metric $d^\ba_n$). Hence $X_1$ can be covered by at most $\#(\beta)$ many balls of radius $\epsilon$ in metric $d^\ba_n$. Let $\gamma>0$ so that $\exp(\gamma)=\prod_{i=1}^k(\#(\alpha_i))^{a_1+\cdots+a_i+1}$. Then $$\#(\beta)\leq \prod_{i=1}^k(\#(\alpha_i))^{\lceil(a_1+\cdots+a_i)n\rceil}\leq \exp(n\gamma),$$ which implies the result of the lemma.
Let $Z\subseteq X_1$, $s\geq 0$, $\epsilon, \delta>0$. Taking $f=\chi_Z$ and $c_i\equiv 1$ in the definition , we see that $\W^{\ba,s}_{\Phi, N,\epsilon}(Z)\leq \Lambda^{\ba, s}_{\Phi, N,\epsilon}(Z)$ for each $N\in \N$. In the following, we prove that $\Lambda^{\ba, s+\delta}_{\Phi, N,6\epsilon}(Z)\leq \W^{\ba, s}_{\Phi, N,\epsilon}(Z)$ when $N$ is large enough.
Let $\gamma>0$ be given as in Lemma \[lem-51\]. Assume that $N\geq 2$ such that $$\label{e-gamma}
n^2(n+1)e^{\gamma-n\delta}\le 1,\quad \forall\; n\geq N.$$ Let $\{(n_i, A_i, c_i)\}_{i\in \mathcal I}$ be a family so that $\I\subseteq \N$, $A_i\in {\mathcal T}^\ba_{n_i,\epsilon} $, $0<c_i<\infty$, $n_i\geq N$ and $$\label{e-gez}
\sum_{i\in \I}c_i\chi_{A_i}\geq \chi_Z.$$ We show below that $$\label{e-key}
\Lambda^{\ba, s+\delta}_{\Phi, N,6\epsilon}(Z)\leq \sum_{i\in \I}c_i\exp\left(-n_is+\frac{1}{a_1}\sup_{x\in A_j}\log \phi_{\lceil a_1n_j\rceil}(x)\right) ,$$ which implies $\Lambda^{\ba, s+\delta}_{\Phi, N,6\epsilon}(Z)\leq \W^{\ba,s}_{\Phi, N,\epsilon}(Z)$.
To prove , we write $\I_n=\{i\in \I:\; n_i=n\}$, $$g_n(x)=(\phi_{\lceil a_1n\rceil}(x))^{1/a_1},\quad g_n(E)=\sup_{x\in E} g_n(x)$$ for $n\in \N$, $x\in X_1$ and $E\subseteq X_1$. Moreover set $$\begin{split}
Z_{n,t}&=\Big\{x\in Z:\; \sum_{i\in \I_{n}}c_i\chi_{A_i}(x)>t\Big\}.
\end{split}$$ We claim that $$\label{e-2.10}
\Lambda^{\ba,s+\delta}_{\Phi, N,\epsilon}(Z_{n,t})\leq \frac{1}{t n^2}\sum_{i\in \I_n}c_i \exp(-ns) g_n(A_i),\quad \forall \; n\geq N, \;0<t<1.$$
To prove the claim, assume that $n\geq N$ and $0<t<1$. Set $D=\frac{1}{n}\log g_n(Z_{n,t})$. For $\ell=1,\ldots, n$ and $i\in \I_n$, write $$Z_{n,t}^\ell=\left\{x\in Z_{n,t}: \frac{1}{n}\log g_n(x)\in \Big(D-\frac{\gamma \ell}{n}, D-\frac{\gamma (\ell-1)}{n}\Big]\right\},\quad A_{i,\ell}:=A_i\cap Z_{n,t}^\ell,$$ and $$Z_{n,t}^0=\left\{x\in Z_{n,t}: \frac{1}{n}\log g_n(x)\leq D-\gamma\right\},\quad A_{i,0}=A_i\cap Z_{n,t}^0.$$ For $\ell=0, 1,\ldots, n$, write $\I_{n,\ell}=\{i\in \I_n:\; A_{i,\ell}\neq \emptyset\}$; then $$Z_{n, t}^{\ell}= \Big\{x\in X_1:\; \sum_{i\in \I_{n,\ell}}c_i\chi_{A_{i,\ell}}(x)>t\Big\}.$$ Hence by Lemma \[lem-2.0\], $Z_{n, t}^{\ell}$ can be covered by no more than $\frac{1}{t}\sum_{i\in \I_{n,\ell}} c_i$ balls with center in $\bigcup_{i\in \I_n}A_{i,\ell}$ and radius $6\epsilon$ (in metric $d_n^\ba$). It follows that for $\ell=1,\ldots, n$, $$\label{e-im}
\begin{split}
\Lambda^{\ba,s+\delta}_{\Phi, N,6\epsilon}(Z_{n, t}^{\ell})&\leq e^{-n(s+\delta)} (\frac{1}{t}\sum_{i\in \I_{n,\ell}} c_i) g_n(Z_{n,t}^\ell)\leq e^{-n(s+\delta)} e^{\gamma} \frac{1}{t}\sum_{i\in \I_{n,\ell}} c_i g_n(A_{i,\ell})\\
&\leq e^{\gamma-n\delta} \frac{1}{t}\sum_{i\in \I_{n}} c_i e^{-ns} g_n(A_{i}).\\
\end{split}$$ We still need to estimate $\Lambda^{\ba,s+\delta}_{\Phi, N,6\epsilon}(Z_{n, t}^{0})$. By Lemma \[lem-51\], $X_1$ (and thus $Z_{n, t}^{0}$) can be covered by no more than $\exp(n\gamma)$ balls of radius $6\epsilon$ (in metric $d_n^\ba$). Hence $$\label{e-im1}
\begin{split}
\Lambda^{\ba,s+\delta}_{\Phi, N,6\epsilon}(Z_{n, t}^{0})&\leq \exp(n\gamma) e^{-n(s+\delta)}g_n(Z_{n, t}^{0})\leq \exp(n\gamma) e^{-n(s+\delta)} \exp (n (D-\gamma))\\
&\leq e^{-n(s+\delta)} \exp (nD)\leq e^{-n\delta} \frac{1}{t}\sum_{i\in \I_{n}} c_i e^{-ns} g_n(A_{i}),
\end{split}$$ where the last inequality uses the following arguments: since $\exp (nD)=g_n(Z_{n,t})$, for any $u<\exp (nD)$, there exists $x\in Z_{n,t}$ so that $g_n(x)\geq u$; however since $x\in Z_{n,t}$ we have $\sum_{i\in \I_n: \; A_i\ni x}c_i\geq t$, which implies $$\frac{1}{t}\sum_{i\in \I_{n}} c_i g_n(A_{i})\geq \frac{1}{t}\sum_{i\in \I_{n}: A_i\ni x } c_i g_n(A_{i})\geq \frac{1}{t}\sum_{i\in \I_{n}: A_i\ni x } c_i u\geq u.$$
Combining -, we have $$\label{e-im2}
\begin{split}
\Lambda^{\ba,s+\delta}_{\Phi, N,6\epsilon}(Z_{n, t})\leq \sum_{\ell=0}^n \Lambda^{\ba,s+\delta}_{\Phi, N,6\epsilon}(Z_{n, t}^{\ell})&\leq
(n+1)e^{\gamma-n\delta} \frac{1}{t}\sum_{i\in \I_{n}} c_i e^{-ns} g_n(A_{i})\\
&\leq \frac{1}{n^2t}\sum_{i\in \I_{n}} c_i e^{-ns} g_n(A_{i}),
\end{split}$$ where in the last inequality we use . This finishes the proof of .
To complete the proof of Proposition \[pro-2.1\], notice that $\sum_{n=N}^\infty n^{-2}\leq \sum_{n=2}^\infty n^{-2}\leq 1$; hence if $x\not\in
\bigcup_{n\geq N} Z_{n, n^{-2}t}$, then $$\sum_{i\in \I}c_i\chi_{A_i}(x)=\sum_{i\in \bigcup_{n=N}^\infty\I_n}c_i\chi_{A_i}(x)\leq \sum_{n=N}^\infty\sum_{i\in \I_n}c_i\chi_{A_i}(x)\leq \sum_{n=N}^\infty n^{-2}t\leq t<1,$$ thus $x\not\in Z$ by . Therefore $Z\subseteq \bigcup_{n\geq N} Z_{n, n^{-2}t}$. By , $$\Lambda^{\ba,s+\delta}_{\Phi, N,6\epsilon}(Z)\leq \sum_{n=N}^\infty \Lambda^{\ba,s+\delta}_{\Phi, N,6\epsilon}(Z_{n, n^{-2}t})\leq
\frac{1}{t}\sum_{n=N}^\infty\sum_{i\in \I_{n}} c_i e^{-ns} g_n(A_{i})\leq \frac{1}{t}\sum_{i\in \I} c_i e^{-n_is} g_{n_i}(A_{i}).$$ Letting $t\uparrow 1$, we have $$\Lambda^{\ba,s+\delta}_{\Phi, N,6\epsilon}(Z)\leq \sum_{i\in \I} c_i e^{-n_is} g_{n_i}(A_{i}),$$ that is, holds. This finishes the proof of Proposition \[pro-2.1\].
Proof of Lemma \[lem-Frost\] {#S-3}
----------------------------
It is easy to see that Lemma \[lem-Frost\] follows directly from Proposition \[pro-2.1\] and the following lemma.
\[pro-3.1\] Let $s\geq 0$, $N\in \N$ and $\epsilon>0$. Suppose that $c:=\W^{\ba,s}_{\Phi, N,\epsilon}(X_1)>0$. Then there is a Borel probability measure $\mu$ on $X_1$ such that for any $n\geq N$, $x\in X_1$, and any compact $K\subset B^\ba_n(x,\epsilon)$, $$\mu(K)\leq \frac{1}{c}e^{-ns} g_n(K),$$ where $$g_n(z)=(\phi_{\lceil a_1n\rceil}(z))^{1/a_1},\quad g_n(K)=\sup_{z\in K} g_n(z).$$
Here we adopt the idea employed by Howroyd in his proof of the Frostman lemma in compact metric spaces (cf. [@How95 Theorem 2]). Clearly $c<\infty$. We define a function $p$ on the space $C(X_1)$ of continuous real-valued functions on $X_1$ by $$\label{zx-eq}
p(f)=(1/c)\W^{\ba,s}_{\Phi, N,\epsilon}(f).$$
Let ${\bf 1}\in C(X_1)$ denote the constant function ${\bf 1}(x)\equiv 1$. It is easy to verify that
1. $p(f+g)\le p(f)+p(g)$ for any $f,g\in C(X_1)$.
2. $p(tf)=tp(f)$ for any $t\ge 0$ and $f\in C(X_1)$.
3. $p({\bf 1})=1$, $0\leq p(f)\leq \|f\|_\infty$ for any $f\in C(X_1)$, and $p(g)=0$ for $g\in C(X_1)$ with $g\le 0$.
By the Hahn-Banach theorem, we can extend the linear functional $t\mapsto t p({\bf 1})$, $t\in \R$, from the subspace of the constant functions to a linear functional $L:\;
C(X_1)\to \R$ satisfying $$L({\bf 1})=p({\bf 1})=1 \text{ and }-p(-f)\le L(f)\le p(f) \text{ for any }f\in C(X_1).$$ If $f\in C(X_1)$ with $f\ge 0$, then $p(-f)=0$ and so $L(f)\ge 0$. Hence combining the fact $L({\bf 1})=1$, we can use the Riesz representation theorem to find a Borel probability measure $\mu$ on $X_1$ such that $L(f)=\int f d\mu$ for $f\in C(X_1)$.
Now let $x\in X_1$ and $n\geq N$. Suppose that $K$ is a compact subset of $B^\ba_n(x,\epsilon)$. Let $\delta>0$. Since $g_n$ is upper semi-continuous, there exists an open set $B^\ba_n(x,\epsilon)\supset V\supset K$ such that $g_n(V)\leq g_n(K)+\delta$.
By the Uryson lemma, there exists $f\in C(X_1)$ such that $0
\le f\le 1$, $f(y)=1$ for $y\in K$, and $f(y)=0$ for $y\in X_1\backslash V$. Then $\mu(K)\le L(f)\le p(f)$. Since $f\leq \chi_{V}$ and $n\geq N$, we have $\W^{\ba, s}_{\Phi, N,\epsilon}( f)\leq e^{-ns}g_n(V)$ and thus $p(f)\le \frac{1}{c} e^{-sn}g_n(V)$. Therefore $$\mu(K)\le \frac{1}{c}e^{-ns}g_n(V)\leq \frac{1}{c}e^{-ns} (g_n(K)+\delta).$$ Letting $\delta\to 0$, we have $\mu(K)\le \frac{1}{c}e^{-ns}g_n(K).$ This completes the proof of the lemma.
The proof of Theorem \[thm-1.1\]: Lower bound {#s-4}
=============================================
In this section, we prove the lower bound part of Theorem \[thm-1.1\]. The following weighted version of Brin-Katok theorem plays a key role in our proof.
\[thm-4.1\] For each ergodic measure $\mu\in \mathcal{M}(X_1,T_1)$, we have $$\lim_{\epsilon \rightarrow 0} \liminf_{n\rightarrow +\infty} \frac{-\log \mu(B_n^{\bf a}(x,\epsilon))}{n}=
\lim_{\epsilon \rightarrow 0} \limsup_{n\rightarrow +\infty}
\frac{-\log \mu(B_n^{\bf a}(x,\epsilon))}{n}=h_\mu^{\bf a}(T_1)$$ for $\mu$-a.e. $x\in X_1$.
We shall postpone the proof of Theorem \[thm-4.1\] to Appendix \[s-a\]. In the following we prove the lower bound part of Theorem \[thm-1.1\] for sub-additive potentials rather than additive potentials.
\[thm-3.3\] Let $\Phi=\{\log \phi_n\}_{n=1}^\infty$ be a sub-additive potential on $X_1$. Then $$P^\ba (T_1,\Phi)\ge
\sup\left\{\Phi_*(\mu)+h^\ba_\mu(T_1):\; \mu\in \M(X_1,T_1), \Phi_*(\mu)\neq -\infty \right\}.$$
By Jacobs’ theorem (cf. [@Wal82 Theorem 8.4]) and Proposition A.1.(3) in [@FeHu10], if $\mu=\int_{\E(X_1, T_1)} m \;d\tau(m)$ is the ergodic decomposition of an element $\mu$ in $\M(X_1, T_1)$, then $$h^\ba_\mu(T_1)=\int_{\E(X_1, T_1)} h^\ba_m(T_1) \;d\tau(m),\quad \Phi_*(\mu)=\int_{\E(X_1, T_1)} \Phi_*(m)\;d\tau(m).$$ Hence to prove the proposition, it suffices to show that $$\label{e-0.2}
P^\ba (T_1,\Phi)\geq \Phi_*(\mu)+\min\{\delta^{-1}, \;h^\ba_\mu(T_1)-\delta\}-\delta$$ for any $\delta>0$ and any ergodic $\mu\in \M(X_1,T_1)$ with $\Phi_*(\mu)\neq -\infty$.
For this purpose, we fix $\delta>0$ and an ergodic measure $\mu$ on $X_1$ with $\Phi_*(\mu)\neq -\infty$. Write $$H:=\min\{\delta^{-1}, \;h^\ba_\mu(T_1)-\delta\}.$$ By Theorem \[thm-4.1\], we can choose $\epsilon>0$ so that $$\label{e-2.1}
\liminf_{n\to \infty}\frac{-\log \mu(B^\ba_n(x,\epsilon))}{n}>H \quad \mbox{ for $\mu$-a.e. }x\in X_1.$$ Since $\Phi$ is sub-additive, by Kingman’s subadditive ergodic theorem (cf. [@Wal82 p. 231] and [@FeHu10 Proposition A.1.]), we have $$\lim_{n\to \infty} \frac{1}{n} \log \phi_n(x)=\Phi_*(\mu)$$ for $\mu$-a.e. $x\in X_1$. Hence there exists a large $N\in \N$ and a Borel set $E_N\subset X_1$ with $\mu(E_N)>1/2$ such that for any $x\in E_N$ and $n\geq N$, $$\label{e-2.2}
\mu(B^\ba_n(x,\epsilon))<\exp(-nH),\quad \log \phi_{\lceil a_1n\rceil}(x)\geq a_1n\Phi_*(\mu)-a_1n\delta.$$
Now assume that $\Gamma
=\{(n_j, A_j)\}_i$ is a countable collection so that $n_j\geq N$, $A_j\in {\mathcal T}^\ba_{n_j,\epsilon/2}$ (cf. for the definition) and $\bigcup_jA_j=X_1$. By definition, for each $j$, there exists $x_j\in X$ so that $A_j\subseteq B^\ba_{n_j}(x_j,\epsilon/2)$. Set $$\I:=\{j:\; A_j\cap E_N\neq \emptyset\}.$$ For $j\in \I$, pick $y_j\in A_j\cap E_N$; then we have $$A_j\subseteq B_{n_j}^\ba(x_j,\epsilon/2)\subseteq B_{n_j}^\ba(y_j,\epsilon)$$ and thus $$\mu(A_j)\leq \mu(B_{n_j}^\ba(y_j,\epsilon))\leq \exp(-n_jH);$$ moreover, $$\frac{1}{a_1} \sup_{x\in A_j} \log \phi_{\lceil a_1n_j \rceil}(x) \geq \frac{1}{a_1}\log \phi_{\lceil a_1n_j \rceil}(y_j)\geq n_j\Phi_*(\mu)-n_j\delta.$$ Set $s:=\Phi_*(\mu)+H-\delta$. Then for any $j\in \I$, $$\exp\left(-sn_j+\frac{1}{a_1}\sup_{x\in A_j}\phi_{\lceil a_1n_j\rceil}(x)\right )
\geq {\mu(A_j)} \exp \left(n_j \left(-s+\Phi_*(\mu)+H-\delta\right)\right)=\mu(A_j).$$ Summing over $j\in \I$, we have $$\begin{split}
\sum_{j\in \I}\exp \left(-sn_j+\frac{1}{a_1} \sup_{x\in A_j} \phi_{\lceil a_1n_j \rceil}(x)\right)&\geq \sum_{j\in \I}\mu(A_j)
\geq \mu\left(\bigcup_{j\in \I} A_j\right)
\geq \mu(E_N)\geq 1/2.
\end{split}$$ It follows that $\Lambda_{\Phi, \epsilon}^{\ba, s}(X_1)\geq \Lambda_{\Phi, N, \epsilon}^{\ba, s}(X_1)\geq 1/2$, and thus $$P^\ba(T_1,\Phi)\geq P^\ba (T_1,\Phi, X_1,\epsilon/2)\geq s= \Phi_*(\mu)+\min\{\delta^{-1}, \;h^\ba_\mu(T_1)-\delta\}-\delta,$$ as desired.
The proof of Theorem \[thm-1.1\]: upper bound {#s-5}
=============================================
In this section, we prove the upper bound in Theorem \[thm-1.1\], that is, for any $f\in C(X_1)$ and $\delta>0$, there exists $\mu\in \M(X_1, T_1)$ such that $$P^\ba(T_1, f)\leq h^\ba_\mu(T_1)+\int_{X_1} f d\mu+\delta.$$
Before proving the above result, we first give some lemmas.
\[lem-en.0\] Let $(X, T)$ be a TDS and $\mu\in \M(X)$. Let $\alpha=\{A_1,\ldots, A_M\}$ be a Borel partition of $X$ with cardinality $M$. Write for brevity $$h(n):=H_{\frac{1}{n}\sum_{i=0}^{n-1}\mu\circ T^{-i}}(\alpha), \quad h(n,m):=H_{\frac{1}{m}\sum_{i=n}^{m+n-1}\mu\circ T^{-i}}(\alpha).$$ for $n, m\in \mathbb{N}$. Then
- $h(n)\leq \log M$ and $h(n,m)\leq \log M$ for $n,m\in \N$.
- $|h(n+1)-h(n)|\leq \frac{1}{n+1}\log \left(3M^2(n+1)\right)$ for all $n\in \mathbb{N}$.
- $\left|h(n+m)-\frac{n}{n+m}h(n)-\frac{m}{n+m}h(n,m)\right|\leq \log 2$ for all $n, m\in \N$.
\(i) is obvious. Now we turn to the proof of (ii). It is well known (see e.g. [@Wal82 Theorem 8.1] and the proof therein) that for any finite Borel partition $\beta$ of $X$, $\nu_1,\nu_2\in \M(X)$ and $p\in [0,1]$, $$\label{eq-en-es}
\begin{aligned}
0&\le H_{p\nu_1+(1-p)\nu_2}(\beta)-pH_{\nu_1}(\beta)-(1-p)H_{\nu_2}(\beta)\\
&\le -(p\log p+(1-p)\log (1-p))\\
&\le \log 2.
\end{aligned}$$ Let $n\in \mathbb{N}$. Applying and (i), we have $$\begin{aligned}
| h & (n+1) -h(n)|\\
&=\Big| h(n+1)-\frac{n}{n+1}h(n)-\frac{1}{n+1}H_{\mu\circ T^{-n}}(\alpha)-\frac{1}{n+1}h(n)+\frac{1}{n+1}H_{\mu\circ T^{-n}}(\alpha)\Big|\\
&\le \Big| h(n+1)-\frac{n}{n+1}h(n)-\frac{1}{n+1}H_{\mu\circ T^{-n}}(\alpha)\Big|+\frac{2}{n+1}\log M\\
&\le -\frac{n}{n+1}\log \frac{n}{n+1}-\frac{1}{n+1}\log \frac{1}{n+1}+\frac{2}{n+1}\log M\\
&\le \frac{1}{n+1}\log \big(3 M^2(n+1)\big),\end{aligned}$$ where we use the fact $(1+1/n)^n<e<3$ in the last inequality. This proves (ii).
Finally, since $$\frac{1}{n+m}\sum_{i=0}^{n+m-1}\mu\circ T^{-i}=\frac{n}{n+m}\left (\frac{1}{n}\sum_{i=0}^{n-1}\mu\circ T^{-i}\right)+\frac{m}{n+m}\left(\frac{1}{m}\sum_{i=n}^{n+m-1}\mu\circ T^{-i}\right)$$ for $n,m\in \mathbb{N}$, (iii) follows from .
\[lem-en.1\] Let $(X, T)$ be a TDS and $\mu\in \M(X)$. For $\epsilon>0$ and $\ell,M\in \N$, let $H_\bullet(\epsilon, M; \ell)$ be defined as in . Then the following statements hold.
1. For all $n\in \mathbb{N}$, $$\begin{aligned}
\Big|H_{\frac{1}{n} \sum_{i=0}^{n-1} \mu\circ T^{-i}} & (\epsilon, M;\ell)-H_{\frac{1}{n+1}\sum_{i=0}^{n}\mu\circ T^{-i}}(\epsilon, M;\ell) \Big|\\
& \le \frac{1}{\ell(n+1)}\log \big(3M^{2\ell}(n+1)\big).\end{aligned}$$
2. For all $n, m\in \N$, $$\label{e-inequality}
\begin{aligned}
\frac{n}{n+m}& H_{\frac{1}{n}\sum_{i=0}^{n-1}\mu\circ T^{-i}}(\epsilon, M;\ell)
+\frac{m}{n+m}H_{\frac{1}{m}\sum_{i=n}^{n+m-1}\mu\circ T^{-i}}(\epsilon, M;\ell)\\
&\leq H_{\frac{1}{n+m}\sum_{i=0}^{m+n-1}\mu\circ T^{-i}}(\epsilon, M;\ell)+\frac{\log 2}{\ell}.
\end{aligned}$$
The statements directly follow from the definition of $H_{\bullet}(\epsilon, M;\ell)$ and Lemma \[lem-en.0\].
\[lem-6.1\] Let $\nu\in \M(X)$ and $M\in \N$. Suppose $\xi=\{A_1,\ldots,A_j\}$ is a Borel partition of $X$ with $j\leq M$. Then for any positive integers $n,\ell$ with $n\geq 2\ell$, we have $$\frac 1n H_\nu\left(\bigvee_{i=0}^{n-1}T^{-i}\xi\right)\leq \frac
1\ell
H_{\nu_n}\left(\bigvee_{i=0}^{\ell-1}T^{-i}\xi\right)+\frac{2\ell}{n}\log
M,$$ where $\nu_n=\frac{1}{n}\sum_{i=0}^{n-1}\nu\circ T^{-i}$.
The following lemma is a slight variant of by Kenyon and Peres.
\[lem-KP\] Let $p\in \N$. Let $u_j: \N\to \R$ ($j=1,\ldots, p$) be bounded functions with $$\lim_{n\to \infty} |u_j(n+1)-u_j(n)|=0.$$ Then for any positive numbers $c_1,\ldots, c_p$ and $r_1,\ldots, r_p$, $$\limsup_{n\to +\infty}\sum_{j=1}^p (u_j(\lceil c_jn\rceil)-u_j(\lceil r_jn \rceil))\geq 0.$$
For the convenience of reader, we give a proof by adapting the argument of Kenyon and Peres in [@KePe96].
For $j=1,\ldots, p$, extend $u_j$ in a piecewise linear fashion to a bounded continuous function on $[1,+\infty)$. Then for each $1\leq j\leq p$, $$\label{eq-KP}
\lim_{t\rightarrow +\infty} \sup\left\{ |u_j(x)-u_j(y)|:x,y\ge t,\; |x-y|\leq \max_{1\leq i\leq p}\max\{c_i,r_i, 1\}\right\}=0.$$ Take a positive number $M$ so that $$\label{e-M}
M>\max_{1\le j\le p}\{ |\log c_j|+|\log r_j|+1\}.$$ Then for every $w>M$, $$\begin{aligned}
& \Big| \int_M^w \sum_{j=1}^p \big( u_j(e^{x+\log c_j})-u_j(e^{x+\log r_j})\big)dx \Big| \\
&\quad =\left| \sum_{j=1}^p \left[ \int_{M+\log c_j}^{w+\log c_j} u_j(e^x)dx - \int_{M+\log r_j}^{w+\log r_j} u_j(e^x)dx \right] \right|\\
&\quad \leq \sum_{j=1}^p \left| \int_{M+\log c_j}^{w+\log c_j} u_j(e^x)dx - \int_{M+\log r_j}^{w+\log r_j} u_j(e^x)dx \right|\\
&\quad = \sum_{j=1}^p \Big|\int^{w+\log c_j}_{w+\log r_j} u_j(e^x) dx-\int^{M+\log c_j}_{M+\log r_j} u_j(e^x) dx\Big|,\end{aligned}$$ Since each $u_j$ is bounded, the sum in the right-hand side of the last ‘$=$’ above is uniformly bounded. It follows that $$\limsup_{x\rightarrow +\infty}\sum_{j=1}^p \big( u_j(e^{x+\log c_i})-u_j(e^{x+\log r_j})\big)\ge 0.$$ Setting $t=e^x$, one has $$\limsup_{t\rightarrow +\infty}\sum_{j=1}^p(u_j( c_it)-u_j(r_j t ))\ge 0.$$ Combining the above inequality with , we have $$\begin{aligned}
\limsup_{n\rightarrow +\infty}& \sum_{j=1}^p(u_j(\lceil c_jn\rceil)-u_j(\lceil r_j n\rceil)\\
&=\limsup_{n\rightarrow +\infty}\sum_{j=1}^p(u_j(c_jn)-u_j(r_j n))
\\
&= \limsup_{t\rightarrow +\infty}\sum_{j=1}^p(u_j( c_j\lceil t\rceil )-u_j(r_j \lceil t \rceil ))\\
&= \limsup_{t\rightarrow +\infty}\sum_{j=1}^p(u_j( c_jt)-u_j(r_j t ))\ge 0,\end{aligned}$$ which completes the proof of the lemma.
Suppose that $P^{\ba}(T_1, f)>0$. Fix $0<s<s'<P^\ba (T_1,f)$. Let $\Phi=\{\log \phi_n\}_{n=1}^\infty$ be the additive potential generated by $f$, that is, $\phi_n(x)=\exp(S_nf(x))$ where $S_nf(x):=\sum_{i=0}^{n-1}f(T_1^ix)$. Take $\epsilon_0>0$ such that $$\label{star}
\sup\{|f(x)-f(y)|:\; x,y\in X_1,\; d_1(x,y)\leq \epsilon_0\}<(s'-s)a_1/(1+a_1).$$ By Lemma \[lem-Frost\], there exist $\nu\in \M(X_1)$, $\epsilon\in (0,\epsilon_0)$, and $N\in \N$ such that $$\label{e-6.1}
\begin{split}
\nu(B^\ba_n(x,\epsilon))&\leq \sup_{y\in B^\ba_n(x,\epsilon)}\exp\left(-s'n + \frac{1}{a_1}S_{\lceil a_1n\rceil}f(y)\right)\\
&\leq
\exp\left(-sn + \frac{1}{a_1}S_{\lceil a_1n\rceil}f(x)\right)
\end{split}$$ for any $n\geq N$ and $x\in X_1$, where in the last inequality we use .
By continuity, there exists $\tau\in (0, \epsilon)$ such that for any $1\le i<j\le k$, if $x_i,y_i\in X_i$ satisfy $d_i(x_i,y_i)<\tau$, then $$d_j(\pi_{j-1}\circ \cdots\circ \pi_i(x_i),\pi_{j-1}\circ \cdots\circ \pi_i(y_i))<\epsilon.$$ Take $M_0\in \mathbb{N}$ with $\mathcal{P}_{X_i}(\tau,M_0)\neq \emptyset$ for $i=1,\ldots,k$, where $\mathcal{P}_{X_i}(\tau,M_0)$ is defined as in . Now fix $M\in \N$ with $M\ge M_0$. Let $\alpha_i\in \mathcal{P}_{X_i}(\tau,M)$ for $i=1,\ldots, k$. Set $\beta_i=\tau_{i-1}^{-1}\alpha_i$ and write for brevity that $$t_0(n)=0, \quad t_i(n)=\lceil (a_1+\ldots +a_{i})n\rceil$$ for $n\in \N$ and $i=1,\ldots, k$. Then for any $n\in \N$ and $x\in X_1$, we have $$\label{e-relation}
\bigvee_{i=1}^k\bigvee_{j=t_{i-1}(n)}^{t_i(n)-1}T^{-j}_1\beta_i(x)\subseteq B^\ba_n(x,\epsilon).$$ Now assume that $n\geq N$. By and , $$\label{e-6.2}
\nu\Big(\bigvee_{i=1}^k\bigvee_{j=t_{i-1}(n)}^{t_i(n)-1}T^{-j}_1\beta_i(x)\Big)\leq \exp\left(-sn + \frac{1}{a_1}S_{\lceil a_1n\rceil}f(x)\right)$$ for any $x\in X_1$. It follows that $$\begin{aligned}
H_{\nu}\Big(\bigvee_{i=1}^k\bigvee_{j=t_{i-1}(n)}^{t_i(n)-1}T^{-j}\beta_i\Big) &= -\int \log \nu\Big(\bigvee_{i=1}^k\bigvee_{j=t_{i-1}(n)}^{t_i(n)-1}T^{-j}_1\beta_i(x)\Big) d\nu(x)\\
& \geq sn -\int \frac{1}{a_1}S_{\lceil a_1n\rceil}f(x) d\nu(x).\end{aligned}$$ Hence $$\label{e-6.2}
\sum_{i=1}^k H_{\nu}\Big(\bigvee_{j=t_{i-1}(n)}^{t_i(n)-1}T^{-j}_1\beta_i\Big)\geq sn -\int \frac{1}{a_1}S_{\lceil a_1n\rceil}f(x) d\nu(x).$$
Now fix $\ell\in \N$. By Lemma \[lem-6.1\], the left-hand side of is bounded from above by $$\sum_{i=1}^k \frac{t_i(n)-t_{i-1}(n)}{\ell}
H_{w_{i,n}}\Big(\bigvee_{j=0}^{\ell-1}T^{-j}_1\beta_i\Big)+2k\ell\log M,$$ where $$w_{i,n}:=\frac{ \sum_{j=t_{i-1}(n)}^{t_i(n)-1} \nu\circ T^{-j}_1} {t_{i}(n)-t_{i-1}(n)}.$$ Hence by and the definition of $H_\bullet(\tau, M;\ell)$ (cf. ), we have $$\label{e-2014'}
\begin{aligned}
\sum_{i=1}^k & (t_i(n)-t_{i-1}(n))
H_{w_{i,n}\circ \tau_{i-1}^{-1}}(\tau, M; \ell)\\
& \geq sn -\frac{\lceil a_1n\rceil}{a_1}\int f d w_{1,n}-2k\ell\log M.
\end{aligned}$$
Define $\nu_{m}=\frac{\sum_{j=0}^{m-1}\nu\circ T^{-j}_1}{m}$ for $m\in \mathbb{N}$. For $i=1,\ldots,k$, we have $$\nu_{m}\circ \tau^{-1}_{i-1}=\frac{\sum_{j=0}^{m-1}(\nu\circ \tau^{-1}_{i-1})\circ T^{-j}_i}{m}, \, w_{i,n}\circ \tau_{i-1}^{-1}=\frac{ \sum_{j=t_{i-1}(n)}^{t_i(n)-1} (\nu\circ \tau_{i-1}^{-1})\circ T^{-j}_i} {t_{i}(n)-t_{i-1}(n)}$$ and $$\begin{aligned}
\label{e-2014-0}
\nu_{t_i(n)}\circ \tau^{-1}_{i-1}=\frac{t_{i-1}(n)}{t_i(n)}\nu_{t_{i-1}(n)}\circ \tau^{-1}_{i-1}+\frac{t_i(n)-t_{i-1}(n)}{t_i(n)}w_{i,n}\circ \tau_{i-1}^{-1}.\end{aligned}$$
Applying Lemma \[lem-en.1\](2) to the measure $\nu\circ \tau_{i-1}^{-1}$ (more precisely, in , we replace the terms $T$, $\mu$, $n$, $m$ by $T_i$, $\nu\circ \tau_{i-1}^{-1}$, $t_{i-1}(n)$, $t_i(n)-t_{i-1}(n)$, respectively), we have $$\begin{aligned}
\frac{t_{i-1}(n)}{t_i(n)} H_{\nu_{t_{i-1}(n)}\circ \tau_{i-1}^{-1}} & (\tau,M,\ell)+ \frac{t_i(n)-t_{i-1}(n)}{t_i(n)} H_{w_{i,n}\circ \tau_{i-1}^{-1}}(\tau, M; \ell)\\
&\le H_{\nu_{t_i(n)}\circ \tau_{i-1}^{-1}}(\tau, M;\ell)
+\frac{\log 2}{\ell}.\end{aligned}$$ That is, $$\begin{aligned}
t_i(n)H_{\nu_{t_i(n)}\circ \tau_{i-1}^{-1}} & (\tau, M;\ell)-t_{i-1}(n)H_{\nu_{t_{i-1}(n)}\circ \tau_{i-1}^{-1}}(\tau,M,\ell)\\
&\ge (t_i(n)-t_{i-1}(n))
H_{w_{i,n}\circ \tau_{i-1}^{-1}}(\tau, M; \ell)-\frac{t_i(n)\log 2}{\ell}.\end{aligned}$$ Combining the above inequality with , we have $$\label{e-2014}
\begin{aligned}
\Theta_n: = & \sum_{i=1}^k \left(t_i(n)H_{\nu_{t_i(n)}\circ \tau_{i-1}^{-1}}(\tau, M;\ell)-t_{i-1}(n)H_{\nu_{t_{i-1}(n)}\circ \tau_{i-1}^{-1}}(\tau,M,\ell)\right)\\
\geq & sn -\frac{t_1(n)}{a_1}\int f d \nu_{t_1(n)}-2k\ell\log M-\frac{k t_{k}(n)\log 2}{\ell}.
\end{aligned}$$
Write $g_i(n):=H_{\nu_{n}\circ \tau_{i-1}^{-1}}(\tau, M;\ell)$. Then by Lemma \[lem-en.1\](1), $$\label{e-2014-1}
|g_i(n)-g_i(n+1)|\leq \frac{1}{\ell(n+1)}\log \big( 3M^{2\ell}(n+1)\big).$$ Set $$\gamma(n):=\sum_{i=2}^k t_i(n) (g_i(t_i(n))-g_i(t_1(n))) -\sum_{i=2}^k t_{i-1}(n)(g_i(t_{i-1}(n))-g_i(t_1(n))).$$ Then we have $$\Theta_n=\gamma(n)+ \sum_{i=1}^k (t_i(n)-t_{i-1}(n))g_i(t_1(n)),$$ where $\Theta_n$ is defined as in . Hence by , we have $$\label{e-2014-2}
\begin{aligned}
\sum_{i=1}^k & \frac{t_i(n)-t_{i-1}(n)}{n} g_i(t_1(n))+\frac{t_1(n)}{a_1n} \int f d \nu_{t_1(n)} \\
& \geq -\frac{\gamma(n)}{n}+s-\frac{2k\ell\log M}{n}-\frac{k t_{k}(n)\log 2}{n\ell}.
\end{aligned}$$
Define $$w(n)=\sum_{i=2}^k (a_1+\cdots+a_{i-1})(g_i(t_{i-1}(n))-g_i(t_1(n)))-\sum_{i=2}^k (a_1+\cdots+ a_i) (g_i(t_i(n))-g_i(t_1(n))).$$ Then we have $\limsup_{n\to \infty} w(n)\geq 0$ by applying Lemma \[lem-KP\], in which we take $p=2k-2$, $$u_j(n)=\left\{ \begin{array}{ll}
(a_1+\cdots+a_{j})g_{j+1}(n) & \mbox{ if } 1\leq j\leq k-1,\\
-(a_1+\cdots+a_{j-k+2})g_{j-k+2}(n) & \mbox{ if }k \leq j\leq 2k-2,
\end{array}
\right.$$ and $$c_j=\left\{ \begin{array}{ll}
a_j & \mbox{ if } 1\leq j\leq k-1,\\
a_{j-k+2} & \mbox{ if }k \leq j\leq 2k-2,
\end{array}
\right.$$ and $r_j=1$ for all $j$; the condition $\lim_{n\to \infty}|u_j(n+1)-u_j(n)|=0$ fulfils, thanks to .
Since $g_i$’s are bounded functions, we have $$\limsup_{n\to \infty}\frac{-\gamma(n)}{n}=\limsup_{n\to \infty}w(n)\geq 0.$$ Hence letting $n\to \infty$ in and taking the upper limit, we obtain $$\label{e-2014-3}
\limsup_{n\to \infty}\left( \sum_{i=1}^k a_i g_i(t_1(n))+ \int f dv_{t_1(n)}\right) \geq s-\frac{k (a_1+\cdots+a_k) \log 2}{\ell}.$$ Take a subsequence $(n_j)$ of the natural numbers so that the left-hand side of equals $$\lim_{j\to \infty}\left( \sum_{i=1}^k a_i H_{\nu_{t_1(n_j)} \circ \tau_{i-1}^{-1}}(\tau, M;\ell) + \int f d\nu_{t_1(n_j)}\right)$$ and moreover, $\nu_{t_1(n_j)}$ converges to an element $\lambda\in \M(X_1, T_1)$ in the weak\* topology. Since the map $H_\bullet(\tau, M;\ell)$ is upper semi-continuous on $\M(X_1)$ (see Lemma \[usc\]), we have $$\label{e-2014-4}
\sum_{i=1}^k a_i H_{\lambda \circ \tau_{i-1}^{-1}}(\tau, M;\ell) + \int f d\lambda\geq s-\frac{k (a_1+\cdots+a_k) \log 2}{\ell}.$$
Define $$\E:=\left\{(M,\ell,\delta): M,\ell\in \mathbb{N},\delta>0 \text{ with }M\ge M_0, \ell \ge \frac{k (a_1+\cdots+a_k) \log 2}{\delta}\right\}$$ and $$\Omega_{M,\ell, \delta}:=\left\{\eta\in \M(X_1, T_1): H^\ba_\eta(\tau,M;\ell)+\int f d\eta\geq s-\delta\right\},$$ where $H^\ba_\eta(\tau,M;\ell):=\sum_{i=1}^k a_i H_{\eta \circ \tau_{i-1}^{-1}}(\tau, M;\ell)$. Then by , $\Omega_{M,\ell, \delta}$ is a non-empty compact set whenever $(M,\ell,\delta)\in \E$. However $$\Omega_{M_1,\ell_1, \delta_1}\cap \Omega_{M_2,\ell_2,\delta_2}\supseteq \Omega_{M_1+M_2,\ell_1\ell_2, \min\{\delta_1,\delta_2\}}$$ for any $(M_1,\ell_1,\delta_1),(M_2,\ell_2,\delta_2)\in \E$. It follows (by finite intersection property) that $$\bigcap_{(M,\ell,\delta)\in \E} \Omega_{M,\ell, \delta}\neq \emptyset.$$ Take $\mu_s\in \bigcap_{(M,\ell,\delta)\in \E} \Omega_{M,\ell, \delta}$. Then $$h^\ba_{\mu_s}(T_1,\tau)+\int f d\mu_s\geq s,$$ where $h^\ba_{\mu_s}(T_1,\tau):=\sum_{i=1}^k a_i h_{\mu_s\circ \tau_{i-1}^{-1}}(T_i,\tau)$. Since the map $\theta\in \mathcal{M}(X_1,T_1)\mapsto
h_\theta^{\ba}(T_1,\tau)$ is upper semi-continuous (see Lemma \[usc\]), we can find $\mu\in \M(X_1, T_1)$ such that $$h^\ba_{\mu}(T_1,\tau)+\int f d\mu \geq P^\ba_W(T_1, f,\epsilon) -\omega_\epsilon(f)$$ by letting $s\nearrow P^\ba_W(T_1, f,\epsilon)$. Since $h^\ba_{\mu}(T_1)\geq h^\ba_{\mu}(T_1,\tau)$, this completes the proof of the proposition.
Sub-additive case {#s-6}
=================
In this section, we extend Theorem \[thm-1.1\] to sub-additive potentials, under the following two additional assumptions: (1) $\htop(T_1)<\infty$ and (2) the entropy maps $\theta\in \mathcal{M}(X_i,T_i)\mapsto h_\theta(T_i)$, $i=1,2,\cdots,k$, are upper semi-continuous.
Let $f:X_1\rightarrow [-\infty,+\infty)$ be an upper semicontinuous function. Define $\Psi=\{\log \psi_n\}_{n=1}^\infty$ by $\psi_n(x)=\exp(\sum_{j=0}^{n-1}f(T_1^jx))$. In this case, $\Psi$ is additive. We just define $$P^{\bf a} (T_1,f):=P^{\bf a}
(T_1,\Psi).$$
\[usc-est\] Assume that $\htop(T_1)<\infty$ and the entropy maps $\theta\in \mathcal{M}(X_i,T_i)\mapsto h_\theta(T_i)$, $i=1,2,\cdots,k$, are upper semi-continuous. Let $f:X_1\to
[-\infty,+\infty)$ be a upper semicontinuous function. Then there exists $\mu\in \mathcal{M}(X_1,T_1)$ such that $$h_\mu^{\bf a}(T_1)+\int_{X_1} f d \mu \ge P^{\bf a} (T_1,f).$$
For $g\in C(X_1)$ with $g\ge f$, we define $$\mathcal{M}_g=\Big\{ \nu\in \mathcal{M}(X_1,T_1):h_\nu^{\bf
a}(T_1)+\int_{X_1} g d\nu \ge P^{\bf a}
(T_1,f)\Big\}.$$ Notice that, under the assumptions of the lemma, the entropy map $\nu\in \mathcal{M}(X_1,T_1)\mapsto h^{\ba}_\nu(T_1)$ is a bounded upper semi-continuous function. Hence by Theorem \[thm-1.1\], there exists $\mu_g\in
\mathcal{M}(X_1,T_1)$ such that $$h_{\mu_g}^{\bf a}(T_1)+\int_{X_1} g d \mu_g\ge P^{\bf a}
(T_1,g) \ge P^{\bf a} (T_1,f).$$ Thus $\mu_g\in \mathcal{M}_g$. Since $\nu\in \mathcal{M}(X_1,T_1)\mapsto
\int_{X_1}g d\nu$ is a bounded continuous non-negative valued function on $\mathcal{M}(X_1,T_1)$, the mapping $\nu\in
\mathcal{M}(X_1,T_1)\mapsto h_\nu^{\bf a}(T_1)+\int_{X_1}g
d\nu$ is a bounded upper semicontinuous non-negative valued function on $\mathcal{M}(X_1,T_1)$. Thus $\mathcal{M}_g$ is a non-empty closed subset of $\mathcal{M}(X_1,T_1)$.
Now put $$\mathcal{M}_f:=\bigcap_{g\in C(X_1),g\ge f} \mathcal{M}_g.$$ Note that $\mathcal{M}_{g_1}\cap \mathcal{M}_{g_2}\supseteq
\mathcal{M}_{\min\{g_1,g_2\}}$ for any $g_1,g_2\in C(X_1)$ with $g_1\ge f$, $g_2\ge f$, and each $\mathcal{M}_g$ is a non-empty closed subset of the compact metric space $\mathcal{M}(X_1,T_1)$. Hence $\mathcal{M}_f\neq \emptyset$, by the finite intersection property characterization of compactness. Take any $\mu\in
\mathcal{M}_f$. Then $$h_{\mu}^{\bf a}(T_1)+\int_{X_1} g d \mu\ge P^{\bf a} (T_1,f)$$ for any $g\in C(X_1)$ with $g\ge f$. Moreover, since $0\le
h_{\mu}^{\bf a}(T_1)<\infty$, we have $$h_{\mu}^{\bf a}(T_1)+\inf \limits_{g\in C(X_1),g\ge f} \int_{X_1} g d \mu\ge P^{\bf a} (T_1,f).$$ Finally by Lemma \[appro\], $\inf \limits_{g\in C(X_1),g\ge f}
\int_{X_1} g d \mu=\int_{X_1}f d \mu$ and thus $$h_{\mu}^{\bf a}(T_1)+ \int_{X_1} f d \mu\ge P^{\bf a} (T_1,f).$$ This completes the proof of the lemma.
\[estimate-1\] Let $\Phi=\{\log \phi_n\}_{n=1}^\infty$ be a sub-additive potential on $X_1$. If for $\ell\in \mathbb{N}$ and $M\in
\mathbb{N}$, let $f_{\ell,M}(x)=\max\{\frac{1}{\ell}\log \phi_\ell(x),-M\}$ for $x\in X_1$, then $f_{\ell,M}: X_1\rightarrow \mathbb{R}$ is a bounded upper semi-continuous function and $$P^{\bf
a}(T_1,f_{\ell,M}) \ge P^{\bf
a}(T_1,\Phi).$$
Let $\ell\in \mathbb{N}$ and $M\in \mathbb{N}$. Let $f_{\ell,M}=\max\{ \frac{1}{\ell}\log \phi_\ell,-M\}$. It is clear that $f_{\ell,M}: X_1\rightarrow \mathbb{R}$ is a bounded upper semi-continuous function since $\frac{1}{\ell}\log
\phi_\ell:X_1\rightarrow [-\infty,+\infty)$ is upper semi-continuous.
Let $\phi_0(x)\equiv 1$ for $x\in X_1$ and $$D:=D(\ell)=\sup \limits_{x\in X_1,\; i\in\{0,1,\cdots,\ell-1\}} \log
\phi_i(x).$$ Then $0\le D<\infty$. For $x\in X_1$ and $n\ge
2\ell$, we have $$\begin{aligned}
\log \phi_n(x)&\le \log
\phi_i(x)+\left(\sum_{j=0}^{[\frac{n-i}{\ell}]-1}\log
\phi_\ell(T_1^{j\ell+i}x)\right)+\log
\phi_{n-i-[\frac{n-i}{\ell}]\ell}\left(T_1^{i+[\frac{n-i}{\ell}]\ell}x\right)\\
&\le 2D+\sum_{j=0}^{[\frac{n-i}{\ell}]-1}\log \phi_\ell(T_1^{j\ell+i}x)\end{aligned}$$ for each $i\in\{0,1,\ldots,\ell-1\}$, using the sub-additivity of $\Phi=\{\log
\phi_n\}_{n=1}^\infty$, where $[a]$ denotes the greatest integer $\leq a$. Summing $i$ from $0$ to $\ell-1$, we obtain $$\begin{aligned}
\log \phi_n(x) &\le
2D+\sum_{i=0}^{\ell-1}\sum_{j=0}^{[\frac{n-i}{\ell}]-1}\frac{1}{\ell}\log
\phi_\ell(T^{j\ell+i}x)=2D+\sum_{j=0}^{n-\ell} \frac{1}{\ell}\log \phi_\ell(T_1^jx)\\
&\le 2D+\sum_{j=0}^{n-\ell} f_{\ell,M}(T_1^jx)\le
C+\sum_{j=0}^{n-1}f_{\ell,M}(T_1^jx)\end{aligned}$$ where $C=2D+\ell M\in [0,+\infty)$.
Define $\Psi=\{\log \psi_n\}_{n=1}^\infty$ by $\psi_n(x)=\exp\left(\sum_{j=0}^{n-1}f_{\ell,M}(T_1^jx)\right)$. Then $$\label{gg-eq-1}
\phi_n(x)\le e^C \psi_n(x), \qquad \forall\; x\in X_1,\; n\ge 2\ell,$$ This implies that for any $\epsilon>0$, $s\in \mathbb{R}$ and $N\ge 2a_1\ell$, $$\mathcal{M}^{{\bf a},s}_{\Phi, N,\epsilon}(X_1)\le e^{\frac{C}{a_1}}\cdot \mathcal{M}^{{\bf a}, s}_{\Psi, N,
\epsilon}(X_1).$$ Hence $\mathcal{M}^{{\bf a},s}_{\Phi,
\epsilon}(X_1)\le e^{\frac{C}{a_1}} \mathcal{M}^{{\bf a}, s}_{\Psi,
\epsilon}(X_1)$ for $\epsilon>0$, $s\in \mathbb{R}$. It follows that $$P^{\bf
a}(T_1,\Phi,X_1,\epsilon)\le P^{\bf
a}(T_1,\Psi,X_1,\epsilon)=P^{\bf a}(T_1,f_{\ell,M}, X_1,\epsilon) .$$ Letting $\epsilon \to 0$, we are done.
Assume that $\htop(T_1)<\infty$ and the entropy maps $\theta\in \mathcal{M}(X_i,T_i)\mapsto h_\theta(T_i),$ $i=1,2,\cdots,k$, are upper semi-continuous. Let $\Phi=\{\log \phi_n\}_{n=1}^\infty$ be a sub-additive potential on $X_1$. Then $$P^{\bf a}(T_1,\Phi)=\sup\{ h_\mu^{\bf a}(T_1)+\Phi_*(\mu): \mu\in \mathcal{M}(X_1,T_1)\},$$ and moreover the supremum is attainable.
By Proposition \[thm-3.3\], it is sufficient to show that there exists $\mu\in \mathcal{M}(X_1,T_1)$ such that $P^{\bf a}(T_1,\Phi)\le h_\mu^{\bf a}(T_1)+\Phi_*(\mu)$.
For $n,M\in \mathbb{N}$, let $f_n(x)=\frac{1}{n}\log \phi_n(x)$ and $f_{n,M}(x)=\max\{ \frac{1}{n}\log \phi_n(x),-M\}$ for $x\in
X_1$. Then $f_{n,M}$ is a bounded upper semi-continuous function. Define $$\mathcal{M}_{n,M}=\left\{ \nu\in \mathcal{M}(X_1,T_1):
h_\nu^{\bf a}(T_1)+\int_{X_1}f_{n,M} \, d \nu\ge P^{\bf
a}(T_1,\Phi)\right\}.$$ By Lemma \[usc-est\], there exists $\mu_{n,M}\in \mathcal{M}(X_1,T_1)$ such that $$h_{\mu_{n,M}}^{\bf a}(T_1)+\int_{X_1}f_{n,M} \, d \mu_{n,M}\ge P^{\bf
a}(T_1,f_{n,M}) \ge P^{\bf
a}(T_1,\Phi),$$ where the last inequality comes from Lemma \[estimate-1\]. Thus $\mu_{n,M}\in \mathcal{M}_{n,M}$. By the assumption, we know that the function $h_{\bullet}^{\bf
a}(T_1)$ is bounded, upper semi-continuous and non-negative on $\mathcal{M}(X_1,T_1)$. Notice that $\nu\in
\mathcal{M}(X_1,T_1)\mapsto \int_{X_1}f_{n,M} d\nu$ is also an upper semi-continuous function from $\mathcal{M}(X_1,T_1)$ to $\mathbb{R}$. Hence $\nu\in \mathcal{M}(X_1,T_1)\mapsto
h_\nu^{\bf a}(T_1)+\int_{X_1}f_{n,M} d\nu$ is upper semi-continuous. Thus $\mathcal{M}_{n,M}$ is a non-empty closed subset of $\mathcal{M}(X_1,T_1)$. Moreover since $\mathcal{M}_{n,1}\supseteq \mathcal{M}_{n,2}\supseteq \cdots$ and $\inf_{M\in \mathbb{N}}\int_{X_1}f_{n,M}d\nu=\int_{X_1} f_n d\nu$ for any $\nu\in \mathcal{M}(X_1,T_1)$, one has $\mathcal{M}_n=\bigcap_{M\in \mathbb{N}}\mathcal{M}_{n,M}$ is a non-empty closed subset of $\mathcal{M}(X_1,T_1)$.
Now put $$\mathcal{M}_{\Phi}:=\bigcap_{n\in \mathbb{N}} \mathcal{M}_n.$$ Since $\int_{X_1} f_{n_1n_2} d \nu \le \min\{ \int_{X_1} f_{n_1} d
\nu, \int_{X_1} f_{n_2} d \nu\}$ for $\nu\in \mathcal{M}(X_1,T_1)$, we have $\mathcal{M}_{n_1}\cap \mathcal{M}_{n_2}\supseteq
\mathcal{M}_{n_1n_2}$ for any $n_1,n_2\in \mathbb{N}$. Moreover since each $\mathcal{M}_n$ is a non-empty closed subset of the compact metric space $\mathcal{M}(X_1,T_1)$, one has $\mathcal{M}_{\Phi}\neq \emptyset$ by the finite intersection property characterization of compactness. Take any $\mu\in
\mathcal{M}_{\Phi}$. Then $$h_{\mu}^{\bf a}(T_1)+\int_{X_1} f_n d \mu\ge P^{\bf a} (T_1,\Phi)$$ for any $n\in \mathbb{N}$. Moreover, since $0\le h_{\mu}^{\bf
a}(T_1)<\infty$, we have $$h_{\mu}^{\bf a}(T_1)+\inf \limits_{n\in \mathbb{N}} \frac{1}{n}\int_{X_1} \log \phi_n d \mu\ge P^{\bf a} (T_1,\Phi).$$ Finally since $\inf \limits_{n\in \mathbb{N}} \frac{1}{n}\int_{X_1}
\log \phi_n d \mu=\Phi_*(\mu)$ and thus $$h_{\mu}^{\bf a}(T_1)+ \Phi_*(\mu)\ge P^{\bf a} (T_1,\Phi).$$ This finishes the proof of the Theorem.
Final remarks and examples
==========================
In this section we give some remarks, examples and questions.
{#s-7.1}
In [@BaFe12; @Fen11], Barral and the first author defined weighted topological pressure for factor maps between subshifts in a different way, motivated from the study of multifractal analysis on affine Sierpinski gaskets [@BaMe07; @BaMe08; @Kin95; @Ols98] and a question of Gatzouras and Peres [@GaPe96] on the uniqueness of invariant measures of full dimension on certain affine invariant sets. The approach is based on the following lemma, which is derived from the relativized variational principle of Ledrappier and Walters [@LeWa77] and its sub-additive extension [@ZhCa08].
\[lem-BF\][@BaFe12; @Fen11] Assume that $(X, T)$ and $(Y, S)$ are subshifts over finite alphabets and $\pi:X\to Y$ is a factor map. Let $f\in C(X)$ (or more general, a subadditive potential on $X$). Then there exists a sub-additive potential $\Phi_f=(\log \phi_n)_{n=1}^\infty$ on $Y$ such that for any $\nu\in \M(Y,S)$, $$\sup_{\mu\in \M(X,T),\; \mu\circ \pi^{-1}=\nu} \left(\int f d\mu+h_\mu(T)-h_\nu(S)\right)=\Phi_*(\nu):=\lim_{n\rightarrow +\infty}
\frac{1}{n} \int \log \phi_n d \nu.$$
According to above lemma, for given $a_1, a_2>0$, one has $$\begin{aligned}
&\sup_{\mu\in \M(X,T), \; \mu\circ \pi^{-1}=\nu}\left(\int f d\mu+a_1h_\mu(T)+a_2h_\nu(S)\right)\\
&\qquad\mbox{}=\sup_{\nu\in \M(Y,S)} \left\{ (a_1+a_2)h_\nu(S)+\sup_{\mu\in \pi^{-1}\nu} a_1\left(\int \frac{1}{a_1}f d\mu+h_\mu(T)-h_\nu(S)\right)\right\}\\
&\qquad\mbox{}=\sup_{\nu\in \M(Y,S)} \{(a_1+a_2)h_{\nu}(S)+(\Phi_{a_1^{-1}f})_*(\nu)\}\\
&\qquad\mbox{}=(a_1+a_2) P\left(S, \frac{a_1}{a_1+a_2}\Phi_{a_1^{-1}f}\right).\end{aligned}$$ where the last equality follows from the sub-additive thermodynamic formalism (see e.g. [@CFH08]). Hence in [@BaFe12; @Fen11], $P^{(a_1,a_2)}(T,f)$ was defined in terms of sub-additive topological pressure in the subshift case.
However, Lemma \[lem-BF\] does not extend to factor maps between general topological dynamical systems. Below we will give a counter example. Hence the approach in [@BaFe12; @Fen11] in defining weighted topological pressure does not extend to general topological dynamical systems.
\[ex-1\] Let $X=\{ (x,y,z)\in \mathbb{R}^3: -1\le x \le 1, y^2+z^2=x^2\}$ be a cone surface. Define $T:X\rightarrow X$ by $$T((x,x\cos \theta,x\sin \theta))=(x,x\cos(2\theta), x\sin(2\theta)), \quad x\in [-1,1].$$ Let $Y=[-1,1]$ and $S:Y\rightarrow Y$ be the identity. Set $\pi: X\rightarrow Y$ by $\pi((x,y,z))= x$. Then $(Y, S)$ is a factor of $(X, T)$ associated with the factor map $\pi$. Take $f\in C(X)$ with $f\equiv 0$. Suppose that Lemma \[lem-BF\] extends to this case, that is, there exists a sub-additive potential $\Phi$ on $Y$ such that for any $\nu\in \M(Y,S)$, $$\label{e-BF}
\sup_{\mu\in \pi^{-1}\nu} (h_\mu(T)-h_\nu(S))=\Phi_*(\nu).$$ In what follows we derive a contradiction.
We first claim that the mapping $$\label{e-map}\nu\in \M(Y,S)\mapsto \sup_{\mu\in \pi^{-1}\nu} (h_\mu(T)-h_\nu(S))$$ is not upper semi-continuous. To see this, for $t\in Y$, let $\nu_t=\delta_t$ (the Dirac measure at $t$). Clearly $\delta_t\in \M(Y, S)$ and when $t\rightarrow 0$, $\delta_t\rightarrow\delta_0$ in the weak-star topology. However one can check that $$\sup_{\mu\in \pi^{-1}\delta_t} \big( h_\mu(T)-h_{\nu_t}(S) \big)=\begin{cases} \log 2, &\text{ for } t\neq 0\\
0, &\text{ if } t=0 \end{cases}.$$ Hence the mapping in is not upper semi-continuous. Therefore by , $\nu\mapsto \Phi_*(\nu)$ is not upper semi-continuous on $\M(Y,S)$. But this contradicts the fact that $\nu\mapsto \Phi_*(\nu)$ is always upper semi-continuous (see e.g. [@FeHu10 Proposition A.1.(2)]).
{#s-7.2}
Using Corollary \[cor-entropy\], we can extend Kenyon-Peres’ variational principle and its higher dimensional version to a particular class of skew product expanding maps on the $k$-torus $\T^k:=\R^k/\Z^k$ ($k\geq 2$).
To see this, let $2\leq m_1\leq m_2\leq \ldots\leq m_k$ be integers. For $i=1, \ldots, k-1$, let $\phi_i$ be $C^1$ real-valued functions on $\T^{i}$. Define $T_1: \T^k\to \T^k$ by $$T_1((x_1,\ldots,x_k))=(m_1x_1, m_2x_2+\phi_1(x_1), \ldots, m_kx_k+\phi_{k-1}(x_1,\ldots, x_{k-1})).$$ This transformation can be viewed as a skew product of the maps $$x_i\mapsto m_ix_i,\quad (i=1,\ldots,k).$$
Let $K\subset \T^k$ be a $T_1$-invariant compact set. Let $\tau_i$ $(i=1,\ldots, k-1$) be the canonical projection from $\T^{k}$ to $\T^{k-i}$, i.e. $$\tau_i(x_1,\ldots, x_k)=(x_1,\ldots, x_{k-i}).$$ Set $X_1= K$ and $X_i=\tau_{i-1}(K)$ for $2\leq i\leq k$. Define $T_i: X_i\to X_i$ ($i=2,\ldots, k$) by $$T_i((x_1,\ldots,x_i))=(m_1x_1, m_2x_2+\phi_1(x_1), \ldots, m_ix_i+\phi_{i-1}(x_1,\ldots, x_{i-1})).$$ Then $(X_{i+1}, T_{i+1})$ is the factor of $(X_i, T_i)$ associated with the factor map $\pi_i: X_i\to X_{i+1}$, which is defined by $$(x_1,\ldots, x_{k+1-i})\mapsto (x_1,\ldots, x_{k-i}).$$ Define $\ba=(a_1,\ldots, a_k)$ with $$a_1=\frac{1}{\log m_k},\quad a_i=\frac{1}{\log m_{k+1-i}}-\frac{1}{\log m_{k+2-i}} \quad \mbox{ for }i=2,\ldots, k.$$
It is direct to check that there exist two constants $C_1, C_2>0$ (depending on $\phi_i$’s) such that for any $\epsilon>0$ and $x\in \T^k$, $$\label{e-equiv}
C_2B_{e^{-n}\epsilon}(x)\subset B_n^{\ba}(x, \epsilon)\subset C_1 B_{e^{-n}\epsilon}(x).$$ Hence from the definition of $\htop^{\ba}(\cdot)$, we see that $\htop^{\ba}(T_1, K)=\dim_H K$. Applying Corollary \[cor-entropy\], we have $$\label{e-var''}
\dim_HK=\htop^{\ba}(T_1, K)=\sup_{\mu\in \M(X_1, T_1)} h^\ba_\mu(T_1),$$ where the supremum is attainable at some ergodic $\mu\in \M(X_1, T_1)$. Moreover by and Theorem \[thm-4.1’\], we have $\dim_H\mu=h^\ba_\mu(T_1)$ for each ergodic $\mu\in \M(X_1, T_1)$. Hence there exists an ergodic $\mu\in \M(X_1, T_1)$ of full Hausdorff dimension, i.e. $$\label{e-end}
\dim_H\mu=\dim_HK.$$ This extends the work of Kenyon and Peres [@KePe96]. We remark that was also proved by Luzia [@Luz06] for a more general class of skew product expanding maps on $\T^2$.
{#section}
In [@FeHu12], the authors proved a variational principle for topological entropies for arbitrary Borel subsets. We remark that this principle also holds for weighted topological entropies, by applying Lemma \[pro-3.1\] and following the arguments in [@FeHu12].
In the end we pose several questions about possible extensions of Theorem \[thm-1.1\]: does this result remain valid for $\Z^d$-actions? and moreover does it admit a relativized or randomized version? is there an analogous topological extension of the dimensional result on Gatzouras-Lalley self-affine carpets [@LaGa92]?
A weighted version of the Brin-Katok theorem {#s-a}
============================================
The main result in this appendix is the following weighted version of the Brin-Katok theorem. It is needed in our proof of the lower bound of Theorem.
\[thm-4.1’\] For each ergodic measure $\mu\in \mathcal{M}(X_1,T_1)$, we have $$\lim_{\epsilon \rightarrow 0} \liminf_{n\rightarrow +\infty} \frac{-\log \mu(B_n^{\bf a}(x,\epsilon))}{n}=
\lim_{\epsilon \rightarrow 0} \limsup_{n\rightarrow +\infty}
\frac{-\log \mu(B_n^{\bf a}(x,\epsilon))}{n}=h_\mu^{\bf a}(T_1)$$ for $\mu$-a.e. $x\in X_1$.
When $\ba=(1,0,\ldots, 0)$, the above result reduces to the Brin-Katok theorem on local entropy [@BrKa83].
The proof of Theorem \[thm-4.1’\] is based on the following weighted version of the Shannon-McMillan-Breiman theorem.
\[SMB\] Let $(X,\mathcal{B},\mu,T)$ be a measure preserving dynamical system and $k\ge 1$. Let $\alpha_1, \ldots,
\alpha_k$ be $k$ countable measurable partitions of $(X,\mathcal{B},\mu)$ with $H_\mu(\alpha_i)<\infty$ for each $i$, and ${\bf a}=(a_1,\ldots,a_k)\in \mathbb{R}^k$ with $a_1>0$ and $a_i\ge 0$ for $i\ge 2$. Then $$\label{fh-eq-1} \lim_{N\rightarrow +\infty}\frac{1}{N}I_\mu
\Big(\bigvee_{i=1}^k (\alpha_i)_0^{\lceil(a_1+\cdots+a_i)N\rceil-1}
\Big)(x)=\sum_{i=1}^k a_i\mathbb{E}_\mu(F_i|\mathcal{I}_\mu)(x)$$ almost everywhere, where $$F_i(x):=I_\mu
\Big(\bigvee_{j=i}^{k}\alpha_j\big|\bigvee_{n=1}^\infty
T^{-n}(\bigvee_{j=i}^{k}\alpha_j)\Big)(x), \quad i=1,\ldots,k$$ and $\mathcal{I}_\mu=\{ B\in \mathcal{B}:\; \mu(B\triangle T^{-1}B)=0\}$. In particular, if $T$ is ergodic, we have $$\lim_{N\rightarrow +\infty}\frac{1}{N}I_\mu \Big(\bigvee_{i=1}^k
(\alpha_i)_0^{\lceil(a_1+\cdots+a_i)N\rceil-1} \Big)(x)=\sum_{i=1}^k
a_ih_\mu(T,\bigvee_{j=i}^{k}\alpha_j)$$ almost everywhere.
When $k=1$ and $a_1=1$, Proposition \[SMB\] reduces to the classical Shannon-McMillan-Breiman theorem (see e.g. [@Par81 Theorem 7]). We remark that a variant of Proposition \[SMB\], for certain particular partitions, was proved by Kenyon and Peres (cf. [@KePe96 Lemmas 3.1 and 4.4]) in the case that $\mu$ is ergodic. For completeness and for the convenience of the reader, we will provide a full proof of Proposition \[SMB\] in the end of this section, by adapting the argument by Kenyon and Peres in [@KePe96].
The following result is a direct corollary of Proposition \[SMB\].
\[c-SMB\] Let $(X,\mathcal{B},\mu,T)$ be an ergodic measure preserving dynamical system and $k\ge 1$. If $\alpha_1,\ldots, \alpha_k$ are $k$ countable measurable partitions of $(X,\mathcal{B},\mu)$ with $\alpha_1\succeq\alpha_2\succeq\cdots\succeq\alpha_k$ and $H_\mu(\alpha_i)<\infty$, $i=1,\ldots,k$, and ${\bf
a}=(a_1,\ldots,a_k)\in \mathbb{R}^k$ with $a_1>0$ and $a_i\ge
0$ for $i\ge 2$, then $$\lim_{N\rightarrow +\infty}\frac{1}{N}I_\mu \Big(\bigvee_{i=1}^k
\Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})N\rceil}^{\lceil(a_1+\cdots+a_i)N\rceil-1}T^{-j}\alpha_i\Big)
\Big)(x)=\sum_{i=1}^k a_ih_\mu(T,\alpha_i)$$ almost everywhere, where we make the convention $a_0=0$.
We just adapt the proof of Brin and Katok [@BrKa83] for their local entropy formula.
We first prove the upper bound. Let $\epsilon>0$. Let $\alpha_i$ be a finite Borel partition of $X_i$, $i=1,\ldots,k$, with $\text{diam}(\alpha_i)<\epsilon$. Then $$B_n^{\bf a}(x,\epsilon)\supseteq \bigcap_{i=1}^k (\tau_{i-1}^{-1}\alpha_i)_0^{\lceil (a_1+\cdots+a_i)n\rceil-1}(x)$$ for $x\in X_1$. Hence by Proposition \[SMB\], for $\mu$-a.e $x\in X_1$ we have $$\begin{aligned}
\limsup_{n\rightarrow +\infty} &\frac{-\log \mu(B_n^{\bf
a}(x,\epsilon))}{n}\le \limsup_{n\rightarrow +\infty}
\frac{-\log \mu\Big(\bigcap \limits_{i=1}^k (\tau_{i-1}^{-1}\alpha_i)_0^{\lceil (a_1+\cdots+a_i)n\rceil-1}(x)\Big)}{n}\\
&=\limsup_{n\rightarrow +\infty} \frac{I_\mu\Big(\bigvee \limits_{i=1}^k (\tau_{i-1}^{-1}\alpha_i)_0^{\lceil (a_1+\cdots+a_i)n\rceil-1}\Big)(x)}{n}=\sum_{i=1}^k a_ih_\mu\Big(T_1,\bigvee_{j=i}^k \tau_{j-1}^{-1}\alpha_j\Big)\\
&=\sum_{i=1}^k a_ih_\mu\Big(T_1,\tau_{i-1}^{-1}\Big(\alpha_i\vee \bigvee_{j=i+1}^k \pi_{i}^{-1}\circ \cdots \circ \pi_{j-1}^{-1}\alpha_j\Big)\Big)\\
&=\sum_{i=1}^k a_ih_{\mu\circ \tau_{i-1}^{-1}}\Big(T_i,\alpha_i\vee
\bigvee_{j=i+1}^k
\pi_{i}^{-1}\circ \cdots \circ \pi_{j-1}^{-1}\alpha_j\Big)\\
&\le \sum_{i=1}^k a_i h_{\mu\circ \tau_{i-1}^{-1}}(T_i)=h_\mu^{\bf
a}(T_1).\end{aligned}$$ Letting $\epsilon\to 0$ in the above inequality, we have $$\lim_{\epsilon \rightarrow 0} \limsup_{n\rightarrow +\infty}
\frac{-\log \mu(B_n^{\bf a}(x,\epsilon))}{n}\le h_\mu^{\bf a}(T_1).$$ This completes the proof of the upper bound.
Next we prove the lower bound. It is sufficient to show that for any $\delta>0$, there exist $\epsilon>0$ and a measurable subset $D$ of $X_1$ such that $\mu(D)>1-3\delta$ and $$\liminf_{n\rightarrow +\infty} \frac{-\log
\mu(B_n^{\bf a}(x,\epsilon))}{n}\ge \min\left\{\frac{1}{\delta}, h^{\bf
a}_{\mu}(T_1)-\delta\right\}-2(1+a_1+\cdots+a_k)\delta$$ for any $x\in D$.
Fix $\delta>0$. We are going to find such $\epsilon$ and $D$. First, we find a finite Borel partition $\alpha_i=\{
A^i_1,A^i_2,\ldots,A^i_{u_i}\}$ of $X_i$, $i=1,\ldots,k$, such that
- $\alpha_i\succeq \pi_i^{-1}(\alpha_{i+1})$ for $i=1,\ldots,k-1$.
- $\sum_{i=1}^k a_ih_{\mu\circ \tau_{i-1}^{-1}}(T_i,\alpha_i)\ge \min\{\frac{1}{\delta}, h_\mu^{\bf a}(T_1)-\delta\}$.
- $\mu\circ \tau_{i-1}^{-1}(\partial \alpha_i)=0$ for $i=1,\ldots,k$.
Let $M=\max\{u_i:\; 1\leq i\leq k\}$ and $\Lambda=\{1,\ldots,M\}$. Given $m\in \mathbb{N}$, for ${\bf s}=(s_i)_{i=0}^{m-1},{\bf
t}=(t_i)_{i=0}^{k-1}\in \Lambda^{\{0,1,\cdots,m-1\}}$, the [*Hamming distance*]{} between ${\bf s}$ and ${\bf t}$ is defined to be the following value $$\frac{1}{m}\#\left\{ i\in \{0,1,\cdots,m-1\}: s_i\neq t_i\right\}.$$ For ${\bf s}\in \Lambda^{\{0,1,\cdots,m-1\}}$ and $0<\tau\le 1$, let $Q({\bf s},\tau)$ be the total number of those ${\bf t}\in
\Lambda^{\{0,1,\cdots,m-1\}}$ so that the Hamming distance between ${\bf s}$ and ${\bf t}$ does not exceed $\tau$. Clearly, $$Q_m(\tau):=\max \limits_{{\bf s}\in
\Lambda^{\{0,1,\cdots,m-1\}}}Q({\bf s},\tau)\le \binom{m}{\lceil
m\tau \rceil}M^{\lceil m\tau \rceil}.$$ By the Stirling formula, there exists a small $\delta_1>0$ and a positive constant $C:=C(\delta,M)>0$ such that $$\label{esti-1}
\binom{m}{\lceil m\delta_1\rceil}M^{\lceil m\delta_1\rceil}\le
e^{\delta m+C}$$ for all $m\in \mathbb{N}$.
For $\eta>0$, set $$\begin{aligned}
&U^i_\eta(\alpha_i)=\{x\in X_1:\; B(\tau_{i-1}x,\eta)\not \subseteq
\alpha_i(\tau_{i-1}x)\}, \quad i=1,\ldots,k.\end{aligned}$$ Then $\bigcap_{\eta>0} U^i_\eta(\alpha_i)=\tau_{i-1}^{-1}(\partial
\alpha_i)$, and hence $\mu(U^i_\eta(\alpha_i))\rightarrow \mu(\tau_{i-1}^{-1}(\partial
\alpha_i))=0$ as $\eta \to 0$. Therefore, we can choose $\epsilon>0$ such that $\mu(U^i_\eta(\alpha_i))< \delta_1$ for any $0<\eta\le \epsilon$ and $i=1,\ldots,k$.
By the Birkhoff ergodic theorem, for $\mu$-a.e. $x\in X_1$, we have $$\begin{aligned}
\lim_{n\rightarrow +\infty} & \frac{1}{\lceil
(a_1+\cdots+a_k)n \rceil}\sum \limits_{i=1}^k \sum_{j=\lceil
(a_0+\cdots+a_{i-1})n\rceil}^{\lceil
(a_1+\cdots+a_i)n\rceil-1}\chi_{U^i_\epsilon(\alpha_i)}(T^j_1x)\\
&=\frac{1}{(a_1+\cdots+a_k)} \sum \limits_{i=1}^k
a_i\mu(U^i_\epsilon(\alpha_i))<\delta_1,\end{aligned}$$ where we take the convention $a_0=0$. Thus we can find a large natural number $\ell_0$ such that $\mu(A_\ell)>1-\delta$ for any $\ell\ge \ell_0$, where $$\begin{aligned}
A_\ell &=\left\{ x\in X_1:\;\frac{1}{\lceil (a_1+\cdots+a_k)n \rceil}\sum
\limits_{i=1}^k \sum_{j=\lceil (a_0+\cdots+a_{i-1})n\rceil}^{\lceil
(a_1+\cdots+a_i)n\rceil-1}\chi_{U^i_\epsilon(\alpha_i)}(T^j_1x)\le
\delta_1 \text{ for all }n\ge \ell\right\}.\end{aligned}$$ Since $\tau_{0}^{-1}\alpha_1\succeq
\tau_{1}^{-1}\alpha_2\succeq\cdots\succeq \tau_{k-1}^{-1}\alpha_k$, we have $$\begin{aligned}
\lim_{n\rightarrow +\infty}& \frac{-\log \mu
\Big(\bigvee \limits_{i=1}^k \Big(
\bigvee \limits_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big)(x)\Big)}{n}\\
&=\sum_{i=1}^k a_i h_\mu(T_1,\tau_{i-1}^{-1}\alpha_i)=\sum_{i=1}^k
a_i h_{\mu\circ \tau_{i-1}^{-1}}(T_i,\alpha_i)
$$ almost everywhere by Corollary \[c-SMB\]. Hence we can find a large natural number $\ell_1$ such that $\mu(B_\ell)>1-\delta$ for any $\ell\ge \ell_1$, where $B_\ell$ is the set of all points $x\in
X_1$ such that $$\label{esti-3}\frac{-\log \mu
\Big(\bigvee_{i=1}^k \Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big)(x)\Big)}{n}\ge \sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)-\delta$$ for all $n\ge \ell$.
Fix $\ell\ge \max\{ \ell_0,\ell_1\}$. Let $E=A_\ell\cap B_\ell$. Then $\mu(E)>1-2\delta$. For $x\in X_1$ and $n\in
\mathbb{N}$, the unique element $$C(n,x)=(C_j(n,x))_{j=0}^{\lceil
(a_1+\cdots+a_k)n\rceil-1}$$ in $\Lambda^{\{0,1,\cdots,\lceil
(a_1+\cdots+a_k)n\rceil-1\}}$ satisfying that $T_1^jx\in
\tau_{i-1}^{-1}(A^i_{C_j(n,x)})$ for $\lceil(a_0+\cdots+a_{i-1})n\rceil\le j\le
\lceil(a_1+\cdots+a_i)n\rceil-1$, $i=1,\ldots,k$, is called the [*$(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $x$*]{}. Since each point in one atom $A$ of $\bigvee_{i=1}^k \Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big)$ has the same $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name, we define $$C(n,A):=C(n,x)$$ for any $x\in A$, which is called the [*$(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $A$*]{}.
Now if $y\in B_n^{\bf a}(x,\epsilon)$, then for $i=1,\ldots,k$ and $\lceil(a_0+\cdots+a_{i-1})n\rceil\le
j\le \lceil(a_1+\cdots+a_i)n\rceil-1$, either $T_1^jx$ and $T_1^jy$ belong to the same element of $\tau_{i-1}^{-1}\alpha_i$ or $T_1^jx\in U^i_\epsilon(\alpha_i)$. Hence if $x\in E$, $n\ge \ell$ and $y\in B_n^{\bf a}(x,\epsilon)$, then the Hamming distance between $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $x$ and $y$ does not exceed $\delta_1$. Furthermore, $B_n^{\bf a}(x,\epsilon)$ is contained in the set of points $y$ whose $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name is $\delta_1 $-close to $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $x$. It is clear that the total number $L_n(x)$ of such $(\{\alpha_i\}_{i=1}^k,{\bf
a};n)$-names admits the following estimate: $$\begin{aligned}
L_n(x)&\le \binom{\lceil(a_1+\cdots+a_k)n\rceil}{\lceil\lceil(a_1+\cdots+a_k)n\rceil\delta_1\rceil}
M^{\lceil\lceil(a_1+\cdots+a_k)n\rceil\delta_1\rceil}\\
&\le
e^{\delta \lceil(a_1+\cdots+a_k)n\rceil+C}\\
&\le
e^{(a_1+\cdots+a_k)\delta n+C+\delta}\end{aligned}$$ where the second inequality comes from . More precisely, we have shown that for any $x\in E$ and $n\ge \ell$, $$\label{esti-2}\begin{aligned}
B^a_n(x,\epsilon)&\subseteq \{ y\in X_1:
C(n,y)\text{ is $\delta_1 $-close to } C(n,x)\}\\
&=\bigcup \Big\{ A\in \bigvee_{i=1}^k \Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big): C(n,A)\text{ is $\delta_1 $-close to } C(n,x)\Big\}
\end{aligned}$$ and $$\label{esti-2-1}\begin{aligned}
&\#\Big\{ A\in \bigvee_{i=1}^k \Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big): C(n,A)\text{ is $\delta_1 $-close to } C(n,x)\Big\}\\
&\le e^{(a_1+\cdots+a_k)\delta n+C+\delta}.
\end{aligned}$$
Now for $n\in \mathbb{N}$, let $E_n$ denote the set of points $x$ in $E$ such that there exists an element $A$ in $\bigvee_{i=1}^k \Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big)$ with $$\mu(A)>e^{\big(-\sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)+(2+a_1+\cdots+a_k)\delta \big)n}$$ and the $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $A$ is $\delta_1$-close to the $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $x$. It is clear that if $x\in E\setminus E_n$, then for each $A\in \bigvee_{i=1}^k \Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big)$ whose $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name is $\delta_1$-close to the $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $x$, one has $$\mu(A)\le
e^{\big(-\sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)+(2+a_1+\cdots+a_k)\delta \big)n}.$$ In the following, we wish to estimate the measure of $E_n$ for $n\ge
\ell$.
Let $n\ge \ell$. Put $$\mathcal{F}_n=\left\{ A\in \bigvee_{i=1}^k
\Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big): \mu(A)>e^{\big(-\sum \limits_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)+(2+a_1+\cdots+a_k)\delta
\big)n}\right\}.$$ Obviously, $$\#\mathcal{F}_n\le e^{\big(\sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)-(2+a_1+\cdots+a_k)\delta \big)n}$$ since $\mu(X_1)=1$.
Let $x\in E_n$. On the one hand since $x\in B_\ell$, $$\mu
\Big(\bigvee_{i=1}^k \Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big)(x)\Big)\le e^{\big(-\sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)+\delta \big)n}$$ by . On the other hand by the definition of $E_n$, there exists $A\in
\mathcal{F}_n$ with the $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $A$ is $\delta_1$-close to the $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $x$, that is the $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $A$ is $\delta_1$-close to the $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $$\Big(\bigvee_{i=1}^k
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big)(x).$$ According to this, we have $$\label{esti-cc}
E_n\subset \bigcup \{B:B\in \mathcal{G}_n\}$$ where $\mathcal{G}_n$ denotes the set all elements $B$ in $\bigvee_{i=1}^k \Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big)$ satisfying $\mu(B)\le e^{\big(-\sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)+\delta \big)n}$ and the $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $B$ is $\delta_1$-close to the $(\{\alpha_i\}_{i=1}^k,{\bf a};n)$-name of $A$ for some $A\in
\mathcal{F}_n$.
Since for each $A\in \mathcal{F}_n$, the total number of $B$ in $\bigvee_{i=1}^k \Big(
\bigvee_{j=\lceil(a_0+\cdots+a_{i-1})n\rceil}^{\lceil(a_1+\cdots+a_i)n\rceil-1}T_1^{-j}\tau_{i-1}^{-1}\alpha_i
\Big)$, whose $(\{\alpha_i\},{\bf a};n)$-name is $\delta_1$-close to the $(\{\alpha_i\},{\bf a};n)$-name of $A$, is upper bounded by $$\binom{\lceil(a_1+\cdots+a_k)n\rceil}{\lceil\lceil(a_1+\cdots+a_k)n\rceil\delta_1\rceil}
M^{\lceil\lceil(a_1+\cdots+a_k)n\rceil\delta_1\rceil}\le
e^{(a_1+\cdots+a_k)\delta n+C+\delta}.$$ Hence $$\#\mathcal{G}_n\le e^{(a_1+\cdots+a_k)\delta n+C+\delta}\cdot
(\# \mathcal{F}_n)\le e^{\left(\sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)-2\delta\right)n+C+\delta}.$$ Moreover $$\mu(E_n)\le e^{\big(-\sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)+\delta \big)n}\cdot
(\#\mathcal{G}_n)\le e^{-\delta n+C+\delta}$$ by and the definition of $\mathcal{G}_n$.
Next we take $\ell_2\ge \ell$ so that $\sum_{n=\ell_2}^\infty
e^{-\delta n+C+\delta}<\delta$. Then $ \mu(\bigcup_{n\ge
\ell_2}E_n)<\delta$. Let $D=E\setminus \bigcup_{n\ge \ell_2}E_n$. Then $\mu(D)>1-3\delta$. For $x\in D$ and $n\ge \ell_2$, since $x\in E\setminus E_n$, one has $$\begin{aligned}
\mu(B_n^{\bf a}(x,\epsilon))&\le e^{(a_1+\cdots+a_k)n+C+\delta}\cdot
e^{\big(-\sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)+(2+a_1+\cdots+a_k)\delta \big)n}\\
&=e^{\big(-\sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)+2(1+a_1+\cdots+a_k)\delta
\big)n+C+\delta}\end{aligned}$$ by , and the definition of $E_n$. Thus for $x\in D$, $$\begin{aligned}
\liminf_{n\rightarrow +\infty} \frac{-\log \mu(B_n^{\bf
a}(x,\epsilon))}{n}&\ge \sum_{i=1}^k a_i h_{\mu\circ
\tau_{i-1}^{-1}}(T_i,\alpha_i)-2(1+a_1+\cdots+a_k)\delta\\
&\ge\min\left\{\frac{1}{\delta}, h^{\bf
a}_{\mu}(T_1)-\delta\right\}-2(1+a_1+\cdots+a_k)\delta.\end{aligned}$$ This finishes the proof of Theorem \[thm-4.1’\].
In the remaining part of this section, we provide a full proof of Proposition \[SMB\]. First we give two lemmas.
\[Martingale\] Let $(X,\mathcal{B},\mu,T)$ be a measure preserving dynamical system. Let $\alpha,\beta$ be two countable measurable partitions of $(X,\mathcal{B},\mu)$ with $H_\mu(\alpha)<\infty,
H_\mu(\beta)<\infty$ and $\mathcal{A}$ a sub-$\sigma$-algebra of $\mathcal{B}$. Let $I_\mu(\cdot|\cdot)$ denote the conditional information of $\mu$. Then we have the following:
- $I_\mu(\alpha|\mathcal{A})\circ
T=I_\mu(T^{-1}\alpha|T^{-1}\mathcal{A})$.
- $I_\mu(\alpha\vee \beta|\mathcal{A})=I_\mu(\alpha|\mathcal{A})+I_\mu(\beta|\alpha\vee
\mathcal{A})$. In particular, $H_\mu(\alpha\vee
\beta|\mathcal{A})=H_\mu(\alpha|\mathcal{A})+H_\mu(\beta|\alpha\vee
\mathcal{A})$.
- If $\mathcal{A}_1\subset \mathcal{A}_2\subset \cdots$ is an increasing sub-$\sigma$-algebra of $\mathcal{B}$ with $\mathcal{A}_n\uparrow \mathcal{A}$, then $I_\mu(\alpha|\mathcal{A}_n)$ converges almost everywhere and in $L^1$ to $I_\mu(\alpha|\mathcal{A})$. In particular, $\lim_{n\rightarrow +\infty}
H_\mu(\alpha|\mathcal{A}_n)=H_\mu(\alpha|\mathcal{A})$.
\[Breiman\] Let $(X,\mathcal{B},\mu,T)$ be a measure preserving dynamical system and $F_n\in L^1(X,\mathcal{B},\mu)$ be a sequence that converges almost everywhere and in $L^1$ to $F\in
L^1(X,\mathcal{B},\mu)$ and $\int_X \sup_k |F_n(x)|
d\mu(x)<+\infty$. If $f:\mathbb{N}\rightarrow \mathbb{N}$ satisfies $f(n)\ge n$ for all $k\in \mathbb{N}$, then $$\lim \limits_{n\rightarrow
+\infty}\frac{1}{n}\sum_{j=0}^{n-1}F_{f(n)-j}(T^jx)=\mathbb{E}_\mu(F|\mathcal{I}_\mu)(x)$$ almost everywhere and in $L^1$, where $\mathcal{I}_\mu=\{ B\in
\mathcal{B}: \mu(B\Delta T^{-1}B)=0\}$ and $\mathbb{E}_\mu(F|\mathcal{I}_\mu)$ stands for the conditional expectation of $F$ given $\mathcal{I}_\mu$.
This is a slight variant of Maker’s ergodic theorem [@Mak40]. For the convenience of the reader, we give a detailed proof. Since $F\in L^1(X,\mathcal{B},\mu)$, by Birkhoff’s ergodic theorem, we have $$\lim \limits_{n\rightarrow
+\infty}\frac{1}{n}\sum_{j=0}^{n-1}F(T^jx)=\mathbb{E}_\mu(F|\mathcal{I}_\mu)(x)$$ almost everywhere and in $L^1$. Since $$\frac{1}{n}\sum_{j=0}^{n-1}F_{f(n)-j}(T^jx)=\frac{1}{n}\sum_{j=0}^{n-1} F(T^jx)+\frac{1}{n}\sum_{j=0}^{n-1}(F_{f(n)-j}(T^jx)-F(T^jx)),$$ it is suffices to show that $$\lim \limits_{n\rightarrow
+\infty}\frac{1}{n}\sum_{j=0}^{n-1}|F_{f(n)-j}(T^jx)-F(T^jx)|=0$$ almost everywhere and in $L^1$. Set $Z_m(x)=\sup_{j\ge
m}|F_j(x)-F(x)|$ for $m\in \N$. Then $0\le Z_m(x)\le \sup_n |F_n(x)|+|F(x)|$ and $Z_m(x)\to 0$ as $m\rightarrow +\infty$ almost everywhere. Since $\sup_n|F_n(x)|+|F(x)|\in L^1(X,\mathcal{B},\mu)$, we have $\lim_{m\rightarrow +\infty} \int Z_m(x) d\mu(x)=0$ by Lebesgue’s dominated convergence theorem. Then we have $\mathbb{E}_\mu(Z_m|\mathcal{I}_\mu)\rightarrow 0$ as $m\rightarrow
+\infty$ almost everywhere and in $L^1$ (cf. [@Bil95 Theorem 34.2]).
Now let $m\in \mathbb{N}$. For $n>m+1$, $$\begin{aligned}
\frac{1}{n} & \sum_{j=0}^{n-1} |F_{f(n)-j}(T^jx)-F(T^jx)|\\
&\le
\frac{1}{n}\sum_{j=n-m}^{n-1}|F_{f(n)-j}(T^jx)-F(T^jx)|+\frac{1}{n}\sum_{j=0}^{n-m-1}
Z_m(T^jx)\\
&\le
\frac{1}{n}\sum_{j=n-m}^{n-1}Z_1(T^jx)+\frac{n-m}{n}\Big(\frac{1}{n-m}\sum_{j=0}^{n-m-1}
Z_m(T^jx)\Big).\end{aligned}$$
Letting $n\rightarrow +\infty$ and using Birkhoff’s ergodic theorem we have $$\limsup_{n\rightarrow +\infty}\frac{1}{n}\sum_{j=0}^{n-1}
|F_{f(n)-j}(T^jx)-F(T^jx)|\le
\mathbb{E}_\mu(Z_m|\mathcal{I}_\mu)(x)$$ almost everywhere. Since $\mathbb{E}_\mu(Z_m|\mathcal{I}_\mu)\to 0 $ almost everywhere and in $L^1$ as $m\to \infty$, we have $$\limsup_{n\rightarrow +\infty}\frac{1}{n}\sum_{j=0}^{n-1} |F_{f(n)-j}(T^jx)-F(T^jx)|=0$$ almost everywhere and in $L^1$, as desired.
Our proof is adapted from the arguments of Kenyon and Peres in [@KePe96 Lemmas 3.2, 4.4].
First we show that for any $a>0$, $b\ge 0$ and a countable measurable partition $\beta$ of $(X,\mathcal{B},\mu)$ with $H_\mu(\beta)<\infty$, $$\label{fh-eq-2}
\lim_{N\rightarrow +\infty}\frac{1}{N}I_\mu \Big(
\beta_{\lceil aN \rceil}^{\lceil (a+b)N \rceil-1}
\Big)(x)=b\mathbb{E}_\mu(G|\mathcal{I}_\mu)(x)$$ almost everywhere, where $G(x):=I_\mu
\Big(\beta|\bigvee_{n=1}^\infty T^{-n}\beta \Big)(x)$.
If $b=0$, then $\beta_{\lceil aN\rceil}^{\lceil (a+b)N \rceil-1}=\{X, \emptyset\}\ (\text{mod}\ \mu)$ for each $N\in\mathbb{N}$ and so holds. Now assume that $b>0$. Note that $$I_\mu \Big( \bigvee_{n=\lceil aN \rceil}^{\lceil(a+b)N\rceil-1}T^{-n}\beta \Big)(x)=I_\mu
\Big(\bigvee_{n=0}^{\lceil(a+b)N\rceil-1}T^{-n}\beta \Big)(x)-I_\mu(
\bigvee_{n=0}^{\lceil aN\rceil-1}T^{-n}\beta|
\bigvee_{n=\lceil aN \rceil}^{\lceil (a+b)N\rceil-1}T^{-n}\beta\Big)(x).$$ By the Shannon-McMillan-Breiman theorem, is equivalent to $$\label{fh-eq-3}
\lim_{N\rightarrow +\infty}\frac{1}{N}I_\mu( \bigvee_{n=0}^{\lceil
aN\rceil-1}T^{-n}\beta| \bigvee_{n=\lceil aN\rceil}^{\lceil
(a+b)N\rceil-1}T^{-n}\beta\Big)(x)=a\mathbb{E}_\mu(G|\mathcal{I}_\mu)(x)$$ almost everywhere.
Note that $$\begin{aligned}
I_\mu & \Big( \bigvee_{n=0}^{\lceil aN\rceil-1}T^{-n}\beta|
\bigvee_{n=\lceil aN\rceil}^{\lceil(a+b)N\rceil-1}T^{-n}\beta\Big)(x)\\
&=I_\mu\Big(\beta|\bigvee_{n=1}^{\lceil
(a+b)N\rceil-1}T^{-n}\beta\Big)(x)+I_\mu\Big(\bigvee_{n=1}^{\lceil aN
\rceil-1}T^{-n}\beta|
\bigvee_{n=\lceil aN \rceil}^{\lceil(a+b)N\rceil-1}T^{-n}\beta\Big)(x)\\
&=I_\mu\Big(\beta|\bigvee_{n=1}^{\lceil(a+b)N
\rceil-1}T^{-n}\beta\Big)(x)+I_\mu\Big(\bigvee_{n=0}^{\lceil aN
\rceil-2}T^{-n}\beta|
\bigvee_{n=\lceil aN \rceil-1}^{\lceil (a+b)N \rceil-2}T^{-n}\beta\Big)(Tx)\\
&\qquad\qquad \vdots \\
&=\sum_{j=0}^{\lceil aN
\rceil-1}I_\mu\Big(\beta|\bigvee_{n=1}^{[(a+b)N]-1-j}T^{-n}\beta\Big)(T^jx).\end{aligned}$$ Write $G_k(x)=I_\mu(\beta|\bigvee_{n=1}^{k-1}T^{-n}\beta)(x)$ for $k\in \mathbb{N}$ and $x\in X$. Then $$\label{fh-eq-4}
I_\mu( \bigvee_{n=0}^{\lceil aN \rceil-1}T^{-n}\beta|
\bigvee_{n=\lceil aN \rceil}^{\lceil (a+b)N
\rceil-1}T^{-n}\beta\Big)(x)=\sum_{j=0}^{\lceil
aN\rceil-1}G_{\lceil(a+b)N\rceil-j}(T^jx).$$ Since $\bigvee_{n=1}^{k-1} T^{-n}\beta\uparrow \bigvee_{n=1}^\infty
T^{-n}\beta$ when $k\rightarrow +\infty$, $G_k\in
L^1(X,\mathcal{B},\mu)$ is a sequence that converges almost everywhere and in $L^1$ to $G\in L^1(X,\mathcal{B},\mu)$ by Lemma \[Martingale\]. As $H_\mu(\beta)<\infty$, we have $\int_X \sup_k |G_k(x)| d\mu(x)\le
H_\mu(\beta)+1<\infty$ by Chung’s lemma [@Chung]. By and Lemma \[Breiman\], $$\begin{aligned}
\lim_{N\rightarrow +\infty} & \frac{1}{N}I_\mu\Big( \bigvee_{n=0}^{\lceil
aN \rceil-1}T^{-n}\beta \Big| \bigvee_{n=\lceil aN \rceil}^{\lceil (a+b)N
\rceil-1}T^{-n}\beta\Big)(x)\\
&=a\lim_{N\rightarrow
+\infty}\frac{1}{\lceil aN\rceil }\sum_{j=0}^{\lceil
aN\rceil-1}G_{\lceil(a+b)N\rceil-j}(T^jx)\\
&=a\mathbb{E}_\mu(G|\mathcal{I}_\mu)(x)\end{aligned}$$ almost everywhere. Hence holds, so does .
Now we are ready to prove , by induction on $k$. For $k=1$, reduces to the Shannon-McMillan-Breiman theorem. Assume that holds for $k=\ell$ ($\ell\ge 1$). We show below that it holds for $k=\ell+1$.
Let $k=\ell+1$. Write $\beta_i=\bigvee_{j=i}^{\ell+1}\alpha_i$ for $i=1,\ldots,\ell+1$. Then $\beta_1\succeq
\beta_2\succeq\cdots\succeq \beta_{\ell+1}$ and $F_i(x)=I_\mu(\beta_i|\bigvee_{n=1}^{+\infty}T^{-n}\beta_i)(x)$ for $i=1,\ldots,\ell+1$. Note that $$\label{fh-eq-eq}
\bigvee_{i=1}^{\ell+1}
(\alpha_i)_0^{\lceil(a_1+\cdots+a_i)N\rceil-1}=\Big(
\bigvee_{i=1}^\ell (\beta_i)_0^{\lceil(a_1+\cdots+a_i)N\rceil-1}
\Big) \vee (\beta_{\ell+1})_{\lceil
(a_1+\cdots+a_\ell)N\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})N\rceil-1}.$$
By the induction assumption and , we have $$\label{fh-eq-5}
\begin{aligned} &\lim_{N\rightarrow
+\infty}\frac{1}{N}I_\mu \Big( \bigvee_{i=1}^\ell (\beta_i)_0^{\lceil(a_1+\cdots+a_i)N\rceil-1}\Big)(x)=\sum_{i=1}^\ell a_i\mathbb{E}_\mu(F_i|\mathcal{I}_\mu)(x) \text{ and }\\
&\lim_{N\rightarrow +\infty}\frac{1}{N}I_\mu\Big(
(\beta_{\ell+1})_{\lceil (a_1+\cdots+a_\ell)N\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})N\rceil-1})\Big)(x)=a_{\ell+1}
\mathbb{E}_\mu(F_{\ell+1}|\mathcal{I}_\mu)(x)
\end{aligned}$$ almost everywhere. Next we use the idea employed by Algoet and Cover [@AC] in their elegant “sandwich” proof of the Shannon-McMillan-Breiman theorem. For $\mu$-a.e. $x\in X$, we define $$Z_m(x)=\frac{\mu \Big( \bigvee \limits_{i=1}^\ell (\beta_i)_0^{\lceil(a_1+\cdots+a_i)m\rceil-1}(x)\Big)\cdot
\mu\Big( (\beta_{\ell+1})_{\lceil
(a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1}(x) \Big)} {\mu\Big(\Big(
\bigvee \limits_{i=1}^\ell
(\beta_i)_0^{\lceil(a_1+\cdots+a_i)m\rceil-1} \vee
(\beta_{\ell+1})_{\lceil (a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1}\Big)(x)\Big)}$$ for all $m\in \mathbb{N}$. Then for $\mu$-a.e. $x\in X$, $Z_m(x)>0$ for all $m\in \mathbb{N}$.
Since $$\begin{aligned}
\int_X Z_m(x)d \mu(x)&=\sum_{A\in
\bigvee \limits_{i=1}^\ell (\beta_i)_0^{\lceil(a_1+\cdots+a_i)m\rceil-1} \atop{B\in
(\beta_{\ell+1})_{\lceil (a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1}}} \int_{A\cap
B}\frac{\mu(A)\mu(B)}{\mu(A\cap B)}d\mu(x)\\
&=\sum_{A\in
\bigvee \limits_{i=1}^\ell (\beta_i)_0^{\lceil(a_1+\cdots+a_i)m\rceil-1} \atop{B\in
(\beta_{\ell+1})_{\lceil (a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1}}} \mu(A)\mu(B)\\
&=1,\end{aligned}$$ the series $\sum_{m=1}^\infty \mu(\{x\in X: Z_m(x)\ge e^{\epsilon
m}\})$ converges for every $\epsilon>0$ and the Borel-Canteli Lemma implies that $\limsup_{N\rightarrow +\infty} \frac{1}{N} \log
Z_N(x)\le 0$ for $\mu$-a.e. $x\in X$. Using the definition of $Z_m$, and , we obtain $$\limsup_{N\rightarrow +\infty}\frac{1}{N}I_\mu \Big(\bigvee_{i=1}^{\ell+1}
(\alpha_i)_0^{\lceil(a_1+\cdots+a_i)N\rceil-1}\Big)(x)\le
\sum_{i=1}^{\ell+1}a_i \mathbb{E}_\mu(F_i|\mathcal{I}_\mu)(x)$$ for $\mu$-a.e. $x\in X$.
Conversely, by and the induction assumption, we have $$\label{fh-eq-6}
\begin{aligned} &\lim_{N\rightarrow +\infty}\frac{1}{N}I_\mu\Big(
(\beta_{i})_{\lceil (a_1+\cdots+a_\ell)N\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})N\rceil-1}\Big)(x)=a_{\ell+1}
\mathbb{E}_\mu(F_{i}|\mathcal{I}_\mu)(x), \ i=\ell,\; \ell+1
\text{ and }\\
& \lim_{N\rightarrow +\infty} \frac{1}{N}I_\mu \Big(
\bigvee_{i=1}^{\ell-1}
(\beta_i)_0^{\lceil(a_1+\cdots+a_i)N\rceil-1}\vee
(\beta_\ell)_0^{\lceil (a_1+\cdots+a_\ell+a_{\ell+1})N\rceil -1}
\Big)(x)\\
&\quad =(a_\ell+a_{\ell+1})\mathbb{E}_\mu(F_\ell|\mathcal{I}_\mu)(x)+\sum_{i=1}^{\ell-1}
a_i\mathbb{E}_\mu(F_i|\mathcal{I}_\mu)(x)
\end{aligned}$$ almost everywhere. Then for $\mu$-a.e. $x\in X$, we define $$\begin{aligned}
R_m(x)=&\frac{\mu\Big(\Big( \bigvee \limits_{i=1}^\ell
(\beta_i)_0^{\lceil(a_1+\cdots+a_i)m\rceil-1} \vee
(\beta_{\ell+1})_{\lceil (a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1}\Big)(x)\Big)} {
\mu\Big(\Big( \bigvee \limits_{i=1}^{\ell-1}
(\beta_i)_0^{\lceil(a_1+\cdots+a_i)N\rceil-1}\vee
(\beta_\ell)_0^{\lceil (a_1+\cdots+a_\ell+a_{\ell+1})N\rceil -1}
\Big)(x)\Big)}\\
&\times \frac{\mu\Big( (\beta_{\ell})_{\lceil
(a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1}(x) \Big)}{\mu\Big(
(\beta_{\ell+1})_{\lceil (a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1}(x)\Big) }\end{aligned}$$ for all $m\in \mathbb{N}$. Then for $\mu$-a.e. $x\in X$, $R_m(x)>0$ for all $m\in \mathbb{N}$.
Since $\beta_\ell\succeq \beta_{\ell+1}$, we have $$\begin{aligned}
\int_X R_m(x)d \mu(x)&=\sum_{A\in
\bigvee \limits_{i=1}^\ell
(\beta_i)_0^{\lceil(a_1+\cdots+a_i)m\rceil-1}
\atop{ B\in (\beta_{\ell+1})_{\lceil (a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1} \atop{ C\in
(\beta_{\ell})_{\lceil (a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1}}}} \int_{A\cap
B\cap C}\frac{\mu(A\cap B)\mu(B\cap C)}{\mu(A\cap B\cap C)\mu(B)}d\mu(x)\\
&=\sum_{A\in \bigvee \limits_{i=1}^\ell
(\beta_i)_0^{\lceil(a_1+\cdots+a_i)m\rceil-1}
\atop{ B\in (\beta_{\ell+1})_{\lceil (a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1} \atop{ C\in
(\beta_{\ell})_{\lceil (a_1+\cdots+a_\ell)m\rceil}^{\lceil
(a_1+\cdots+a_\ell+a_{\ell+1})m\rceil-1}}}} \frac{\mu(A\cap B)\mu(B\cap C)}{\mu(B)}\\
&=1\end{aligned}$$ for $m\in \mathbb{N}$. Thus the series $\sum_{m=1}^\infty \mu(\{x\in
X: R_m(x)\ge e^{\epsilon m}\})$ converges for every $\epsilon>0$ and the Borel-Canteli Lemma implies that $\limsup_{N\rightarrow +\infty}
\frac{1}{N} \log R_N(x)\le 0$ for $\mu$-a.e. $x\in X$. Using the definition $R_N$, and , we have $$\liminf_{N\rightarrow +\infty}\frac{1}{N}I_\mu
\Big(\bigvee_{i=1}^{\ell+1}
(\alpha_i)_0^{\lceil(a_1+\cdots+a_i)N\rceil-1}\Big)(x)\ge
\sum_{i=1}^{\ell+1}a_i \mathbb{E}_\mu(F_i|\mathcal{I}_\mu)(x)$$ for $\mu$-a.e. $x\in X$. for $\mu$-a.e. $x\in X$. This completes the proof of Proposition \[SMB\].
[**Acknowledgements**]{}. The first author was partially supported by RGC grants in the Hong Kong Special Administrative Region, China (projects CUHK401112, CUHK401013). The second author was partially supported by NNSF (11225105, 11371339, 11431012).
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|
OUT–4102–74\
hep-th/9806174\
1 July 1998
[ **Solving differential equations for 3-loop diagrams:\
relation to hyperbolic geometry and knot theory** ]{}
[**Abstract**]{}In hep-th/9805025, a result for the symmetric 3-loop massive tetrahedron in 3 dimensions was found, using the lattice algorithm PSLQ. Here we give a more general formula, involving 3 distinct masses. A proof is devised, though it cannot be accounted as a derivation; rather it certifies that an Ansatz found by PSLQ satisfies a more easily derived pair of partial differential equations. The result is similar to Schläfli’s formula for the volume of a bi-rectangular hyperbolic tetrahedron, revealing a novel connection between 3-loop diagrams and 1-loop boxes. We show that each reduces to a common basis: volumes of ideal tetrahedra, corresponding to 1-loop massless triangle diagrams. Ideal tetrahedra are also obtained when evaluating the volume complementary to a hyperbolic knot. In the case that the knot is positive, and hence implicated in field theory, ease of ideal reduction correlates with likely appearance in counterterms. Volumes of knots relevant to the number content of multi-loop diagrams are evaluated; as the loop number goes to infinity, we obtain the hyperbolic volume of a simple 1-loop box. $^1$) D.Broadhurst@open.ac.uk; http://physics.open.ac.uk/$\;\widetilde{}$dbroadhu
Introduction
============
In [@CTet] we studied the 3-loop 3-dimensional tetrahedral Feynman diagram $$C(a,b):=\frac{1}{\pi^6}\int\int\int\frac{d^3\bk_1d^3\bk_2d^3\bk_3}
{(k_1^2+a^2)(k_2^2+1)(k_3^2+1)(k_{2,3}^2+b^2)(k_{1,3}^2+1)(k_{1,2}^2+1)}
\label{cab}$$ with $k_n^2:=|\bk_n|^2$ and $k_{i,j}^2:=|\bk_i-\bk_j|^2$. For the totally symmetric tetrahedron, with $a=b=1$, we found a simple reduction to a Clausen integral: $${C(1,1)\over2^{5/2}}=
-\int_{2\alpha}^{4\alpha}d\theta\log(2\sin{\mbox{$\frac{1}{2}$}}\theta)\,;
\quad\alpha:=\arcsin{\mbox{$\frac{1}{3}$}}\,,\label{ans}$$ thus obtaining an exact dilogarithmic result for the diagram evaluated numerically in [@GKM].
The discovery route for (\[ans\]) was based on a dispersion relation for the more general Feynman tetrahedron (\[cab\]), with masses $a$ and $b$ on non-adjacent lines and unit masses on the other 4 lines. This was derived by applying the methods of [@mas; @sixth] in 3 dimensions. In this paper, we reduce $C(a,b)$ to 7 dilogarithms, for $a^2+b^2>4$, and to 8 Clausen values, for $a^2+b^2<4$. In the latter case, (\[ans\]) results by use of the classical formula [@BBP] $$\pi=2\arcsin{\mbox{$\frac{1}{3}$}}+4\arcsin{\mbox{$\frac{1}{\sqrt3}$}}\,,\label{class}$$ which reduces the Clausen values to only 2.
Section 2 gives the 3-loop results. In Sections 3 and 4 we examine connections, via hyperbolic geometry, to very different types of diagrams, in 4 dimensions: massive box diagrams, with only 1 loop, studied in [@DD], and massless diagrams with more than 6 loops, studied in [@BK15; @BGK]. Remarkably, the infinite-loop limit of the hyperbolic volumes of knots entailed by the latter recovers a simple case of the former. Section 5 gives our conclusions.
Solving the vacuum differential equations
=========================================
In [@CTet], we reduced (\[cab\]) to dispersive integrals of the form $\int dx\, P(x,X)\log Q(x,X)$ where $P$ and $Q$ are rational algebraic functions of $x$ and of the square root, $X$, of a quadratic function of $x$. Section 8.1.2 of [@Lewin] shows that every integral of this form may be reduced to dilogarithms, albeit with the possibility of complex arguments. Pursuing the methods of [@Lewin], one readily establishes that $C(a,b)$ is reducible to real dilogarithms for $a^2+b^2>4$. Implementing the algorithm of [@Lewin], in [Reduce]{}, we obtained a formidably complicated result, involving 2 square roots: $\sqrt{a^2+b^2-4}$ and $\sqrt{2b(b+2)}$. The appearance of the former is to be expected; the characteristics of the result clearly change when $a^2+b^2-4$ changes sign. The appearance of the latter is a gratuitous consequence of the dispersive derivation; it may be removed by consideration of $C(b,a)=C(a,b)$, but then $\sqrt{2a(a+2)}$ appears. Clearly there must exist a result involving neither $\sqrt{2a(a+2)}$ nor $\sqrt{2b(b+2)}$. How to achieve this is problematic.
One strategy for removing a bogus square root is to differentiate the dilogarithms that involve it and then to combine the resultant logarithms, to show that the differential is free of the unwanted square root. In this case, [Reduce]{} showed that the dispersion relation for the Feynman tetrahedron $C(a,b)$ yields the partial differential equation $$\begin{aligned}
\frac{b\sqrt{a^2+b^2-4}}{4}\,\frac{\partial}{\partial a}\,
\frac{a\sqrt{a^2+b^2-4}}{4}\,C(a,b)&=&\log\left(\frac{a+2}{a+b+2}\right)
+\frac{b}{a+2}\log\left(\frac{a+b+2}{b+2}\right)\nonumber\\&&{}
+\frac{2b}{a^2-4}\log\left(\frac{a+2}{4}\right),\label{pde}\end{aligned}$$ which entails only the physical square root, easily traceable to a tree diagram for elastic scattering [@CTet]. A second partial differential equation immediately follows from the symmetry $C(a,b)=C(b,a)$ of the diagram. We checked that the pair agrees with results in [@AKR], obtained by the methods of [@AVK], more recently espoused in [@ER].
Systematic re-integration of (\[pde\]), by the methods of [@Lewin], still produced 20 dilogarithms, with 8 of these entailing the unwanted square root $\sqrt{2b(b+2)}$. Accordingly, we resorted to an alternative strategy, by evaluating $C(a,b)$ numerically at an arbitrarily chosen transcendental point, $a=\exp(1),~b=\pi$, and then using the lattice algorithm PSLQ [@PSLQ] to search for a rational linear combination of dilogarithms of a character suggested by those parts of the analytical 20-dilogarithm result that did not involve the bogus square root $\sqrt{2b(b+2)}$. After much trial and error, in search spaces of dimensions as large as 80, to accommodate the possibility of many products of logs, we found a simple log-free fit to the single numerical datum: $$\begin{aligned}
{\mbox{$\frac{1}{8}$}}abc\,C(a,b)&=&
{\rm Li}_2\left(-\frac{p}{m}\right)+{\rm Li}_2\left(1-\frac{4}{m}\right)
+{\rm Li}_2\left(1-\frac{m}{a+2}\right)
+{\rm Li}_2\left(1-\frac{m}{b+2}\right)\nonumber\\
&-&{\rm Li}_2\left(-\frac{m}{p}\right)-{\rm Li}_2\left(1-\frac{4}{p}\right)
-{\rm Li}_2\left(1-\frac{p}{a+2}\right)
-{\rm Li}_2\left(1-\frac{p}{b+2}\right)\label{dans}\end{aligned}$$ with a dilogarithm ${\rm Li}_2(x):=-\int_0^x(dy/y)\log(1-y)$ and $$c:=\sqrt{a^2+b^2-4}\,,\quad p:=a+b+2+c\,,\quad m:=a+b+2-c\,.$$ Ansatz (\[dans\]) is manifestly symmetric in $(a,b)$ and fits the datum to 360-digit precision.
It was then a routine application of computer algebra to prove that (\[dans\]) is correct, by showing that it satisfies the partial differential equation (\[pde\]). Hence the r.h.s. of (\[dans\]) may differ from the required result only by a function of $b$. But by symmetry it thus differs only by a function of $a$, and hence only by a constant. Since the r.h.s. and l.h.s. both vanish when $c=0$, the constant must vanish. Hence (\[dans\]) is proven to be correct, though no analytical [*derivation*]{} of it has yet been obtained. To our knowledge, this is the first time that a lattice algorithm, such as PSLQ, has been used to find a previously unknown solution to a pair of partial differential equations.
We note that one of the 8 dilogarithms in (\[dans\]) may be removed, using [@Lewin] $$0={\rm Li}_2(-p/m)+{\rm Li}_2(-m/p)+{\mbox{$\frac{1}{6}$}}\pi^2+{\mbox{$\frac{1}{2}$}}\log^2(p/m)\,.$$ No further reduction was found by PSLQ, with transcendental values of $a$ and $b$. With rational values of $\{a,b,c\}$, considerable simplification was obtained. For example $$224\,C(14,8)={\rm Li}_2\left({\mbox{$\frac{3}{5}$}}\right)+{\mbox{$\frac{1}{12}$}}\pi^2-\log5\log{\mbox{$\frac{9}{5}$}}$$ was spectacularly reduced by PSLQ to a single dilogarithm. It remains an open question whether (\[dans\]) may be reduced to fewer than 7 dilogs, in the general case. We suspect not.
Reduction to Clausen values
---------------------------
The result (\[dans\]) clearly entails only real dilogarithms when $a^2+b^2>4$. When $a^2+b^2<4$, it may be reduced, by application of Eq (A.2.5.1) of [@Lewin], to Clausen values of the form $${\rm Cl}_2(\theta):=-\int_0^\theta d\phi\log\left|2\sin{\mbox{$\frac{1}{2}$}}\phi\right|
=\sum_{n>0}\frac{\sin(n\theta)}{n^2}\,.$$ Since the imaginary part of a dilog yields 3 Clausen values, plus the product of an angle and log, the result (\[dans\]) might be expected to be rather complicated, involving up to 16 terms. Transforming to the regime where $\gamma:=\sqrt{4-a^2-b^2}$ is real, one finds that $$\begin{aligned}
{\mbox{$\frac{1}{16}$}}ab\gamma\,C(a,b)&=&{\mbox{$\frac{1}{2}$}}\left\{
{\rm Cl}_2(4\phi)
+{\rm Cl}_2(2\phi_a+2\phi_b-2\phi)
+{\rm Cl}_2(2\phi_a-2\phi)
+{\rm Cl}_2(2\phi_b-2\phi)\right.\nonumber\\&&\left.{}
-{\rm Cl}_2(2\phi_a+2\phi_b-4\phi)
-{\rm Cl}_2(2\phi_a)
-{\rm Cl}_2(2\phi_b)
-{\rm Cl}_2(2\phi)\right\}\label{cla}\end{aligned}$$ is log-free and involves only 8 Clausen values, with arguments formed from $$\phi:=\arctan\frac{\gamma}{a+b+2}\,,\quad
\phi_a:=\arctan\frac{\gamma}{a}\,,\quad
\phi_b:=\arctan\frac{\gamma}{b}\,,\label{phi}$$ which are related by $$\cos\phi_a\cos\phi_b
=\cos(\phi_a+\phi_b-2\phi)\,.
\label{key}$$ The freedom from logs is highly non-trivial, entailing the multiplicative relation $$\left(1-\frac{4}{m}\right)\left(1-\frac{4}{p}\right)=
\left(1-\frac{m}{a+2}\right)\left(1-\frac{m}{b+2}\right)
\left(1-\frac{p}{a+2}\right)\left(1-\frac{p}{b+2}\right)$$ between 6 of the arguments of the 8 dilogarithms of (\[dans\]). Had it been known in advance that neither (\[dans\]) nor (\[cla\]) entails logs, while each reduces to only 8 terms, the process of constructing a viable symmetric Ansatz would have been greatly simplified. We offer this observation as a guide to future work.
The symmetric tetrahedron
-------------------------
To obtain a result for $C(1,1)$, we use the specific values of the angles (\[phi\]), namely $\phi=\alpha$, $\phi_a=\phi_b=\frac14\pi+\frac12\alpha$, with $\alpha:=\arcsin\frac13$ appearing as the only non-trivial angle, by virtue of (\[class\]). Then using ${\rm Cl}_2(\pi)=0$, and the general identity [@Lewin] $${\mbox{$\frac{1}{2}$}}{\rm Cl}_2(\pi-2\alpha)
={\rm Cl}_2({\mbox{$\frac{1}{2}$}}\pi-\alpha)
-{\rm Cl}_2({\mbox{$\frac{1}{2}$}}\pi+\alpha)\,,$$ one finds that only the first and last terms in (\[cla\]) survive, giving $$\frac{C(1,1)}{8\sqrt2}={\mbox{$\frac{1}{2}$}}\left\{{\rm Cl}_2(4\alpha)-{\rm Cl}_2(2\alpha)
\right\}\approx0.01537\,,
\label{sym}$$ in agreement with (\[ans\]). The tiny value will be seen to be significant.
Connection to 1-loop diagrams
=============================
In [@DD], Andrei Davydychev and Bob Delbourgo considered an apparently very different problem, namely the massive 1-loop box diagram in 4 dimensions, which yields a result uncannily similar to (\[dans\],\[cla\]), in the case of a common mass on the internal lines and a common norm for the external 4-momenta. Then there are three kinematic variables, which may be taken as Mandelstam’s $\{s,t,u\}$. The internal mass provides the scale, here set to unity. In certain kinematic regimes, $\{s,t,u\}$ may be transformed to the 3 non-trivial dihedral angles, $\{\psi_1,\psi_2\,\psi_3\}$, of a bi-rectangular tetrahedron in a 3-space of constant curvature [@DD]. This is one of the 4 congruent parts that result from dissection of a tetrahedron with a symmetry that derives from the common internal mass. The result then entails its volume, which is a Schläfli [@Schl; @Cox] function.
After the results (\[dans\]) and (\[cla\]), for the 3-loop vacuum diagram, were communicated to Andrei Davydychev, he made the intriguing suggestion that (\[cla\]), for the case $a^2+b^2<4$, might be reducible from 8 real Clausen values to 7, as is the case [@DD] for the box diagram, in restricted kinematic regimes. If this were the case, one might hope to cap the ‘magic’ feat in [@magic], where a 2-loop vacuum diagram was transformed to a massless 1-loop triangle diagram, in a dimension differing by 2 units. In the present case, such a conjuring act would entail a more remarkable connection, between diagrams whose loop numbers differ by 2, while their spacetime dimensions differ only by unity. We now examine this issue.
Geometric and non-geometric boxes
---------------------------------
From [@DD], we obtained a simple conversion of $\{s,t,u\}$ to $\{\psi_1,\psi_2,\psi_3\}$ as follows. Let $$v:={4\over s}\,,\quad w:={4\over t}\,,\quad
x:=\frac{8}{s+t+u-8}\label{kin}$$ be a re-parametrization of Mandelstam space. Then the dihedral angles satisfy $${1-w\over\tan^2\psi_1}={\tan^2\psi_2\over1-x^2}={1-v\over\tan^2\psi_3}
={1\over\tan^2\delta}=G:={vw\over x^2}-(1-v)(1-w)
\label{psi}$$ where $G$ derives from a Gram determinant and $\delta$ is an auxiliary angle, with $$\tan\delta\cos\psi_1\cos\psi_3
=D(\psi_1,\psi_2,\psi_3):=
\sqrt{\cos^2\psi_2-\sin^2\psi_1\sin^2\psi_3}\,.\label{delta}$$ The box diagram evaluates to $$B(s,t,u):=\frac{N(\psi_1,\psi_2,\psi_3)}{D(\psi_1,\psi_2,\psi_3)}\,,
\label{box}$$ with a numerator that is a Schläfli function [@DD]: $$\begin{aligned}
N(\psi_1,\psi_2,\psi_3)&:=&{\mbox{$\frac{1}{2}$}}\left\{
{\rm Cl}_2(2\psi_1+2\delta)-{\rm Cl}_2(2\psi_1-2\delta)
+{\rm Cl}_2(2\psi_3+2\delta)-{\rm Cl}_2(2\psi_3-2\delta)
\right.\nonumber\\&&\left.{}
-{\rm Cl}_2(\pi-2\psi_2+2\delta)+{\rm Cl}_2(\pi-2\psi_2-2\delta)
+2{\rm Cl}_2(\pi-2\delta)\right\}\,.\label{schl}\end{aligned}$$ When $\{1-v,1-w,1-x^2,G\}$ are all positive, $\{\psi_1,\psi_2,\psi_3,\delta\}$ are all real and (\[schl\]) is 4 times the volume of a bi-rectangular tetrahedron in hyperbolic space, since the full tetrahedron may be dissected into 4 congruent bi-rectangular parts [@DD].
In the case that $\{1-v,1-w,1-x^2,G\}$ are all negative, $\delta$ is imaginary, while $\{\psi_1,\psi_2,\psi_2\}$ are real. Then both the numerator and denominator of (\[box\]) are pure imaginary and we obtain a geometric interpretation that entails the volume of a tetrahedron in spherical space. For the residual sign possibilities, there is [*no*]{} interpretation in terms of real geometry. Indeed, unitarity often requires the amplitude to be complex. Thus vanishing of the Gram determinant of the external momenta, at $G=0$, is emphatically [*not*]{} the signal for the geometry to change from one sign of curvature to the other. If one has a real geometry at some point $\{s,t,u\}$ near $G=0$, then there is no real geometry at a neighbouring point, with the opposite sign of $G$, since there (\[psi\]) forces $\{\psi_1,\psi_2,\psi_3\}$ to be imaginary.
By way of examples of geometric and non-geometric behaviour, we consider $B_0(s,t):=B(s,t,-s-t)$, with light-like external momenta. In the hyperbolic regime, we obtain $$B_0(4,4)=4{\rm Cl}_2({\mbox{$\frac{1}{2}$}}\pi)\,,\quad
B_0(6,6)=\frac{5{\rm Cl}_2({\mbox{$\frac{1}{3}$}}\pi)}{\sqrt3}\,,\quad
B_0({\mbox{$\frac{16}{3}$}},{\mbox{$\frac{16}{3}$}})=\frac{3{\rm Cl}_2(2\alpha)
+6{\rm Cl}_2({\mbox{$\frac{1}{2}$}}\pi-\alpha)}{2\sqrt2}\label{al}$$ with the first example giving 4 times Catalan’s constant, while the second is a rational multiple of a constant found in the 2-loop 4-dimensional vacuum diagram with 3 equal masses [@BV], which enjoys a ‘magic’ connection [@magic] to a massless 1-loop triangle diagram. The final example entails $\alpha:=\arcsin\frac13$, though in a manner markedly different from (\[ans\]).
Non-geometric results are obtainable from the instructive duality relation $$B_0(s,t)-B_0(\lambda/s,\lambda/t)
=\frac{2}{\sqrt{\lambda}}\,\arccos\left({s\over2}-1\right){\rm arccosh}
\left({\lambda\over2s}-1\right)
+\left\{s\leftrightarrow t \right\}\,,\label{dual}$$ with $\lambda:=4s+4t-st$. It was proven by analytic continuation of (\[schl\]), after the discovery by PSLQ that $$B_0({\mbox{$\frac{8}{3}$}},{\mbox{$\frac{8}{3}$}})-B_0({\mbox{$\frac{16}{3}$}},{\mbox{$\frac{16}{3}$}})
=\frac{3({\mbox{$\frac{1}{2}$}}\pi-\alpha)\log3}{2\sqrt2}\,,$$ with product terms familiar from [@CTet; @sixth; @poly]. When one box in (\[dual\]) is geometric, the products of angles and logs show that its dual is not. Since (\[cla\]) has no such product, it cannot be such a non-geometric box. We now consider whether it might be geometric.
Obstacles to a single 3-loop vacuum volume
------------------------------------------
Analytical considerations and numerical investigations, alike, suggest that no geometric interpretation as a single tetrahedral volume, and hence no relation to a single 1-loop diagram, is obtainable for the 3-loop 3-dimensional vacuum diagram $C(a,b)$.
The argument against a real tetrahedral volume in spherical space is compelling: the formula for such a volume involves the real parts of complex ${\rm Li}_2$ values [@DD; @Cox]. In contrast, our result (\[dans\]) entails purely real ${\rm Li}_2$ values when $a^2+b^2>4$. Thus the simplicity of the vacuum diagram seems to preclude a geometric interpretation in a space of positive curvature, since any such interpretation would be too complicated, analytically speaking.
We argue that there is no interpretation as a single volume in hyperbolic space, for $a^2+b^2<4$. Here we are guided by the fact that all attempts to reduce the Clausen values in (\[cla\]) from 8 to 7, as would be required by (\[schl\]), met with abject failure.
Since no-go claims based on analysis are notoriously fallible, we also investigated the situation empirically, using PSLQ. The first step was clear: is there a simple integer relations between the 8 Clausen values in (\[cla\])? PSLQ replied with an emphatic [*no*]{}, by proving that any integer relation would entail a coefficient in excess of $10^{30}$.
Then we considered relations between Clausen values generated by Abel’s identity for 5 dilogarithms [@Lewin]. Since the imaginary part of a dilogarithm generates 3 Clausen values, the generic relation will entail 15 Clausen values. The symmetric form of the result is $$0=\sum_{6\ge k\ge1}\theta_k=\sum_{6\ge k\ge1}\sin\theta_k
\quad\Longrightarrow\quad
0=\sum_{6\ge j>k\ge1}{\rm Cl}_2(\theta_j+\theta_k)\,,\label{15}$$ with 6 angles, whose values and sines sum to zero, producing 15 Clausen values, which also sum to zero. From (\[key\],\[15\]) we derived 3 relations between Clausen values whose arguments are linear combinations of $\{\phi,\phi_a,\phi_b\}$. A pair is formed by $$\begin{aligned}
0&=&2{\rm Cl}_2(2\phi)-4{\rm Cl}_2(2\phi_b)+{\rm Cl}_2(4\phi_b)
+2{\rm Cl}_2(2\phi_b-2\phi)-2{\rm Cl}_2(2\phi_a-2\phi)
\nonumber\\&&{}+{\rm Cl}_2(2\phi_a-4\phi)+2{\rm Cl}_2(2\phi_a+2\phi_b-2\phi)
-{\rm Cl}_2(2\phi_a+4\phi_b-4\phi)\label{first}\end{aligned}$$ and its $a\leftrightarrow b$ transform, while the third is symmetric: $$\begin{aligned}
0&=&2{\rm Cl}_2(2\phi)-2{\rm Cl}_2(2\phi_a-2\phi)-2{\rm Cl}_2(2\phi_b-2\phi)
+{\rm Cl}_2(2\phi_a-4\phi)+{\rm Cl}_2(2\phi_b-4\phi)\nonumber\\&&{}
-2{\rm Cl}_2(2\phi_a+2\phi_b-2\phi)+2{\rm Cl}_2(2\phi_a+2\phi_b-4\phi)
-{\rm Cl}_2(4\phi_a+4\phi_b-8\phi)\nonumber\\&&{}
+{\rm Cl}_2(2\phi_a+4\phi_b-4\phi)+{\rm Cl}_2(4\phi_a+2\phi_b-4\phi)\,.
\label{third}\end{aligned}$$
The next step was to engage PSLQ to search for more relations. At the arbitrarily chosen transcendental point $a=\exp(-1)$, $b=1/\pi$, we computed, to 360-digit precision, 44 Clausen values of the form ${\rm Cl}_2(2j\phi_a+2k\phi-2n\phi)$, with non-negative integers bounded by $j<3$, $k<3$, $n<5$, $j+k+n>0$. PSLQ found only the 3 known relations. Moreover, it proved that any other relation would involve an integer in excess of $10^5$. Enlarging the search space to include angles in which the coefficients of $\phi_a$ and $\phi_b$ differ in sign, we found no new relation. It is easy to show that the 3 proven relations do not enable a reduction of (\[cla\]) to less than 8 Clausen values. Hence, a reduction to 7 real Clausen values, as required for a single Schläfli function, would seem to require a non-linear transformation of angles, for which we have seen no precedent.
Reduction of diagrams to ideal tetrahedra
-----------------------------------------
The difficulty in relating (\[cla\]) to a geometric box is more apparent when one writes it in terms of [*differences*]{} of volumes of ideal hyperbolic tetrahedra. An ideal tetrahedron has all its vertices at infinity and is specified by 2 dihedral angles, $\theta_1$ and $\theta_2$, at adjacent edges. The dihedral angle at the edge adjacent to these is $\theta_3:=\pi-\theta_1-\theta_2$. Each remaining edge has a dihedral angle equal to that at its opposite edge. The volume of such a ideal tetrahedron is [@DD] $$V(\theta_1,\theta_2):={\mbox{$\frac{1}{2}$}}\sum_{k=1}^3{\rm Cl}_2(\theta_k)
={\mbox{$\frac{1}{2}$}}\{{\rm Cl}_2(2\theta_1)
+{\rm Cl}_2(2\theta_1)-{\rm Cl}_2(2\theta_1+2\theta_2)\}\,.\label{ideal}$$ Thus (\[cla\]) may be written, rather neatly, as $${\mbox{$\frac{1}{16}$}}ab\gamma\,C(a,b)
=V(\phi,\psi_a)+V(\phi,\psi_b)
-V(\phi,\psi_a+\psi_b)-V(\phi,\phi)\,,\label{neat}$$ where $\psi_{a,b}:=\phi_{a,b}-\phi$ are confined to the interval $[\phi,\pi/2-\phi]$, with $\phi$ confined to $[0,\pi/4]$. Similarly, the box volume (\[schl\]) may be written as $$\begin{aligned}
N(\psi_1,\psi_2,\psi_3)
&=&V(\delta+\psi_1,\delta-\psi_1)+V(\delta+\psi_3,\delta-\psi_3)\nonumber\\
&+&V({\mbox{$\frac{1}{2}$}}\pi+\psi_2-\delta,{\mbox{$\frac{1}{2}$}}\pi-\psi_2-\delta)
+V({\mbox{$\frac{1}{2}$}}\pi-\delta,{\mbox{$\frac{1}{2}$}}\pi-\delta)\label{Nis}\end{aligned}$$ with the auxiliary angle $\delta$ given by (\[delta\]).
Both the vacuum result (\[neat\]) and the box volume (\[Nis\]) are non-negative, in their hyperbolic regimes, where the angles are real. Now we consider their zeros and maximum values. The box volume (\[Nis\]) vanishes only for $\cos\psi_2=\sin\psi_1\sin\psi_3$, where the denominator (\[delta\]) of the diagram vanishes, at the boundary of the hyperbolic regime. The maximum volume is $N(0,0,0)=4{\rm Cl}_2(\pi/2)$, achieved in the case of the box diagram $B_0(4,4)$ in (\[al\]), with $D(0,0,0)=1$. In contrast, the vacuum diagram yields a combination (\[neat\]) of ideal tetrahedral volumes that vanishes at $b=0$, where $\psi_a=\phi$ and $\psi_b=\frac12\pi-\phi$, with the last term cancelling the first, and the third cancelling the second; and at $a=0$, with the last cancelling the second, and the third cancelling the first; and at $\gamma=0$, where all terms vanish separately. Its maximum value occurs at the totally symmetric point $a=b=1$, where (\[sym\]) gives a combination of volumes that is more than 200 times smaller than the maximum volume, $N(0,0,0)=3.66386237$, achieved by the box.
From the above, the difficulty of relating the vacuum diagram to a box is glaring. The geometric insight of [@DD] led to the conclusion that every 4-dimensional 1-loop box diagram may be evaluated by dissecting[^1] its associated volume into no more than 6 bi-rectangular parts, each given by a Schläfli function. We have shown that the addition and subtraction of ideal tetrahedra, entailed by the vacuum diagram in (\[neat\]), leads to net volumes that are, typically, two orders of magnitude smaller than the volumes associated with a box diagram, via the additions in (\[Nis\]).
Yet there [*is*]{} a remarkably strong connection between 3-loop vacuum diagrams and 1-loop boxes: both entail [*combinations*]{} of volumes of ideal tetrahedra. We have show this for the vacuum diagram (\[ideal\]). In the more complicated case of an arbitrary box diagram, one may obtain up to 24 ideal tetrahedra, with each of the 6 bi-rectangular constituents [@DD] of a general tetrahedron yielding 4 ideal tetrahedra, via (\[Nis\]). Moreover every such ideal tetrahedron equates to a massless 1-loop triangle diagram [@DD; @magic].
We conclude that 3-loop 3-dimensional vacuum diagrams and 1-loop 4-dimensional boxes do not equate, directly. Rather, they share a common reduction, via hyperbolic geometry, to 1-loop massless 4-dimensional triangle diagrams, i.e. ideal tetrahedra.
Hyperbolic manifolds from multi-loop diagrams
=============================================
The box-diagram value $B_0(4,4)=N(0,0,0)=4{\rm Cl}_2(\pi/2)=
3.66386237$ is familiar in an apparently quite different context: it is the hyperbolic volume complementary to Whitehead’s 2-component link, with 5 crossings [@Adams]. Like the majority of knots and links, this link is hyperbolic, which means that the 3-manifold complementary to it admits a metric of constant negative curvature. The volume of this hyperbolic manifold is then an invariant [@AHW] associated with the link. The Borromean rings have a volume twice as large, namely 8 times Catalan’s constant. Moreover, the numerator $N(\pi/4,0,\pi/4)=\frac52{\rm Cl}_2(\pi/3)$ of the box diagram $B_0(6,6)$ in (\[al\]) is a rational multiple of the volume, $2{\rm Cl}_2(\pi/3)=2.02988231$, of the figure-8 knot, which is the unique knot with 4 crossings.
The common analytical feature of such link invariants and the Feynman diagrams of this paper is the volume, (\[ideal\]), of an ideal tetrahedron. It may be regarded as a real-valued function of a single complex variable [@AMS]: $${\cal V}(z):=\Im\left\{{\rm Li}_2(z)
+\log\left|z\right|\log(1-z)\right\}=V(\arg(z),-\arg(1-z))
\label{vz}$$ with dihedral angles that are the arguments of $\{z,1/(1-z),1-1/z\}$. The symmetries $${\cal V}(1-z)={\cal V}(1/z)={\cal V}(\overline{z})=-V(z)\,,\label{12}$$ where $\overline{z}$ is the complex conjugate of $z$, imply that $\{z,1/\overline{z},1/(1-z),
1-\overline{z},1-1/z,\overline{z}/(\overline{z}-1)\}$ all give the same value for (\[vz\]), while their conjugates give a result differing only in sign. The hyperbolic volume of a knot or link is expressible as a finite number of ideal terms of the form (\[vz\]), with arguments that result from complex roots of polynomials [@Adams; @AHW]. For example, the volume of the figure-8 knot is $2{\cal V}(z)$, with $z(1-z)=1$, while the volume of the Borromean rings is $8{\cal V}(z)$, with $z(1-z)=\frac12$. The sole hyperbolic 5-crossing knot, $5_2$, has a volume, 2.8281220, given by $3{\cal V}(z)$, with $z^3=z-1$. This cubic gives the relation $3\theta_1+\theta_2=\pi$ between the dihedral angles in (\[vz\]). We shall meet it again, at 12 crossings, in the context of 8-loop quantum field theory.
Positive hyperbolic knots at 7 loops
------------------------------------
In [@BK15], Dirk Kreimer and I considered knots with up to 15 crossings, classifying the numerical content of field-theory counterterms up to 9 loops. An account of the wider issues is provided by [@DK]. The knots in question are all positive, i.e. their minimal braidwords involve only positive powers of the generators, $\sigma_k$, of the braid group [@VJ]. A consequence is that no hyperbolic knot is encountered in the analysis of diagrams with less than 7 loops, where only torus knots are encountered. At the 7-loop level one encounters two 10-crossing knots that are both positive and hyperbolic, with braidwords $10_{139}=
\sigma_1^{}\sigma_2^{3}\sigma_1^{3}\sigma_2^{3}$ and $10_{152}=
\sigma_1^{2}\sigma_2^{2}\sigma_1^{3}\sigma_2^{3}$, offering the first possibility to study the reduction to ideal tetrahedra of knots implicated by counterterms. Numerical triangulations were obtained, at 12-digit accuracy, from Jeff Weeks’ program SnapPea [@snap]. We then used PSLQ to identify the relevant polynomials, whose roots were extracted to 50 digits, giving $$\begin{aligned}
V_{10_{139}}&=&{\tt
4.85117075733273756705832705211531247884528302776999
}\label{139}\\
V_{10_{152}}&=&{\tt
8.53606534720560860314418192054932599496499139691401
}\label{152}\end{aligned}$$ as the volumes of the positive 10-crossing knots. SnapPea identified the manifold complementary to $10_{139}$ as isometric to entry m389 in its census. Its volume coincides with that of m391, for the 8-crossing 2-component link labelled $8_2^2$ in the appendices of [@Adams] and [@Rolfsen]. The manifold complementary to $10_{152}$ has a volume greater than any in SnapPea’s census of 6,075 cusped manifolds triangulated by not more than 7 tetrahedra.
We found that (\[139\]) results from a remarkably simple triangulation, $$V_{10_{139}}=4{\cal V}(z)+{\cal V}(z^2+1)\,;\quad z^2+1=z^2(z-1)^2\,,
\label{139is}$$ with a matching condition that makes the dihedral angles of the second term linear combinations of those of the first. This simplicity is in marked contrast to $$\begin{aligned}
V_{10_{152}}&=&{\cal V}(z)+2{\cal V}(z+1)
+{\cal V}(2z-z^2)+{\cal V}(z^2(z-1)^2)
+4{\cal V}\left(\frac{2z-1}{2z-z^2}\right)
\,;\nonumber\\&&{}\quad
z(2z-z^2)^2=(z+1)^2(z-1)\,,\label{152is}\end{aligned}$$ whose quintic produces 15 distinct Clausen values, with angles reducible to linear combinations of the arguments of $\{z^2,(z+1)^2,(z-1)^2,(2z-1)^2\}$. The simpler form of (\[139is\]), with only 2 distinct tetrahedra, and only 2 linearly independent angles, accords with the experience of [@BKP], where it was found that $10_{139}$ is simpler than $10_{152}$ in the field-theory context, since it is more readily obtained from the skeining of link diagrams that encode the intertwining of momenta in 7-loop diagrams.
Positive hyperbolic knots at 8 loops
------------------------------------
Observing the contrasting reductions (\[139is\],\[152is\]) to ideal volumes, we proceeded to 12 crossings, relevant to 8-loop counterterms [@BK15]. Work with John Gracey and Dirk Kreimer [@BGK] had focussed on a pair of positive hyperbolic knots, $12_A:=\sigma_1^{}\sigma_2^{7}\sigma_1^{}\sigma_2^{3}$ and $12_B:=\sigma_1^{}\sigma_2^{5}\sigma_1^{}\sigma_2^{5}$, one of which is associated with the appearance of the irreducible [@DZ; @BBG; @Eul] double Euler sum $\zeta_{9,3}:=\sum_{j>k>0}j^{-9}k^{-3}$ in counterterms, while the other relates to a quadruple sum that cannot be reduced to simpler non-alternating sums, and was found in [@Eul] to entail the alternating Euler sum $U_{9,3}:=\sum_{j>k>0}(-1)^{j+k}j^{-9}k^{-3}$. In [@BGK] we tentatively identified $12_A$ as the knot associated with $\zeta_{9,3}$, by study of counterterms in the large-$N$ limit, at $O(1/N^3)$, where $\zeta_{9,3}$ occurs, but $U_{9,3}$ is absent. Thus we expect $12_A$ to have a simpler reduction to ideal tetrahedra than that for $12_B$.
This expectation was notably confirmed by computation, which gave the volumes $$\begin{aligned}
V_{12_A}&=&{\tt
2.82812208833078316276389880927663494277098131730065
}\label{12a}\\
V_{12_B}&=&{\tt
5.91674573518278869527226015189683245321707317046868
}\label{12b}\end{aligned}$$ with triangulations that SnapPea identified with the manifolds m016 and v2642. The result for $12_A:=\sigma_1^{}\sigma_2^{7}\sigma_1^{}\sigma_2^{3}$ is indeed rather special: the volume is equal to that of manifold m015, for the hyperbolic knot with 5 crossings[^2]: $$V_{12_A}=V_{5_2}=3{\cal V}(z)\,;\quad z^3=z-1\,.\label{12ais}$$ Equalities between volumes of hyperbolic knots are rare, with none occurring at less than 10 crossings. It is intriguing that the knot $12_A$, identified with $\zeta_{9,3}$ in [@BGK], has a triangulation as simple as that for the knot $5_2$. By contrast the result for $12_B:=\sigma_1^{}\sigma_2^{5}\sigma_1^{}\sigma_2^{5}$, $$\begin{aligned}
V_{12_B}&=&4V(\psi_1,\psi_2)
+V(2\psi_1,2\psi_1+2\psi_2)
+2V(3\psi_1+\psi_2,\psi_1-\psi_2)\,;\nonumber\\&&
\psi_1=\arg(z)\,;\quad\psi_2=-\arg(1-z)\,;\quad4z^4=2z^2-2z+1\,,
\label{12bis}\end{aligned}$$ involves 9 distinct Clausen values, with angles coming from the solution to a quartic. As before, the relative ease with which positive hyperbolic knots are obtained from Feynman diagrams is reflected by the relative simplicity of their triangulations. As further confirmation of this trend, we cite the cases of the remaining 5 positive knots with 12 crossings, which were not obtained from skeining counterterms in [@BGK], nor related to Euler sums in [@BK15]. Their volumes exceed that of (\[12b\]), ranging from 7.40 to 13.64, with commensurately complicated triangulations.
It thus appears that Feynman diagrams entail positive knots that are either not hyperbolic, as in the case of torus knots, which suffice through 6 loops, or ‘marginally’ hyperbolic, with a small volume, related to a relatively simple triangulation.
A simple hyperbolic volume at infinite loops
--------------------------------------------
We now study the volume, $V_{2n}$, of the positive $2n$-crossing knot $K_{2n}:=\sigma_1^{}\sigma_2^{2n-5}\sigma_1^{}\sigma_2^{3}$, related to double Euler sums of weight $2n$ in counterterms at $n+2\ge6$ loops [@BK15; @BGK]. We found that this volume is bounded, as $n\to\infty$.
Since $K_{8}=8_{19}$ and $K_{10}=10_{124}$ are the (4,3) and (5,3) torus knots, $V_{8}=V_{10}=0$. At 12 crossings, $V_{12}:=V_{12_A}$ is given by (\[12a\],\[12ais\]); the appendix of [@Adams] shows that no hyperbolic knot from 6 through 9 crossings has a volume as small as this. We found that $K_{14}$, with manifold m223, has the same volume, 4.12490325, as $8_{20}=\sigma_1^{}\sigma_2^{-3}\sigma_1^{}\sigma_2^3$, with manifold m222. In general, the volume of the $2n$-crossing positive knot $K_{2n}:=\sigma_1^{}\sigma_2^{2n-5}\sigma_1^{}\sigma_2^{3}$, with $2n\ge12$, coincides with that of the non-positive knot $\sigma_1^{}\sigma_2^{11-2n}\sigma_1^{}\sigma_2^{3}$, formally obtained by $n\to8-n$, and hence having a crossing number that cannot exceed $2n-6$.
The manifolds of $K_{16}$ and $K_{18}$ were identified as s384 and v0959, triangulated by 6 and 7 tetrahedra, respectively; their volumes are not much larger than that of $K_{14}$. Moreover, the trend of $$\begin{array}{ll}
V_{14}={\tt4.124903252}\qquad&V_{30}={\tt5.227842810}\\
V_{16}={\tt4.611961374}\qquad&V_{32}={\tt5.244429225}\\
V_{18}={\tt4.854663387}\qquad&V_{34}={\tt5.257409836}\\
V_{20}={\tt4.993271973}\qquad&V_{36}={\tt5.267755714}\\
V_{22}={\tt5.079718733}\qquad&V_{38}={\tt5.276132543}\\
V_{24}={\tt5.137154054}\qquad&V_{40}={\tt5.283008797}\\
V_{26}={\tt5.177195133}\qquad&V_{42}={\tt5.288721773}\\
V_{28}={\tt5.206190226}\qquad&V_{44}={\tt5.293519248}
\end{array}\label{Kvol}$$ suggests an asymptotic value $$\begin{aligned}
V_\infty&=&3{\rm Cl}_2(2\omega)-3{\rm Cl}_2(4\omega)+{\rm Cl}_2(6\omega)
\label{infty}\\&=&{\tt
5.33348956689811958159342492522130008819676777710528
}\nonumber\end{aligned}$$ with $\omega:=\arctan\sqrt7$, which is equal to the volume $$V_{(\sigma_1^{2}\sigma_2^{-1})^2}=4{\cal V}(z)+2{\cal V}(2z)\,;
\quad2z^2=3z-2\label{s776}$$ of manifold s776, complementary to the 6-crossing 3-component link $6_1^3:=(\sigma_1^{2}\sigma_2^{-1})^2$. To test (\[infty\]), we used SnapPea to evaluate volumes for a selection of crossing numbers from 50 up to 500, corresponding to counterterms with up to 252 loops. The tight bounds $$({\mbox{$\frac{1}{4}$}}n-1)^2\left\{V_\infty-V_{2n}\right\}\in[0.811,0.816]\,;
\quad2n\in[50,500]\,,\label{limits}$$ make a compelling case for the asymptotic behaviour $$V_{2n}=V_\infty-\frac{C}{(\frac14n-1)^2}+O(n^{-4}),
\label{asy}$$ with an invariance under $n\to8-n$, noted above, and a constant $C=0.8160\pm0.0001$.
Thus we come full circle, from an infinite number of loops back to a 1-loop result, since (\[infty\]) relates directly to a 1-loop box, with $$V_\infty=V_{(\sigma_1^{2}\sigma_2^{-1})^2}
=3N({\mbox{$\frac{1}{3}$}}\pi,0,{\mbox{$\frac{1}{3}$}}\pi)={\mbox{$\frac{3}{4}$}}\sqrt7\,B_0(7,7)\label{b7}$$ being 3 times the volume of the light-like equal-mass box diagram of Section 3.1, at $s=t=7$. This complements the link invariants obtained at $s=t=4$ and $s=t=6$ in (\[al\]). Moreover, the 12-crossing 3-component link $(\sigma_1^2\sigma_2^{-2})^3$ has a volume $$\begin{aligned}
V_{(\sigma_1^2\sigma_2^{-2})^3}
&=&6N({\mbox{$\frac{1}{6}$}}\pi,0,{\mbox{$\frac{1}{6}$}}\pi)={\mbox{$\frac{3}{2}$}}\sqrt{15}\,B_0(5,5)
\label{b5}\\&=&{\tt
18.83168336678760750554026296116895115755581340126291
}\nonumber\end{aligned}$$ which is 6 times the volume of the box diagram at $s=t=5$. Thus we now have 4 relations between Feynman diagrams and link invariants.
1. The volume of the figure-8 knot, $4_1$, is $2{\rm Cl}_2(\pi/3)=\frac45N(\pi/4,0,\pi/4)$. This Clausen value occurs in the 2-loop equal-mass vacuum diagram [@BV], the 1-loop massless triangle diagram at its symmetric point [@magic], and the equal-mass light-like box diagram of [@DD] at $s=t=6$.
2. The volumes of the Whitehead link, $5_1^2$, and the Borromean rings, $6_2^3:=(\sigma_1^{}\sigma_2^{-1})^3$, are multiples of Catalan’s constant, ${\rm Cl}_2(\pi/2)=\frac14N(0,0,0)$. This Clausen value results at the simultaneous threshold values $s=t=4$ of the box.
3. At $s=t=5$ we obtain the volume of the link $(\sigma_1^2\sigma_2^{-2})^3$ in (\[b5\]).
4. At $s=t=7$ we obtain the volume of the link $6_1^3:=(\sigma_1^{2}\sigma_2^{-1})^2$ in (\[b7\]). This is also the infinite-loop limit of the hyperbolic volumes of the knots $\sigma_1^{}\sigma_2^{2n-5}\sigma_1^{}\sigma_2^{3}$, associated in [@BK15; @BGK] with the appearance in counterterms [@DK; @BKP], at $n+2\ge6$ loops, of irreducible double Euler sums [@Eul; @BBBL] of weight $2n$.
There are further cases of knots and links whose volumes entail a single Schläfli function, and hence a single box diagram. Harnessing PSLQ to SnapPea, we obtained $$\begin{aligned}
V_{9_{41}}&=&10N({\mbox{$\frac{2}{5}$}}\pi,{\mbox{$\frac{1}{10}$}}\pi,{\mbox{$\frac{1}{5}$}}\pi)=
10{\rm Cl}_2({\mbox{$\frac{2}{5}$}}\pi)+5{\rm Cl}_2({\mbox{$\frac{4}{5}$}}\pi)\label{9_41}\\&=&{\tt
12.09893602599078738356455696387624160295557377848341
}\nonumber\\
V_{10_{123}}&=&10N({\mbox{$\frac{3}{10}$}}\pi,{\mbox{$\frac{1}{5}$}}\pi,{\mbox{$\frac{1}{10}$}}\pi)=
15{\rm Cl}_2({\mbox{$\frac{2}{5}$}}\pi)+5{\rm Cl}_2({\mbox{$\frac{4}{5}$}}\pi)\label{10_123}\\&=&{\tt
17.08570948298286127690097484048365482503835960943063
}\nonumber\end{aligned}$$ for the volumes of the knots $9_{41}$ and $10_{123}$, and $$\begin{aligned}
V_{(\sigma_1^2\sigma_2^{-1})^3}&=&
6N({\mbox{$\frac{1}{4}$}}\pi,{\mbox{$\frac{1}{6}$}}\pi,{\mbox{$\frac{1}{4}$}}\pi)\label{9_40^2}\\&=&{\tt
12.04609204009437764726837862923359423099605804944500
}\nonumber\\
V_{(\sigma_1^{}\sigma_2^{-2}\sigma_3^{}\sigma_2^{-2})^2}&=&
6N({\mbox{$\frac{1}{6}$}}\pi,{\mbox{$\frac{1}{6}$}}\pi,{\mbox{$\frac{1}{6}$}}\pi)\label{4link}\\&=&{\tt
16.59129969483175048405984013396780188163367504042159
}\nonumber\end{aligned}$$ for the 9-crossing 2-component link $9_{40}^2:=(\sigma_1^2\sigma_2^{-1})^3$ and the 12-crossing 4-component link $(\sigma_1^{}\sigma_2^{-2}\sigma_3^{}\sigma_2^{-2})^2$. At 8 crossings, we found that $$\begin{aligned}
V_{8_{18}}&=&3{\rm Cl}_2(2\beta)+12{\rm Cl}_2({\mbox{$\frac{1}{2}$}}\pi+\beta)
\label{8_18}\\&=&{\tt
12.35090620915820017473630443842615201419925670412000
}\nonumber\\
V_{8_{21}}&=&{\mbox{$\frac{1}{2}$}}V_\infty+{\mbox{$\frac{1}{3}$}}V_{8_{18}}
\label{8_21}\\&=&{\tt
6.783713519835126515708813942086034048831469456592638
}\nonumber\end{aligned}$$ with $\beta:=\arcsin\frac{\sqrt2-1}{2}$. Integer relations between volumes, as in (\[8\_21\]), appear to be fairly common; we cite $V_{7^2_6}=V_\infty+V_{5^2_1}$ as another example, with the infinite-loop limit of the knots of [@BK15] here appearing as the difference in volume of a pair of 2-component links.
Conclusions
===========
The volumes of ideal hyperbolic tetrahedra play (at least) 6 roles in field theory.
1. They result from the evaluation of 3-loop 3-dimensional vacuum diagrams, where their volumes tend to cancel, making the maximum [@CTet] value (\[sym\]) remarkably small.
2. They also result from 1-loop 4-dimensional box diagrams [@DD], where their volumes tend to add, giving $O(10^2)$ times the volume of 3-loop vacuum diagrams.
3. Each ideal volume corresponds to a massless 1-loop triangle diagram [@DD].
4. Each ideal volume also corresponds to a massive 2-loop vacuum diagram [@magic].
5. The ease with which the volume of a positive hyperbolic knot is reduced to ideal volumes is indicative of the ease with which the knot results from skeining momentum flow in counterterms [@DK; @BKP].
6. The family of knots $\sigma_1^{}\sigma_2^{2n-5}\sigma_1^{}\sigma_2^{3}$, associated with multiple zeta values [@DZ; @Eul] in counterterms [@BK15; @BGK] at $n+2\ge6$ loops, yields a hyperbolic volume, at infinite loops, which is 3 times that for a simple 1-loop box.
Conclusion 1 was obtained via (\[dans\]), for a 3-loop vacuum diagram, with 3 distinct masses, in 3 dimensions. Its analytic continuation to the hyperbolic regime, $a^2+b^2<4$, is given by (\[cla\]), which may expressed, as in (\[neat\]), in terms of 4 volumes of ideal tetrahedra, 2 of which enter with minus signs. Conclusions 2–4 result from the work in [@DD; @magic], which we here extended by exposing the duality relation (\[dual\]) and showing how the additions in (\[Nis\]), for box diagrams, tend to produce results two orders of magnitude greater than those from the cancellations in (\[neat\]), for 3-loop vacuum diagrams. Conclusion 5 is based on contrasting (\[139is\]) with (\[152is\]), at 7 loops, and (\[12ais\]) with (\[12bis\]), at 8 loops. Conclusion 6 is based on the strong numerical evidence (\[limits\]) for the asymptote (\[b7\]), corresponding to the volume of the link $6_1^3:=(\sigma_1^{2}\sigma_2^{-1})^2$, which is 3 times that of the light-like equal-mass box diagram at $s=t=7$.
The discovery (\[dans\]), which sparked these hyperbolic connections, is now proven, though it was not derived, in the traditional sense; instead it was inferred by numerical investigation and then verified by routine differentiation w.r.t. masses. Similarly empirical methods led to (\[dual\],\[b7\]). Such procedures prompt a question: what is served by mathematical proof? The result (\[ans\]) was discovered in [@CTet] at modest numerical precision, and then checked to 1,000 digits. There was no shadow of doubt that it was correct, though unproven. Now it is proven, yet by a method as thoroughly empirical as that which enabled its discovery. More important than the proof itself is the route to it, since discovery of (\[dans\]), with 3 distinct masses, provides fertile ground for conjectures on behaviour with more mass scales, or in 4 dimensions. A comparable situation was apparent in [@sixth; @poly], where the results themselves, again from PSLQ, were more illuminating than the [*post hoc*]{} proofs found for some of them. As Michael Atiyah has remarked [@MA]: if possession is nine tenths of the law, discovery is nine tenths of the proof.
[**Acknowledgments:**]{} I thank David Bailey, for implementing PSLQ, Andrei Davydychev, for suggesting a relation of (\[dans\]) to geometry, Dirk Kreimer, for tuition in knot theory, Al Manoharan, for converting SnapPea to Windows95, Arttu Rajantie, for the stimulus to solve PDEs for vacuum diagrams, and Don Zagier for stressing the importance of (\[vz\]).
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[^1]: We discount the possibility that this dissection might entail subtraction of bi-rectangular volumes in the totally symmetric case (\[sym\]).
[^2]: Section 5.3 of [@Adams] gives an excellent introduction to hyperbolic knots. Unfortunately, Fig 5.29 is misdrawn, depicting $\sigma_1^{}\sigma_2^{-7}\sigma_1^{}\sigma_2^{3}$, with manifold v0960, instead of the positive knot $12_A:=\sigma_1^{}\sigma_2^{7}\sigma_1^{}\sigma_2^{3}$.
|
---
abstract: '= 11pt = 0.65in = 0.65in'
author:
- '**Carsten Dominik**'
- '**Jürgen Blum**'
- '**Jeffrey N. Cuzzi**'
- '**Gerhard Wurm**'
title: '**Growth of Dust as the Initial Step Toward Planet Formation**'
---
**INTRODUCTION**
================
The growth of dust particles by aggregation stands at the beginning of planet formation. Whether planetesimals form by incremental aggregation, or through gravitational instabilities in a dusty sublayer, particles have to grow and settle to the midplane regardless. On the most basic level, the physics of such growth is simple: Particles collide because relative velocities are induced by random and (size-dependent) systematic motions of grains and aggregates in the gaseous nebula surrounding a forming star. The details are, however, highly complex. The physical state of the disk, in particular the presence or absence of turbulent motions, set the boundary conditions. When particles collide with low velocities, they stick by mutual attractive forces, be it simple van der Waals attraction or stronger forces (molecular dipole interaction in polar ices, or grain-scale long-range forces due to charges or magnetic fields). While the lowest velocities create particle shapes governed by the motions alone, larger velocities contribute to shaping the aggregates by restructuring and destruction. The ability to internally dissipate energy is critical in the growth through intermediate pebble and boulder sizes. In this review we will concentrate on the physical properties and growth characteristics of these small and intermediate sizes, but also make some comments on the formation of planetesimals.
Relative velocities between grains in a protoplanetary disk can be caused by a variety of processes. For the smallest grains, these are dominated by Brownian motions, that provide relative velocities in the mm/s to cm/s range for (sub)micron sized grains. Larger grains show systematic velocities in the nebula because they decouple from the gas, settle vertically, and drift radially. At 1AU in a solar nebula, these settling velocities reach m/s for cm-sized grains. Radial drift becomes important for even larger particles and reaches 10’s of m/s for m-sized bodies. Finally, turbulent gas motions can induce relative motions between particles. For details see for example *Weidenschilling* (1977; 1984), *Weidenschilling and Cuzzi* (1993), *Cuzzi and Hogan* (2003).
The timescales of growth processes and the density and strength of aggregates formed by them, will depend on the structure of the aggregates. A factor of overriding importance for dust–gas interactions (and therefore for the timescales and physics of aggregation), for the stability of aggregates, and for optical properties alike is the structure of aggregates as they form through the different processes.
The interaction of particles with the nebula gas is determined primarily by their gas drag stopping time $\tstop$ which is given by $$\label{eq:3}
\tstop = \frac{mv}{F_\mathrm{fric}}
=\frac{3}{4\cs\rhog} \frac{m}{\sigma}$$ where $m$ is the mass of a particle, $v$ its velocity relative to the gas, $\sigma$ the average projected surface area, $\rhog$ is the gas density, $\cs$ is the sound speed, and $F_\mathrm{fric}$ is the drag force. The second equal sign in eq.(\[eq:3\]) holds under the assumption that particles move at sub-sonic velocities and that the mean free path of a gas molecule is large compared to the size of the particle (Epstein regime). In this case, the stopping time is proportional to the ratio of mass and cross section of the particle. For spherical non-fractal (i.e. compact or porous) particles of radius $\agr$ and mass density $\rhos$, this can be written as $\tstop =
\agr \rhos / c \rhog$. Fractal particles are characterized by the fact that the average density of a particle depends on size in a powerlaw fashion, with a power (the fractal dimension $\Df$) smaller than 3. $$\label{mass-size}
m(\agr) \propto \agr^{\Df} ~.$$ For large aggregates, this value can in principle be measured for individual particles. For small particles, it is often more convenient to measure it using sizes and masses of a distribution of particles.
Fractal particles generally have large surface-to-mass ratios; in the limiting case of long linear chains ($\Df=1$) of grains with radii $\agr_0$, $\sigma/m$ approaches the constant value $3\pi/(16\agr_0\rhos)$. This value differs from the value for a single grain $3/(4\agr_0\rhos)$ by just a factor $\pi/4$. Fig. \[fig:sigma\] shows how the cross section of particles varies with their mass for different fractal dimensions. It shows that for aggregates made of 10000 monomers, the surface-to-mass ratio can easily differ by a factor of 10. An aggregate made from 0.1 particles with a mass equivalent to a 10 particle consists of 10$^6$ monomers and the stopping time could vary by a factor of order 100. Just how far the fractal growth of aggregates proceeds is really not yet known.
This review is organized as follows: In section \[sec:dust-aggr-exper\] we cover the experiments and theory describing the basic growth processes of dust aggregates. In section \[sec:part-gas-inter\] we discuss particle-gas interactions and the implications for inter-particle collision velocities as well as planetesimal formation. In section \[sec:global-disk-models\] we describe recent advances in the modeling of dust aggregation in protoplanetary disks and observable consequences.
**DUST AGGREGATION EXPERIMENTS AND THEORY** {#sec:dust-aggr-exper}
===========================================
Interactions between individual dust grains
-------------------------------------------
\[adhesion force\] Let us assume that the dust grains are spherical in shape and that they are electrically neutral and non-magnetic. In that case, two grains with radii $\agr_1$ and $\agr_2$ will always experience a short-range attraction due to induced dielectric forces, e.g. van der Waals interaction. This attractive force results in an elastic deformation leading to a flattening of the grains in the contact region. An equilibrium is reached when the attractive force equals the elastic repulsion force. For small, hard grains with low surface forces, the equilibrium contact force is given by (*Derjaguin et al.*, 1975) $$\label{dmt}
\fcontact = 4 \pi \gamma_\mathrm{s} R ~,$$ where $\gamma_\mathrm{s}$ and $R$ denote the specific surface energy of the grain material and the local radius of surface curvature, given by $R = \agr_1
\agr_2 / (\agr_1 + \agr_2)$, respectively. Measurements of the separation force between pairs of $\rm SiO_2$ spheres with radii $\agr$ between 0.5 and 2.5 (corresponding to reduced radii $R = 0.35
\ldots 1.3$) confirm the validity of Eq. \[dmt\] (*Heim et al.*, 1999).
Possibly the most important parameter influencing the structure of aggregates resulting from low velocity collisions is the resistance to rolling motion. If this resistance is very strong, both aggregate compaction and internal energy dissipation in aggregates would be very difficult. Resistance to rolling first of all depends strongly on the geometry of the grains. If grains contain extended flat surfaces, contact made on such locations could not be moved by rolling - any attempt to roll them would inevitably lead to breaking the contact. In the contact between round surfaces, resistance to rolling must come from an asymmetric distribution of the stresses in the contact area. Without external forces, the net torque exerted on the grains should be zero. *Dominik and Tielens* (1995) showed that the pressure distribution becomes asymmetric, when the contact area is slightly shifted with respect to the axis connecting the curvature centers of the surfaces in contact. The resulting torque is $$\label{eq:1}
M = 4\fcontact \left(\frac{a_\mathrm{contact}}{a_{\mathrm{contact,0}}}\right)^{3/2} \xi$$ where $a_\mathrm{contact,0}$ is the equilibrium contact radius, $a_\mathrm{contact}$ the actual contact radius due to pressure in the vertical direction, and $\xi$ is the displacement of the contact area due to the torque. In this picture, energy dissipation, and therefore friction, occurs when the contact area suddenly readjusts after it has been displaced because of external forces acting on the grains. The friction force is proportional to the pull-off force $\fcontact$.
*Heim et al.* (1999) observed the reaction of a chain of dust grains using a long-distance microscope and measured the applied force with an Atomic Force Microscope (AFM). The derived rolling-friction forces between two $\rm SiO_2$ spheres with radii of $\agr= 0.95~\rm \mu m$ are $F_\mathrm{roll} = (8.5 \pm 1.6) \cdot
10^{-10}$ N. If we recall that there are two grains involved in rolling, we get for the rolling-friction energy, defined through a displacement of an angle $\pi/2$ $$\label{eroll}
E_\mathrm{roll} = \pi \agr F_\mathrm{roll} = {\rm O}(10^{-15} {\rm J}) ~.$$ Recently, the rolling of particle chains has been observed under the scanning electron microscope while the contact forces were measured simultaneously (*Heim et al.*, 2005).
The dynamical interaction between small dust grains was derived by *Poppe et al.* (2000a) in an experiment in which single, micrometer-sized dust grains impacted smooth targets at various velocities ($0 \ldots 100$m/s) under vacuum conditions. For spherical grains, a sharp transition from sticking with an efficiency of $\beta
\approx 1$ to bouncing (i.e. a sticking efficiency of $\beta =
0$) was observed. This threshold velocity is $v_\mathrm{s} \approx 1.2~ \rm
m/s$ for $\agr = 0.6$ and $v_\mathrm{s} \approx 1.9~ \rm
m/s$ for $\agr = 0.25$. It decreases with increasing grain size. The target materials were either polished quartz or atomically-smooth (surface-oxidized) silicon. Currently, no theoretical explanation is available for the threshold velocity for sticking. Earlier attempts to model the low-velocity impact behavior of spherical grains predicted much lower sticking velocities (*Chokshi et al.*, 1993). These models are based upon impact experiments with “softer” polystyrene grains (*Dahneke*, 1975). The main difference becomes visible when studying the behavior of the rebound grains in non-sticking collisions. In the experiments by *Dahneke* (1975) and also in those by *Bridges et al.* (1996) using macroscopic ice grains, the behavior of grains after a bouncing collision was a unique function of the impact velocity, with a coefficient of restitution (rebound velocity divided by impact velocity) always close to unity and increasing monotonically above the threshold velocity for sticking. For harder, still spherical, $\rm SiO_2$ grains (*Poppe et al.*, 2000a), the *average* coefficient of restitution decreases considerably with increasing impact velocity. In addition to that, *individual* grain impacts show considerable scatter in the coefficient of restitution.
The impact behavior of irregular dust grains is more complex. Irregular grains of various sizes and compositions show an overall decrease in the sticking probability with increasing impact velocity. The transition from $\beta = 1$ to $\beta = 0$, however, is very broad so that even impacts as fast as $v \approx 100~ \rm m/s$ can lead to sticking with a moderate probability.
Dust aggregation and restructuring
----------------------------------
In recent years, a number of laboratory and microgravity experiments have been carried out to derive the aggregation behavior of dust under conditions of young planetary systems. To be able to compare the experimental results to theoretical predictions and to allow numerical modelling of growth phases that are not accessible to experimental investigation, “ideal” systems were studied, in which the dust grains were monodisperse (i.e. all of the same size) and initially non-aggregated. Whenever the mean collision velocity between the dust grains or aggregates is much smaller than the sticking threshold (see section \[sticking efficiency\]), the aggregates formed in the experiments are “fractal”, i.e. $\Df<3$ (*Wurm and Blum*, 1998; *Blum et al.*, 1999; *Blum et al.*, 2000; *Krause and Blum*, 2004). The precise value of the fractal dimension depends on the specific aggregation process and can reach values as low as $\Df = 1.4$ for Brownian-motion driven aggregation (*Blum et al.*, 2000; *Krause and Blum*, 2004; *Paszun and Dominik*, 2006), $\Df = 1.9$ for aggregation in a turbulent gas (*Wurm and Blum*, 1998), or $\Df = 1.8$ for aggregation by gravitationally driven sedimentation in gas (*Blum et al.*, 1999). It is inherent to a dust aggregation process in which aggregates with low fractal dimensions are formed that the mass distribution function is rather narrow (quasi-monodisperse) at any given time. In all realistic cases, the mean aggregate mass $\bar{m}$ follows either a power law with time $t$, i.e. $\bar{m} \propto t^\gamma$ with $\gamma > 0$ (*Krause and Blum*, 2004) or grows exponentially fast, $\bar{m} \propto
\exp(\delta t)$ with $\delta >0$ (*Wurm and Blum*, 1998) which can be verified in dust-aggregation models (see Section \[modelling\]).
As predicted by *Dominik and Tielens* (1995, 1996, 1997), experiments have shown that at collision velocities near the velocity threshold for sticking (of the individual dust grains), a new phenomenon occurs (*Blum and Wurm*, 2000). Whereas at low impact speeds, the aggregates’ structures are preserved in collisions (the so-called “hit-and-stick” behavior), the forming aggregates are compacted at higher velocities. In even more energetic collisions, the aggregates fragment so that no net growth is observable. The different stages of compaction and fragmentation are depicted in Fig. \[fig:restruct\].
\[modelling\] The evolution of grain morphologies and masses for a system of initially monodisperse spherical grains that are subjected to Brownian motion has been studied numerically by *Kempf et al.* (1999). The mean aggregate mass increases with time following a power law (see Section \[laboratory aggregation\]). The aggregates have fractal structures with a mean fractal dimension of $\Df = 1.8$. Analogous experiments by *Blum et al.* (2000) and *Krause and Blum* (2004), however, found that the mean fractal dimension was $\Df = 1.4$. Recent numerical work by *Paszun and Dominik* (2006) showed that this lower value is caused by Brownian rotation (neglected by *Kempf et al.* (1999)). More chain-like dust aggregates can form if the mean free path of the colliding aggregates becomes smaller than their size, i.e. if the assumption of ballistic collisions breaks down and a random-walk must be considered for the approach of the particles. The fractal dimension of thermally aggregating dust grains is therefore dependent on gas pressure and reaches an asymptotic value of $\Df =
1.5$ for the low density conditions prevailing through most of the presolar nebula. Only in the innermost regions the densities are high enough to cause deviations.
The experimental work reviewed in Section \[laboratory aggregation\] can be used to test the applicability of theoretical dust aggregation models. Most commonly, the mean-field approach by *Smoluchowski* (1916) is used for the description of the number density $n(m,t)$ of dust aggregates with mass $m$ as a function of time $t$. Smoluchowski’s rate equation reads in the integral form $$\begin{aligned}
\label{smoluchowski1}
\frac{\partial n(m,t)}{\partial t} & = & \frac{1}{2} \int_0^m
K(m',m-m')\\
&& \cdot n(m',t)~n(m-m',t)~{\rm d}m' \nonumber\\
&&- n(m,t)\int_0^\infty
K(m',m)~n(m',t)~{\rm d}m'~. \nonumber\end{aligned}$$ Here, $K(m_1,m_2)$ is the reaction kernel for aggregation of the coagulation equation \[smoluchowski1\]. The first term on the rhs. of Eq. \[smoluchowski1\] describes the rate of sticking collisions between dust particles of masses $m'$ and $m-m'$ whose combined masses are $m$ (gain in number density for the mass $m$). The second term denotes a loss in the number density for the mass $m$ due to sticking collisions between particles of mass $m$ and mass $m'$. The factor 1/2 in the first term accounts for the fact that each pair collision is counted twice in the integral. In most astrophysical applications the gas densities are so low that dust aggregates collide ballistically. In that case, the kernel in Eq. \[smoluchowski1\] is given by $$K(m_1,m_2) = \beta(m_1,m_2;v) ~ v(m_1,m_2) ~ \sigma(m_1,m_2) ~,
\label{kernel1}$$ where $\beta(m_1,m_2;v)$, $v(m_1,m_2)$, and $\sigma(m_1,m_2)$ are the sticking probability, the collision velocity, and the cross section for collisions between aggregates of masses $m_1$ and $m_2$, respectively.
A comparison between numerical predictions from Eq. \[smoluchowski1\] and experimental results on dust aggregation was given by *Wurm and Blum* (1998) who investigated dust aggregation in rarefied, turbulent gas. Good agreement for both the mass distribution functions and the temporal behavior of the mean mass was found when using a sticking probability of $\beta(m_1,m_2;v)=1$, a mass-independent relative velocity between the dust aggregates and the expression by *Ossenkopf* (1993) for the collision cross section of fractal dust aggregates. *Blum* (2006) showed that the mass distribution of the fractal aggregates observed by *Krause and Blum* (2004) for Brownian-motion driven aggregation can also be modelled in the transition regime between free-molecular and hydrodynamic gas flow.
Analogous to the experimental findings for the collisional behavior of fractal dust aggregates with increasing impact energy (*Blum and Wurm* (2000), see Section \[laboratory aggregation\]), *Dominik and Tielens* (1997) showed in numerical experiments on aggregate collisions that with increasing collision velocity the following phases can be distinguished: hit-and-stick behavior, compaction, loss of monomer grains, and complete fragmentation (see Fig. \[fig:restruct\]). They also showed that the outcome of a collision depends on the impact energy, the rolling-friction energy (see Eq. \[eroll\] in Section \[rolling force\]) and the energy for the breakup of single interparticle contacts (see Section \[adhesion force\]). The model by *Dominik and Tielens* (1997) was quantitatively confirmed by the experiments of *Blum and Wurm* (2000) (see Fig. \[fig:restruct\]).
To analyze observations of protoplanetary disks and model the radiative transfer therein, the optical properties of particles are important (*McCabe et al.*, 2003; *Ueta and Meixner*, 2003; *Wolf*, 2003). Especially for particle sizes comparable to the wavelength of the radiation, the shape and morphology of a particle are of major influence for the way the particle interacts with the radiation. With respect to this, it is important to know how dust evolution changes the morphology of a particle. As seen above, in most cases dust particles are not individual monolithic solids but rather aggregates of primary dust grains. Numerous measurements and calculations have been carried out on aggregates (e.g. *Kozasa et al.*, 1992; *Henning and Stognienko*, 1996; *Wurm and Schnaiter*, 2002; *Gustafson and Kolokolova*, 1999; *Wurm et al.*, 2004a; *Min et al.*, 2005). No simple view can be given within the frame of this paper. However, it is clear that the morphology and size of the aggregates will strongly influence the optical properties.
Long range forces may play a role in the aggregation process, if grains are either electrically charged or magnetic. Small iron grains may become spontaneously magnetic if they are single domain (*Nuth et al.*, 1994; *Nuth and Wilkinson*, 1995), typically at sizes of a few tens of nanometers. Larger grains containing ferromagnetic components can be magnetized by an impulse magnetic field generated during a lightning discharge (*Túnyi et al.*, 2003). For such magnetized grains, the collisional cross section is strongly enhanced compared to the geometrical cross section (*Dominik and Nübold*, 2002). Aggregates formed from magnetic grains remain strong magnetic dipoles, if the growth process keeps the grain dipoles aligned in the aggregate (*Nübold and Glassmeier*, 2000). Laboratory experiments show the spontaneous formation of elongated, almost linear aggregates, in particular in the presence of an external magnetic field (*Nübold et al.*, 2003). The relevance of magnetic grains to the formation of macroscopic dust aggregates is, however, unclear.
Electric charges can be introduced through tribo-electric effects in collisions, through which electrons and/or ions are exchanged between the particles (*Poppe et al.*, 2000b; *Poppe and Schräpler*, 2005; *Desch and Cuzzi*, 2000). The number of separated elementary charges in a collision between a dust particle and a solid target with impact energy $E_\mathrm{c}$ can be expressed by (*Poppe et al.*, 2000b; *Poppe and Schräpler*, 2005) $$\label{charging}
N_e \approx \left(\frac{E_\mathrm{c}}{10^{-15} {\rm J}}\right)^{0.8} ~.$$ The cumulative effect of many non-sticking collisions can lead to an accumulation of charges and to the build-up of strong electrical fields at the surface of a larger aggregate. In this way, impact charging could lead to electrostatic trapping of the impinging dust grains or aggregates (*Blum*, 2004). Moreover, impact charging and successive charge separation can cause an electric discharge in the nebula gas. For nebula lighting (*Desch and Cuzzi*, 2000) a few hundred to thousand elementary charges per dust grain are required. This corresponds to impact velocities in the range $20
\ldots 100 ~\rm m/s$ (*Poppe and Schräpler*, 2005) which seems rather high for mm particles.
Electrostatic attraction by dipole-dipole forces has been seen to be important for grains of several hundred micron radius (chondrule size) forming clumps that are centimeters to tens of centimeters across (*Marshall et al.*, 2005; *Ivlev et al.*, 2002). Spot charges distributed over the grain surfaces lead to a net dipole of the grains, with growth dynamics very similar to that of magnetic grains. Experiments in microgravity have shown spontaneous aggregation of particles in the several hundred micron size regime (*Marshall and Cuzzi*, 2001; *Marshall et al.*, 2005; *Love and Pettit*, 2004). The aggregates show greatly enhanced stability, consistent with cohesive forces increased by factors of 10$^3$ compared to the normal van der Waals interaction. Based on the experiments, for weakly charged dust grains, the electrostatic interaction energy at contact for the charge-dipole interaction is in most cases larger than that for the charge-charge interaction. For heavily-charged particles, the mean mass of the system does not grow faster than linearly with time, i.e. even slower than in the non-charged case for Brownian motion (*Ivlev et al.*, 2002; *Konopka et al.*, 2005).
Growth and compaction of large dust aggregates
----------------------------------------------
Macroscopic dust aggregates can be created in the laboratory by a process termed random ballistic deposition (RBD, *Blum and Schräpler*, 2004). In its idealized form, RBD uses individual, spherical and monodisperse grains which are deposited randomly but unidirectionally on a semi-infinite target aggregate. The volume filling factor $\phi=0.11$ of these aggregates, defined as the fraction of the volume filled by dust grains, is identical to ballistic particle-cluster aggregation which occurs when a bimodal size distribution of particles (aggregates of one size and individual dust grains) is present and when the aggregation rates between the large aggregates and the small particles exceed those between all other combinations of particle sizes. When using idealized experimental parameters, i.e. monodisperse spherical $\rm SiO_2$ grains with $0.75 ~ \rm \mu m$ radius, *Blum and Schräpler* (2004) measured a mean volume filling factor for their macroscopic (cm-sized) RBD dust aggregates of $\phi = 0.15$, in full agreement with numerical predictions (*Watson et al.* 1997). Relaxing the idealized grain morphology resulted in a decrease of the volume filling factor to values of $\phi = 0.10$ for quasi-monodisperse, irregular diamond grains and $\phi = 0.07$ for polydisperse, irregular $\rm SiO_2$ grains (*Blum*, 2004).
Static uniaxial compression experiments with the macroscopic RBD dust aggregates consisting of monodisperse spherical grains (*Blum and Schräpler*, 2004) showed that the volume filling factor remains constant as long as the stress on the sample is below $\sim 500 ~ \rm
N~m^{-2}$. For higher stresses, the volume filling factor monotonically increases from $\phi = 0.15$ to $\phi = 0.34$. Above $\sim 10^5 ~ \rm N~m^{-2}$, the volume filling factor remains constant at $\phi = 0.33$. Thus, the compressive strength of the uncompressed sample is $\Sigma \approx 500 ~ \rm N~m^{-2}$. These values differ from those derived with the models of *Greenberg et al.* (1995) and *Sirono and Greenberg* (2000) by a factor of a few. The compressive strengths of the macroscopic dust aggregates consisting of irregular and polydisperse grains was slightly lower at $\Sigma \sim 200 ~ \rm
N~m^{-2}$. The maximum compression of these bodies was reached for stresses above $\sim 5 \cdot 10^5 ~ \rm N~m^{-2}$ and resulted in volume filling factors as low as $\phi = 0.20$ (*Blum*, 2004). As a maximum compressive stress of $\sim 10^5 \ldots 10^6~ \rm N~m^{-2}$ corresponds to impact velocities of $\sim 15 \ldots 50 ~\rm m/s$ which are typical for meter-sized protoplanetary dust aggregates, we expect a maximum volume filling factor for these bodies in the solar nebula of $\phi = 0.20 \ldots 0.34$. *Blum and Schräpler* (2004) also measured the tensile strength of their aggregates and found for slightly compressed samples ($\phi = 0.23$) $T = 1,000 ~ \rm
N~m^{-2}$. Depending on the grain shape and the size distribution, the tensile strength decreased to values of $T \sim 200 ~ \rm
N~m^{-2}$ for the uncompressed case (*Blum*, 2004).
*Sirono* (2004) used the above continuum properties of macroscopic dust aggregates, i.e. compressive strength and tensile strength, to model the collisions between protoplanetary dust aggregates. For sticking to occur in an aggregate-aggregate collision, *Sirono* (2004) found that the impact velocity must follow the relation $$\label{sound speed}
v < 0.04 \sqrt{\frac{{\rm d \Sigma(\phi)}}{{\rm d
\rho(\phi)}}}~,$$ where $\rho(\phi) = \rho_0 \cdot \phi$ is the mass density of the aggregate and $\rho_0$ denotes the mass density of the grain material. Moreover, the conditions $\Sigma(\phi) < Y(\phi)$ and $\Sigma(\phi) < T(\phi)$ must be fulfilled. For the shear strength, *Sirono* (2004) applies $Y(\phi) = \sqrt{2 \Sigma(\phi)
T(\phi) / 3}$. A low compressive strength of the colliding aggregates favors compaction and, thus, damage restoration which can otherwise lead to a break-up of the aggregates. In addition, a large tensile strength also prevents the aggregates from being disrupted in the collision.
*Blum and Schräpler* (2004) found an approximate relation between compressive strength and volume filling factor $$\label{relcompvff}
\Sigma(\phi) = \Sigma_\mathrm{s} \left( \phi - \phi_0 \right)^{0.8} ~,$$ which is valid in the range $\phi_0 = 0.15 \le \phi \le 0.21$. Such a scaling law was also found for other types of macroscopic aggregates, e.g. for jammed toner particles in fluidized bed experiments (*Valverde et al.*, 2004). For the aggregates consisting of monodisperse $\rm SiO_2$ spheres, the scaling factor $\Sigma_\mathrm{s}$ can be determined to be $\Sigma_\mathrm{s} = 2.9 \cdot 10^4 ~ \rm N~m^{-2}$. If we apply Eq. \[relcompvff\] to Eq. \[sound speed\], we get, with $\rho(\phi) = \rho_0 \cdot \phi$ and $\rho_0 = 2 \cdot 10^3 ~ {\rm
kg~m^{-3}}$, for the impact velocity of low-density dust aggregates $$\label{sound speed 2}
v < 0.04 \sqrt{\frac{0.8 \Sigma_\mathrm{s}}{\rho_0 (\phi -
\phi_0)^{0.2}}} \approx 0.14 ~ (\phi -\phi_0)^{-0.1} ~ {\rm m/s}~.$$ Although the function in Eq. \[sound speed 2\] goes to infinity for $\phi \rightarrow \phi_0$, for all practical purposes the characteristic velocity is strongly restricted. For volume filling factors $\phi \ge 0.16$ we get $v < 0.22 ~ {\rm m/s}$. Thus, following the SPH simulations by *Sirono* (2004), we expect aggregate sticking in collisions for impact velocities $v \lesssim
0.2 ~ {\rm m/s}$.
Let us now consider recent results in the field of high-porosity aggregate collisions. *Langkowski and Blum* (unpublished data) performed microgravity collision experiments between 0.1-1 mm-sized (projectile) RBD aggregates and 2.5 cm-sized (target) RBD aggregates. Both aggregates consisted of monodisperse spherical $\rm SiO_2$ grains with radii of $\agr = 0.75 ~ \rm \mu m$. In addition to that, impact experiments with high-porosity aggregates consisting of irregular and/or polydisperse grains were performed. The parameter space of the impact experiments by Langkowski and Blum encompassed collision velocities in the range $0 < v < 3 ~ {\rm m/s}$ and projectile masses of $10^{-9} ~{\rm kg} \le m \le 5 \cdot 10^{-6} ~ {\rm kg}$ for all possible impact parameters (i.e. from normal to tangential impact). Surprisingly, through most of the parameter space, the collisions did lead to sticking. The experiments with aggregates consisting of monodisperse spherical $\rm SiO_2$ grains show, however, a steep decrease in sticking probability from $\beta=1$ to $\beta=0$ if the tangential component of the impact energy exceeds $\sim 10^{-6}$ J (see the example of a non-sticking impact in Fig. \[fig:impact\]a). Other materials also show the tendency towards lower sticking probabilities with increasing tangential impact energies. As these aggregates are “softer”, the decline in sticking probability in the investigated parameter space is not complete. When the projectile aggregates did not stick to the target aggregate, considerable mass transfer from the target to the projectile aggregate takes place during the impact (*Langkowski and Blum*, unpublished data). Typically, the mass of the projectile aggregate was doubled after a non-sticking collision (see Fig. \[fig:impact\]a).
The occurrence of sticking in aggregate-aggregate collisions at velocities $\gtrsim 1~\rm m/s$ is clearly in disagreement with the prediction by *Sirono* (2004) (see Eq. \[sound speed\]). In addition, the evaluation of the experimental data shows that the condition for sticking, $\Sigma(\phi) < Y(\phi)$, seems not to be fulfilled for high-porosity dust aggregates. This means that the continuum aggregate model by *Sirono* (2004) is still not precise enough to fully describe the collision and sticking behavior of macroscopic dust aggregates.
The experiments described above indicate that at velocities above approximately 1m/s, collisions turns from sticking to bouncing, at least for oblique impacts. At higher velocities one would naively expect that bouncing and eventually erosion will continue to dominate, and this is also observed in a number of different experiments (*Colwell*, 2003; *Bridges et al.* 1996, *Kouchi et al.*, 2002; *Blum and Münch*, 1993; *Blum and Wurm*, 2000).
Growth models which assume sticking at velocities $\gg 1 ~\rm
m/s$ are therefore often considered to be impossible (e.g. *Youdin*, 2004). As velocities $\ga 10\rm ~m/s$ clearly occur for particles that have exceeded m-size, this is a fundamental problem for the formation of planetesimals.
However, recent experiments (*Wurm et al.*, 2005) have studied impacts of mm-sized compact dust aggregates onto cm-sized compact aggregate targets at impact velocities between 6 and 25 $\rm m/s$. Compact aggregates can be the result of previous sticking or non-sticking collisions (see Sections \[physical properties of aggregates\] and \[low\_velocity\_collisions\]). Both projectile and target consisted of $\rm \mu m$-sized dust particles. In agreement with the usual findings at lower impact velocities around a few $\rm
m/s$, the projectiles just rebound, slightly fracture or even remove some parts of the target. However, as the velocity increases [*[above]{}*]{} a threshold of 13 $\rm m/s$, about half of the mass of the projectile rigidly sticks to the target after the collision while essentially no mass is removed from the target (see Fig. \[fig:impact\]b). Obviously, higher collision velocities can be favorable for growth, probably by destroying the internal structure of the porous material and dissipating energy in this way.
Only about half of the impactor contributes to the growth of the target in the experiments. The other half is ejected in the form of small fragments, with the important implication that these collisions both lead to net growth of the target and return small particles to the disk. This keeps dust abundant in the disk over a long time. For the specific experiments by *Wurm et al.* (2005), the fragments were evenly distributed in size up to 0.5 mm. In a certain sense, the disk might thus quickly turn into a “debris disk” already at early times. We will get back to this point in section \[sec:glob-settl-aggr\].
PARTICLE-GAS INTERACTION {#sec:part-gas-inter}
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Above we have seen that small solid particles grow rapidly into aggregates of quite substantial sizes, while retaining their fractal nature (in the early growth stage) or a moderate to high porosity (for later growth stages). From the properties of primitive meteorites, we have a somewhat different picture of nebula particulates - most of the solids (chondrules, CAIs, metal grains, etc) were individually compacted as the result of unknown melting processes, and were highly size-sorted. Even the porosity of what seem to be fine-grained accretion rims on chondrules is 25% or less (*Scott et al.*, 1996; *Cuzzi*, 2004; *Wasson et al.*, 2005). Because age-dating of chondrules and chondrites implies a delay of a Myr or more after formation of the first solids, it seems possible that, in the asteroid formation region at least, widespread accretion to parent body sizes did not occur until after the mystery melting events began which formed the chondrules.
It may be that conditions differed between the inner and outer solar system. Chondrule formation might not have occurred at all in the outer solar system where comet nuclei formed, so some evidence of the fractal aggregate growth stage may remain in the granular structure of comet nuclei. New results from Deep Impact imply that comet Tempel 1 has a porosity of 60-80% (*A’Hearn et al.*, 2005)! This value is in agreement with similar porosities found in several other comets (*Davidsson*, 2006). Even in the terrestrial planet/asteroid belt region, there is little reason to doubt that growth of aggregates started well before the chondrule formation era, and continued into and (probably) throughout it. Perhaps, after chondrules formed, previously ineffective growth processes might have dominated (sections 3.2 and 3.3).
Radial and vertical evolution of solids
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\[sec:prior-to-midplane\] The nebula gas (but not the particles) experiences radial pressure gradients because of changing gas density and temperature. These pressure gradients act as small modifications to the central gravity from the star that dominates orbital motion, so that the gas and particles orbit at different speeds and a gas drag force exists between them which constantly changes their orbital energy and angular momentum. Because the overall nebula pressure gradient force is outward, it counteracts a small amount of the inward gravitational force and the gas generally orbits more slowly than the particles, so the particles experience a headwind which saps their orbital energy, and the dominant particle drift is inward. Early work on gas-drag related drift was by *Whipple* (1972), *Adachi et al.* (1976), and *Weidenschilling* (1977). Analytical solutions for how particles interact with a non-turbulent nebula having a typically outward pressure gradient were developed by *Nakagawa et al.* (1986). For instance, the ratio of the pressure gradient force to the dominant central gravity is $\eta \sim 2 \times 10^{-3}$, leading to a net velocity difference between the gas and particles orbiting at Keplerian velocity $\vkep$ of $\eta \vkep$ (see, e.g. *Nakagawa et al.* 1986). However, if local radial maxima in gas pressure exist, particles will drift towards their centers from both sides, possibly leading to radial bands of enhancement of solids (see section 3.4.1).
Small particles generally drift slowly inwards, at perhaps a few cm/s; even this slow inexorable drift has generated some concern over the years, as to how CAIs (early, high-temperature condensates) can survive over the apparent 1-3 Myr period between their creation and the time they were incorporated into chondrite meteorite parent bodies. This concern, however, neglected the role of turbulent diffusion (see Section \[sec:role-of-turb\]). Particles of meter size drift inwards very rapidly - 1 AU/century. It has often been assumed that these particles were “lost into the sun”, but more realistically, their inward drift first brings them into regions warm enough to evaporate their primary constituents, which then become entrained in the more slowly evolving gas and increase in relative abundance as inward migration of solids supplies material faster than it can be removed. Early models describing significant global redistribution of solids relative to the nebula gas by radial drift were presented by *Morfill and V[ö]{}lk* (1984) and *Stepinski and Valageas* (1996, 1997); these models either ignored midplane settling or made simplifying approximations regarding it, and did not emphasize the potential for enhancing material in the vapor phase. Indeed, however, because of the large mass fluxes involved, this “evaporation front” effect can alter the nebula composition and chemistry significantly (*Cuzzi et al.*, 2003; *Cuzzi and Zahnle*, 2004; *Yurimoto and Kuramoto*, 2004; *Krot et al.*, 2005; *Ciesla and Cuzzi*, 2005); see also *Cyr et al.* (1999) for a discussion; however, the results of this paper are inconsistent with similar work by Supulver and *Lin* (2000) and *Ciesla and Cuzzi* (2005). This stage can occur very early in nebula history, long before formation of objects large enough to be meteorite parent bodies.
\[sec:role-of-turb\] The presence or absence of gas turbulence plays a critical role in the evolution of nebula solids. There is currently no widespread agreement on just how the nebula gas may be maintained in a turbulent state across all regions of interest, if indeed it is (*Stone et al.*, 2000, *Cuzzi and Weidenschilling*, 2005). Therefore we discuss both turbulent and non-turbulent situations. For simplicity we will treat turbulent diffusivity $\cal D$ as equal to turbulent viscosity $\nu_\mathrm{T} = \alpha c H$, where $c$ and $H$ are the nebula sound speed and vertical scale height, and $\alpha \ll 1$ is a non-dimensional scaling parameter. Evolutionary timescales of observed protoplanetary nebulae suggest that $10^{-5} < \alpha < 10^{-2}$ in some global sense. The largest eddies in turbulence have scale sizes $H \sqrt{\alpha}$ and velocities $v_\mathrm{turb} = c \sqrt{\alpha}$ (*Shakura et al.*, 1978; *Cuzzi et al.*, 2001).
Particles respond to forcing by eddies of different frequency and velocity as described by *Völk et al.* (1980) and *Markiewicz et al.* (1991), determining their relative velocities with respect to the gas and to each other. The diffusive properties of MRI turbulence, at least, seem not to differ in any significant way from the standard homogeneous, isotropic models in this regard (*Johansen and Klahr*, 2005). Analytical solutions for resulting particle velocities in these regimes were derived by *Cuzzi and Hogan* (2003). These are discussed in more detail below and by *Cuzzi and Weidenschilling* (2005).
Vertical turbulent diffusion at intensity $\alpha$ maintains particles of stopping time $\tstop$ in a layer of thickness $h \sim H \sqrt{\alpha
/\Omega \tstop}$ (*Dubrulle et al.*, 1995; *Cuzzi et al.*, 1996), or a solid density enhancement $H/h = \sqrt{\Omega \tstop
/ \alpha}$ above the average value. For particles of 10 cm size and smaller and $\alpha > \alpha_\mathrm{min}=10^{-6}(\agr/1{\rm cm})$ (*Cuzzi and Weidenschilling*, 2005), the resulting layer is much too large and dilute for collective particle effects to dominate gas motions, so radial drift and diffusion continue unabated. Outward radial diffusion relieves the long-standing worry about “loss into the sun" of small particles, such as CAIs, which are too small to sediment into any sort of midplane layer unless turbulence is vanishingly small ($\alpha \ll \alpha_\mathrm{min}$), and allows some fraction of them to survive over 1 to several Myrs after their formation as indicated by meteoritic observations (*Cuzzi et al.*, 2003). A similar effect might help explain the presence of crystalline silicates in comets (*Bockelée-Morvan*, 2002; *Gail*, 2004).
\[sec:dense-midplane-physics\] When particles [*are*]{} able to settle to the midplane, the particle density gets large enough to dominate the motions of the local gas. This is the regime of [*collective effects*]{}; that is, the behavior of a particle depends indirectly on how all other local particles combined affect the gas in which they move. In regions where collective effects are important, the mass-dominant particles can drive the entrained gas to orbit at nearly Keplerian velocities (if they are sufficiently well coupled to the gas), and thus the headwind the gas can exert upon the particles diminishes from $\eta \vkep$ (section 3.1.1). This causes the headwind-driven radial drift and all other differential particle velocities caused by gas drag to diminish as well.
The particle mass loading $\rhop/\rhog$ cannot increase without limit as particles settle, even if the global turbulence vanishes, and the density of settled particle layers is somewhat self-limiting. The relative velocity solutions of *Nakagawa et al.* (1986) apply in particle-laden regimes once $\rhop/\rhog$ is known, but do not provide for a fully self consistent determination of $\rhop/\rhog$ in the above sense; this was addressed by *Weidenschilling* (1980) and subsequently *Cuzzi et al.* (1993), *Champney et al.* (1995), and *Dobrovolskis et al.* (1999). The latter numerical models are similar in spirit to the simple analytical solutions of *Dubrulle et al.* (1995) mentioned earlier, but treat large particle, high mass loading regimes in globally nonturbulent nebulae which the analytic solutions cannot address. Basically, as the midplane particle density increases, local, entrained gas is accelerated to near-Keplerian velocities by drag forces from the particles. Well above the dense midplane, the gas still orbits at its pressure-supported, sub-Keplerian rate. Thus there is a vertical shear gradient in the orbital velocity of the gas, and the velocity shear creates turbulence which stirs the particles. This is sometimes called “self-generated turbulence". Ultimately a steady-state condition arises where the particle layer thickness reaches an equilibrium between downward settling and upward diffusion. This effect acts to block a number of gravitational instability mechanisms in the midplane (section \[sec:instab\]).
Relative velocities and growth in turbulent and nonturbulent nebulae {#sec:relat-veloc-growth}
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In both turbulent and nonturbulent regimes, particle relative velocities drive growth to larger sizes. Below we show that relative velocities in both turbulent and nonturbulent regimes are probably small enough for accretion and growth to be commonplace and rapid, at least until particles reach meter size or so. We only present results here for particles up to a meter or so in size, because the expression for gas drag takes on a different form at larger sizes. As particles grow, their mass per unit area increases so they are less easily influenced by the gas, and “decouple” from it. Their overall drift velocities and relative velocities all diminish roughly in a linear fashion with particle radius larger than a meter or so (*Cuzzi and Weidenschilling*, 2005).
We use particle velocities relative to the gas as derived by *Nakagawa et al.* (1986) for a range of local particle mass density relative to the gas density (their equations 2.11, 2.12, and 2.21) to derive particle velocities relative to each other in the same environment; all relative velocities scale with $\eta \vkep$.
For simplicity we will assume particles which differ by a factor of three in radius; *Weidenschilling* (1997) finds mass accretion to be dominated by size spreads on this order; the results are insensitive to this factor. Relative velocities for particles of radii $\agr$ and $\agr/3$, in the absence of turbulence and due only to differential, pressure-gradient-driven gas drag, are plotted in the top two panels of Fig. \[fig:vrel\]. In the top panel we show cases where collective effects are negligible (particle density $\rhop$ $\ll$ gas density $\rhog$). Differential vertical settling (shown at different heights $z$ above the midplane, as normalized by the gas vertical scale height $H$) dominates relative velocities and particle growth high above the midplane $(z/H
> 0.1)$, and radial relative velocities dominate at lower elevations. Except for the largest particles, relative velocities for particles with this size difference are much less than $\eta \vkep$; particles closer in mass would have even smaller relative velocities.
![\[fig:vrel\]Relative velocities between particles of radii $\agr$ and $\agr/3$, in nebulae which are non-turbulent (top two panels) or turbulent (bottom two panels), for a minimum mass solar nebula at 2.5AU. In the top panel, the particle density $\rhop$ is assumed to be much smaller than the gas density $\rhog$: $\rhop/\rhog \ll 1$. Shown are the radial (solid line) and azimuthal (dotted line) components of the relative velocities. The dashed curves show the vertical relative velocities, which depend on height above the midplane and are shown for different values of $z/H$. For nonturbulent cases, particles settle into dense midplane layers (section \[sec:dense-midplane-physics\]), so a more realistic situation would be $\rhop/\rhog \ga 1$ or even $\gg 1$ (*Cuzzi et al.*, 1993); thus in the second panel we show relative radial and angular velocities for three different values of $\rhop/\rhog =$ 0, 1, and 10. For these high mass loadings, $z/H$ must be small, so the vertical velocities are smaller than the radial velocities. In the third panel we show relative velocities for the same particle size difference due only to turbulence, for several values of $\alpha$. In the bottom panel, we show the quadrature sum of turbulent and non-turbulent velocities, assuming $z/H=0.01$.](dominiketal_fig4.eps){width="8.2cm"}
Moreover, in a dense midplane layer, when collective effects dominate (section \[sec:dense-midplane-physics\]), all these relative velocities are reduced considerably from the values shown. In the second panel we show radial and azimuthal relative velocities for several values of $\rhop/\rhog$. When the particle density exceeds the gas density, collective effects reduce the headwind, and all relative velocities diminish.
Relative velocities in turbulence of several different intensities, as constrained by the nebula $\alpha$ (again for particles of radii $\agr$ and $\agr/3$), are shown in the two bottom panels. In the second panel from the bottom, relative velocities are calculated as the difference of their velocities relative to the turbulent gas, neglecting systematic drifts and using analytical solutions derived by *Cuzzi and Hogan* (2003; their equation 20) to the formalism of *Völk et al.* (1980). Here, the relative velocities are forced by turbulent eddies with a range of size scales, having eddy turnover times ranging from the orbit period (for the large eddies) to much smaller values (for the smaller eddies), and scale with $v_\mathrm{turb}
= c \alpha^{1/2}$.
In the bottom panel we sum the various relative velocities in quadrature to get an idea of total relative velocities in a turbulent nebula in which particles are also evolving by systematic gas-pressure-gradient driven drift. This primarily increases the relative velocities of the larger particles in the lower $\alpha$ cases.
Overall, keeping in mind the critical velocities for sticking discussed in section 2 ($\sim$ m/s), and that particle surfaces are surely crushy and dissipative, one sees that for particles up to a meter or so, growth by sticking is plausible even in turbulent nebulae for a wide range of $\alpha$. Crushy aggregates will grow by accumulating smaller crushy aggregates as described in earlier sections ([*eg.*]{} *Weidenschilling*, 1997, for the laminar case). After this burst of initial growth to roughly meter size, however, the evolution of solids is very sensitive to the presence or absence of global nebula turbulence, as described in sections 3.3-3.4 below. Meter-size particles inevitably couple to the largest eddies, with $v_\mathrm{turb} \ga$ several meters per second, and would destroy each other if they were to collide. We refer to this as the fragmentation limit. However, if particles can somehow grow their way past 10 meters in size, their survival becomes more assured because all relative velocities, such as shown in Fig. \[fig:vrel\], decrease linearly with $\agr \rhos$ for values larger than shown in the plot due to the linear decrease of the area/mass ratio.
The role of gas in protoplanetary disks is not restricted to generate relative velocities between two bodies which then collide. The gas also plays an important role [*during*]{} individual collisions. A large body which moves through the disk faces a headwind and collisions with smaller aggregates take place at its front (headwind) side. Fragments are thus ejected against the wind and can be driven back to the surface by the gas flow.
For small bodies the gas flow can be regarded as free molecular flow. Thus streamlines end on the target surface and the gas drag is always towards the surface. Whether a fragment returns to the surface depends on its gas-particle coupling time (i.e. size and density) and on the ejection speed and angle. Whether reaccretion of enough fragments for net growth occurs, eventually depends on the distribution of ejecta parameters, gas density, and target size. It was shown by *Wurm et al.* (2001) that growth of a larger body due to impact of dust aggregates entrained in a head wind is possible for collision velocities above 12m/s. At 1 AU a 30-cm body in a disk model according to *Weidenschilling and Cuzzi* (1993) can grow in a collision with small dust aggregates even if the initial collision is rather destructive.
*Sekiya and Takeda* (2003) and *Künzli and Benz* (2003) showed that the mechanism of aerodynamic reaccretion might be restricted to a maximum size due to a change in the flow regime from molecular to hydrodynamic. Fragments are then transported around the target rather then back to it. *Wurm et al.* (2004b) argue that very porous targets would allow some flow going through the body, which would still allow aerodynamic reaccretion, but this strongly depends on the morphology of the body (*Sekiya and Takeda*, 2005). As the gas density decreases outwards in protoplanetary disks, the maximum size for aerodynamic reaccretion increases. However, the minimum size also increases and the mechanism is only important for objects which have already grown beyond the fragmentation limit in some other way - [*e.g.*]{} by immediate sticking of parts of larger particles as discussed above (*Wurm et al.*, 2005).
Planetesimal formation in a midplane layer
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Based on relative velocity arguments such as given above, *Weidenschilling* (1988, 1997) and *Dullemond and Dominik* (2004, 2005) find that growth to meter size is rapid (100-1000 yr at 1 AU; 6-7$\times 10^4$ yrs at 30 AU) whether the nebula is turbulent or not. Such large particles settle towards the midplane within an orbit period or so. However, in turbulence, even meter-sized particles are dispersed sufficiently that the midplane density remains low, and growth remains slow. A combination of rapid radial drift, generally erosive, high-velocity impacts with smaller particles, and occasional destructive collisions with other meter-sized particles frustrates growth beyond meter-size or so under these conditions.
In [*nonturbulent*]{} nebulae, even smaller particles can settle into fairly thin midplane layers and the total particle densities can easily become large enough for collective effects to drive the entrained midplane gas to Keplerian, diminishing both headwind-induced radial drift and relative velocities. In this situation, meter-sized particles quickly grow their way out of their troublesome tendency to drift radially (*Cuzzi et al.*, 1993); planetesimal-sized objects form in only $10^3 - 10^4$ years at 1 AU (*Weidenschilling*, 2000), and a few times $10^5$ years at 30 AU (*Weidenschilling*, 1997). However, such robust growth may, in fact, be too rapid to match observations of several kinds (see section \[sec:glob-settl-aggr\] and chapters by *Dullemond et al.* and *Natta et al.*)
\[sec:instab\] While to some workers the simplicity of “incremental growth” by sticking in the dense midplane layer of a nonturbulent nebula is appealing, past uncertainty in sticking properties has led others to pursue instability mechanisms for particle growth which are insensitive to these uncertainties. Nearly all instability mechanisms discussed to date (*Safronov*, 1969, 1991; *Goldreich and Ward*, 1973; *Ward*, 1976, 2000; *Sekiya*, 1983, 1998; *Goodman and Pindor*, 2000; *Youdin and Shu*, 2002) occur [*only*]{} in nebulae where turbulence is essentially absent, and particle relative velocities are already very low. Just how low the global turbulence must be depends on the particle size involved, and the nebula $\alpha$ (sections \[sec:prior-to-midplane\] and \[sec:dense-midplane-physics\]).
Classical treatments (the best known is *Goldreich and Ward*, 1973) assume that gas pressure plays no role in gravitational instability, being replaced by an effective pressure due to particle random velocities (below we note this is not the case). Particle random velocities act to puff up a layer and reduce its density below the critical value, which is always on the order of the so-called Roche density $\rho^* \sim M_{\odot}/R^3$ where $R$ is the distance to the central star; different workers give constraints which differ by factors of order unity ([*cf.*]{} *Goldreich and Ward*, 1973, *Weidenschilling*, 1980; *Safronov*, 1991; *Cuzzi et al.*, 1993). These criteria can be traced back through Goldreich and *Ward* (1973) to *Goldreich and Lynden-Bell* (1965), *Toomre* (1964), *Chandrasekhar* (1961) and *Jeans* (1928), and in parallel through *Safronov* (1960), *Bel and Schatzman* (1958), and *Gurevitch and Lebedinsky* (1950). Substituting typical values one derives a formal, nominal requirement that the local particle mass density must exceed about $10^{-7}$ g cm$^{-3}$ at 2 AU from a solar mass star even for [*marginal*]{} gravitational instability - temporary gravitational clumping of small amplitude - to occur. This is about $10^3$ times larger than the gas density of typical minimum mass nebulae, requiring enhancement of the solids by a factor of about $10^5$ for a typical average solids-to-gas ratio. From section \[sec:role-of-turb\] we thus require the particle layer to have a thickness $h < 10^{-5} H$, which in turn places constraints on the particle random velocities $h
\Omega$ and on the global value of $\alpha$.
Even assuming global turbulence to vanish, *Weidenschilling* (1980, 1984) noted that turbulence stirred by the very dense particle layer itself will puff it up to thicknesses $h$ that precluded even this marginal gravitational instability. This is because turbulent eddies induced by the vertical velocity profile of the gas (section \[sec:dense-midplane-physics\]) excite random velocities in the particles, diffusing the layer and preventing it from settling into a sufficiently dense state. Detailed two-phase fluid models by *Cuzzi et al.* (1993), *Champney et al.* (1995), and *Dobrovolskis et al.* (1999) confirmed this behavior.
It is sometimes assumed that ongoing, but slow, particle growth to larger particles, with lower relative velocities and thus thinner layers (section 3.2), can lead to $\rhop \sim \rho^*$ and gravitational instability can then occur. However, merely achieving the formal requirement for marginal gravitational instability does not inevitably lead to planetesimals. For particles which are large enough to settle into suitably dense layers for [*marginal*]{} instability under self-generated turbulence (*Weidenschilling*, 1980; *Cuzzi et al.*, 1993) random velocities are not damped on a collapse timescale, so incipient instabilities merely “bounce” and tidally diverge. This is like the behavior seen in Saturn’s A ring, much of which is gravitationally unstable by these same criteria (*Salo*, 1992; *Karjalainen and Salo*, 2004). Direct collapse to planetesimals is much harder to achieve, requiring much lower relative velocities, and is unlikely to have occurred this way (*Cuzzi et al.* 1994, *Weidenschilling*, 1995; *Cuzzi and Weidenschilling*, 2005). Recent results by *Tanga et al.* (2004) assume an artificial damping by gas drag and find gravitationally bound clumps form which, while not collapsing directly to planetesimals, retain their identity for extended periods, perhaps allowing for slow shrinkage; this is worth further numerical modeling with more realistic damping physics, but still presumes a globally laminar nebula.
For very small particles ($\agr<$ 1mm; the highly relevant chondrule size), a different type of instability comes into play because the particles are firmly trapped to the gas by their short stopping times, and the combined system forms a single “one-phase” fluid which is stabilized against producing turbulence by its vertical density gradient (*Sekiya*, 1998; *Youdin and Shu*, 2002; *Youdin and Chiang*, 2004; *Garaud and Lin*, 2004). Even for midplane layers of such small particles to [*approach*]{} a suitable density for this to occur requires nebula turbulence to drop to what may be implausibly low values ($\alpha < 10^{-8}$ to $10^{-10}$). Moreover, such one-phase layers, with particle stopping times $\tstop$ much less than the dynamical collapse time $(G
\rhop)^{-1/2}$, cannot become “unstable” and collapse on the dynamical timescale as normally envisioned, because of [*gas*]{} pressure support, which is usually ignored (*Sekiya*, 1983; *Safronov*, 1991). *Sekiya* (1983) finds that particle densities must exceed $10^4 \rho^*$ for such particles to undergo instability and actually collapse. While especially difficult on one-phase instabilities by definition, this obstacle should be considered for any particle with stopping time much shorter than the dynamical collapse time - that is, pretty much anything smaller than a meter for $\rhop \sim \rho^*$.
A slower “sedimentation” from axisymmetric rings (or even localized blobs of high density, which might form through fragmentation of such dense, differentially rotating rings), has also been proposed to occur under conditions normally ascribed to marginal gravitational instability (*Sekiya*, 1983; *Safronov*, 1991; *Ward*, 2000), but this effect has only been modeled under nonturbulent conditions where, as mentioned above, growth can be quite fast by sticking alone. In a turbulent nebula, diffusion (or other complications discussed below, such as large vortices, spiral density waves, etc) might preclude formation of all but the broadest-scale “rings” of this sort, which have radial scales comparable to $H$ and grow only on extremely long timescales.
Planetesimal formation in turbulence
------------------------------------
A case can be made that astronomical, asteroidal, and meteoritic observations require planetesimal growth to stall at sizes much smaller than several km, for something like a million years (*Dullemond and Dominik*, 2005; *Cuzzi and Weidenschilling*, 2005; *Cuzzi et al.*, 2005). This is perhaps most easily explained by the presence of ubiquitous weak turbulence ($\alpha > 10^{-4}$). Once having grown to meter-size, particles couple to the largest, most energetic turbulent eddies, leading to mutual collisions at relative velocities on the order of $
v_\mathrm{turb} \sim \sqrt{\alpha} c \sim 30$ m/s, which are probably disruptive, stalling incremental growth by sticking at around a meter in size. Astrophysical observations supporting this inference are discussed in the next section. In principle, planetesimal formation could merely await cessation of nebula turbulence and then happen all at once; pros and cons of this simple concept are discussed by *Cuzzi and Weidenschilling* (2005). The main difficulty with this concept is the very robust nature of growth in dense midplane layers of nonturbulent nebulae, compared to the very extended duration of $10^6$ years which apparently characterized meteorite parent body formation (chapter by *Wadhwa et al.*). Furthermore, if turbulence merely ceased at the appropriate time for parent body formation to begin, particles of all sizes would settle and accrete together, leaving unexplained the very well characterized chondrite size distributions we observe. Alternately, several suggestions have been advanced as to how the meter-sized barrier might be overcome even in ongoing turbulence, as described below.
The speedy inward radial drift of meter-sized particles in nebulae where settling is precluded by turbulence might be slowed if they can be, even temporarily, trapped by one of several possible fluid dynamical effects. It has been proposed that such trapping concentrates them and leads to planetesimal growth as well.
Large nebula gas dynamical structures such as systematically rotating vortices (not true turbulent eddies) have the property of concentrating large boulders near their centers (*Barge and Sommeria*, 1995; *Tanga et al.*, 1996; *Bracco et al.*, 1998; *Godon and Livio*, 2000; *Klahr and Bodenheimer*, 2006). In some of these models the vortices are simply prescribed and/or there is no feedback from the particles. Moreover, there are strong vertical velocities present in realistic vortices, and the vortical flows which concentrate m-size particles are not found near the midplane, where the m-sized particles reside (*Barranco and Marcus*, 2005). Finally, there may be a tendency of particle concentrations formed in modeled vortices to drift out of them and/or destroy the vortex (*Johansen et al.*, 2004).
Another possibility of interest is the buildup of solids near the peaks of nearly axisymmetric, localized radial pressure maxima, which might for instance be associated with spiral density waves (*Haghighipour and Boss*, 2003a,b; *Rice et al.*, 2004). *Johansen et al.* (2006) noted boulder concentration in radial high pressure zones of their full simulation, but (in contrast to above suggestions about vortices), saw no concentration of meter-sized particles in the closest thing they could resolve in the nature of actual turbulent eddies. Perhaps this merely highlights the key difference between systematically rotating (and often artificially imposed) vortical fluid structures, and realistic eddies in realistic turbulence.
Overall, models of boulder concentration in large-scale fluid structures will need to assess the tendency for rapidly colliding meter-sized particles in such regions to destroy each other, in the real turbulence which will surely accompany such structures. For instance, breaking spiral density waves are themselves potent drivers for strong turbulence (*Boley et al.*, 2005).
Another suggestion for particle growth beyond a meter in turbulent nebulae is motivated by observed size-sorting in chondrites. *Cuzzi et al.* (1996, 2001) have advanced the model of turbulent concentration of chondrule-sized (mm or smaller diameter) particles into dense zones, that ultimately become the planetesimals we observe. This effect, which occurs in genuine, 3D turbulence (both in numerical models and laboratory experiments), naturally satisfies meteoritics observations in several ways under quite plausible nebula conditions. It offers the potential to leapfrog the problematic meter-size range entirely and would be applicable (to differing particle types) throughout the solar system (see *Cuzzi and Weidenschilling*, 2005 and *Cuzzi et al.*, 2005 for reviews). This scenario faces the obstacle that the dense, particle-rich zones which certainly [*do*]{} form are far from solid density, and might be disrupted by gas pressure or turbulence before they can form solid planetesimals. As with dense midplane layers of small particles, gas pressure is a formidable barrier to gravitational instability on a dynamical timescale in dense zones of chondrule-sized particles formed by turbulent concentration. However, as with other small-particle scenarios, sedimentation is a possibility on longer timescales than that of dynamical collapse. It is promising that *Sekiya* (1983) found that zones of these densities, while “incompressible” on the dynamical timescale, form stable modes. Current studies are assessing whether the dense zones can survive perturbations long enough to evolve into planetesimals.
Summary of the situation regarding planetesimal formation
---------------------------------------------------------
As of the writing of this chapter, the path to planetesimal formation remains unclear. In nonturbulent nebulae, a variety of options seem to exist for growth which - while not on dynamical collapse timescales, is rapid on cosmogonic timescales ($\ll 10^5$ years). However, this set of conditions and growth timescales seems to be at odds with asteroidal, meteoritic, and astronomical observations of several kinds (*Russell et al.*, 2006; *Dullemond and Dominik*, 2005; *Cuzzi and Weidenschilling*, 2006; *Cuzzi et al.*, 2005; chapter by *Wadhwa et al.*). The alternate set of scenarios - growth beyond a meter or so in size in turbulent nebulae - are perhaps more consistent with the observations but are still incompletely developed beyond some promising directions. The challenge is to describe quantitatively the rate at which planetesimals form under these inefficient conditions.
GLOBAL DISK MODELS WITH SETTLING AND AGGREGATION {#sec:global-disk-models}
================================================
Globally modeling a protoplanetary disk including dust settling, aggregation, radial drift and mixing, along with radiative transfer solutions for the disk temperature and spectrum form a major numerical challenge, because of the many orders of magnitude that have to be covered both in time scales (inner disk versus outer disk, growth of small particles versus growth of large objects) and particle sizes. Further numerical difficulties result from the fact that small particles may contribute significantly to the growth of larger bodies, and careful renormalization schemes are necessary to treat these processes correctly and in a mass-conserving manner (*Dullemond and Dominik*, 2005). Further difficulties arise from uncertainty about the strength and spatial extent of turbulence during the different evolutionary phases of a disk. A complete model covering an entire disk and the entire growth process along with all relevant disk physics is currently still out of reach. Work so far has therefore either focused on specific locations in the disk, or has used parametrized descriptions of turbulence with limited sets of physical growth processes. However, these “single slice” models have the problem that radial drift can become so large for m-sized objects, that these leave the slice on a time scale of a few orbital times (*Weidenschilling*, 1977; section \[sec:prior-to-midplane\]). Nevertheless, important results have come forth from these efforts, that test underlying assumptions of the models.
For the spectral and imaging appearance of disks, there are two main processes that should produce easily observable results: particle settling and particle growth. Particle settling is due to the vertical component of gravity acting in the disk on the pressure-less dust component (section \[sec:role-of-turb\]). Neglecting growth for the moment, settling leads to a vertical stratification and size sorting in the disk. Small particles settle slowly and should be present in the disk atmosphere for a long time, while large particles settle faster and to smaller scale heights. While in a laminar nebula this is a purely time dependent phenomenon, this result is permanent in a turbulent nebula as each particle size is spread over its equilibrium scale height (*Dubrulle et al.*, 1995). From a pure settling model, one would therefore expect that *small dust grains* will increasingly dominate dust emission features (cause strong feature-to-continuum ratios) as large grains disappear from the surface layers.
Grain growth may have the opposite effect. While vertical mixing and settling still should lead to a size stratification, particle growth can become so efficient that all small particles are removed from the gas. In this case, dust emission features should be characteristic for *larger particles* (i.e. no or weak features, see chapter by *Natta et al.*). At the same time, the overall opacity decreases dramatically. This effect can become significant, as has been realized already early on (*Weidenschilling*, 1980, 1984; *Mizuno*, 1989). In order to keep the small particle abundance at realistic levels and the dust opacity high, *Mizuno et al.* (1988) considered a steady inflow of small particles into a disk. However, disks with signs of small particles are still observed around stars that seem to have completely removed their parental clouds, so this is not a general solution for this problem. In the following we discuss the different disk models documented in the literature. We begin with a discussion of earlier models focusing on specific regions of the solar system.
Models limited to specific regions in the solar system
------------------------------------------------------
Models considering dust settling and growth in a single vertical slice have a long tradition, and have been reviewed in previous Protostars and Planets III (*Cuzzi and Weidenschilling*, 1993). We therefore refrain from an in-depth coverage and only recall a few of the main results. The global models discussed later are basically similar calculations, with higher resolution, and for a large set of radii.
*Weidenschilling* has studied the aggregation in laminar (*Weidenschilling*, 1980, 2000) and turbulent (*Weidenschilling*, 1984, 1988) nebulae, focusing on the region of terrestrial planet formation, in particular around 1AU. These papers contain the basic descriptions of dust settling and growth under laminar and turbulent conditions. They show the occurrence of a rain-out after particles have grown to sizes where the settling motion starts to exceed the thermal motions. *Nakagawa et al.* (1981, 1986) study settling and growth in vertical slices, also concentrating on the terrestrial planet formation regions. They find that within 3000 years, the midplane is populated by cm-sized grains. *Weidenschilling* (1997) studied the formation of comets in the outer solar system with a detailed model of a non-turbulent nebula, solving the coagulation equation around 30AU. In these calculations, growth initially proceeds by Brownian motion, without significant settling, for the first 10000yrs. Then, particles become large enough and start to settle, so that the concentration of solids increases quickly after $5\times10^4$yr. The particle layer reaches the critical density where the layer gravitational instability is often assumed to occur, but first the high velocity dispersion prevents the collapse. Later, a transient density enhancement still occurs, but due to the small collisional cross section of the typically 1m-sized bodies, growth must still happen in individual 2-body collisions.
Dust aggregation during early disk evolution
--------------------------------------------
*Schmitt et al.* (1997) implemented dust coagulation in an $\alpha$ disk model. They considered the growth of PCA in a one-dimensional disk model, i.e. without resolving the vertical structure of the disk. The evolution of the dust size distribution is followed for 100 years only. In this time, at a radius of 30AU from the star, first the smallest particles disappear within 10 years, due to Brownian motion aggregation. This is followed by a self-similar growth phase during which the volume of the particles increases by 6 orders of magnitude. Aggregation is faster in the inner disk, and the decrease in opacity followed by rapid cooling leads to a *thermal gap* in the disk around 3AU. Using the CCA particles, aggregation stops in this model after the small grains have been removed. For such particles, longer timescales are required to continue the growth.
Global models of dust aggregation during the prestellar collapse stage and into the early disk formation stage are numerically feasible because the growth of particles is limited. *Suttner et al.* (1999) and *Suttner and Yorke* (2001) study the evolution of dust particles in protostellar envelopes, during collapse, and the first 10$^4$ years of dynamical disk evolution, respectively. These very ambitious models include a radiation hydrodynamic code that can treat dust aggregation and shattering using an implicit numerical scheme. They find that during a collapse phase of 10$^3$ years, dust particles grow due to Brownian motion and differential radiative forces, and can be shattered by high velocity collisions cause by radiative forces. During early disk evolution, they find that at 30AU from the star within the first pressure scale height from the midplane, small particles are heavily depleted because the high densities lead to frequent collisions. The largest particles grow by a factor of 100 in mass. Similar results are found for PCA particles, while CCA particles show accelerated aggregation because of the enhanced cross section in massive particles. Within 10$^4$ years, most dust moves to the size grid limit of 0.2mm. While aggregation is significant near the midplane (opacities are reduced by more than a factor 10), the overall structure of the model is not yet affected strongly, because at the low densities far from the midplane aggregation is limited and changes in the opacity are only due to differential advection.
Global settling models
----------------------
Settling of dust without growth goes much slower than settling that is accelerated by growth. However, even pure settling calculations show significant influence on the spectral energy distributions of disks. While the vertical optical depth is unaffected by settling alone, the height at which stellar light is intercepted by the disk surface changes. *Miyake and Nakagawa* (1995) computed the effects of dust settling on the global SED and compared these results with IRAS observations. They assume that after the initial settling and growth phase, enough small particles are left in the disk to provide an optically thick surface and follow the decrease of the height of this surface, concluding that this is consistent with the life-times of T Tauri disks, because the settling time of a 0.1 grain within a single pressure scale height is of order 10Myr. However, the initial settling phase does lead to strong effects on the SED, because settling times at several pressure scale heights are much shorter. *Dullemond and Dominik* (2004) show that settling from a fully mixed passive disk leads to a decrease of the surface height in 10$^{4}$–10$^{5}$ years, and can even lead to self-shadowed disks (see chapter by *Dullemond et al.*).
Global models of dust growth {#sec:glob-settl-aggr}
----------------------------
*Mizuno* (1989) computes global models including evaporation, and a steady state assumption using small grains continuously raining down from the ISM. The vertical disk structure is not resolved, only a single zone in the midplane is considered. He finds that the Rosseland mean opacity decreases, but then stays steady due to the second generation grains.
*Kornet et al.* (2001) model the global gas and dust disk by assuming that at a given radius, the size distribution of dust particles (or planetesimals) is exactly monodisperse, avoiding the numerical complications of a full solution of the Smoluchowski equation. They find that the distribution of solids in the disk after 10$^7$ years depends strongly on the initial mass and angular momentum of the disk.
*Ciesla and Cuzzi* (2005) model the global disk using a four-component model: Dust grains, m-size boulders, planetesimals and the disk gas. This model tries to capture the main processes happening in a disk: growth of dust grains to m-sized bodies, the migration of m-sized bodies and the resulting creation of evaporation fronts, and the mixing of small particles and gas by turbulence. The paper focuses on the distribution of water in the disk, and the dust growth processes are handled by assuming timescales for the conversion from one size to the next. Such models are therefore mainly useful for the chemical evolution of the nebula and need detailed aggregation calculations as input.
The most complete long-term integrations of the equations for dust settling and growth are described in recent papers by *Tanaka et al.* (2005, henceforth THI05) and *Dullemond and Dominik* (2005, henceforth DD05). These papers implement dust settling and aggregation in individual vertical slices through a disk, and then use many slices to stitch together an entire disk model, with predictions for the resulting optical depth and SED from the developing disk. Both models have different limitations. THI05 consider only laminar disk models, so that turbulent mixing and collisions between particles driven by turbulence are not considered. Their calculations are limited to compact solid particles. DD05’s model is incomplete in that it does not consider the contributions of radial drift and differential angular velocities between different particles. But in addition to calculations for a laminar nebula, they also introduce turbulent mixing and turbulent coagulation, as well as PCA and CCA properties for the resulting dust particles. THI05 use a two-layer approximation for the radiative transfer solution, while DD05 run a 3D Monte-Carlo radiative transfer code to compute the emerging spectrum of the disk. Both models find that aggregation proceeds more rapidly in the inner regions of the disk than in the outer regions, quickly leading to a region of low optical depth in the inner disk.
Both calculations find that a bi-modal size distribution is formed, with large particles in the midplane, formed by rainout (the fast settling of particles after their settling time has decreased below their growth time) and continuing to grow quickly, and smaller particles remaining higher up in the disk and then slowly trickling down. In the laminar disk, growth stops in the DD05 calculations at cm sizes because radial drift was ignored. in THI05, particles continue to grow beyond this regime.
The settling of dust causes the surface height of the disk to decrease, reducing the overall capacity of the disk to reprocess stellar radiation. THI05 find that at 8 AU from the star the optical depth of the disk at 10 reaches about unity after a bit less than 10$^6$yrs. In the inner disk, the surface height decreases to almost zero in less than 10$^6$yrs. The SED of the model shows first a strong decrease at wavelength of 100 and longer, within the first 10$^5$yrs. After that the near-IR and mid-IR radiation also decreases sharply. THI05 consider their results to be roughly consistent with the observations of decreasing fluxes at near-IR and mm wavelengths in disks.
The calculations by DD05 show a more dramatic effect, as shown in Fig.\[fig-sed\]. In the calculations for a laminar disk, here the surface height already significantly decreases in the first 10$^4$ years, then the effect on the SED is initially strongest in the mid-IR region. After 10$^6$yrs, the fluxes have dropped globally by at least a factor of 10, except for the mm regime, which is affected greatly only after a few times 10$^6$ years.
In the calculations for a turbulent disk (DD05), the depletion of small grains in the inner disk is strongly enhanced. This result is caused by several effects. First, turbulent mixing keeps the particles moving even after they have settled to the mid-plane, allowing them to be mixed up and rain down again through a cloud of particles. Furthermore, vertical mixing in the higher disk regions mixes low-density material down to higher densities, where aggregation can proceed much faster. The material being mixed back up above the disk is then largely deprived of solids, because the large dust particles decouple from the gas and stay behind, settling down to the mid plane. The changes to the SED caused by coagulation and settling in a turbulent disk are dramatic, and clearly inconsistent with the observations of disks around T Tauri stars that indicate lifetimes of up to 10$^7$ years. DD05 conclude that ongoing particle destruction must play an important role, leading to a steady-state size distribution for small particles (section \[physical properties of aggregates\]).
The role of aggregate structure
-------------------------------
Up to now, most solutions for the aggregation equation in disks are still based on the assumption of compact particles resulting from the growth process. However, at least for the small aggregates formed initially, this assumption is certainly false. First of all, if aggregates are fluffy, with large surface-to-mass ratios, it will be much easier to keep these particles in the disk surface where they can be observed as scattering and IR emitting grains. Observations of the 10 silicate features show that in many disks, the population emitting in this wavelength range is dominated by particles larger than interstellar (*van Boekel et al.*, 2005; *Kessler-Silacci et al.*, 2005). When modelled with compact grains, the typical size of such grains is several microns, with corresponding settling times less than a Myr. When modelled with aggregates, particles have to be much larger to produce similar signatures (flattened feature shapes, e.g. *Min et al.*, 2005).
When considering the growth time scales, in particular in regions where settling is driving the relative velocities, the timescales are surprisingly similar to the case of compact particles (*Safronov*, 1969, *Weidenschilling*, 1980). While initially, fluffy particles settle and grow slowly because of small settling velocities, the larger collisional cross section soon leads to fast collection of small particles, and fluffy particles reach the mid-plane as fast as compact grains, and with similar masses collected.
[|p|p|p|]{} & &\
\
Dust particles stick in collisions with less than $\sim 1$ m/s velocity due to van der Waals force or hydrogen bonding. For low relative velocities ($\ll 1$m/s) a cloud of dust particles evolves into fractal aggregates ($\Df<2$) with a quasi-monodisperse mass distribution. Due to the increasing collision energy, growing fractal aggregates can no longer keep their structures so that non-fractal (but very porous) aggregates form (still at $v\ll 1$m/s). Macroscopic aggregates have porosities $>65$% when collisional compaction, and not sintering or melting occurs. & At what aggregate size does compaction happen in a nebula environment? When do collisions between macroscopic aggregates result in sticking? Some experiments show no sticking at rather low impact velocities, while others show sticking at high impact speeds. How important are special material properties: organics, ices, magnetic and electrically charged particles? What are the main physical parameters (e.g. velocity, impact angle, aggregate porosity/material/shape/mass) determining the outcome of a collision? & More empirical studies in collisions between macroscopic aggregates required. Macroscopic model for aggregate collisions (continuum description) based on microscopic model and experimental results. Develop recipes for using the microphysics in large scale aggregation calculations. Develop aggregation models that treat aggregate structure as a *variable* in a self-consistent way.\
\
Particle velocities and relative velocities in turbulent and nonturbulent nebulae are understood; values are $<1$ m/s for $\agr\rho
< 1-3$ g cm$^{-2}$ depending on alpha. Radial drift decouples large amounts of solids from the gas and migrates it radially, changing nebula mass distribution and chemistry Turbulent diffusion can offset inward drift for particles of cm size and smaller, relieving the “problem” about age differences between CAIs and chondrules. & What happens to dust aggregates in highly mass-loaded regions in the solar nebula, e.g. midplane, eddies, stagnation points? Is the nebula turbulent? If so, how does the intensity vary with location and time? Can purely hydrodynamical processes produce self-sustaining turbulence in the terrestrial planet formation zone? Can large-scale structures (vortices, spiral density waves) remain stable long enough to concentrate boulder-size particles? Can dense turbulently concentrated zones of chondrule-size particles survive to become actual planetesimals? & Relative velocities in highly mass-loaded regions in the solar nebula, e.g. midplane, eddies, stagnation points. Improve our understanding of turbulence production processes at very high nebula Reynolds numbers. Model effects of MRI-active upper layers on dense, non-ionized gas in magnetically dead zones. Model the evolution of dense strengthless clumps of particles in turbulent gas. Model collisional processes in boulder-rich vortices and high-pressure zones. Model evolution of dense clumps in turbulent gas.\
\
Small grains are quickly depleted by incorporation into larger grains. Growth timescales are short for small compact and fractal grains alike. Vertical mixing and small grain replenishment are necessary to keep the observed disk structures (thick/flaring). & What is the role of fragmentation for the small grain component? Are the “small” grains seen really large, fluffy aggregates? Are the mm/cm sized grains seen in observations compact particles, or much larger fractal aggregates? What is the global role of radial transport? & Study the optical properties of *large* aggregates, fluffy and compact. Implement realistic opacities in disk models to produce predictions and compare with observations. Construct truly global models including radial transport. More resolved disk images at many wavelengths, to better constrain models.\
SUMMARY AND FUTURE PROSPECTS
============================
A lot has been achieved in the last few years, and our understanding of dust growth has advanced significantly. There are a number of issues where we now have clear answers. However, a number of major controversies remain, and future work will be needed to address these before we can come to a global picture of how dust growth in protoplanetary disks proceeds and which of the possible ways toward planetesimals are actually used by nature. In table \[tab:overview\] on the following page we summarize our main conclusions and questions, and note some priorities for research in the near future.
**Acknowledgments.** We thank the referee (Stu Weidenschilling) for valuable comments on the manuscript. This work was partially supported by JNC’s grant from the Planetary Geology and Geophysics program. CD thanks Kees Dullemond for many discussions, Dominik Paszun for preparing figure 1. JNC thanks Andrew Youdin for a useful conversation regarding slowly evolving, large scale structures.
**REFERENCES**
|
---
abstract: 'Chu Spaces and Channel Theory are well established areas of investigation in the general context of category theory. We review a range of examples and applications of these methods in logic and computer science, including Formal Concept Analysis, distributed systems and ontology development. We then employ these methods to describe human object perception, beginning with the construction of uncategorized object files and proceeding through categorization, individual object identification and the tracking of object identity through time. We investigate the relationship between abstraction and mereological categorization, particularly as these affect object identity tracking. This we accomplish in terms of information flow that is semantically structured in terms of local logics, while at the same time this framework also provides an inferential mechanism towards identification and perception. We show how a mereotopology naturally emerges from the representation of classifications by simplicial complexes, and briefly explore the emergence of geometric relations and interactions between objects.'
author:
- |
Chris Fields\
23 Rue des Lavandières\
11160 Caunes Minervois, FRANCE\
fieldsres@gmail.com\
\
and\
\
James F. Glazebrook\
Department of Mathematics and Computer Science\
Eastern Illinois University, 600 Lincoln Ave.\
Charleston, IL 61920–3099, USA\
jfglazebrook@eiu.edu\
Adjunct Faculty\
Department of Mathematics\
University of Illinois at Urbana–Champaign\
Urbana, IL 61801, USA
title: |
A mosaic of Chu spaces and Channel Theory with applications to\
Object Identification and Mereological Complexity
---
**Keywords**: Chu space, Information Channel, Infomorphism, Formal Concept Analysis, Distributed System, Event File, Ontology, Mereological Complexity, Colimit.
Introduction
============
Category theory provides a language and a range of conceptual tools towards the general study of complexity, originally in a mathematical framework, and later applied extensively to computer science, artificial intelligence, the life sciences, and the study of ontologies (reviewed by @Baianu2006 [@EV2007; @Goguen1; @Goguen2; @Healy1; @Healy2; @Poli1; @Poli2; @Rosen]). This follows a tradition in conceiving of a range of descriptive methods in analytical philosophy as first advocated by F. @Brentano and then later by E. @Husserl, and others (surveyed by e.g. @Simons [@Smith2003]).
One particular categorical concept is that of a *Chu space*, which entered computer science as a representable model of linear logic originally formulated by @Barr1 [@Barr2] and @Seely. An advantage of using Chu spaces is their flexibility in adapting to a wide range of interpretations and applications. They are more general than topological spaces, and they can be represented in straightforward *object-attribute* rectangular/matrix-like arrays (the rows consisting of object names, and the columns consisting of attribute names; so an $[ij]$-entry simply means that an object $o_i$ has an attribute $a_j$). From the observational perspective, the attributes are taken to provide information about the structural and dynamical configurations of, and between objects. Following earlier developments of the theory, Chu spaces emerged with importance in areas dealing with machine learning and data mining; these include (but are not limited to) parallel programming algorithms, information retrieval, concurrent computation automata, physical systems, local logics, formal concept analysis (see below), the semantics of observation-measurement problems, decision theory, and ontological engineering [@Abramsky1; @Allwein; @Barr1; @Barwise1; @BHL; @Pratt1; @Pratt2; @Pratt3; @ZS]. Accordingly, one may find a variety of interpretations of Chu-space representations, including the object/attribute criteria used to define informational relationships, depending on the chosen context.
During information processing, the various channels of information assimilation may possess intrinsic qualities that influence the type of inferences they derive from the basic premise that “X being A carries the information that Y is B” [@Dretske1]. As a step towards conceptualizing information flow within a logical environment in category-theoretic terms, the basic elements of Chu spaces have been adapted to the concept of *Classifications*, as the latter are expressed in terms of *Tokens* and *Types* [@Barwise1; @Barwise2; @Barwise3; @Barwise4]. The resulting framework of *Channel Theory* casts information flow within a logical and distributed systems environment. An *infomorphism*, as a reformulated Chu morphism (in a sense ‘dual’) constitutes a pivotal concept of Channel Theory, by defining a channel through which the information represented by one classification is re-represented in another.
We have two broad aims in this paper: 1) to assemble the main concepts and tools of Chu spaces and Channel Theory in one place, and briefly review some of their applications, and 2) to apply these concepts and tools to develop category-theoretic descriptions of three interdependent cognitive processes, the construction of object files [@Kahneman] and object tokens [@Zimmer], the binding of type and token information in object categorization [@Martin07; @Keifer12], and the recognition and categorization of mereologically-complex individuals. While the notion of “entry-level” categories and the extension of such categories both upward and downward in an abstraction (or type) hierarchy has been intensively investigated both experimentally [@Clarke15] and theoretically (@sowa06; see also §\[cognitive\] below), the representation of mereological complexity has received far less attention. Mereological categorization can be functionally dissociated from abstraction-based categorization in humans, e.g. in high-functioning autism where “weak central coherence”, and hence deficit understanding of mereological complexity may be displayed alongside normal or even superior abstraction ability [@Happe06; @Booth:16]. How abstraction-based types and mereological types are related, and how their implementations in humans are related, thus remain to be worked out. We are also interested in how tokens representing individual, re-identifiable objects, i.e. object tokens as discussed in §\[object-token\], are able to participate both as such and as instances of classified types in both hierarchies simultaneously. How, for example, can an object token representing a particular dog be both an instance of the entry-level category \[dog\], as well as more abstract categories such as \[mammal\] or \[animal\], while at the same time being represented as both an individual entity with mereological complexity at multiple scales and as a proper component of even more complex entities? As both abstraction and mereology contribute to the construction of prior probabilities and to the regulation of precision or attention within Bayesian classifiers [@Friston2], the question of how these representations interact – from an implementation perspective, how they cross-modulate each other – is crucial to understanding both how mereologically complex objects are identified as individual entities and how identifiable individual entities are recognized as being mereologically complex.
The first part of the paper addresses the initial aim of the tool assembly. We begin by defining and reviewing some of the basic properties of Chu spaces in §\[chu\]. Although Chu spaces have been traditionally applied to fields such as those listed above, they also have a number of other significant applications of interest here. How Chu spaces can be implemented within Formal Concept Analysis and Domain Theory (e.g. to represent information systems and approximable concepts following @HZ [@KHZ; @Scott1982; @ZS]) is reviewed in §\[FCA\]. In §\[topology\] we discuss representations of spaces (and representations by spaces), spatial coarse-graining and finite sampling of information [@GP1; @Sorkin1]; we then review the representation of sampled information by simplicial complexes constructed “above” the sampled space in §\[simplicial\]. The following two sections, §\[channel-I\] and §\[channel-II\] establish a similar working account of Channel Theory. We survey a number of motivating examples and applications, including Distributed Systems [@Barwise1] in §\[distributed\], Cognizance Classification [@SS1; @SS2] in §\[cognizance\], and the flow of information in Ontology Comparison and Alignment [@Kalfoglou1; @Schorlemmer; @Schorlemmer2005] in §\[ontologies\]. The category-theoretic concepts of *cocone* and *colimit* (e.g. @Awodey) naturally arise in both Chu space and Channel Theory descriptions; we review these concepts in §\[colimits\] with illustrative examples.
The second part of the paper presents new results. We begin in §\[cognitive\] with brief reviews of perception, categorization and attention as neurocognitive processes and of multi-layer recurrent network models (e.g. @Friston2 [@Grossberg13]) of these processes. In §\[tt-flow\], we re-describe perception and categorization, using the Chu space and Channel Theory tools assembled in the first part, in a way that makes explicit the dualities between dynamic and static properties, individuals and categories, and states and events. We capture these dualities in a “cone-cocone diagram” that formalizes the inferential steps required to link object tokens together to produce a “history” of a persistent object. We then turn our attention, in §\[mereological\], to mereological categorization and to the key mereotopological question of how the *boundaries* between the components of a complex object are defined. As scenes are themselves mereological complexes, the boundary construction required for object segmentation emerges as the simplest case of inter-object boundary definition.
Category theory is in essence a theory of dualities. Applying the tools of Chu spaces and Channel Theory to cognition, and in particular to object identification, emphasizes the role of concepts and processes representable as category-theoretic duals in cognitive processing. The roles of complementary information flows at all scales, from on-center/off-surround networks to the dorsal and ventral attention systems to the interplay of memory and prediction that constructs object histories, exemplify such duality. By taking object identities and object persistence for granted, AI systems have largely neglected the problem of object re-identification that lies at the heart of the frame problem [@Fields13; @Fields16]. Taking this problem and the dual organization required to solve it into account suggests reconceptualizations of learning and memory as overarching dual processes.
Part I: Category-theoretic Concepts and Tools {#part-i-category-theoretic-concepts-and-tools .unnumbered}
=============================================
Chu spaces and Chu transforms {#chu}
=============================
Basic definitions for objects and attributes {#chu-defs}
--------------------------------------------
\[chu-def-1\] *A (dyadic or two-valued) Chu space* ${\mathsf{C}}$ consists of a triple $(C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ where $C_{{\mathsf{o}}}$ is a set of *objects*, $C_{{\mathsf{a}}}$ is a set of *attributes*, along with a *satisfaction relation* (or *evaluation*) $\Vdash_{{\mathsf{C}}} \subseteq C_{{\mathsf{o}}} \times C_{{\mathsf{a}}}$.
For observational purposes, we may regard the “attributes” as providing information about the structural and dynamical configurations of and between the “objects.” Two objects can be distinguished if but only if there is at least one attribute that they do not share. Otherwise, objects are said to be equivalent. This sense of equivalence formalizes Leibniz’ principle of ”identity of indiscernibles.” The “objects” and “attributes” can equally well be thought of as “states” and “events,” with “states” distinguished by the “events” that can occur in them or, as we will see, in terms of “tokens” and “types” or other similar pairs of concepts.
\[chu-def-2\] A *morphism* or *Chu transform* of a Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ to a Chu space ${\mathsf{D}}= (D_{{\mathsf{o}}}, \Vdash_{{\mathsf{D}}}, D_{{\mathsf{a}}})$ is a pair of functions $(f_{{\mathsf{a}}}, f_{{\mathsf{o}}})$ with $f_{{\mathsf{o}}}: C_{{\mathsf{o}}} {\longrightarrow}D_{{\mathsf{o}}}$, and $f_{{\mathsf{a}}}: D_{{\mathsf{a}}} {\longrightarrow}C_{{\mathsf{a}}}$, such that for any $x \in C_{{\mathsf{o}}}$, and $y \in D_{{\mathsf{a}}}$, we have $f_{{\mathsf{o}}}(x) \Vdash_{{\mathsf{D}}} y$, if and only if $x \Vdash_{{\mathsf{C}}} f_{{\mathsf{a}}}(y)$.
If ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ is a Chu space, then ${\mathsf{C}}^{\perp} = (C_{{\mathsf{a}}}, \Vdash_{{\mathsf{C}}}^{\rm{op}}, C_{{\mathsf{o}}})$ is the *dual space* of ${\mathsf{C}}$ in which the roles of objects and attributes are interchanged. This sense of duality allows us to think, for example, of attributes being distinguished by the objects to which they apply, events being distinguished by the states in which they participate, or types being distinguished by the tokens they include. Chu-space duality will provide, in §\[tt-flow\], the key to representing recurrent networks in a fully-symmetric way.
\[multivalued-1\] More generally, for some set ${\mathsf{K}}$, we could also speak of *a ${\mathsf{K}}$-valued Chu space* ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ with a satisfaction relation (evaluation) $\Vdash_{{\mathsf{C}}}: C_{{\mathsf{o}}} \times C_{{\mathsf{a}}} {\longrightarrow}{\mathsf{K}}$, with relation $\Vdash_{{\mathsf{C}}}(a,b)$ an element of ${\mathsf{K}}$.
Chu flows {#chu-flow}
---------
What is the information preserved when switching between Chu spaces that are tied by a Chu transform? Let a *Chu flow* [@VBent], cf. @Barwise1 be specified by a “flow formula” constructed from the elements of the following schema:
$$x \Vdash a ~ \vert~ \neg (x \Vdash a) ~ \vert ~ \wedge ~\vert \vee \vert ~ \exists x ~ \vert ~\forall a.$$
Any such formula $\psi(a_1, \ldots, a_k, x_1, \ldots, x_m)$ specifies which objects $x_i$ have which attributes $a_i$ in the Chu space in which it applies. @VBent has shown that for finite Chu spaces ${\mathsf{C}}$ and ${\mathsf{D}}$, the existence of a Chu transform ${\mathsf{C}}{\longrightarrow}{\mathsf{D}}$ is equivalent to every flow formula valid in ${\mathsf{C}}$ being valid in ${\mathsf{D}}$ as well. The transform ${\mathsf{C}}{\longrightarrow}{\mathsf{D}}$ can, in this case, be viewed as “transporting” the information encoded in valid flow formulas from ${\mathsf{C}}$ to ${\mathsf{D}}$; it can thus be thought of informally as a “channel” from ${\mathsf{C}}$ to ${\mathsf{D}}$ and as implicitly providing a sense of “spatial” and/or “temporal” separation between ${\mathsf{C}}$ and ${\mathsf{D}}$. These informal notions will be made more precise in §\[channel-I\].
A given flow formula $\psi(a_1, \ldots, a_k, x_1, \ldots, x_m)$ can give rise to useful relations between $k$ objects and $m$ types. For instance [@VBent]: $$\begin{aligned}
\forall x (\neg a_1 \in x \wedge a_2 \in x )~~ &\subset~~\text{object~inclusion}, \\
\forall x (\neg a_1 \in x \vee a_2 \in x )~~ &\boxminus ~~\text{object~incompatibility}, \\
\exists a (a \in x_1 \wedge a \in x_2 ) ~~&~{\mathsf{o}}~~~\text{type~overlap}. \\
\end{aligned}$$
Biextensional collapse {#bi-ext}
----------------------
Following @Pratt2 we define a pair of maps relative to power sets ${\mathcal P}(\cdot)$ as follows: $$\label{pratt-1}
\begin{aligned}
\hat{{\alpha}} &: C_{{\mathsf{o}}} {\longrightarrow}{\mathcal P}(C_{{\mathsf{a}}}) ~\text{with}~ \hat{{\alpha}}(x) = \{a \in C_{{\mathsf{a}}} : x \Vdash_{{\mathsf{C}}} {\mathsf{a}}\} \\
\hat{\omega} &: C_{{\mathsf{a}}} {\longrightarrow}{\mathcal P}(C_{{\mathsf{o}}}) ~\text{with}~ \hat{\omega}({\mathsf{a}}) = \{x \in C_{{\mathsf{o}}}: x \Vdash_{{\mathsf{C}}} {\mathsf{a}}\}.
\end{aligned}$$ Given $X \subseteq C_{{\mathsf{o}}}$, and $A \subseteq C_{{\mathsf{a}}}$, the above two maps extend to the following maps, respectively [@ZS]: $$\label{pratt-2}
\begin{aligned}
{\alpha}&: {\mathcal P}(C_{{\mathsf{o}}}) {\longrightarrow}{\mathcal P}(C_{{\mathsf{a}}}) ~\text{with}~ {\alpha}(x) = \{{\mathsf{a}}: \forall x \in X ~x \Vdash_{{\mathsf{C}}} {\mathsf{a}}\} \\
\omega &: {\mathcal P}(C_{{\mathsf{a}}}) {\longrightarrow}{\mathcal P}(C_{{\mathsf{o}}}) ~\text{with}~ \omega(A) = \{x: \forall {\mathsf{a}}\in A ~ x \Vdash_{{\mathsf{C}}} {\mathsf{a}}\}.
\end{aligned}$$
A Chu space ${\mathsf{C}}$ is said to be *extensional* if $\hat{\omega}$ is injective, and *separable* if $\hat{{\alpha}}$ is injective. If ${\mathsf{C}}$ is both extensional and separable, then let us say it is *biextensional*. In fact, any Chu space can be turned into a biextensional type, provided the lack of injectivity of ${\alpha}$ and $\omega$ can be factored out in a suitable sense. This creates a *biextensional collapse* of a Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$, namely the Chu space $${\widehat}{{\mathsf{C}}} = ({\widehat}{C}_{{\mathsf{o}}}, \Vdash_{{\widehat}{{\mathsf{C}}}}, {\widehat}{C}_{{\mathsf{a}}})
= (\hat{{\alpha}}(C_{{\mathsf{o}}}), \Vdash_{{\widehat}{{\mathsf{C}}}}, \hat{\omega}(C_{{\mathsf{a}}})),$$ where $\hat{{\alpha}}(x) \Vdash_{{\widehat}{{\mathsf{C}}}} \hat{\omega}(a)$, if and only if $x \Vdash_{{\mathsf{C}}} {\mathsf{a}}$. In essence this means that in the biextensional collapse any repetitions in the rows of objects (tokens) and columns of attributes (types) are factored out, thus removing unnecessary repetitions in the content of information and hence minimizing the amount of processing units in a given algorithm.
Formal Concept Analysis and Computation in Chu spaces {#FCA}
=====================================================
Category theory can be viewed as a unified language for handling conceptual complexities in both mathematics and computer science. Chu spaces and Chu flows provide a natural way of representing both the structure and processing of information and have been used to investigate the semantic foundations and design of data structures and programming languages. The examples that follow illustrate these applications and introduce concepts that will prove useful later.
Concept lattices and approximable concepts {#FCA-1}
------------------------------------------
*Formal Concept Analysis* (FCA) is an approach to the semantics of symbolic data structures that studies the clustering of attributes into partially ordered sets that give rise to a *concept lattice* [@Ganter]. *Domain Theory* (DT) for programming languages is concerned with higher-order relations between concepts that involve partial information and successive approximation, and with the question of when information can be approximated by finitely representable *approximable concepts* [@ZS] (cf. *formal contexts* described in @HZ). A central idea of FCA is the distinction between the ‘extension’ of a concept as consisting of all objects belonging to that concept, and the ‘intension’ of the concept as consisting of all attributes common to all objects belonging to that concept. Defining a concept in FCA thus involves identifying a collection of attributes which agrees with the ‘intension of the extension’. Note that the idea of ‘intension’ in FCA captures the philosophical notion of an “essential property” that all members (here, objects) of a category (here, a concept) must have.
This FCA notion of ‘concept’ has been shown to be intrinsic to a Chu space [@KHZ; @ZS]; indeed each Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ has an associated complete lattice $\mathcal{L} {\mathsf{C}}$ of formal concepts associated with ${\mathsf{C}}$. @ZS [Th. 4.1] have further shown that for every complete lattice $D$ of formal (in the sense of FCA) concepts, there is a Chu space ${\mathsf{C}}$ such that $D$ is order-isomorphic to $\mathcal{L} {\mathsf{C}}$. The following definition(s) then characterize the differences between ‘formal’ (in the sense of FCA) and ‘approximable’ (in the sense of DT) concepts.
Let $P,Q$ be sets, and $\mathcal{A} \subseteq \mathcal{P}(P)$, $\mathcal{B} \subseteq \mathcal{P}(Q)$ (recall that $\mathcal{P}(\cdot)$ denotes the power set). Any pair of functions $s: \mathcal{A} {\longrightarrow}\mathcal{B}$, $t: \mathcal{B} {\longrightarrow}\mathcal{A}$, is called a *Galois connection*, if for each $X \in \mathcal{A}$ and $Y \in \mathcal{B}$, $s(X) \supseteq Y$ if and only if $X \subseteq t(Y)$. With respect to a Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$, we recall from the two associated functions (depending on ${\mathsf{C}}$, so $\alpha = \alpha_{{\mathsf{C}}}$ and $\omega = \omega_{{\mathsf{C}}}$ ): $$\begin{aligned}
{\alpha}&: {\mathcal P}(C_{{\mathsf{o}}}) {\longrightarrow}{\mathcal P}(C_{{\mathsf{a}}}) ~\text{with}~ {\alpha}(x) = \{{\mathsf{a}}: \forall x \in X ~x \Vdash_{{\mathsf{C}}} {\mathsf{a}}\} \\
\omega &: {\mathcal P}(C_{{\mathsf{a}}}) {\longrightarrow}{\mathcal P}(C_{{\mathsf{o}}}) ~\text{with}~ \omega(A) = \{x: \forall {\mathsf{a}}\in A ~ x \Vdash_{{\mathsf{C}}} {\mathsf{a}}\}.
\end{aligned}$$ The pair of maps $(\alpha, \omega)$ forms such a Galois connection [@Ganter]: i) the set of attribute (object) concepts of $P$ forms a closure system, i.e. a family of subsets closed under intersection [@Caspard03]; the attribute (object) concepts of ${\mathsf{C}}$ under set inclusion form a complete lattice; and, iii) the lattice of attribute concepts, and the lattice of object concepts are anti-isomorphic to each other. We then have:
$~$
- A subset $A \subseteq C_{{\mathsf{a}}}$ is called an *(formal) concept (of attributes)*, if it is a fixed point of $\alpha \circ \omega$, i.e. $\alpha(\omega(A)) = A$. Dually, a subset $X \subseteq C_{{\mathsf{o}}}$ is called a *(formal) concept (of objects)* if it is a fixed point of $\omega \circ \alpha$. For each object $x \in C_{{\mathsf{o}}}$, the set of its attributes $\alpha\{x\}$ is a concept.
- A subset $A \subseteq C_{{\mathsf{a}}}$ is called an *approximable concept*, if for every finite subset $X \subseteq A$, we have $\alpha (\omega(X)) \subseteq A$.
Note that (1) above allows “single-object concepts”; these will become important in §\[tt-flow\] as representations of object tokens [@Fields3], and for a hierarchial iteration of the idea in §\[mereological\].
A *complete algebraic lattice* (henceforth, an *algebraic lattice*) is a partial order which is both a complete lattice and a directed complete partial order (dcpo).
We have now the following basic representation theorem for approximable concepts [@ZS Th. 6.3]:
For any Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$, the set of its approximable concepts $\mathcal{A}{\mathsf{C}}$ under inclusion forms an algebraic lattice. Conversely, for every algebraic lattice $D$, there is a Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ such that $D$ is order-isomorphic to $\mathcal{A}{\mathsf{C}}$.
Chu spaces as information systems {#chu-info-1}
---------------------------------
An *information system* with “states” consisting of finite subsets of tokens selected from some set $A$ can be defined in terms of an underlying Chu space as follows. Let $\mathsf{Fin}(A)$ be the set of finite subsets of $A$ and choose a subset $\mathsf{Con} \subset \mathsf{Fin}(A)$ and a relation $\vdash$ (see @Scott1982 for details). Interpret the information states $x$ (i.e. elements of $\mathsf{Con}$) as objects, the tokens $a \in A$ as attributes, and let $x \Vdash a$ if and only if $a$ is a member of $x$. In this case, the subset $\mathsf{Con}$ on $A$ is called the *consistency predicate*, and $\vdash$ the *entailment relation*. Following @ZS, a Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ gives rise to an information system $(A_{{\mathsf{C}}}, \mathsf{Con}_{{\mathsf{C}}}, \vdash_{{\mathsf{C}}})$ via the assignment $A_{{\mathsf{C}}} = C_{{\mathsf{a}}}$, $X \vdash_{{\mathsf{C}}} {\mathsf{a}}$, if ${\mathsf{a}}\in {\alpha}_{{\mathsf{C}}} \circ \omega_{{\mathsf{C}}}(X)$, and a consistency predicate $\mathsf{Con}_{{\mathsf{C}}}$ for which every subset of $C_{{\mathsf{a}}}$ is consistent. @ZS [Th. 4.6] have shown, for a given Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ with $C_{{\mathsf{a}}}$ finite, a state $X \subset C_{{\mathsf{a}}}$, taken to be a *concept*, is equivalent to $X$ being a state of the derived information system $(A_{{\mathsf{C}}}, \mathsf{Con}_{{\mathsf{C}}}, \vdash_{{\mathsf{C}}})$. Intuitively, a Chu morphism in Definition \[chu-def-2\] correlating the objects and attributes of ${\mathsf{C}}$ to those of some other Chu space ${\mathsf{D}}$ is a correlation between the respective information systems. Such a morphism similarly maps sequences of flow formulas valid in ${\mathsf{C}}$ to sequences of flow formulas valid in ${\mathsf{D}}$, and hence correlates information *processes* in the respective information systems.
Ordered dynamical systems and computation {#ordered}
-----------------------------------------
The stage is now set to develop a general notion of computation for arbitrary dynamical systems with discrete states. As a sequence of *measurements* of any arbitrary dynamical system can itself be considered a dynamical system with discrete states [@Fields89], nothing is lost by assuming discreteness. Artificial neural networks (ANNs) are such systems [@Nauck], as are Turing machines, cellular automata, etc.
Consider a quadruple $\langle S, \mathsf{ns}, \leq, {\mathcal T}\rangle$, where $S$ is the state space of an information system as characterized above, $\mathsf{ns}$ is the next-state function, $\leq$ is a partial order and ${\mathcal T}: {\mathcal L}{\longrightarrow}S$ is a mapping where ${\mathcal L}$ denotes a propositional (‘factual’) language [@Leitgeb1]. The map ${\mathcal T}$ assigns some proposition $\phi$ of ${\mathcal L}$ to each state $s \in S$; hence it represents the (stipulated) semantics of $S$. The action of $\mathsf{ns}$, in this case, produces a sequence of propositions $\phi_0, \phi_1, \phi_2, ...$ and so can be interpreted as (in general, nonmonotonic) inference. If this sequence converges to some stable state $\psi$, the action of $\mathsf{ns}$ has “halted” and the proposition $\psi$ can be interpreted as the “result” of the action of $\mathsf{ns}$ on $\phi_0$. The design perspective in which $\mathsf{ns}$ is stipulated and the reverse engineering or debugging perspective in which $\mathsf{ns}$ must be discovered are both clearly supported within this picture.
Recasting the above in the language of ${\mathsf{K}}$-valued Chu spaces (\[multivalued-1\]) provides a representation of computations with imprecise inputs, outputs or both. Recalling the attribute symbol $\Vdash$, let us define $$\begin{cases} s \Vdash^t \phi ~ &\text{iff} ~ {\mathcal T}(\phi) = s ~\text{(precise state information)}, \\
s \Vdash \phi ~ &\text{iff} ~ {\mathcal T}(\phi) \leq s ~\text{(imprecise state information)}. \\
\end{cases}$$ The computational interpretation is straightforward: $s \Vdash^t \phi$ if and only if $\phi$ completely specifies the system state, whereas $s \Vdash \phi$ if and only if the system state is described by $\phi$ as well as some other propositions in ${\mathcal L}$. A computation with an initial state $s \Vdash^t \phi$ and a final state $s^\prime \Vdash \psi$, for example, would provide an ambiguous answer ($\psi$ together with other propositions) to a precise question ($\phi$).
This Chu/information space representation of computation has been adapted to capture Bayesian inference in a connectionist context [@Dayan; @McClelland1]; we develop this representation further in §\[tt-flow\]. The close relationship between Chu flows and infomorphisms as defined within Channel Theory [@Barwise1] and their application to problems such as ontology alignment are considered in §\[channel-I\] and §\[channel-II\], respectively. In particular, state space systems will be further described in the context of Channel Theory in §\[states-1\] and §\[states-2\]
Topology of information and observation {#topology}
=======================================
Propositions used to describe the world are semantically related; in the limit, all propositions in any language form a connected semantic network [@sowa06]. Observations or, more precisely, finitely-specifiable observational outcomes are similarly related. Considering an information system to be defined merely over a *set* of propositions provides no means of capturing such relations. It is, therefore, useful to introduce additional structure, with the addition of topological structure a natural first step. Doing this allows a structured notion of sampling the information encoded in a Chu space, and particularly the idea of a *finite sample of attributes* (FSA) for an object or collection of objects.
The Sorkin perspective {#Sorkin}
----------------------
One approach to developing an information topology is via a notion of *causality*; this has been pursued by @Sorkin1 [@Sorkin2] through the development of *causal sets*. While the motivation in this case has been to model the fundamental structure of spacetime in a way that could produce the continuum of macroscopic geometry as an emergent ‘classical limit’ (see [@Raptis1; @Sorkin1; @Sorkin2] for details of the mathematical physics application domain), the techniques employed are generally applicable to approximating a class of highly structured or idealized spaces by means of taking a certain limit of less complex, more user-friendly spaces. The point is to represent non-spatial information in a spatial form as means of ‘visualization’, and to consider variation in data and observations in the context of such representations. The use of spatial dimensions as a way of “displaying” information in a meaningful way on both the input and output sides of connectionist systems (and more generally, ANNs) is an example of this approach. In this case, the very complex, essentially causal relations between information computed by a learning algorithm are approximated, on an imposed spatial array of output “units” that have no intrinsic spatial relationships, in a way that makes them meaningful to external observers [@Rogers Ch. 8]. An input array similarly approximates causal relations in “the world” when the array geometry is assigned semantic significance, e.g. in computer vision applications.
Fundamental causality relations between objects $x,y,z$ can be expressed in terms of an order relation ‘$\prec$’:
- $x \prec y \prec z~\Rightarrow~ x \prec z$ (Transitivity: if $y$ is the outcome of $z$, and $x$ is the outcome of $y$, then $x$ is the outcome of $z$).
- $x \prec y$ and $y \prec x ~ \Rightarrow x = y$ (Symmetry).
- Let $[[x,y]]$ denote the cardinality of the number of elements $z$ between $x$ and $y$ such that $x \prec z \prec y$, then $[[x,y]] < \infty$ (Discreteness).
These relations can also be expressed in terms of a (locally finite) *partially ordered set* as we do below; we then apply the inherent sense of causality to the structure provided by Chu spaces. This is achieved by approximating a highly structured space by a spatial model based on simplicial complexes and related posets as developed in @GP1 [@GP2; @GP3], which we survey in part here, and in §\[simplicial\].
The Sorkin poset $P_{{\mathcal F}}$
-----------------------------------
Suppose we are given a topological space $X$, viewed as a space of ‘observables’, and let us observe $X$ from a finite family of open sets (FFOS) ${\mathcal F}$, not necessarily covering $X$. This will represent *a set of observations made on $X$*, where objects are observed in relationship to their attributes. In this way, the FFOS partitions $X$ into the ‘attributes’, and $X$ can then be regarded as a union of ‘zones’ (see below) in which two points lie in the same zone if they share the very same attributes; in other words, they consist of clusters of points in the same open set of the FFOS, and thus cannot be distinguished by the corresponding set of observations.
We can define an equivalence relation “$\sim_{{\mathcal F}}$”, by $x \sim_{{\mathcal F}} x'$, if and only if for all $U \in {\mathcal F}, ~ x \in U$ if and only if $x' \in U$. Thus two points are *equivalent* if all the observations from ${\mathcal F}$ attribute the same positive or negative result on both of them. This is simply another way of stating the causality relations above. We can factor out by this equivalence relation to obtain a quotient mapping: $$\label{quotient-1}
\pi_{{\mathcal F}}: X {\longrightarrow}X_{{\mathcal F}} = X/\sim$$ where the quotient $X_{{\mathcal F}}$ can be regarded as encoding the observational data on $X$ in a way that organizes that data by “merging” equivalent observations.
The space $X_{{\mathcal F}}$ has topological type $T_0$[^1] and corresponds to a *partially ordered set* (*poset*) denoted $P_{{\mathcal F}}$ and constructed as follows. We take $[x]_{{\mathcal F}}$ to be the equivalence class of $x \in X$, with $[x]_{{\mathcal F}} \leq [y]_{{\mathcal F}}$ if and only if for every open set $U \in {\mathcal F}$, if $y \in U$, then $x \in U$. For practical purposes we consider the family ${\mathcal F}$ as finite, and $X_{{\mathcal F}}$ is a finite $T_0$-space. Each point $[x]_{{\mathcal F}}$ is contained in a minimal open set $U_{[x]}$ of $X_{{\mathcal F}}$, and $[x]_{{\mathcal F}} \leq [y]_{{\mathcal F}}$ if and only if $x \in U_{[y]}$. The resulting poset $P_{{\mathcal F}}$ contains much of the essential observational (or causal data) on $X$. Besides organizing that data, this poset will serve as a means of ‘measurement’ (though not point-dependent) for gauging whether ‘objects’ and ‘attributes’ (or, ‘tokens’ and ‘types’) are seen as proximate to each other, or in contrast, are actually very far apart. Its structure is motivated by the ideas of @Sorkin1, as adopted by @GP1, describing how certain types of spaces can be approximated by ‘inverse limits’ of more regular spaces.
Observe that the FFOS ${\mathcal F}$ determines a secondary topology $\tau({\mathcal F})$ on $X$ which is just the topology generated by ${\mathcal F}$. If $\tau(X)$ denotes the original topology on $X$, then $\tau({\mathcal F}) \subseteq \tau(X)$ with the closure with respect to $\tau({\mathcal F})$ interpreted as a proximity between ‘zones’ (see below). Let $\tau(P_{{\mathcal F}})$ denote the quotient topology on $P_{{\mathcal F}}$ such that the map $$\label{quotient-2}
\pi_{{\mathcal F}}: (X, \tau({\mathcal F})) {\longrightarrow}(P_{{\mathcal F}}, \tau(P_{{\mathcal F}})),$$ is continuous (and then is seen to be an open map). We summarize the nomenclature in the following:
\[sorkin-poset\] Given $X$ and a FFOS ${\mathcal F}$, we say that the pair $(P_{{\mathcal F}}, \pi_{{\mathcal F}})$ is a *Sorkin model of $X$ relative to ${\mathcal F}$*, in which case $P_{{\mathcal F}}$ is called the *Sorkin poset* for $(X, {\mathcal F})$. Given $x \in P_{{\mathcal F}}$, the corresponding subset $\pi_{{\mathcal F}}^{-1}(x) \subseteq X$ is called the *zone determined by x*, which in general will be neither an open nor closed subset of $X$.
Given two FFOSs ${\mathcal F}$ and ${\mathcal G}$ of a topological space $X$, we say that ${\mathcal F}$ is *a Sorkin refinement* of ${\mathcal G}$ if ${\mathcal G}\subseteq \tau({\mathcal F})$.
From @GP1 [Prop. 11] we observe that ${\mathcal F}$ is a refinement of ${\mathcal G}$, if and only if there exists a continuous surjective map $\pi_{{\mathcal F}{\mathcal G}} : P_{{\mathcal F}} {\longrightarrow}P_{{\mathcal G}}$ such that the following diagram commutes $$\xymatrix@C=4pc{ (X, \tau(X)) \ar[dr]_{\pi_{{\mathcal G}}} \ar[r]^{\pi_{{\mathcal F}}} & (P_{{\mathcal F}}, \tau(P_{{\mathcal F}}))
\ar[d]^{\pi_{{\mathcal F}{\mathcal G}}} \\ & (P_{{\mathcal G}}, \tau(P_{{\mathcal G}}))}$$ that is, $\pi_{{\mathcal G}} = \pi_{{\mathcal F}{\mathcal G}} \circ \pi_{{\mathcal F}}$.
A Chu FSA
---------
We start with a basic observation that any space $X$ along with a FFOS ${\mathcal F}$ can be formulated in terms of a Chu space ${\mathsf{C}}= (X, \in, {\mathcal F})$ (namely, $X$ is the set of objects, $\in = \Vdash$, and ${\mathcal F}$ is the set of attributes). Thus an object $x \in X$ satisfies an attribute $U \in {\mathcal F}$, if $x \in U$. This type of Chu space is said to be *normal* [@Pratt2]. Thinking back to §\[bi-ext\], we see that the quotient map $\pi_{{\mathcal F}}$ in is simply the universal map to the biextensional collapse of ${\mathsf{C}}= (X, \in , {\mathcal F})$, and that the Chu space ${\mathsf{C}}_{P_{{\mathcal F}}} = ({\mathcal F}, \in, \tau(P_{{\mathcal F}}))$ is itself biextensional, observing that the poset structure on $P_{{\mathcal F}}$ is given by $$\hat{{\alpha}}(x) \leq \hat{{\alpha}}(y) \Longleftrightarrow \forall a \in C_{{\mathsf{a}}} ~(y \Vdash_{{\mathsf{C}}} a \Rightarrow x \Vdash_{{\mathsf{C}}} a) \Longleftrightarrow
\hat{{\alpha}}(x) \supseteq \hat{{\alpha}}(y).$$ To avoid possible complications, we assume, as in [@GP1], that ${\mathcal F}$ is suitably ‘sampled’ and extensional (briefly, ${\mathcal F}$ has no repetitive columns). Accordingly, we obtain a Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ consisting of a finite sample of attributes ${\mathcal F}$ resulting in a pair $({\mathsf{C}}, {\mathcal F})$, entitled a *Chu FSA*. Given $({\mathsf{C}}, {\mathcal F})$, we call ${\mathsf{C}}_{\vert {\mathcal F}} = (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, {\mathcal F})$ the *corestriction* of $({\mathsf{C}}, {\mathcal F})$.
Putting a topology on a Chu space
---------------------------------
The next step is to put a topology on a Chu space ${\mathsf{C}}$. Thus, we commence by saying that ${\mathsf{C}}$ is *topologically closed* if the attributes $C_{{\mathsf{a}}}$ is a topology on objects $C_{{\mathsf{o}}}$, meaning that ${\mathsf{C}}$ is normal, and $C_{{\mathsf{a}}}$ includes all unions and finite intersections. Without too much loss of generality, we assume that ${\mathsf{C}}$ is biextensional. Thus given ${\mathsf{C}}$, we have a topologically closed Chu space $$\tau({\mathsf{C}}) = (C_{{\mathsf{o}}}, \in, \tau(C_{{\mathsf{a}}})),$$ which is naturally a topological closure of ${\mathsf{C}}$. Furthermore, there is a universal Chu morphism $\tau: \tau({\mathsf{C}}) {\longrightarrow}{\mathsf{C}}$, with $\tau_{{\mathsf{o}}}: C_{{\mathsf{o}}} {\longrightarrow}C_{{\mathsf{o}}}$ the identity, and $\tau_{{\mathsf{a}}}: C _{{\mathsf{a}}} {\longrightarrow}\tau(C_{{\mathsf{a}}})$ the inclusion. The point here is that $\tau({\mathsf{C}})$ contains the same informational (or observational) structure as the original ${\mathsf{C}}$, and in $\tau({\mathsf{C}})$ the information has been encoded by means of the propositional operations of geometric logic, and sampled via ${\mathcal F}\subset C_{{\mathsf{a}}}$. Hence as proposed by @GP1 [@GP3], *a Sorkin model ${\mathsf{C}}_{{\mathcal F}}$ for $({\mathsf{C}}, {\mathcal F})$ is defined to be the biextensional collapse$\backslash$Sorkin poset of ${\mathsf{C}}_{\vert {\mathcal F}}$*. @GP1 [Prop. 18] have also shown that any row $x$ in ${\mathsf{C}}_{{\mathcal F}}$ consists of $n$ entries $0$ or $1$, and hence corresponds to a flow formula (§\[chu-flow\]): $$(x \Vdash a_{i_1}) \wedge \cdots \wedge (x \Vdash a_{i_k}) \wedge \neg (x \Vdash a_{i_{k+1}}) \wedge
\cdots \wedge \neg (x \Vdash a_{i_n}),$$ in turn showing that the rows of ${\mathsf{C}}_{{\mathcal F}}$ can be considered to encode the elementary flow formulae: $$\exists x (\bigwedge_{i \in {\mathcal F}_1} (x \Vdash a_i) \wedge \bigwedge_{i \in {\mathcal F}_2} \neg (x \Vdash a_i)),$$ for given partitions $({\mathcal F}_1, {\mathcal F}_2)$ of ${\mathcal F}$ .
Given Chu spaces ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$ and ${\mathsf{D}}= (D_{{\mathsf{o}}}, \Vdash_{{\mathsf{D}}}, D_{{\mathsf{a}}})$, we say that ${\mathsf{C}}$ is a *Sorkin refinement of ${\mathsf{D}}$* if there exists a Chu transform $\phi: \tau({\mathsf{C}}) {\longrightarrow}{\mathsf{D}}$, which is the identity on objects ($\phi_{{\mathsf{o}}}(x) = x$). Further, any Chu space is a Sorkin refinement of itself, the Sorkin refinement is transitive, and if ${\mathsf{C}}$ is both extensional and a Sorkin refinement of ${\mathsf{D}}$, then the map $\phi$ is uniquely determined [@GP1 Prop. 20].
Introducing simplicial methods on Chu spaces {#simplicial}
============================================
Given a topology on an information space, algebraic methods can be used to extend the topology into a geometry. The resulting (discrete) geometry provides a natural representation of sets of observations made at different resolutions or scales, and hence a natural way to represent coarser- to finer-grained approximations of the topology. Such approximations will provide, in §\[mereotop\], the basis for a mereotopology of “parts” of objects.
Simplicial complexes: basic definitions {#simpl-def}
---------------------------------------
We first introduce simplicial complexes as representations of observational data, following @Cordier [@Friedman; @Goerss] and @Spanier.
\[simpl-def-1\] A *simplicial complex* $K$ consists of a set $K_0$ of objects called the *vertices* and a set of finite, non-empty subsets of $K_0$ called the *simplices*. The latter satisfy the condition that if $\sigma \subset K_0$ is a simplex, and if $\tau \subset \sigma$ (with $\tau \neq
\emptyset$), then $\tau$ is also a simplex. Simplicial complexes are objects in a category denoted $\mathbf{Simpl}$.
Simplicial complexes over an information space provide the structure needed to define an information geometry. To each simplicial complex $K$ is associated the *polyhedron* or *geometric realization* of $K$, denoted $\vert K \vert$, formed from the set of all functions $K_0 {\longrightarrow}[0,1]$ satisfying:
- if $\alpha \in \vert K \vert$, then the set $\{ v \in K_0: \alpha(v) \neq 0 \}$ is a simplex of $K$;
- $\sum_{v \in K_0} \alpha(v) = 1$.
Here each function $\alpha$ can be thought of as “picking out” a subset of vertices to be the vertices of some particular polyhedron. These functions are normalized so that they “pick out” each vertex to the same extent.
For any simplex $s \in K$, there is an associated set $\vert s \vert =\{ \alpha \in \vert K \vert: \alpha (v) \neq 0 \Rightarrow v \in s \}$ as well as a set $\langle s \rangle =\{\alpha \in \vert K \vert: \alpha (v) \neq 0 \Leftrightarrow v \in s \}$. Often $\alpha(v)$ is called the *$v^{th}$ barycentric coordinate of $\alpha$*, and the mapping $\vert K \vert {\longrightarrow}[0,1]$ defined by $p_v(\alpha) = \alpha(v)$ is the *$v^{th}$ barycentric projection of $\alpha$*. With these coordinates, a metric $d$ can be defined on $K$ as given by $$d(\alpha,\beta) =\big( \sum_{v \in K_0} (p_v(\alpha) - p_v(\beta))^2\big)^{\frac{1}{2}}.$$ This distance $d(\alpha,\beta)$ measures the number of vertices shared between the polyhedra “picked out” by $\alpha$ and $\beta$, normalized to account for differences in the numbers of vertices of the two polyhedra.
\[simpl-def-2\] If $K$ and $L$ are two simplicial complexes, a simplicial mapping $f: K {\longrightarrow}L$ is a map $f_0: K_0 {\longrightarrow}L_0$ of vertex sets that preserves simplices, meaning that if $\sigma \subset K_0$ is a simplex of $K$, then its image $f(\sigma) \subset L_0$ is a simplex of $L$.
Any simplicial complex $K$ gives rise, in a straightforward way, to a poset, namely the poset of its “faces.” The elements of this poset are the simplices of $K$, arranged according to the rule $\sigma \leq \rho$ if $\sigma$ is a face of $\rho$, that is, if $\sigma \subseteq \rho$ as subsets of the vertex set $K_0$. Note that unions of “adjoining” faces are faces with this definition. We will tacitly employ the (contravariant) functor relating the categories $\mathbf{Simpl} {\longrightarrow}\mathbf{Sets}$ to speak of a *simplicial set* corresponding to its underlying structure as a simplicial complex.
Simplicial homotopy
-------------------
Two simplicial maps $f,g: X {\longrightarrow}Y$ of simplicial sets $X,Y$ are said to be *homotopic* if there exists a simplicial map $H: X \times I {\longrightarrow}Y$ (here $I=[0,1]$ the closed unit interval) such that $H_{\vert X \times \{0\}} = g$, and $H_{\vert X \times \{1\}} = f$. In other words, we have $g= H \circ i_0$, and $f = H \circ i_1$, with respect to inclusion maps $i_0: X \times \{0\} \hookrightarrow X \times I$, and $i_1: X \times \{1\} \hookrightarrow X \times I$. This is summarized by the following commutative diagram $$\xymatrix@C=5pc{X \times \{1\} \ar[d]_{i_1} \ar[dr]^f \\ X \times I \ar[r]^{H} & Y
\\ X \times \{0\} \ar[u]^{i_0} \ar[ur]_{g} }$$ so that we have $H(x,0) = f(x)$ and $H(x,1) = g(x)$.[^2]
The nerve of a relation
-----------------------
For a space $X$ and open cover ${\mathcal F}$ of $X$, the *Čech nerve $N({\mathcal F})$ of ${\mathcal F}$* is defined as the simplicial complex whose vertices are the (open) sets in ${\mathcal F}$ and for which $\{U_0, \ldots, U_n \}$ is an $n$-simplex of $N({\mathcal F})$ if and only if $\bigcap^n_{i=0} U_i \neq \emptyset$. Intuitively, the Čech nerve is the simplicial complex over ${\mathcal F}$ comprising only *connected* simplices. As pointed out by @GP1, the face poset $P_{{\mathcal F}}$ as defined above bears a close relation with $N({\mathcal F})$, but they need not be identified. In a dual sense, there is the *Vietoris complex $V({\mathcal F})$ of $(X, {\mathcal F})$* in which the vertices are simply the points of $X$, and $\langle x_0, \ldots, x_n\rangle$ is an $n$-simplex if there exists a $U \in {\mathcal F}$ that contains them all, that is, $\{x_0, \ldots, x_n \} \subseteq U$.
@Dowker provides an abstraction in this setting, given a *relation* ${\mathcal R}\subseteq X \times Y$ from $X$ to $Y$. A simplicial complex $K_{{\mathcal R}}$, called *the nerve of the relation* can be specified by: i) the vertices of $K_{{\mathcal R}}$ are those elements $x \in X$ for which there exists a $y$ such that $(x,y) \in {\mathcal R}$, and ii) the set $\{x_0, \ldots, x_n\} \in X$ is an $n$-simplex if and only if there exists some $y$ such that $(x_i, y) \in {\mathcal R}$, for $0 \leq i \leq n$. From this it can be deduced that $N({\mathcal F})$ and $V({\mathcal F})$ each provide the same information about the open cover ${\mathcal F}$ up to homotopy.
Let us exemplify some of these concepts for the basic case of the circle $S^1$, following @Porter1. Here we take an open covering ${\mathcal F}=\{U_1, U_2, U_3 \}$, relative to polar coordinates, with $$\begin{aligned}
U_1 &= \big( - \frac{2\pi}{3}, \frac{2 \pi}{3} \big); \\
U_2 &= ``\big(0, - \frac{2\pi}{3} \big)\text{''} ~{\rm{i.e.}} ~ \big( 0, \pi \big] \cup \big( - \pi, - \frac{2\pi}{3} \big);\\
U_3 &= ``\big(\frac{2 \pi}{3}, 0 \big)\text{''} ~{\rm{i.e.}} ~ \big( \frac{2 \pi}{3}, \pi \big] \cup \big( - \pi, 0 \big).
\end{aligned}$$ Every point of $S^1$, with the exception of $0, \frac{2 \pi}{3}$ and $-\frac{2 \pi}{3}$, is in exactly two of these, with a total of six equivalence classes. Choosing three representatives for the non-singleton classes gives the following minimal open sets: $$U_0 = U_1, ~ ~ U_{\frac{2 \pi}{3}} = U_2, ~ ~ U_{-\frac{2 \pi}{3}} = U_3$$ $$U_{\frac{\pi}{3}} = U_1 \cap U_2 := U_{12}, ~ ~ U_{-\frac{\pi}{3}} = U_1 \cap U_3 := U_{13}, ~ ~ U_{\pi} = U_2 \cap U_3 := U_{23}.$$ Now we have a partially ordered set with associated Hasse diagram $$\xymatrix{1 \ar@{-}[d] \ar@{-}[drr] & 2 \ar@{-}[dl] \ar@{-}[d] & \ar@{-}[dl] 3 \ar@{-}[d] \\
12 & 23 & 13
}$$ showing that $S^1_{{\mathcal F}}$ has 6 points, and the homotopy type of the former is that of $S^1$.
If we set $X = S^1$ and take the open cover ${\mathcal F}=\{U_1, U_2, U_3 \}$ as above, the vertices of $N({\mathcal F}) = \langle U_1\rangle, \langle U_2 \rangle, \langle U_3 \rangle$, and the 1-simplices of $N({\mathcal F}) = \langle U_1, U_2 \rangle, \langle U_1, U_3 \rangle, \langle U_2, U_3 \rangle$. Thus, $N({\mathcal F})$ may be represented schematically by the diagram $$\xymatrix{& 2 & \\
1 \ar@{-}[ur]^{1,2} \ar@{-}[rr]_{1,3} & & 3 \ar@{-}[ul]_{2,3}
}$$ Recalling that any simplicial complex determines a poset by subset inclusion of simplices, it can be seen that the resulting poset is the opposite of that representing $X_{{\mathcal F}}$.
The Čech and Vietoris nerves of a Chu space {#chu-nerve}
-------------------------------------------
In the context of a Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$, the *Čech nerve* is the simplicial complex denoted $N({\mathsf{C}})$ with vertex set $C_{{\mathsf{a}}}$ and where a (non-empty) subset $\{a_0, \ldots a_p\}$ of $C_{{\mathsf{a}}}$ is a $p$-simplex if there is an object $x \in C_{{\mathsf{o}}}$ satisfying $x \Vdash_{{\mathsf{C}}} a_i$, for $0 \leq i \leq p$. This is motivated by the fundamental principle that for simplicial complexes, *the nerve can be viewed as a set of instructions serving to construct (an approximation of) a space by fitting together the individual geometric simplicies*.[^3] At the same time, the associated *Vietoris nerve* $V({\mathsf{C}})$ is, in this context, the Čech nerve of the dual space ${\mathsf{C}}^{\perp}$. Given that some $\{a_0, \ldots a_p\}$ comprises a simplex, the latter can be symbolized as $\langle a_0, \ldots a_p \rangle$.
As pointed out by @GP1 [§3] there may be possible complications in dealing with induced Chu morphisms, since the set $C_{{\mathsf{a}}}$ may be infinitely large. To remedy this situation, it is necessary to finitely sample the attributes by restricting consideration to a subset ${\mathcal F}$ of $C_{{\mathsf{a}}}$. Thus following @GP1 [Prop. 4], if $f= (f_{{\mathsf{o}}}, f_{{\mathsf{a}}}): {\mathcal P}= (P_{{\mathsf{o}}}, \Vdash_{{\mathcal P}}, P_{{\mathsf{a}}}) {\longrightarrow}{\mathcal Q}= (Q_{{\mathsf{o}}}, \Vdash_{{\mathcal Q}}, Q_{{\mathsf{a}}})$ is a morphism of Chu spaces, and ${\mathcal F}$ is an open cover (the *observations*) representing a finite sample of the attributes of ${\mathcal Q}$, i.e we have ${\mathcal F}\subseteq Q_{{\mathsf{a}}}$ and finite, then there is an induced map $$f= (f_{{\mathsf{o}}}, f_{{\mathsf{a}}}): {\mathcal P}= (P_{{\mathsf{o}}}, \Vdash_{{\mathcal P}}, f_{{\mathsf{a}}}({\mathcal F})) {\longrightarrow}{\mathcal Q}= (Q_{{\mathsf{o}}}, \Vdash_{{\mathcal Q}}, {\mathcal F}).$$ Furthermore, there are also induced simplicial maps given as follows: firstly, with respect to the Vietoris nerve, we have $$\label{v-simpl-1}
V(f): V(P_{{\mathsf{o}}}, \Vdash_{{\mathcal P}}, f_{{\mathsf{a}}}({\mathcal F})) {\longrightarrow}V (Q_{{\mathsf{o}}}, \Vdash_{{\mathcal Q}}, {\mathcal F}),$$ given by $V(f) \langle p_0, \ldots, p_n \rangle = \langle f_{{\mathsf{o}}}(p_0), \ldots, f_{{\mathsf{o}}}(p_n) \rangle$, and secondly, for any choice of splitting, the function $f_{{\mathsf{a}}}: {\mathcal F}{\longrightarrow}f_{{\mathsf{a}}}({\mathcal F})$ (recall ${\mathcal F}\subseteq Q_{{\mathsf{a}}}$) induces a simplicial map with respect to the Čech nerve $$\label{c-simpl-1}
N(f): N(P_{{\mathsf{o}}}, \Vdash_{{\mathcal P}}, f_{{\mathsf{a}}}({\mathcal F})) {\longrightarrow}N (Q_{{\mathsf{o}}}, \Vdash_{{\mathcal Q}},f_{{\mathsf{a}}}({\mathcal F})),$$ given by $N(f) \langle f_{{\mathsf{a}}}(q_0), \ldots, f_{{\mathsf{a}}}(q_n) \rangle = \langle q_0, \ldots, q_n \rangle$.
The Chu FSA and induced morphisms between nerves {#Chu-FSA}
------------------------------------------------
The above simplicial procedures show that, for any Chu FSA $({\mathsf{C}}, {\mathcal F})$, there are two associated simplicial complexes $N({\mathsf{C}}_{\vert {\mathcal F}})$ and $V({\mathsf{C}}_{\vert {\mathcal F}})$, along with the associated posets of their faces. Recall that we took ${\mathsf{C}}_{{\mathcal F}}$ to denote the biextensional collapse$\backslash$Sorkin poset of ${\mathsf{C}}_{\vert {\mathcal F}}$. Let ${\widehat}{{\mathcal F}}$ denote a corresponding family of attributes. Again assuming ${\mathsf{C}}_{\vert {\mathcal F}}$ is extensional (no repeated columns in ${\mathcal F}$) then [@GP1 Th. 21], the quotient map $$\pi_{{\mathcal F}}: {\mathsf{C}}_{\vert {\mathcal F}} {\longrightarrow}{\mathsf{C}}_{{\mathcal F}},$$ exists, and there is an induced isomorphism $$\pi^N_{{\mathcal F}}: N({\mathsf{C}}, {\mathcal F}) {\overset {\cong}{{\longrightarrow}}} N({\mathsf{C}}_{\vert {\mathcal F}}, {\widehat}{{\mathcal F}}),$$ of simplicial complexes.
Intuitively, representing a set of observations of an FSA of a Chu space by a simplicial complex renders them a coarse-graining of the underlying Chu space, with the “grain size” determined by the number of vertices in the Čech nerve. The adjacency relations implicit in the Čech nerve provide this coarse-graining with a local geometry.
An excursion into Channel Theory I {#channel-I}
==================================
Classifications: Tokens and Types {#classifications-1}
---------------------------------
Situation Theory and Channel Theory [@Barwise1] provide a structure for describing informational relations in the setting of information flow through systems distributed across space and time. They provide a conceptual and schematic generalization of the ontological notion of information as causal connection introduced by @Dretske1. An assumption lending realism to the approach is that the channels through which information flows may have implicit or unknown properties that influence the nature of the inferences they implement.
We focus here on Channel Theory, the fundamental concept of which is the idea of a classification relating tokens to the types that encompass them.
\[class-def\] A *classification* ${\mathcal A}= \langle \rm{Tok}({\mathcal A}), \rm{Typ}({\mathcal A}), \Vdash_{{\mathcal A}} \rangle$ consists of a set $\rm{Tok}({\mathcal A})$ consisting of the *tokens of ${\mathcal A}$*, a set $\rm{Typ}({\mathcal A})$ consisting of the *types of ${\mathcal A}$*, and a classification relation $$\Vdash_{{\mathcal A}} \subseteq \rm{Tok}({\mathcal A}) \times \rm{Typ}({\mathcal A}),$$ that classifies tokens to types.
A classification ${\mathcal A}= \langle \rm{Tok}({\mathcal A}), \rm{Typ}({\mathcal A}), \Vdash_{{\mathcal A}} \rangle$ has the structure of a Chu space, that is, via the assignment $(\rm{Tok}({\mathcal A}), \Vdash_{{\mathcal A}}, \rm{Typ}({\mathcal A})) \mapsto (\rm{object}, \Vdash, \rm{attribute})$ or, as is more typical in @Barwise1, the ‘dual’ form $(\rm{Tok}({\mathcal A}), \Vdash_{{\mathcal A}}, \rm{Typ}({\mathcal A})) \mapsto (\rm{attribute}, \Vdash, \rm{object})$. As will be seen in §\[flip\] below, these interpretations are interchangeable. Let us also keep in mind that for Chu spaces, “objects” and “attributes” can be aptly replaced by terms such as “events” and “states”, with $\Vdash$ then interpreted as selecting the events that occur in a given state or, alternatively, the states participating in a given event.
\[multivalued-2\] As for a ${\mathsf{K}}$-valued Chu space in \[multivalued-1\], we may speak of a *${\mathsf{K}}$-valued Classification* ${\mathcal A}= \langle \rm{Tok}({\mathcal A}), \rm{Typ}({\mathcal A}), \Vdash_{{\mathcal A}} \rangle$, with a classification relation $\Vdash_{{\mathcal A}} \subseteq \rm{Tok}({\mathcal A}) \times \rm{Typ}({\mathcal A}) {\longrightarrow}{\mathsf{K}}$, where $\Vdash_{{\mathcal A}}(a,b)$ is an element of ${\mathsf{K}}$.
Let ${\mathcal A}= \langle Pts, \mathbf{U}, \Vdash_{{\mathcal A}} \rangle$ where $Pts$ denotes points of a topological space, $\mathbf{U}$ denotes the open sets of that space, and $x \Vdash_{{\mathcal A}} U$ if and only if $x \in U$. Thus, points are classified by the open sets in which they are contained. The open sets may be classified by the points within them by reversing $\Vdash_{{\mathcal A}}$.
Following @Allwein1, let $$\mathbf{FOL} = \langle Models, Sentences, \Vdash_{\mathbf{FOL}} \rangle,$$ where $Sentences$ are sentences in First Order Logic (FOL). $Models$ are models of FOL sentences, and $x \Vdash_{\mathbf{FOL}} S$ if and only if $x$ is a model of the sentence $S$. Here, there are various internal relations holding on both the set of sentences and that of models, but none are imposed as external conditions in this case without further modification. One could also reverse matters, by taking the Types to be Models, and the Tokens as Sentences, so that Sentences in this case would be classified by Models.
Infomorphisms {#tokens-1}
-------------
Here we recall the idea of a Chu morphism in order to link the information between two given classifications ${\mathcal A}= \langle \rm{Tok}({\mathcal A}), \rm{Typ}({\mathcal A}), \Vdash_{{\mathcal A}} \rangle$ and ${\mathcal B}= \langle \rm{Tok}({\mathcal B}), \rm{Typ}({\mathcal B}), \Vdash_{{\mathcal B}} \rangle$. In this case it is useful to define “switching relations” $\overrightarrow{f}: \rm{Typ}({\mathcal A}) {\longrightarrow}\rm{Typ}({\mathcal B})$ and $\overleftarrow{f}: \rm{Tok}({\mathcal B}) {\longrightarrow}\rm{Tok}({\mathcal A})$ that can be specified by introducing the Channel Theory concept of an *infomorphism*. Specifically:
\[infomorph-1\] Given two classifications ${\mathcal A}= \langle \rm{Tok}({\mathcal A}), \rm{Typ}({\mathcal A}), \Vdash_{{\mathcal A}} \rangle$ and ${\mathcal B}= \langle \rm{Tok}({\mathcal B}), \rm{Typ}({\mathcal B}), \Vdash_{{\mathcal B}} \rangle$, an *infomorphism* $f: {\mathcal A}\rightleftarrows {\mathcal B}$, is a pair of contravariant maps
- $\overrightarrow{f}: \rm{Typ}({\mathcal A}) {\longrightarrow}\rm{Typ}({\mathcal B})$
- $\overleftarrow{f}: \rm{Tok}({\mathcal B}) {\longrightarrow}\rm{Tok}({\mathcal A})$
such that for all $b \in \rm{Tok}({\mathcal B})$, and for all $a \in \rm{Typ}({\mathcal A})$, we have $$\overleftarrow{f}(b) \Vdash_{{\mathcal A}} a, ~\text{if and only if}~ b \Vdash_{{\mathcal B}} \overrightarrow{f}(a).$$ This last condition may be schematically represented by: $$\label{info-diagram-1}
\xymatrix@!C=3pc{\rm{Typ}({\mathcal A}) \ar[r]^{\overrightarrow{f}} & \rm{Typ}({\mathcal B}) \ar@{-}[d]^{\Vdash_{{\mathcal B}}} \\
\rm{Tok}({\mathcal A}) \ar@{-}[u]^{\Vdash_{{\mathcal A}}} & \rm{Tok}({\mathcal B}) \ar[l]_{\overleftarrow{f}}}$$
Note that this definition, given in @Barwise1, employs the ‘dual’ interpretation of types as objects and tokens as attributes. Interpreting tokens as objects and types as attributes yields infomorphisms with the usual Chu-morphism arrow directions.
In the context of situations, ‘attributes’ can be interpreted as statements of ‘situation types’. In the Dretske spirit, to say that “$x$ is $T_1$” transmits information that “$y$ is $T_2$” can be represented as an informorphism representing these classification statements. Here the content of information such as $(T_1,T_2)$ is defined as the ‘type’, and the carrier of the respective types, such as $(x,y)$, is defined as the ‘token’.
Let $\mathbf{M} = \langle Messages, Contents, \Vdash_{\mathbf{M}} \rangle$ where Messages are classified by their Contents [@Allwein1]. Suppose we have another such classification $\mathbf{M}' = \langle Messages', Contents', \Vdash_{\mathbf{M}'} \rangle$. An infomorphism $f: \mathbf{M} {\longrightarrow}\mathbf{M'}$ may represent a function decoding messages from $\mathbf{M}'$ to messages in $\mathbf{M}$, so that whatever can be noted about the translation, may be mapped into something noted in the original message. That is, $m^f \Vdash_{\mathbf{M}} C \Leftrightarrow m \Vdash_{\mathbf{M}'} C^f$.
Here is an example from decision theory [@Allwein §2.3]. Let $\mathbf{s}$ be a classification of propositional logic and its model states, and let $f$ represent a decision which evaluates a state $s$ and the agent making the decision (e.g. “either walk home, or take the bus home”). Let $\mathbf{O}$ be the classification of outcomes, and let $s^f$ represent a particular outcome of the decision of choosing either option (either “walk home” or “take a bus home”). A proposition, denoted $Q$ over outcomes (in accordance with a slogan such as “Keeping Fit”) characterizes them, and let $Q^f$ be the proposition categorizing all of the states in which $Q$ is satisfied. Thus, with respect to the above scheme of infomorphisms, we set the classification ${\mathcal A}= \mathbf{O}$, and ${\mathcal B}= \mathbf{S}$, and thus leads to $$\label{info-diagram-2}
\xymatrix@!C=3pc{{\rm{Typ}}(\mathbf{O}) \ar[r]^{\overrightarrow{f}} & {\rm{Typ}}(\mathbf{S}) \ar@{-}[d]^{\Vdash_{\mathbf{S}}} \\
{\rm{Tok}}(\mathbf{O}) \ar@{-}[u]^{\Vdash_{\mathbf{O}}} & {\rm{Tok}}(\mathbf{S}) \ar[l]_{\overleftarrow{f}}}$$ in which case the infomorphism condition is expressed by $s \Vdash_{\mathbf{S}} Q^f$ if and only if $s^f \Vdash_{\mathbf{O}} Q$.
\[tokens-states\] It is worth noting that in the framework of infomorphisms, there is a natural mapping between tokens $A$ and the set of informational states: $$A {\longrightarrow}{\mathsf{S}}(A).$$ For instance, as pointed out by @Barwise1 [§2.5], the truth classification of a first order language $L$ is the classification whose types are the sentences of $L$, and the tokens are the $L$-structures. In which case, the classification relation is defined by $N \Vdash {\varphi}$, if and only if ${\varphi}$ is true in the structure of $L$ [@Barwise1 Example 4.3].
Information channels
--------------------
An *information channel* $\mathbf{Chan}$ consists of an indexed family $\{f_i: {\mathcal A}_i \rightleftarrows \mathbf{C}\}_{i \in \mathcal{I}}$ of infomorphisms having a common codomain $\mathbf{C}$ called the *core of the channel $\mathbf{Chan}$*: $$\xymatrix{&\mathbf{C} & \\
{\mathcal A}_1 \ar[ur]^{f_1} & {\mathcal A}_2 \ar[u]_{f_2} & \ldots ~{\mathcal A}_i \ar[ul]_{f_i}~ \ldots
}$$ The core $\mathbf{C}$ is essentially a carrier of information flow between the $f_i$ and hence between the classifications ${\mathcal A}_i$, and is itself a classification in the above sense. The tokens $\rm{Tok}(\mathbf{C})$ of $\mathbf{C}$ are called *connections*. A connection $c$ is said to *connect* the tokens $f_i(c)$ of the classifications ${\mathcal A}_i$ for $i \in \mathcal{I}$ (note that tokens are mapped from $\mathbf{C}$ to the ${\mathcal A}_i$ in the ‘dual’ interpretation of @Barwise1). A channel with index set $\{0, \ldots, n-1\}$ is called an *$n$-ary* channel. Composing information channels amounts to taking their limit and the channels themselves may be refinable by straightforward categorical means [@Barwise1].
The above definition extends the intuitive picture of a channel as a wire connecting two agents (i.e. classifiers) to the idea of a blackboard or other shared memory via which multiple classifiers exchange information. The shared memory $\mathbf{C}$ being itself a classifier provides it with a structure that can affect how information is written to and read from it. Consistent with the essentially causal notion of information of @Dretske1, the connections between the tokens of different classifiers are purely functional; no overarching semantics is assumed. How such a semantics can be constructed, *post hoc*, given a channel is discussed in §\[ontologies\] below.
\[idealization\] For instance, if we have a binary channel $\mathbf{Chan} = \{f: {\mathcal A}\rightleftarrows \mathbf{C}, g: {\mathcal B}\rightleftarrows \mathbf{C} \}$, then the local logic (see §\[local\] below) on ${\mathcal B}$ induced by $\mathbf{Chan}$, is the logic ${\mathsf{Lg}}_{\mathbf{Chan}}({\mathcal B}) = g^{-1}[f [{\mathsf{Lg}}({\mathcal A})]]$ (in @Barwise1 [14.1] classifications ${\mathcal A}$ and ${\mathcal B}$ are interpreted as “idealization” and “reality”, respectively). This induced logic can be characterized by [@Barwise1 Prop. 14.2]:
- A partition $\langle \Gamma, \Delta \rangle$ of $\rm{Typ}({\mathcal B})$ is consistent in ${\mathsf{Lg}}_{\mathbf{Chan}}({\mathcal B})$ if and only if $\langle f^{-1}[g[\Gamma]], f^{-1}[g[\Delta]] \rangle$ is the state description of some $a \in \rm{Tok}({\mathcal A})$.
- A token $b \in \rm{Tok}({\mathcal B})$ is normal in ${\mathsf{Lg}}_{\mathbf{Chan}}({\mathcal B})$, if and only if it is connected to some token $a \in \rm{Tok}({\mathcal A})$.
Cocone of infomorphisms {#info-cocone}
-----------------------
A network of infomorphisms between classifications admits a limit classification that gathers all of the information in the network into a single classification (a *cone*) with projections back down to the individual classifications [@Barwise1]. There is a dual notion which we will describe as follows. A channel is an instance of the more general category-theoretic concept of a *cocone* being the core classification. To motivate the construction, consider any finite directed graph with vertex labels $1, 2, ..., n$ and edge labels $f_{ij}$. Considering such a graph to represent a network of communicating agents is, from a category-theoretic perspective, invoking a map $G$ (technically, a functor from the category of finite directed graphs to the category of classifications) that constructs a classification $G(i)$ at each vertex and an infomorphism $G(f_{ij})$ at each edge. A *commuting finite cocone* of infomorphisms (e.g. @Barwise1 [@Allwein]) is a finite network of classifications $G(i)$ and infomorphisms $G(f_{ij})$, a vertex classification $\mathbf{C}$, and a collection of infomorphisms $g_i: G(i) {\longrightarrow}\mathbf{C}$. $$\label{cocone-diagram}
\xymatrix{& & &\mathbf{C} & \\
&\ar[urr]^{g_1} G(1) \dots & G(i) \ar[ur]_{g_i} \ar[rr]_{G(f_{ij})} & & G(j) \ar[ul]^{g_j} & \ar[ull]_{g_n} \ldots G(n)
}$$ The commutativity condition is that for all $f_{ij}$, we have $g_i = g_j \circ G(f_{ij})$. The base of the cocone consists of the classifications and infomorphisms constructed by $G$; the *cocone vertex classification* $\mathbf{C}$ together with the maps $g_i$, is a channel. Note that in the complementary sense, a *commuting finite cone* of infomorphisms consists of a finite network of classifications $G(i)$ and infomorphisms $G(f_{ji})$, a vertex classification $\mathbf{C}$, and a collection of infomorphisms $g_i: G(i) {\longrightarrow}\mathbf{C}$. For all $f_{ji}$, we have $g_i = G(f_{ji}) \circ g_j$, and all arrows in the above diagram are reversed.
In short, we have this colimit classification into which there are infomorphisms from each constituent classification, and this colimit contains all of the information that is common to the different component parts of the network. The generalization from channel to cocone will prove useful in the discussion of “minimal covers” of distributed systems in §\[distributed\]. We further characterize cocones and relate them to colimits in the descriptive discussion of §\[colimits\]. We then apply these concepts to model abstraction-based categorization in §\[tt-flow\] and to mereological categorization in §\[mereological\].
The flip of a classification {#flip}
----------------------------
For any classification ${\mathcal A}$, the *flip* of ${\mathcal A}$, is the classification ${\mathcal A}^{\perp}$ whose tokens are the types of ${\mathcal A}$, whose types are the tokens of ${\mathcal A}$, such that ${\alpha}\Vdash_{{\mathcal A}^\perp} a$ if and only if $a \Vdash_{{\mathcal A}} {\alpha}$ (see @Barwise1 [§4.4]). In deciding how to model a classification there may be epistemological questions, e.g. the types in question are given as things or attributes we may know about, and the tokens are those things we wish to have information about. The fact that the flip of a classification is a classification (and both can be treated as Chu spaces) and these behave well under infomorphisms, means that a situation involving types or tokens, can be dualized to tokens or types. For instance, “the type set of a token” dualizes to “the token set of a type”. Effectively, $f: {\mathcal A}\rightleftarrows {\mathcal B}$ is an informorphism if and only if $f^\perp: {\mathcal B}^\perp \rightleftarrows {\mathcal A}^\perp$ is an infomorphism [@Barwise1 Prop. 4.19]. Further, $({\mathcal A}^\perp)^\perp = {\mathcal A}$ with $(f^\perp)^\perp = f$, and $(fg)^\perp = g^\perp f^\perp$ [@Barwise1 Prop. 4.20]. Thus for $f, f^\perp$ respectively, we have the diagrams
$$\xymatrix@!C=3pc{\rm{Typ}({\mathcal A})\ar[r]^{\overrightarrow{f}} & \rm{Typ}({\mathcal B}) \ar@{-}[d]^{\Vdash_{{\mathcal B}}} \\
\rm{Tok}({\mathcal A}) \ar@{-}[u]^{\Vdash_{{\mathcal A}}} & \rm{Tok}({\mathcal B}) \ar[l]_{\overleftarrow{f}}} ~ ~ ~
\xymatrix@!C=3pc{\rm{Tok}({\mathcal B}) \ar[r]^{\overleftarrow{f}} & \rm{Tok}({\mathcal A}) \ar@{-}[d]^{\Vdash^{-1}_{{\mathcal A}}} \\
\rm{Typ}({\mathcal B}) \ar@{-}[u]^{\Vdash^{-1}_{{\mathcal B}}} & \rm{Typ}({\mathcal A}) \ar[l]_{\overrightarrow{f}}}$$
The nerve of a classification
-----------------------------
As any classification is a Chu space, any operation defined for Chu spaces is meaningful for a classification. A finite sample ${\mathcal F}$ of ‘attributes’, for example, becomes a finite sample of ‘tokens’ (or ‘types’). Simplicial complexes are defined as in §\[simpl-def\] and nerves as in §\[chu-nerve\]. The Čech nerve of a classification ${\mathcal A}= \langle \rm{Tok}({\mathcal A}), \rm{Typ}({\mathcal A}), \Vdash_{{\mathcal A}} \rangle$, for example, is the simplicial complex $N({\mathcal A})$ with vertex set $\rm{Tok}({\mathcal A})$, where a (non-empty) subset $\{b_0, \ldots b_p\}$ of $\rm{Tok}({\mathcal A})$ is a $p$-simplex if there is a type $v \in \rm{Typ}({\mathcal A})$ satisfying $v \Vdash_{{\mathcal A}} b_i$, for $0 \leq i \leq p$. The notion of the Vietoris nerve follows in a similar way as in §\[chu-nerve\].
If ${\mathcal F}= \rm{Tok}({\mathcal B})$ is a finite sample of tokens of a classification ${\mathcal B}$, we have the infomorphism : $$f=(\overleftarrow{f}, \overrightarrow{f}): (\overleftarrow{f}({\mathcal F}), \rm{Typ}({\mathcal A}), \Vdash_{{\mathcal A}}) {\longrightarrow}({\mathcal F}, \rm{Typ}({\mathcal B}), \Vdash_{{\mathcal B}}),$$ while for a finite sample ${\mathcal G}\subseteq \rm{Typ}({\mathcal A})$, we have: $$f=(\overleftarrow{f}, \overrightarrow{f}): (\rm{Tok}({\mathcal A}), {\mathcal G}, \Vdash_{{\mathcal A}}) {\longrightarrow}(\rm{Tok}({\mathcal B}), \overrightarrow{f}({\mathcal G}), \Vdash_{{\mathcal B}}),$$ We will henceforth assume that finite samples of tokens (and types) have been taken, so that we may consider, as in , well-defined simplicial maps $$\label{class-nerve-1}
N(f): N({\mathcal A}) {\longrightarrow}N({\mathcal B}),$$ as defined for the Čech nerve of the corresponding Chu spaces, here with respect to finite samples of tokens and types.
Associating a theory with a classification
------------------------------------------
Here and in the following sections we collect together some useful definitions from @Barwise1 and @Barwise2, starting with *sequents* and *theories*:
Let $\Sigma$ be an arbitrary set (which may be viewed as set of *types*). A binary relation $\vdash$ between subsets of $\Sigma$ is called a *consequence relation on $\Sigma$*. A *(Gentzen) sequent* is a pair $I= \langle \Gamma, \Delta \rangle$ of subsets of $\Sigma$ (here it is apt to view $\Gamma$ and $\Delta$ as sets of *situation types*). A sequent $I= \langle \Gamma, \Delta \rangle$ is said to hold of a situation $s$ provided that if $s$ supports every type in $\Gamma$, then it supports some type in $\Delta$. A sequent $I$ is said to be *information* about a set $S$ of situations if it holds at each $s \in S$; here again the causal notion of information flow is evident. Finally, a sequent is called a *partition* of a set $\Sigma'$ if $\Gamma \cup \Delta = \Sigma'$ and $\Gamma \cap \Delta = \emptyset$.
A *theory* is a pair $T= \langle \Sigma, \vdash_T \rangle$, where $\vdash_T$ is a consequence relation on $\Sigma$. A *constraint* of the theory $T$ is a sequent $\langle \Gamma, \Delta \rangle$ of $\Sigma$ for which $\Gamma \vdash_T \Delta$. A sequent $\langle \Gamma, \Delta \rangle$ is *$T$-consistent* if $\Gamma \nvdash_T \Delta$.
Here again, the idea of some aspects of a situation either causally requiring or merely causally allowing other aspects of a situation makes this definition clear.
Each classification has a theory associated with it in the following way (see also Definition \[local-2\] below). A *theory* ${\mathsf{Th}}({\mathcal A}) = (\Sigma_{{\mathcal A}}, \vdash_{{\mathcal A}})$ generated by a classification ${\mathcal A}$, satisfies for all types ${\alpha}$ and all sets $\Gamma, \Gamma', \Delta, \Delta', \Sigma', \Sigma_0, \Sigma_1$ of types [@Barwise1 Prop 9.5]:
- *Identity*: ${\alpha}\vdash {\alpha}$.
- *Weakening*: If $\Gamma \vdash \Delta$, then $\Gamma, \Gamma' \vdash \Delta, \Delta'$.
- *Global cut*: If $\Gamma, \Sigma_0 \vdash \Delta, \Sigma_1$, for each partition $\langle \Sigma_0, \Sigma_1 \rangle$ of $\Sigma$, then $\Gamma \vdash \Delta$.
More generally, we can say that a theory $T = \langle \Sigma, \vdash_T \rangle$ is *regular* if it satisfies the above three conditions.
Local logics {#local}
------------
We can specify a classification of a regular theory $T$ as given by:
- $\rm{Typ}\mathnormal{({\mathsf{Cl}}(T))} = \rm{Typ}\mathnormal{(T)}$.
- $\rm{Tok}\mathnormal{({\mathsf{Cl}}(T)) = \{ \langle \Gamma, \Delta \rangle: \langle \Gamma, \Delta \rangle~
\text{is a $T$ consistent partition of}}~\rm{Typ} \mathnormal{(T) \} }$.
- $\langle \Gamma, \Delta \rangle \vdash_{{\mathsf{Cl}}(T)} {\alpha}$ if and only if ${\alpha}\in \Gamma$.
Indeed, for any regular theory it can be seen that ${\mathsf{Th}}({\mathsf{Cl}}(T)) = T$.
Since we are mainly considering distributed systems, and information processing entails computation within a logical framework, the following system of local logics [@Barwise1 Def. 12.1] is one suited to representing various types of state spaces.
\[local-1\] *A local logic* consists of a triple $({\mathcal L}= \langle \rm{Tok}({\mathcal L}), \rm{Typ}({\mathcal L}), \Vdash_{{\mathcal L}} \rangle, \vdash_{{\mathcal L}}, {\mathsf{N}}_{{\mathcal L}})$ in which we have:
- a classification ${\mathcal L}= \langle \rm{Tok}({\mathcal L}), \rm{Typ}({\mathcal L}), \Vdash_{{\mathcal L}} \rangle$,
- a regular theory ${\rm{Th}}({\mathcal L}) = (\rm{Typ}({\mathcal L}), \vdash_{{\mathcal L}})$, and
- a subset ${\mathsf{N}}_{{\mathcal L}} \subset \rm{Tok}({\mathcal L})$, called *the normal tokens* of ${\mathcal L}$, which satisfy all of the constraints of the theory $\rm{Th}({\mathcal L})$ in (2).
\[local-2\] Let ${\mathcal A}$ be a classification. The *local logic generated by ${\mathcal A}$*, denoted ${{\mathsf{Lg}}}({\mathcal A})$, has classification ${\mathcal A}$, a regular theory ${{\mathsf{Th}}}({\mathcal A}) = (\rm{Typ}({\mathcal A}), \vdash_{{\mathcal A}})$, and all its tokens are normal. A logic is said to be *natural* if it is generated by some classification.
In fact, for any local logic $\mathcal{L}$ on ${\mathcal A}$, we have $\mathcal{L} = {{\mathsf{Lg}}}({\mathcal A})$ by @Barwise1 [Prop. 12.7]. This relationship will be exemplified in Example \[building\] in the context of ontologies.
Intuitively, a local logic is “local” to the classification that generates it. Infomorphisms allow mapping the local logic of one classification to that of another; hence we can think of channels as supporting the flow of locally-defined logical relations between classifications. Recalling from §\[Chu-FSA\] that any classification can be interpreted as defining a coarse-graining and hence a “scale” at which information is being organized and represented, each local logic can be thought of as a “logic at some level of description.” This interpretation is made explicit in §\[cccd-logic-1\] and §\[mereotop\] below. As any classification can also be interpreted as describing a state space (§\[chu-info-1\]), one can further associate a canonical logic ${\mathsf{Lg}}({\mathsf{S}})$ to any state space ${\mathsf{S}}$. Specifically, if ${\mathsf{S}}$ is such a state space with a classification of events ${\mathsf{Evt}}({\mathsf{S}})$, then we can speak of an ${\mathsf{S}}$-logic as a logic $\mathfrak{L}$ on this classification such that ${\mathsf{Lg}}({\mathsf{S}}) \subseteq \mathfrak{L}$, with the intuition that this ${\mathsf{S}}$-logic can accommodate the theory that is implicit to the structure of ${\mathsf{S}}$ [@Barwise1 §16].
In @Barwise2, ${\mathcal L}$ is called an *information context* and $\vdash$ is a binary relation relating sets of situation types. In this case ${\mathsf{N}}_{{\mathcal L}}$ is said to be a set of *normal situations*. Intuitively, these are the situations that the available information concerns. They may comprise all or only some of the situations satisfying the information. For instance, we may start with some set of normal situations accounting for an individual’s experiences to date, and then the information context consists of all the sequents satisfied by, i.e. consistent with, this experience. Stepping outside of the context generates “surprise” in the sense of expectation violation (cf. @Friston2).
Next, we look to what extent an infomorphism between classifications will respect the associated local logics. This is given by the following (@Barwise1 [12.3]):
\[logic-info\] A *logic infomorphism* $f: \mathcal{L}_1 \leftrightarrows \mathcal{L}_2$, consists of a covariant pair $f= \langle f\sphat, f\spcheck \rangle$ of functions satisfying
- $f: {\mathsf{Cl}}(\mathcal{L}_1) \leftrightarrows {\mathsf{Cl}}(\mathcal{L}_2)$ is an infomorphism of classifications.
- $f\sphat : \mathsf{Th}(\mathcal{L}_1) {\longrightarrow}\mathsf{Th}(\mathcal{L}_2)$ is a theory interpretation, and
- $f\spcheck[{\mathsf{N}}_{\mathcal{L}_2}] {\longrightarrow}{\mathsf{N}}_{\mathcal{L}_1}$
For further consequences of this definition, see @Barwise1.
Boolean Classification
----------------------
We can exemplify local logics following @Barwise1 [@Barwise3] in terms of a Boolean classification ${\mathcal A}= \langle S, \Sigma, \Vdash, \wedge, \neg \rangle$. Here we have a set $S$ of *situations* (tokens as objects) and a set $\Sigma$ of *propositions* (types as attributes). This leads to a *Boolean local logic* ${\mathcal L}= \langle {\mathcal A}, \vdash, N \rangle$ where $N \subseteq S$ consists of the normal situations [@Barwise1; @Barwise3]. If $s \in N$ is a normal situation, $\Gamma \vdash \Delta$ and $s \Vdash p$, for all $p \in \Gamma$, then $s \Vdash q$ for some $q \in \Delta$. A partial ordering “$\subseteq$” on local logics ${\mathcal L}_1, {\mathcal L}_2$ on a fixed classification of ${\mathcal A}$ is defined by ${\mathcal L}_1 \subseteq {\mathcal L}_2$, if and only if
- for all sets $\Gamma, \Delta$ of propositions, $\Gamma \vdash_{{\mathcal L}_1} \Delta$ entails $\Gamma \vdash_{{\mathcal L}_2} \Delta$, and
- every situation of ${\mathcal A}$ that is normal in ${\mathcal L}_2$ is also normal in ${\mathcal L}_1$.
Now, if we take any set of sequents $T$, the logic ${\mathsf{Lg}}({\mathcal A}^T)$ generated by $T$ on ${\mathcal A}$:
- has as normal situations all of those situations that satisfy the sequents in $T$,
- has as constraints all sequents satisfied by all situations in $N$, and
- has as normal situations all of the situations of ${\mathcal A}$ satisfying these constraints.
Given a fixed Boolean classification [@Barwise3]:
- If $T_0 \subseteq T_1$, then ${\mathsf{Lg}}({\mathcal A}^{T_0}) \subseteq {\mathsf{Lg}}({\mathcal A}^{T_1})$.
- If $N_0 \supseteq N_1$, then ${\mathsf{Lg}}({\mathcal A}_{N_0}) \subseteq {\mathsf{Lg}}({\mathcal A}_{N_1})$.
Suppose ${\mathcal A}$ is a classification of bird sightings (observations), and $N$ consists of the actual sightings to date. Then ${\mathsf{Lg}}({\mathcal A}_N)$ has as constraints all sequents satisfied by all those bird sightings to date, and the normal situations consist of all bird sightings that satisfy all of these constraints, a set that clearly contains $N$. This logic may entail the constraint BIRD $\vdash$ FLY, a constraint that holds as long as the situations encountered are meaningfully compatible with the elements of $N$. But now, suppose a penguin is observed. It will lie outside of the normal situations since it violates BIRD $\vdash$ FLY. This observation uncovers a new set $N' \supset N$, and accordingly their logics satisfy ${\mathsf{Lg}}({\mathcal A}_{N'}) \subseteq {\mathsf{Lg}}({\mathcal A}_N)$. There will be fewer constraints tenable in ${\mathsf{Lg}}({\mathcal A}_{N'})$ as we can see, since BIRD $\nvdash$ FLY in this new logic.
An excursion into Channel Theory II {#channel-II}
===================================
Classifications and channels have been applied widely in theoretical computer science; we briefly review some of these applications here as motivations for applying these tools to perceptual processing.
The information channel in a MLP network {#mlp}
----------------------------------------
One of the earliest ANNs studied was the *Multilayer Perceptron* (MLP) network [@Rosenblatt; @Rummelhart3]. As with typical ANNs, it has an input layers ($I_i$), hidden layers ($H_i$), and an output layer ($O_i$) with weighted directional (in an MLP, exclusively feedforward) linkages between subsequent layers. @Kikuchi develop a Chu space/Channel Theory representation of a 3-layer MLP, showing how the synaptic weights between layers form a channel; we follow their example closely.
Let $\mathbf{wo}_{ij}$ denote a synaptic weight between the $j$-th neuron in the hidden layer and the $i$-th neuron in the output layer. Similarly, let $\mathbf{wh}_{jk}$ denote a synaptic weight between the $k$-th neuron in the input layer and the $j$-th neuron in the hidden layer. Then for a given state function $f(x)$, the layers $O_i, H_i$ and $I_i$ are related in accordance with $$O_i = f( \sum_j \mathbf{wo}_{ij} H_j) = f(\sum_j \mathbf{wo}_{ij} ~f(\sum_k \mathbf{wh}_{jk} I_k)).$$ It is convenient to regard an MLP with a fixed topology as a map $\mathbf{F}: \mathbf{I} {\longrightarrow}\mathbf{O}$, from the input data space $\mathbf{I} =\{I_i\}$ to the output data space $\mathbf{O} =\{O_i\}$, so that $\mathbf{F}$ is uniquely defined by a point in the parameter space of weights $\mathbf{\Phi} = \{ \langle \mathbf{wh}, \mathbf{wo} \rangle \}$. In this way, a fixed topology on a MLP can be represented as $\mathbf{F}_{\langle \mathbf{wh}, \mathbf{wo} \rangle}$, once given $\langle \mathbf{wh}, \mathbf{wo} \rangle \in \mathbf{\Phi}$.
Next consider the sub-parameter spaces $\mathbf{\Phi}_h = \{\langle \mathbf{wh} \rangle \}$ and $\mathbf{\Phi}_o = \{\langle \mathbf{wo} \rangle \}$, and the following three classifications
- ${\mathcal A}= (\rm{Tok}({\mathcal A}), \rm{Typ}({\mathcal A}), \Vdash_{{\mathcal A}})$ (the states of “cognition” i.e. of $\mathbf{O}$)
- ${\mathcal B}= (\rm{Tok}({\mathcal B}), \rm{Typ}({\mathcal B}), \Vdash_{{\mathcal B}})$ (the states of the “environment” i.e. of $\mathbf{I}$)
- ${\mathcal C}= (\rm{Tok}({\mathcal C}), \rm{Typ}({\mathcal C}), \Vdash_{{\mathcal C}})$ (the states of the network)
where for the tokens $A = \mathbf{\Phi}_h,~ B = \mathbf{\Phi}_o$ and $C = \mathbf{\Phi}$, we define projections $$\begin{aligned}
g_h &: \mathbf{\Phi} {\longrightarrow}\mathbf{\Phi}_h, ~
\langle \mathbf{wh}, \mathbf{wo} \rangle \mapsto \langle \mathbf{wh} \rangle \\
g_o &: \mathbf{\Phi} {\longrightarrow}\mathbf{\Phi}_o, ~
\langle \mathbf{wh}, \mathbf{wo} \rangle \mapsto \langle \mathbf{wo} \rangle
\end{aligned}$$ as well as the obvious respective inclusions $f_h: \mathbf{\Phi}_h {\longrightarrow}\mathbf{\Phi}, ~f_o: \mathbf{\Phi}_o {\longrightarrow}\mathbf{\Phi}$. Thus, we obtain a core (and vertex of a cocone) that is in ${\mathcal C}$ along with an information channel $$\xymatrix{&{\mathcal C}= \langle \mathbf{wh}, \mathbf{wo} \rangle & \\
\langle \mathbf{wh} \rangle \ar[ur]^{f_h} & & \langle \mathbf{wo} \rangle \ar[ul]_{f_o}
}$$ where, as shown by @Kikuchi, an algorithm for modifying $\langle \mathbf{wh}, \mathbf{wo} \rangle$ corresponds to a local logic on ${\mathcal C}$. This method can be developed in terms of Distributed Systems, as explained below in §\[distributed\] (see also Remark \[gnw-remark\]).
Distributed Systems {#distributed}
-------------------
Following the development of @Barwise1 [Ch. 6] we provide an example of Dretske’s “Xerox principle”, namely, that information flow is transitive. Consider two information channels sharing common infomorphisms. Suppose the first channel represents the examination of a map, capturing the notion of a person’s perceptual state carrying information about the map being examined. The second channel represents the informational relationship between the map and the region it depicts. These coupled channels can be illustrated: $$\xymatrix{&\mathbf{B}_1 & & \mathbf{B}_2 \\
{\mathcal A}_1 \ar[ur]^{f_1} & & {\mathcal A}_2\ar[ul]_{f_2}\ar[ur]^{f_3} & & {\mathcal A}_3 \ar[ul]_{f_4}
}$$ Recall that the elements of $\rm{Tok}(\mathbf{B}_1)$ are ‘connections’. In this case the connections are spatio-temporal perceptual events involving persons in $\rm{Tok}({\mathcal A}_1)$ looking at maps in (i.e. elements of) $\rm{Tok}({\mathcal A}_2)$. The connections of the second channel, elements of $\rm{Tok}(\mathbf{B}_2)$, are spatio-temporal events involving making the maps in $\rm{Tok}({\mathcal A}_2)$ to represent various regions in $\rm{Tok}({\mathcal A}_3)$. Under certain circumstances, a person’s perceptual state carries information about a particular mountain, given that the person is reading a map showing that mountain. In this regard, we may reasonably consider ${\mathcal A}_1$ as the *idealized space* of the physical space $A_3$.
The next step is to construct another channel that fits both $\mathbf{B}_1$ and $\mathbf{B}_2$ together. The process is: i) choose a person, ii) go to a map she is reading, and then iii) proceed to the region shown on that map. Here we will restrict to pairs $c=(b_1, b_2)$ ((perceptual event, map-making)), so that $f_2(b_1) = f_3(b_2)= a_2$ holds, i.e. there is just one map in question. In this way, types ${\beta}_1 = f_2({\alpha}_2)$ and ${\beta}_2 = f_3({\alpha}_2))$ are equivalent since they are both translations of $a_2$. This is built into the channel by identifying ${\beta}_1$ and ${\beta}_2$ (cf. the biextensional collapse of a Chu space) and gives rise to a new classification ${\mathcal C}$ having the above tokens and $\rm{Typ}({\mathcal C}) = \rm{Typ}(\mathbf{B}_1 \cup \mathbf{B}_2)$, but identifying types originating from a common type in ${\mathcal A}_2$. Thus we obtain a new channel with core $\mathbf{C}$, connecting $\mathbf{B}_1$ and $\mathbf{B}_2$, as depicted below: $$\xymatrix{& & \mathbf{C} & \\ &\mathbf{B}_1\ar[ur]^{g_1} & & \mathbf{B}_2 \ar[ul]_{g_2} \\
{\mathcal A}_1 \ar[ur]^{f_1} & & {\mathcal A}_2\ar[ul]_{f_2}\ar[ur]^{f_3} & & {\mathcal A}_3 \ar[ul]_{f_4}
}$$ The channel infomorphisms are defined via composition $h_1 = g_1 f_1$, and $h_3= g_2 h_4$, so linking ${\mathcal A}_1$ to ${\mathcal A}_3$.
In general, we have
\[distributed-def\] A *Distributed System* ${\mathcal A}$ consists of an indexed family ${\mathsf{Cl}}({\mathcal A}) = \{{\mathcal A}_i\}_{i \in \mathcal{I}}$ of classifications, together with a set ${\mathsf{Inf}}({\mathcal A})$ of infomorphisms having both domain and codomain in ${\mathsf{Cl}}({\mathcal A})$. Each classification may be taken to support a local logic, along with the core of the channel.
\[minimal\] An information channel $\mathbf{Chan} = \{h_i: {\mathcal A}_i \rightleftarrows \mathbf{C} \}$ *covers* a distributed system ${\mathcal A}$ if ${\mathsf{Cl}}({\mathcal A})
= \{{\mathcal A}_i\}_{i \in \mathcal{I}}$, and for each $i,j \in \mathcal{I}$, and for each infomorphism $f: {\mathcal A}_i \rightleftarrows {\mathcal A}_j$ in ${\mathsf{Inf}}({\mathcal A})$, the following diagram commutes $$\xymatrix{& \mathbf{C} & \\
{\mathcal A}_i \ar[ur]^{h_i} \ar[rr]_{f} & & {\mathcal A}_j \ar[ul]_{h_j}
}$$ $\mathbf{Chan}$ is said to be a *minimal cover* if it covers ${\mathcal A}$, and for every other channel ${\mathcal D}$ covering ${\mathcal A}$ there is a unique infomorphism from $\mathbf{C}$ to $\mathbf{D}$.
A minimal cover of a system ${\mathcal A}$ converts the entire distributed system, consisting of multiple channels, into a single channel. Every distributed system has a minimal cover, and this cover is unique up to isomorphism [@Barwise1 Th. 6.5].
\[gnw-remark\] The constructions of §\[mlp\] and §\[distributed\] can be extended to parallel distributed processing (PDP) and to more general multi-layer, bidirectional ANNs (see e.g. @Dawson1 [@Rogers; @Rummelhart3]). They are also applicable for modelling constructs of the massively parallel, competitively based, distributed system of the *Global Neuronal Workspace* (GNW) as studied in @Baars3 [@Dehaene1; @Dehaene04; @Wallace2005]. Such modelling can be implemented, for example, by the *Learned Intelligent Distribution Agent* (LIDA) architecture [@Baars3; @Franklin1; @Friedlander]. As observed by @Maia, feedforward and feedback projections in connectionist networks can engage selective attention toward more salient inputs, producing yet strong weighting, that can predict which of the competing elements will gain access to the GNW central core (cf. @Friston2 [@Grossberg13; @Shan3], and the relationship to object-event files in §\[episodic\]).
Ontologies
----------
It is often useful, when describing events or processes in some domain, to represent the ontology of the domain explicitly as a type hierarchy. Following @Schorlemmer2005 (cf. @Kalfoglou2 [@Kalfoglou1]):
\[ontology-1\] An *ontology* is a tuple $\mathcal{O} = (C, \leq, \perp, |)$ where
- $C$ is a finite set of concept symbols;
- $\leq$ is a reflexive, transitive, and anti-symmetric relation on $C$ (a partial order);
- $\perp$ is a symmetric and irreflexive relation on $C$ (disjointness);
- $|$ is symmetric relation on $C$ (coverage)
This is a basic working definition by @Schorlemmer2005. In the case of *reference ontologies*, @Kalfoglou2 append this definition with i) a finite set $R$ of relations, and ii) a function $\sigma: R {\longrightarrow}C^{+}$ assigning to each relation its *arity*. This corresponds to the functor $(-)^+$ which sends a set $C$ to the set of finite tuples whose elements are in $C$ (see the example below).
In applications, the concepts in $C$ typically characterize concrete objects in the domain, which are brought into the theory by populating $\mathcal{O}$ with tokens. Let $X$ be a set of objects to be classified in terms of the concept symbols in $C$, via a classification relation $\Vdash_{{\mathcal A}}$; we define a classification ${\mathcal A}= \langle X, C, \Vdash_{{\mathcal A}} \rangle$, where $X = {\rm{Tok}}({\mathcal A})$, and $C = {\rm{Typ}}({\mathcal A})$. The relation $\Vdash_{{\mathcal A}}$ will have to be defined so that $\leq$, $\perp$, and $|$, are respected. This requirements lead to:
A *populated ontology* is a tuple ${\widetilde}{\mathcal{O}} = ({\mathcal A}, \leq, \perp, |)$ such that ${\mathcal A}= \langle X, C, \Vdash_{{\mathcal A}}\rangle$ is an information flow classification, ${\mathcal{O}} = ({\mathcal A}, \leq, \perp, |)$ is an ontology, and for all $x \in X$, and $c, d \in C$, we have:
- if $x \Vdash_{{\mathcal A}} c$, and $c \leq d$, then $x \Vdash_{{\mathcal A}} d$;
- if $x \Vdash_{{\mathcal A}} c$, and $c \perp d$, then $x \nVdash_{{\mathcal A}} d$;
- if $c | d$, then $x \Vdash_{{\mathcal A}} c$, or $x \Vdash_{{\mathcal A}} d$.
A populated ontology ${\widetilde}{\mathcal{O}} = ({\mathcal A}, \leq, \perp, |)$ having ${\mathcal A}= \langle X, C, \Vdash_{{\mathcal A}} \rangle$, determines a local logic $\mathfrak{L} = ({\mathcal A}, \vdash) $, whose theory $(C, \vdash)$, is given by the smallest regular theory (i.e. the smallest theory closed under Identity, Weakening, and Global Cut), such that for all $c, d \in C$, we have: $$\begin{aligned}
c \vdash d &\Leftrightarrow c \leq d \\
c, d \vdash &\Leftrightarrow c \perp d \\
\vdash c, d &\Leftrightarrow c | d
\end{aligned}$$
\[building\] To get an idea of what these last relations mean, take the case of a reference ontology $\mathcal{O} = (C, R, \leq, \perp, |, \sigma)$ as in @Kalfoglou2 [§4], with a set of concepts $C =\{{\mathsf{building,bird,starling}}\}$, the relation $R=\{\mathsf{isHavenFor} \}$, arities $\sigma(\mathsf{is HavenFor}) = \langle\mathsf{building,bird} \rangle$, where the partial order $\leq$, disjointness $\perp$, and coverage $\vert$, are defined by the following lattice: $$\xymatrix@C=4pc{& \blacklozenge \ar@{-}[dl] \ar@{-}[dr] &\\
\mathsf{building}\ar@{-}[ddr] & & \mathsf{bird} \ar@{-}[d] &\\
& & \mathsf{starling} \ar@{-}[dl] &\\
& \lozenge &
}$$ where $\blacklozenge$ is the top and $\lozenge$ is the bottom of the lattice, i.e. $\mathsf{building} \perp \mathsf{bird}$ and $\mathsf{building} \vert \mathsf{bird}$. In this set up, we then have $$\begin{aligned}
{\mathsf{building,bird}} & \vdash \\
{\mathsf{starling}} & \vdash {\mathsf{bird}}\\
&\vdash {\mathsf{building,bird}}
\end{aligned}$$ where the comma on the left-hand side has conjunctive force, whereas on the right-hand side it has disjunctive force. Thus, with respect to set of concepts $C$, the above constraints declare, respectively: nothing is both a building and a bird, all starlings are birds, and everything is either a building or a bird.
With respect to the theory in question, the corresponding sequents are: $$\begin{aligned}
& \langle \{\mathsf{bird,starling} \}, \{ \mathsf{building}\} \rangle \\
&\langle \{\mathsf{building}\}, \{\mathsf{bird,starling} \} \rangle \\
&\langle \{\mathsf{bird}\}, \{\mathsf{building,starling} \} \rangle
\end{aligned}$$
Then we have a classification in terms of the above sequents, as given by:
$\Vdash_{{\mathcal A}}$ $\mathsf{building}$ $\mathsf{bird}$ $\mathsf{starling}$
------------------------------------------------------------------------- --------------------- ----------------- ---------------------
$\langle \{\mathsf{bird,starling} \}, \{ \mathsf{building}\} \rangle$ 0 1 1
$\langle \{\mathsf{building}\}, \{\mathsf{bird,starling} \} \rangle $ 1 0 0
$ \langle \{\mathsf{bird}\}, \{\mathsf{building,starling} \} \rangle$ 0 1 0
Let the above set of sequents be denoted by $X$. Note how the sequents code the classification ${\mathcal A}= \langle X, C, \Vdash_{{\mathcal A}} \rangle$ whereby the left-hand sides of these indicate which columns contain a ‘1’ entries, and the right-hand sides indicate which columns contain ‘0’ entries. Assuming that $X$ consists of normal tokens, as in @Kalfoglou2, we obtain a local logic $\mathcal{L} = ({\mathcal A}, \vdash)$ of the ontology $\mathcal{O}$. Given that $\mathcal{L}$ is a local logic on ${\mathcal A}$, we have by @Barwise1 [Prop. 12.7], that $\mathcal{L} = {{\mathsf{Lg}}}({\mathcal A})$; that is, with regards to Definition \[local-2\], $\mathcal{L}$ is the local logic generated by the classification ${\mathcal A}$. Associated to $\mathcal{O}$ is a local, populated ontology, as shown in @Kalfoglou2 [§4], to which we refer for details.
In order to formalize semantic integration of a collection of agents in Channel Theory, @Kalfoglou1 propose: i) modeling populated ontologies of agents by classifications; ii) defining the channel, its core, and infomorphisms between classifications; iii) defining a logic on the core of the channel; and, iv) distributing the logic to the sum of agent classifications to obtain the required theory for semantic interoperability within the channel. These steps give rise to a *global ontology* for two candidate agents, $A_1$ and $A_2$, requiring semantic integration. This commences with a distributed logic of a channel $\mathcal{C}$ connecting the classifications $\mathbf{A_1}$ and $\mathbf{A_2}$ that model the agents’ populated ontologies ${\widetilde}{\mathcal{O}}_1$ and ${\widetilde}{\mathcal{O}}_2$, respectively: $$\xymatrix{& \mathbf{C} & \\
\mathbf{A}_1 \ar[ur]^{f_1} \ar[rr]_{f} & & \mathbf{A}_2 \ar[ul]_{f_2}
}$$ At the core of the channel $\mathcal{C}$, ${\rm{Typ}}(\mathbf{C})$ covers ${\rm{Typ}}(\mathbf{A}_1)$ and ${\rm{Typ}}(\mathbf{A}_2)$, while elements of ${\rm{Tok}}(\mathbf{C})$ connect tokens from ${\rm{Tok}}(\mathbf{A}_1)$ with those from ${\rm{Tok}}(\mathbf{A}_2)$. Effectively, the global ontology comes about when the logic on the core of the channel is distributed to the sum of classifications $\mathbf{A}_1 + \mathbf{A}_2$, for the total semantic integration of the combined events.
The structure of a typical ontology mapping may thus be seen as follows [@Kalfoglou2]: $$\xymatrix@C=4pc{& \mathcal{O}_{\rm{r}} \ar[dl] \ar[dr] &\\
\mathcal{O}_{\rm{loc}_1} \ar@{.>}[dr] & & \mathcal{O}_{\rm{loc}_2} \ar@{.>}[dl] & \\
& \mathcal{O}_{\rm{glob}}
}$$ where $\mathcal{O}_{\rm{r}}$ is a reference ontology, $\mathcal{O}_{\rm{loc}_1}, \mathcal{O}_{\rm{loc}_2}$ are local ontologies, and $\mathcal{O}_{\rm{glob}}$ is a global ontology.
Quotient channel
----------------
Given an invariant $I = \langle \Sigma, R \rangle$ on a classification ${\mathcal A}$, the *quotient channel of ${\mathcal A}$ by $I$* is the limit of the distributed system depicted by $${\mathcal A}{\overset {\tau_I}{{\longleftarrow}}} {\mathcal A}/{I} {\overset {\tau_I}{{\longrightarrow}}} {\mathcal A}.$$ As a refinement of any other such channel, the quotient channel makes the following diagram commute $$\xymatrix{& \mathbf{C} & \\
{\mathcal A}\ar[ur]^{g_1} & \ar[l]_{\tau_I} {\mathcal A}/{I} \ar[u]^{h} \ar[r]^{\tau_I} & {\mathcal A}\ar[ul]_{g_2}
}$$
An example is a hierarchially modular chain, where at each level of abstraction, the tokens can be inherited, and the resulting infomorphisms are created by systematically composing those from the levels below. As seen in @Franklin1 or @Friedlander, the perceptual memory relationships and actions of a typical LIDA semantic network architecture appear to fit into this pattern.
Ideas such as information flow, formal concepts, conceptual spaces, and local logics can be categorically unified when they are embraced within the abstract axiomatization of an *Institution* [@Goguen1; @Goguen2]. This consists of a functor from an abstract category of ‘Signatures’ to a category of classifications that involves ‘contexts’ linked via the ‘satisfaction relation’ ($\Vdash$). As pointed out in @Kent, information flow is a particular case of FOL (which is thus one Institution), but the classification relation between Tokens and Types abstracts the Institution satisfaction relation between structures and sentences.
State spaces and projections {#states-1}
----------------------------
Recall that a state space is a classification ${\mathsf{S}}$ for which each token is of exactly one type, and where the types of the space are simply the states themselves. Here $a$ is said to be *in state $\sigma$* if $a \Vdash_{{\mathsf{S}}} \sigma$. The space ${\mathsf{S}}$ is *complete* if every state is the state of some token.
A *projection* $f: {\mathsf{S}}_1 \rightrightarrows {\mathsf{S}}_2$ from a state space ${\mathsf{S}}_1$, to a state space ${\mathsf{S}}_2$ is given by a covariant pair of functions, such that for each token $a \in \rm{Tok}({\mathsf{S}}_1)$, we have $f({\rm{state}}_{{\mathsf{S}}_1}(a)) = {\rm{state}}_{{\mathsf{S}}_2}(f(a))$. This amounts to the commutativity of the following diagram: $$\label{proj-1}
\xymatrix@!C=3pc{{\rm{Typ}}({\mathsf{S}}_1) \ar[r]^{f} & {\rm{Typ}}({\mathsf{S}}_2) \\
{\rm{Tok}}({\mathsf{S}}_1) \ar[u]^{{\rm{state}}_{{\mathsf{S}}_1}} \ar[r]_{f} & {\rm{Tok}}({\mathsf{S}}_2) \ar[u]_{{\rm{state}}_{{\mathsf{S}}_2}}}$$
The composition of projections is well-defined, and so too is the Cartesian product $\Pi_{i \in \mathcal{I}}{\mathsf{S}}_i$ of indexed state spaces with natural projections $$\pi_{{\mathsf{S}}_i} : \Pi_{i \in \mathcal{I}}{\mathsf{S}}_i \rightrightarrows {\mathsf{S}}_i,$$ (see [@Barwise1 §8.2]).
The Event Classification {#event-class}
------------------------
Following the development of ideas in @Barwise1 [Ch. 9], studies such as @Kakuda1 define the *Event Classification ${\mathsf{Evt}}({\mathsf{S}})$* associated with a state space ${\mathsf{S}}$ as follows:
- $\rm{Tok}({\mathsf{Evt}}({\mathsf{S}})) = \rm{Tok}({\mathsf{S}})$;
- $\rm{Typ}({\mathsf{Evt}}({\mathsf{S}})) = {\mathcal P}(\rm{Typ}({\mathsf{S}}))$;
- $s \Vdash_{{\mathsf{Evt}}({\mathsf{S}})} {\alpha}$ is defined by $\mathsf{state}_s (s) \in {\alpha}$, for $s \in \rm{Tok}({\mathsf{Evt}}({\mathsf{S}}))$ and ${\alpha}\in \rm{Typ}({\mathsf{Evt}}({\mathsf{S}}))$;
where as before, ${\mathcal P}(\cdot)$ indicates the power set.
Briefly recapping, this says that the space of events $\rm{Evt}({\mathsf{S}})$ associated to ${\mathsf{S}}$ has as its tokens the tokens of ${\mathsf{S}}$ and its types are arbitrary sets of sets of states of ${\mathsf{S}}$. The classification relation $s \Vdash_{\rm{Evt}({\mathsf{S}})} {\alpha}$ above is equivalent to ${\rm{state}}_{{\mathsf{S}}}(s) \in {\alpha}$. Following @Barwise1 [Prop. 8.17], given state spaces ${\mathsf{S}}_1$ and ${\mathsf{S}}_2$, the following are equivalent:
- $f: {\mathsf{S}}_1 \rightrightarrows {\mathsf{S}}_2$ is a projection;
- ${\rm{Evt}}(f): {\rm{Evt}}({\mathsf{S}}_2) \rightleftarrows {\rm{Evt}}({\mathsf{S}}_1)$ is an infomorphism.
In fact, for any state space ${\mathsf{S}}$, the classification $\rm{Evt}({\mathsf{S}})$ is a Boolean classification in which the operations of taking intersection, union, and complement are here conjunction, disjunction, and negation, respectively [@Barwise1 Prop. 8.18].
\[local-3\] Let ${\mathsf{S}}$ be a state space. The *local logic generated by ${\mathsf{S}}$*, denoted ${{\mathsf{Lg}}}({\mathsf{S}})$, has classification $\rm{Evt}({\mathsf{S}})$ , regular theory ${{\mathsf{Th}}}({\mathsf{S}})$, and all of its tokens are normal.
For further relationships see @Barwise1 [12.1 - 12.2].
Taking ${\mathsf{K}}= [0,1]$, the evaluation relation $$\Vdash_{{\mathsf{Evt}}({\mathsf{S}})}: \rm{Tok}({\mathsf{Evt}}({\mathsf{S}})) \times \rm{Typ}({\mathsf{Evt}}({\mathsf{S}})) {\longrightarrow}[0,1]$$ together with (logic) infomorphisms between event classifications, may be compared with the concept of a *perceptual strategy* as described in @Hoffman1.
State space systems {#states-2}
-------------------
Considering the above ingredients we now seek a unifying principle that characterizes the state space model and provides a suitable information channel. This motivates starting with:
A *state-space system* consists of an indexed family $\mathcal{S} = \{f_i: {\mathsf{S}}\rightrightarrows {\mathsf{S}}_i\}_{i \in \mathcal{I}}$ of state-space projections with a common domain ${\mathsf{S}}$, called the *core* of $\mathcal{S}$, to state spaces ${\mathsf{S}}_i$ (for $i \in \mathcal{I}$); ${\mathsf{S}}_i$ is called the $i$th component space of $\mathcal{S}$.
We now consider an “event” $\rm{Evt}$ as a functor that transforms a state-space system into an information channel. Taking a pair of state spaces as an example, we first take projections: $$\xymatrix{& {\mathsf{S}}\ar[dl]_{f_i} \ar[dr]^{f_j} &\\
{\mathsf{S}}_i & & {\mathsf{S}}_j
}$$ Next, applying the functor $\rm{Evt}$ to this diagram yields a family of infomorphisms with a commom domain $\rm{Evt}({\mathsf{S}})$, yielding an information channel: $$\xymatrix{& \rm{Evt}({\mathsf{S}}) & \\
{\rm{Evt}}({\mathsf{S}}_i) \ar[ur]^{{\rm{Evt}}(f_i)} & & {\rm{Evt}}({\mathsf{S}}_j) \ar[ul]_{{\rm{{\mathsf{Evt}}}}(f_j)}
}$$ State space addition produces a further commuting diagram, where for ease of notation, we write $\sigma_i$ for $\sigma_{{\rm{Evt}}({\mathsf{S}}_i)}$, and simply $f$ for $\sum_{k \in \mathcal{I}} {\rm{Evt}}(f_k)$: $$\xymatrix@C=4pc{& \rm{Evt}({\mathsf{S}}) & \\
\rm{Evt}({\mathsf{S}}_i) \ar[ur]^{{\rm{Evt}}(f_i)} \ar[r]_{\sigma_i} & \sum_{k \in \mathcal{I}} \rm{Evt}({\mathsf{S}}_k) \ar[u]^{f} & \ar[l]^{\sigma_j} \rm{Evt}({\mathsf{S}}_j) \ar[ul]_{{\rm{Evt}}(f_j)}
}$$
Following @Kakuda1 [§4], let $T$ be a regular theory, and let ${\mathsf{S}}$ be a state space. A *medium system* denoted $D := \langle
{\mathsf{D}}, {\mathsf{N}}, f, p \rangle$ between $T$ and ${\mathsf{S}}$, consists of the following:
- a state space ${\mathsf{D}}$,
- a subset ${\mathsf{N}}$ of $\rm{Tok}(\mathnormal{D})$,
- an infomorphism $f: {\mathsf{Cl}}(T_{\rm{Typ}(\mathnormal{T})}) \rightleftarrows {\mathsf{Evt}}({\mathsf{D}})$, and
- a projection $p: {\mathsf{D}}\rightrightarrows {\mathsf{S}}$,
where $\langle \rm{Tok}({\mathsf{D}}), \rm{Typ}(\mathnormal{T}), \vdash_{T}, {\mathsf{N}}, \rm{Typ}(\mathnormal{D}), \mathsf{state}_{D}, \overrightarrow{f}, \overleftarrow{f} \rangle$ forms a functional scheme. Here ${\mathsf{D}}$ is called the *medium space of* $D$. @Kakuda1 [§4] define an information channel: $$\xymatrix{& {\mathsf{Evt}}({\mathsf{D}}) & \\
{\mathsf{Cl}}(T_{\rm{Typ}(T)}) \ar[ur]^{f} & & {\mathsf{Evt}}({\mathsf{S}}) \ar[ul]_{{\mathsf{Evt}}(p)}
}$$ through ${\mathsf{Evt}}({\mathsf{D}})$ to ${\mathsf{Evt}}({\mathsf{S}})$.
\[cognizance\]
@SS1 [@SS2] employ the concept of the core of a binary channel, when realized as a classification, in order to describe an agent’s cognition (cf [@Barwise2]). Consider separate source (${\mathcal A}$) and target (${\mathcal B}$) classifications, and represent the agent’s knowledge by a regular theory $T = \langle \Sigma, \vdash \rangle$. The idea is to construct a set of *possible and realizable states* in several steps:
- For source classification ${\mathcal A}$, and target classification ${\mathcal B}$, let $\Omega_{\langle {\mathcal A}, {\mathcal B}\rangle}$ denote the set of all partitions of $\rm{Typ}({\mathcal A}) \cup \rm{Typ}({\mathcal B})$, called *the set of states generated by ${\mathcal A}$ and ${\mathcal B}$*.
- The set of *realizable states generated by ${\mathcal A}$ and ${\mathcal B}$* is given by $$\Omega^R_{\langle {\mathcal A}, {\mathcal B}\rangle} :=\{ \langle \Theta, \Lambda \rangle \in \Omega_{\langle {\mathcal A}, {\mathcal B}\rangle}: \exists a \in {\mathcal A}, ~ \rm{Typ}(a) \subseteq \Theta~ \text{and},~ \rm{Typ}^c(a) \subseteq \Lambda \},$$ where the notation $\rm{Typ}^c(a)$ indicates the complement, i.e. everything not in $\rm{Typ}(a)$.
- The set of *impossible states under the theory $T$* is given by $$\Omega^{IP}_{\langle {\mathcal A}, {\mathcal B}\vert T \rangle} :=\{ \langle \Theta, \Lambda \rangle \in \Omega_{\langle {\mathcal A}, {\mathcal B}\rangle}:
\Theta \vdash_T \Lambda \}.$$ The “impossibility” here is that $\Omega$ constrains a $\Lambda$ with which it is disjoint.
- *The possible states under the theory $T$* is then $$\Omega^{P}_{\langle {\mathcal A}, {\mathcal B}\vert T \rangle} = \Omega_{\langle {\mathcal A}, {\mathcal B}\rangle} \backslash \Omega^{IP}_{\langle {\mathcal A}, {\mathcal B}\vert T \rangle}.$$
- *The possible and realizable states under the theory $T$* is thus $$\Omega^{PR}_{\langle {\mathcal A}, {\mathcal B}\vert T \rangle} = \Omega^P_{\langle {\mathcal A}, {\mathcal B}\vert T \rangle} \cap \Omega^{R}_{\langle {\mathcal A}, {\mathcal B}\rangle}.$$
The *cognizance classification $\mathfrak{C}_{\langle {\mathcal A}, {\mathcal B}, T \rangle}$* generated by ${\mathcal A}, {\mathcal B}$ and $T$ is then given by $$\mathfrak{C}_{\langle {\mathcal A}, {\mathcal B}, T \rangle} := \big\langle \Omega^{PR}_{\langle {\mathcal A}, {\mathcal B}\vert T \rangle}, \rm{Typ}({\mathcal A}) \cup \rm{Typ}({\mathcal B}), \Vdash_{\mathfrak{C}_{\langle {\mathcal A}, {\mathcal B}, T \rangle}} \big\rangle$$ where the relation $\Vdash_{\mathfrak{C}_{\langle {\mathcal A}, {\mathcal B}, T \rangle}}$ is defined as $\langle \Theta, \Lambda \rangle \Vdash_{\mathfrak{C}_{\langle {\mathcal A}, {\mathcal B}, T \rangle}} {\alpha}$ if and only if ${\alpha}\in \Theta$, i.e. the choice of $\Lambda$ can be arbitrary provided it is disjoint from $\Theta$.
On comparing and combining the Shannon Theory of Information with Channel Theory {#shannon}
--------------------------------------------------------------------------------
@Barwise2 recalls *Shannon’s Inverse Relation* between possibilities and information, basically saying that eliminating possibilities from consideration amounts to increasing one’s information and vice-versa. That relationship is fundamental to Dretske’s original goal of developing a semantic theory of information based on possibilities [@Dretske1; @Dretske2]. Though very general as a quantitative theory of communication flow, the original Shannon theory had largely overlooked the question of semantic content. In any Dretske-type theory, the basis of semantic content is in the world, i.e. in the events or situations that signals or states carry information about. By showing how local logics are connected by information networks, channel theory provides a general qualitative theory of information flow in this context. ‘Channels’ in the theory are more general than the traditional idea of the Shannon communication channels [@Cover2006]. In Shannon’s theory, information flow in a channel is defined in terms of reduction of uncertainty about the *type* of event that will occur; it says nothing about the semantics of any specific bit, i.e. about any specific token. Channel theory specifically concerns particular tokens $x$ and statements of the form “$x$ is an $A$.” @Allwein2004 [@Allwein1] create a synthesis of Shannon’s quantitative theory with the Barwise-Seligman qualitative theory to address the question of how specific objects, situations and events carry information about each other.
A classification of possible outcomes of events starts with a probability space $\mathcal{P} = \langle \Omega, \Sigma, \mu \rangle$, where $\Omega$ denotes the set of possible outcomes, $\Sigma$ is a $\sigma$-algebra on $\Omega$ whose members represent events, and $\mu$ is a probability measure on $\sigma$ representing the probability of an event having or being associated with a particular outcome.[^4] We define a classification $\rm{Tok}(\mathcal{P}) = \Omega$, $\rm{Typ}(\mathcal{P}) = \Sigma$, and let $\omega \Vdash_{\mathcal{P}} e$, if and only if $\omega \in e$. In this context, an infomorphism between probability spaces is topologically a continuous map [@Seligman1].
To see how the basic ontology of the Shannon theory can be conveniently embedded in that of Channel theory, note that the former’s basic unit of information is some tuple of a binary relation. The relation is restricted to be of the form $x \Vdash V$, where $\Vdash$ is regarded as a function, and $V$ is the value of the Token $x$. But closer inspection reveals this characterizes a state space in which $V$ is a state and tokens are ignored. In Channel Theory there are states, but tokens are not ignored and types are values, as in §\[classifications-1\]. States may be amalgamated to form Events as in §\[event-class\] and §\[states-2\]. Channel theory permits this by firstly preserving the tokens, and then replacing states with types, whose events are also types: for some event $E$, $x \Vdash E$, just when $x \Vdash s$, for some state $s \in E$ [@Allwein1]. This is basically the embedding of the one theory into the other, and the two theories together admit a certain generalization as follows.
The presence of a sequent in an information channel (as outlined in §\[local\]) effectively represents a logical gate, and this kind of structure can be seen as more general than a Markov structure (cf. the Kolmogorov axioms in @Allwein2004 [@Allwein1]), since sequents enable the information flow to simultaneously support a flow of reasoning. Specifically, probabilities can be assigned to sequents in ${\mathcal A}$ as follows. Suppose we have: $$\label{prob-1}
M \Vdash_{{\mathcal A}} N \qquad \forall x (x \Vdash_{{\mathcal A}} M \Rightarrow x \Vdash_{{\mathcal A}} N),$$ then the sequent relation $\vdash$ can be weakened by removing $\forall$, and instead stating that for any $x$, we have a probability $x \Vdash N$, given $x \Vdash M$; that is, the probability that $x$ satisfies $N$ given it satisfies $M$. This is clearly a conditional probability, so one defines: $$\label{prob-2}
M \Vdash_{{\mathcal A}}^{\mathbf{P}} N := \mathbf{P}( M \vert N).$$ Thus when the sequent’s conditional probability is $p$, say, we have $M \Vdash_{{\mathcal A}}^{p} N$. *A priori*, one must have $x \Vdash_{{\mathcal A}} M$ in order to apply $M \Vdash_{{\mathcal A}} N$ in a argument. The probability of the former holding in ${\mathcal A}$, is $\mathbf{P}(M)$. Then $x \Vdash_{{\mathcal A}} N$ follows from the rule $\mathbf{P}(M) \cdot M \Vdash_{{\mathcal A}}^{\mathbf{P}} N$. Probability axioms for a Countable Classical Propositional Logic are developed in [@Allwein2004] (cf. @Allwein1) to which we refer for details. Note that information flow in distributed systems can be interpreted dynamically; this amounts to causation in an informational context, consistent with the Dretskean nature of the theory. In this respect, the relations between information theory and logic are also conducive to understanding certain relations between causation and computation [@Collier; @Seligman1].
Colimits for piecing together local-to-distributed information: Application to information flow {#colimits}
===============================================================================================
Cocones and Colimits {#cocone-1}
--------------------
To a category theorist, minimal covers as described in §\[distributed\] are familiar as *colimits*, where the channel core $\mathbf{C}$ represents the vertex of a cocone (as was introduced in §\[info-cocone\]; more formally, see e.g. @Awodey [Ch. 5]) and the existence of colimits of Chu spaces (realized here by the classifications) has been verified [@Barr1]. Let us briefly recall this concept with some graphic intuition behind the idea following @Baianu2006 [@Brown2003].
The “input data” for a colimit is a *diagram* $\mathbf{D}$, i.e. a collection of some objects in a category $\mathfrak{C}$, together with some arrows between them, as depicted by: $$\label{colim-1}
\xymatrix@R=3pc{ & . \ar[rr]&&.\ar[dl]&\\ **[l] \mathbf{D} = \qquad \cdot \ar[ur]\ar[dr]& & .
\ar[ul]\ar[rr]\ar[dl] & &.\ar[ul]\\ &.&&&}$$ This generalizes our use of a directed graph in §\[info-cocone\] by allowing the “vertices” and “edges” of $\mathbf{D}$ to be objects and morphisms of an arbitrary category. Next we need ‘functional controls’ comprising a *cocone with base $\mathbf{D}$ and vertex an object $\mathbf{C}$ in $\mathfrak{C}$*,
$$\label{colim-2}
\xymatrixcolsep{3pc} \xymatrixrowsep{2pc}\xymatrix{
&&\mathbf{C}&&\\&&&&\\ &&&&\\
& . \ar@{-}[r]|(0.35)\hole\ar[uuur]&\ar[r]&.\ar[dl]\ar[uuul]&\\
\mathbf{D} \qquad.\ar[ur]\ar[uuuurr]\ar[dr]& & . \ar[ul]|!{[dl];[uuuu]}\hole\ar[rr]\ar[dl]\ar[uuuu]
& &.\ar[ul]\ar[uuuull]\\
&.\ar[uuuuur]&&&}$$
such that each of the triangular faces of the cocone is commutative. The “output” from $\mathbf{D}$ will be an object $\mathsf{colim}(\mathbf{D})$ in our category $\mathfrak{C}$ defined by a special *colimit cocone*, such that any cocone on $\mathbf{D}$ factors *uniquely* through the colimit cocone. Effectively, the commutativity condition on the cocone induces, in the colimit, an interaction of images from different parts of the diagram $\mathbf{D}$. The uniqueness condition makes the colimit the optimal solution to this factorisation problem.
Let us set $$\blacklozenge= \mathsf{colim}(\mathbf{D})$$ where the dotted arrows in the diagram below represent new morphisms which combine to make the colimit cocone:
$$\label{colim-3}
\xymatrixcolsep{3pc} \xymatrixrowsep{2pc}\xymatrix{ &&\mathbf{C}&&\\&&&&\\
**[l]\mathsf{colim}(\mathbf{D})=\blacklozenge\ar@{-->}[rruu]^
\Phi &&&&\\
& .\ar@/_/@{.>}[lu] \ar@{-}[r]|(0.35)\hole\ar[uuur]&\ar[r]&.\ar@{.>}[lllu]\ar[dl]\ar[uuul]&\\
**[l]\mathbf{D} \qquad \cdot \ar@{.>}[uu] \ar[ur]\ar[uuuurr]\ar[dr]& & .
\ar@/^/@{.>}[lluu] \ar[ul]|!{[dl];[uuuu]}\hole \ar[rr]\ar[dl]\ar[uuuu]
& &.\ar[ul]\ar@{.>}[uullll]\ar [uuuull]\\
&.\ar[uuuuur] \ar@{.>}[uuul]&&&}$$
and for which the broken arrow $\Phi$ is constructed by requiring commutativity for all of the triangular faces of the combined diagram. Next, stripping away the ‘old’ cocone results in a factorisation of the cocone via the colimit: $$\label{colim-4}
\xymatrixcolsep{3pc} \xymatrixrowsep{2pc}\xymatrix{ &&\mathbf{C}&&\\&&&&\\
**[l]\mathsf{colim}(\mathbf{D})=\blacklozenge
{\ar@{-->}[rruu]^\Phi} &&&&\\
& .\ar@/_/@{.>}[lu] \ar@{-}[r]&\ar[r]&.\ar@{.>}[lllu]\ar[dl]&\\
**[l]\mathbf{D} \qquad \cdot \ar@{.>}[uu] \ar[ur]\ar[dr]& & .
\ar@/^/@{.>}[lluu] \ar[ul]\ar[rr]\ar[dl]
& &.\ar[ul]\ar@{.>}[uullll]\\
&. \ar@{.>}[uuul]&&&}$$ Intuitively, the process can be seen as follows. The object $\mathsf{colim}(\mathbf{D})$ is pieced together from the diagram $\mathbf{D}$ by means of the colimit cocone. From beyond $\mathbf{D}$, an arbitrary object $\mathbf{C}$ in $\mathfrak{C}$ ‘sees’ $\mathbf{D}$ as mediated through its colimit. This means that if $\mathbf{C}$ is going to interact with all of $\mathbf{D}$, then it does so via $\mathsf{colim}(\mathbf{D})$. The colimit cocone can be thought of as a kind of program: given any cocone on $\mathbf{D}$ with vertex $\mathbf{C}$, the output will be a morphism $$\Phi: \mathsf{colim}(\mathbf{D})\to \mathbf{C}$$ as constructed from the other data. [^5]
@Brown2003 provide an analogy comparing colimits with how an email message can be relayed. Suppose $E$ denotes some email document. This is to be sent via a server $S$, which decomposes $E$ into numerous parts $E_i$ ($i \in \mathcal{I}$, an indexing set), and labels each part $E_i$, so it becomes $E'_i$. These labelled parts $E'_i$ are then sent to a number of servers $S_i$, which then relay these messages as newly labelled messages $E''_i$ to a server $S_C$, for the receiver $C$. The server $S_C$ then combines the $E''_i$ to produce the recovered message $M_E$ at $C$. Breaking the message down and routing it through the $S_i$ appears arbitrary, but the system is designed such that $M_E$ is independent of all choices made at each step of the process.
Many other illustrative examples applying colimits to computer science, social systems, and neuroscience, can be seen in @Baianu2006 [@EV2007; @Healy1; @Healy2].
Coordinated channels in ontologies {#coordinated}
----------------------------------
Suppose we have two prospectively interoperating agents $A_1,A_2$, with each agent $A_i$ ($i=1,2$) having its knowledge represented according to its own conceptualization, as specified in relationship to its ontology $\mathcal{O}_i$, respectively. This means that a concept of $\mathcal{O}_1$ will, *a priori*, be considered semantically distinct from $\mathcal{O}_2$, even if they are equivalent syntactically. However, the behavior of the agents can provide evidence for a meaning common to $A_1$ and $A_2$. Let us us assume that the agents’ ontologies are not open to a third-party inspection. @Kalfoglou1 use a channel to coordinate the populated ontologies (cf. §\[ontologies\]) ${\widetilde}{\mathcal{O}}_1, {\widetilde}{\mathcal{O}}_2$ by capturing the degree of participation of each agent in communicative behaviors. Specifically, i) agent $A_i$ attempts to “explain” a subset of its concepts to other agents, and ii) other agents exchange with $A_i$ some of their own tokens, thus increasing the set of tokens originally available to $A_i$.
To see how this degree of participation can be captured by Channel Theory, @Kalfoglou1 introduce classifications $\mathbf{A}_i =
\langle \rm{Tok}(\mathbf{A}_i), \rm{Typ}(\mathbf{A}_i), \Vdash_{\mathbf{A}_i} \rangle$, corresponding to the agents $A_i$, respectively, along with subclassifications $\mathbf{A}'_i =
\langle \rm{Tok}(\mathbf{A}'_i), \rm{Typ}(\mathbf{A}'_i), \Vdash_{\mathbf{A}'_i} \rangle$, and infomorphisms $g_i: \mathbf{A}'_i {\longrightarrow}\mathbf{A}_i$, for which functions $\hat{g}_i$ and $\check{g}_i$ are the inclusions $\rm{Typ}(\mathbf{A}'_i) \subseteq
\rm{Typ}(\mathbf{A}_i)$ and $\rm{Tok}(\mathbf{A}'_i) \subseteq \rm{Tok}(\mathbf{A}_i)$, respectively. It is from the subclassifications $\mathbf{A}'_i$ arising from the interactions that coordination is established. Thus we have the following information channel with core (i.e. cocone) $\mathbf{C}'$: $$\xymatrix{& &\mathbf{C}' & \\
\mathbf{A}_1 & \ar[l]^{g_1} \mathbf{A}'_1\ar[ur]^{f_1} & & \mathbf{A}'_2\ar[ul]_{f_2} \ar[r]_{g_2} & \mathbf{A}_2
}$$ The optimal coordinated channel that captures semantic integration achieved by the agents is then represented by the colimit $\mathcal{C}' = \mathsf{colim}\{\mathbf{A}'_1 \leftarrow \mathbf{S} \rightarrow \mathbf{A}'_2 \}$ of the diagram linking the subclassifications that model the agents’ participation in the interoperation: $$\xymatrix{& &\mathbf{C}' & \\
\mathbf{A}_1 & \ar[l]^{g_1} \mathbf{A}'_1\ar[ur]^{f_1} & \ar[l]^{h_1} \mathbf{S} \ar[r]_{h_2} & \mathbf{A}'_2\ar[ul]_{f_2} \ar[r]_{g_2} & \mathbf{A}_2
}$$ “Optimality” here means that every other channel induces a map to $\mathcal{C}'$ when commutativity is required.
Part II: Applications to Perception and Cognition {#perception-cognition .unnumbered}
=================================================
Perception, categorization and attention as neurocognitive processes {#cognitive}
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We now turn from the assembly of category-theoretic tools as presented in Part I, to the application of these tools to modelling object recognition and categorization, particularly the recognition and categorization of mereologically-complex entities. We first review the basic cognitive neuroscience of object recognition and categorization in vision, currently the best understood of the sensory channels. We focus on the data structures employed – what @Marr called the algorithmic/representational level of analysis – with some pointers to the relevant implementation-level neuroscience (see @Fields13 for a review of implementation details).
Dual-process vision and object files {#dual-process}
------------------------------------
The primate, and in particular human, visual system comprises two early processing pathways, a dorsal pathway specialized for the rapid processing of location and motion information, and a ventral pathway specialized for static (e.g. shape, size, texture and color) feature information (reviewed by @goodale92 [@scholl08; @Fields11]; see @cloutman for a discussion of interactions between these pathways and @Alain01 and @Sathian12 for evidence that auditory and haptic perception, respectively, have a similar dual-stream organization). Perception of a located, featured object requires processing by both pathways followed by fusion of the intermediate partial representations they produce.
Studies of visual object tracking over short time periods consistently show that trajectory information dominates static feature information in determining object identity [@scholl08; @Fields11]. @Kahneman termed the initial, transient representation of a moving object in visual short-term memory the “object file” (see also @Treisman06). As under ordinary circumstances all objects are effectively moving due to visual saccades, object files are at least typically initiated by dorsal-stream processing. Static feature information extracted from the relevant part of the visual field by ventral-stream processing is then bound to this initially motion-based representation. These processing steps require 50 – 100 ms in humans, much shorter than the time required for reportable visual awareness of the object.
The object file is the fundamental “token” representation of a located, bounded, featured entity that is distinguished from the “background” of a visual scene. It represents where the object is, its visually-identifiable features, and its instantaneous trajectory during the time window $\Delta t$ from object-file initiation to feature binding. All further information about the object is added by downstream processing; in particular, whether the object is novel or something previously encountered, either as a type or as a specific individual, must be computed from information available in memory.
Feature-category binding and object tokens
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Object files are implemented in a content-dependent way across the posterior temporal cortex [@Martin07; @Mahon09; @Fields13]; features of entities perceived as agents or non-agents, for example, are encoded in the lateral or medial, respectively, fusiform gyrus (Fig. 1). This distributed, content-dependent encoding indicates that top-down category information, e.g. agent versus non-agent, is already active in the binding of location and motion information to feature information at the level of the object file.
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> *Fig. 1*: Simplified functional architecture of visual object perception within the temporal lobe. Abbreviations are: MT, medial temporal area; STS, superior temporal sulcus; MTG, medial temporal gyrus; PHC, parahippocampal cortex; V4, visual area 4 (occipital cortex); MFG, medial fusiform gyrus; LFG, lateral fusiform gyrus; PRC, perirhinal cortex; HC, hippocampus; ATP, anterior temporal pole. Solid lines are feedforward; dashed lines feedback. Adapted from @Fields13.
The functional architecture supporting object representation and object-directed attention is already present at birth (see e.g. @Gao15 [@Huang15] for neuroarchitectural and @Johnson15 for behavioral evidence) and its functionality rapidly matures toward adult levels during the first two years. Visual feature identification, segregation of co-moving, conjoined objects, and the complementary process of grouping co-moving, non-conjoined objects, for example, are highly dependent on top-down, memory-driven categorization or, in Bayesian terms, expectation confirmation or disconfirmation. Four- to six-month old human infants, for example, typically do not segregate static or co-moving conjoined objects that older infants, children or adults do segregate, but quickly learn to do so when the objects are separately manipulated [@Johnson15]. Young infants similarly fail to group co-moving, non-conjoined objects (i.e. fail to perform “object completion”) that older infants, children or adults do group, with the exception of point-light walkers exhibiting biological motion, which infants perceive as single entities from the earliest ages tested (@Johnson15; see @Schlesinger12 for a replication of a canonical object completion experiment in the iCub robot).
Young infants exhibit robust object memory, particularly for familiar faces, and emotional responses to objects, again from the earliest ages tested. Feelings of familiarity and their attendant emotions correlate with feature-based object recognition at the level of perirhinal cortex [@Eichenbaum07]. Memory for a particular, re-identifiable object requires a memory-resident representation of the individual object, what @Zimmer have termed the “object token.” Recognizing a novel object as a distinct, individual thing involves encoding a new object token specifically for it. Recognition or re-identification of the same individual object on a later occassion is then a process of matching the current object file to this previously-encoded object token (Fig. 2). This process is, in general, not straightforward, as object features, behaviors, and locations may change between encounters; both feature matching and, after about four years of age, the construction of unobserved and hence fictive causal histories (FCHs) of objects are employed to establish individual object identity across observations [@Fields3]. Enabling object recognition across feature, behavior, and context changes requires object tokens to have a “core” of essential properties that change only slowly through time. The distinction between core and variable properties in object tokens is category-dependent and not well understood.
{width="18cm"}
> *Fig. 2*: Identifying an object as the same individual across time requires matching a current object file to a memory-resident object token. Both feature matching and the construction of unobserved (fictive) causal histories (FCHs) are employed to link object tokens across observations.
@Eichenbaum07 emphasize that categorization by type precedes object-token matching; e.g. an individual person is typically recognized as a person before they are identified as a particular individual person. The feeling of familiarity is generated already at the level of type recognition; @Eichenbaum07 consider the example of recognizing a person as a person, as a specific individual person encountered before, and only after some deliberation as a known, named, individual person represented by a memory-resident object token. How the relationship between categories and object tokens as components of semantic memory is implemented at the neural circuit level, i.e. the details of the circuitry connecting ATP and PRC in Fig. 1, is not well understood. While both categorization and individual recognition are generally assumed to be implemented by some form of Bayesian predictive coding [@Friston2; @Maloney10], how categorization constrains or guides object token matching in particular cases is also not well characterized. The category - to - object token satisfaction relation $\Vdash$ remains, in other words, an empirical question for each particular type-token pair.
Context perception, event files and episodic memories {#episodic}
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Objects are invariably recognized in some context, typically one involving other recognized or at least categorized objects. The spatial “where” information processed by the dorsal stream (cf. Fig. 1) provides the “container” for this context as well as the relative locations and motions of objects within it. Contexts typically, however, also include “how” and “why” information, largely derived by processing pathways in parietal cortex [@Fields13], that represent inferences about mechanical and intentional causation, respectively. As in the case of object categorization and individual object-token encoding, these causal inference capabilities are present in rudimentary form in early infancy, and develop rapidly over the first two years [@Baillargeon12; @Johnson15]. Context assembly has been mapped to parahippocampal cortex, with object token to context binding implemented by the hippocampus [@Eichenbaum07; @Fields13]. @Hommel has termed the fully-bound representation of interacting objects in context an “event file”; these representations mediate event understanding and context-dependent action planning. Event files are the least complex visual representations that typically enter human awareness; hence they can be considered to be implemented by coherent activity at the level of the GNW (cf. Remark \[gnw-remark\] ; for specific GNW-based models of awareness, see @Baars3 [@Dehaene04; @Baars13; @Dehaene14]; see also @Franklin1 [@Franklin12] for discussions of LIDA as a robotic architecture based on GNW design principles).
Event files correspond to “episodes” in episodic memory, again a hippocampus-centered function [@Eichenbaum07; @Rugg13]. As sequences of episodic memories typically contain many of the same “players” – including in particular the self [@Renoult12] – they pose a particular problem for object-token updating. Each episodic memory must contain at least some episode-specific details – e.g. what a particular person was wearing – in addition to the “core” identifying information for each included object token. Linking episodic memories into a temporal sequence requires maintaining this “core” – which it is useful to consider as a “singular category” with just one member – while modifying its “essential” identifying information for the represented object as needed, e.g. updating a person’s age or personality characteristics (Fig. 3). This maintenance process is effectively the construction of a history or model of the individual represented by the object token, the FCH discussed above [@Fields3]. Episodic memory recall and reconsolidation can modify the properties associated with or even the presence of the object tokens referenced by the memory, demonstrating the fragility of such FCHs [@Schwabe14]. Infants are capable of episodic recall over short periods – e.g. the time periods required for experiments assaying causal inference – but have limited recall and reconsolidation ability over longer periods [@Hayne04; @Bauer06]. Hence infants can be expected not to maintain robust object histories until about age four, the age when “childhood amnesia” typically ends.
{width="16cm"}
> *Fig. 3*: a) Object token updating following a new event involving a recognized object. The singular category specifying the object’s “core” identifying criteria constrains the recognition and updating processes. b) Updating the constraint information in a object-specifying singular category given a sequence of episodic memories and a new event. Such updates must be infrequent compared to object token updates to maintain coherent identification criteria. Adapted from @Fields3.
Episodic memories can also reference object tokens for individuated and categorized but otherwise unidentified objects, e.g. “some other people” present at a meeting or “other cars” involved in an accident. These “other” objects may appear in no other episodic memories and have no associated histories; they are represented by effectively one-off object tokens that are required by the data structure but play no other role in the system. The human ability to learn to recognize new individuals indicates that such minimally filled-out object tokens are available for matching new incoming object files; however, their lifetime and the extent and context-dependence of their availability remain poorly understood.
Attention, salience and Bayesian precision {#bayesian}
------------------------------------------
Systems with limited cognitive resources must allocate processing to the inputs most likely to be important. In the current setting, this corresponds to paying attention to some objects and not others. Attentional control in primates is implemented by competing, cross-modulating dorsal (top-down, goal-driven, proactive) and ventral (bottom-up, percept-driven, reactive) attention systems [@Goodale14; @Vossel14]. The “salience network” that controls these attention systems develops in concert with the medial-temporal object recognition network, starting from earliest infancy [@Gao15; @Uddin15].
Baysian predictive coding has long been employed as a model of perceptual processing from early vision through categorization and individual object identification [@Friston2; @Maloney10]. @Bastos12 review structural and functional evidence that predictive coding is implemented at the level of local microcircuits comprising cortical minicolumns, the dominant architectural units in mammalian cortex, as well as at the larger scales of functional networks responsible for trajectory recognition, categorization or object-token matching. This use of the same or similar processing methods at different scales, as well as the overall hierarchical organization of perceptual processing [@VanEssen92], suggests the kind of association between scale and coarse-graining introduced in §\[simplicial\]; we will return to this connection below.
In a Bayesian predictive coding system, attentional control can be modelled by varying the precision assigned to inputs and expectations. In the Bayesian “active inference” framework of @Friston2, relatively high-precision inputs drive the revision of expectations and model reactive, ventral attention, while relatively high-precision expectations drive input-changing behavior and model proactive dorsal attention. The framework also allows direct alterations of precision assignments as inferential outcomes. The adaptive resonance (ART) framework of @Grossberg13 provides a functionally similar model of attentional control, although its motivation and underlying ideas are distinct from those of @Friston2 and its state-updating rules are not Bayesian. When viewed as implementations of constraint hierarchies, however, both Bayesian active inference and ART exhibit the deep duality discussed in §\[CCCD\] below. It is this duality, we will argue, that makes them useful models of attentional control.
Tokens, types and information flow in perception and categorization {#tt-flow}
===================================================================
While the human classification of perceived objects into cognitive categories corresponding to verbally-expressible concepts like “person” or “house” forms part of the motivation for the work of @Dretske1 [@Barwise4; @Barwise1] and others reviewed in Part I, category-theoretic methods have yet to be applied to the analysis of these processes at the level of detail reviewed in §\[cognitive\]. This formal treatment of ontologies discussed in §\[ontologies\], for example, does not explicitly address the question of how ontologies are constructed or maintained through time in the face of new observations. We begin in this section to develop the constructs needed for more complete models. We introduce the idea of a Cone-Cocone Diagram (CCCD) to capture the deep duality evident in the bidirectional flow of constraints between perception and categorization.
Representing object files in a Chu space {#object-file}
----------------------------------------
The fundamental perceptual token, the initial representation of a discrete perceptual entity, is the object file. As outlined in §\[dual-process\] above, an object file binds a collection of static features such as size, shape, texture and color extracted by ventral-stream processing to “instantaneous” (i.e. within $\Delta t$) location and trajectory information extracted by dorsal stream processing. Let $F_1 \dots F_n$ be a finite tuple of static features, each of which can have any one of $m$ distinct values; e.g. if $F_i$ is ‘color’ its distinct values are the colors distinguishable by the visual system of interest. We can then consider a finite binary array $F = [f_{ij}]$, where $f_{ij} = 1$ for some object if and only if feature $F_i$ of that object has its $j^{th}$ possible value. We can similarly consider finite binary arrays $X = [x_{ijk}]$ of discrete instantaneous three-dimensional locations and $V = [v_{ijk}]$ of discrete instantaneous three-dimensional velocities. We will restrict attention to the case in which every object has some value for every perceptible feature, a single instantaneous location, and a single instantaneous velocity. In this case, we can characterize an *object file* as an instance of the finite array $[F,X,V]$. These instances form a finite set $\{O_i \}$.
The most fundamental abstraction implemented by the visual system is *object permanence*, i.e. the maintenance of object identity over time [@Fields17]. At the level of the object file, the relevant timeframe for object permanence is a “view” lasting between half a second and a few seconds. Objects that remain fully or even partially visible during a view are considered to remain “the same thing” while seen. Whether an object that does not remain visible is perceived as remaining “the same thing” during a view depends on the age of the perceiver (less or more than 1 year) and the details of its occluded trajectory. Objects moving sufficiently fast are “seen” as persistent even if their features, e.g. size, shape or color, vary over considerable ranges [@scholl08].
Let $\{C_i \}$ be the finite set of finite (indeed short) sequences of object files that are treated by the cognitive system of interest as indicating object permanence during the course of a single view. The elements of $\{C_i \}$ are then natural “types” relative to the “tokens” in the set $\{O_i \}$ of possible object files; an element $C_i \in \{C_i \}$ can be though of as “associating” a sequence of object files into a single abstracted representation. Hence we can consider $$\label{chu-object}
\mathcal{C}_i = ( \{O_i \},\Vdash_{P},\{C_i \} )$$ to be a Chu space, where here $\Vdash_{P}$ is the empirically-determined relation “consistent with object permanence” defined on sequences of object files. This Chu space clearly describes a Classification in the sense of @Barwise1 as stated in §\[classifications-1\]. Note that an element of $\{C_i \}$ may not be a concept in the sense of §\[FCA-1\], as the value of every feature as well as the position and velocity can, at least in principle, change between every object token contributing to the perception of a persistent object.
The association of object files into an element of $\{C_i \}$ adds top-down, expectation-based information about identity over time to the “raw” information of perception. This added information may, in fact, be incorrect; trajectories that appear to preserve object identity may involve distinct objects, while those that appear not to preserve object identity may involve a single object [@Fields11]. More subtly, association into an element of $\{C_i \}$ also subtracts information by suppressing motion information relative to feature information. While trajectory information dominates feature information in determining which sequences of object files to associate, persistent objects are required to have persistent features, at least during the course of short, single-view interactions [@Baillargeon08]. Conferring persistence on an object converts its motion into a categorizable “behavior” that the object may or may not execute on other occasions.
From object files to object tokens and object histories {#object-token}
-------------------------------------------------------
Persistent objects are the “entities” in the common-sense ontology humans typically develop in late infancy. These entities participate in episodic memories and are represented by object tokens and, if they recur sufficiently often to be recognized as persistent, by singular categories and (largely fictive) histories. As with the abstraction of persistence, these successive levels of abstraction both add and subtract information. Types at one level of abstraction, in particular, become tokens at the next.
As representations of persistent objects, *object tokens* can be identified with elements of the set $\{C_i \}$. Such tokens are, at their construction, already instances of multiple types (Fig. 4). Persistent objects are instances, first, of the types representing their visually-identified features. They are, second, classified automatically by threat detection, agency detection and animacy detection systems active beginning in early infancy [@Fields14]; the presence of a face alone indicates agency to human infants. They are also classified, when possible, into entry-level and then more abstract cognitive categories, an ability also developed in infancy [@Rakison10]. These token - type relationships can be represented as Classifications, as is standard in the literature (e.g. @Barwise1), and as we have surveyed here. Recognition of an object by type generates a feeling of familiarity with the *type*; e.g. seeing a cat generates a feeling of familiarity with cats.
{width="16cm"}
> *Fig. 4*: An object token is classified at construction into multiple types by distinct but cross-modulating processes. These include animacy and agency detection, emotion-mediated threat detection, and entry-level followed by superordinate and subordinate categorization into “type” of object.
Here, however, we are primarily interested in the *re-identification* of individual objects, i.e. the creation of an association indicating identity, and hence persistence over time, between an object token constructed now and one constructed previously. At the object token level, the relevant timeframes for persistence range from the few seconds separating views to the decades separating a high-school graduation from a 50$^{th}$ reunion.
Let $C_i(t_1), C_j(t_2), ... C_k(t_n)$ be a sequence of $n$ object tokens encountered at successive times. Recognizing successive object tokens as tokens of the very same individual thing involves at least the two processes discussed in §\[episodic\] above, i.e. matching to a set of core features composing a singular category and linking via FCH construction, with the uncertainty associated with both increasing with the time between perceptual encounters. Let $D_l[t_1,t_n]$ comprise both the singular category and the FCH that together confer persistence on the object-token sequence $C_i(t_1), C_j(t_2), ... C_k(t_n)$, and let $\{D_l[t_1,t_n] \}$ be the set of all such representations over sequences of elements of $\{C_i \}$ indexed by observation times in the closed interval $[t_1, t_n]$. The elements of $\{D_l[t_1,t_n] \}$ are, once again, natural “types” for the object tokens over which they are defined. Hence we can consider a Chu space or Classification (in the sense of §\[classifications-1\]) ${\mathcal A}_i[t_1,t_n]$ as given by: $$\label{fch-chu}
{\mathcal A}_i[t_1,t_n] : = \langle \{C_i[t_1,t_n] \},\{D_i[t_1,t_n] \}, \Vdash_{P}[t_1,t_n] \rangle$$ where $\Vdash_{P}[t_1,t_n]$ is the empirically-determined relation “consistent with object permanence” defined on sequences of object tokens between $t_1$ and $t_n$.
The set of time-indexed representations $\{D_l \}[t_1,t_n]$ can be conceptualized more abstractly by noting that at each $t_j$, the set of possible object tokens $\{C_i(t_j) \}$ is also the set of types of a classification. For each single time step $t_j \longrightarrow t_k$, the persistence criterion $\Vdash_{P}[t_j,t_k]$ induces maps – what we have called FCHs – between pairs of object tokens that can be consistently considered to be tokens of the same individual object (Fig. 5a). These FCHs, together with the maps (here assumed to be identities) linking the singular categories for persistent objects, can be considered infomorphisms between the underlying classifications at $t_j$ and $t_k$. It is then natural to interpret the set $\{D_l \}[t_j,t_k]$ as a channel between the underlying classifications; this channel comprises, intuitively, the (assumed constant) singular categories and the constructed FCHs (Fig. 5b). Extending the process of linking object tokens by FCHs forward in time results in a hierarchy of channels, with the most temporally-extended channel as the colimit (Fig. 5c). The colimit cocone $\{D_i\}[t_1, t_n]$ admits a vertex classification, which we denote $\mathbf{C}_i$ (with time interval $[t_1, t_n]$ understood). Recall that this $\mathbf{C}_i$ is induced by a complex of infomorphisms: $$\label{info-complex}
\cdots ~{\longrightarrow}{\mathcal A}_i[t_1, t_n] {\longrightarrow}{\mathcal A}_{i+1}[t_1,t_n] {\longrightarrow}~ \cdots$$ as depicted in . We will refer to such diagrams $\{D_i\}[t_1, t_n]$ as “Cocone Diagrams” or CCDs extending for a specified time interval, e.g. $t_1 ... t_n$ in Fig. 5c.
a\) $$\xymatrix@!C=6pc{\rm{SC}_i(t_j) \ar[r]^{\rm{Id}} & \rm{SC}_i(t_k) \\
\rm{OT}_i(t_j) \ar[u]^{\rm{Inst}} & \ar[l]^{\rm{FCH}} \rm{OT}_i(t_k) \ar[u]_{\rm{Inst}}}$$
b\) $$\xymatrix{& \{D_i\}[t_1, t_2] & \\
\{C_i\}(t_1) \ar[ur] & & \{C_i\}(t_2) \ar[ll]^{\rm{FCH}} \ar[ul]}
\xymatrix{& \{D_i\}[t_3, t_4] & ~ ~\\
\{C_i\}(t_3) \ar[ur] & & \{C_i\}(t_4) \ar[ll]^{\rm{FCH}} \ar[ul]}$$
c\) $$\xymatrix@!C=3pc{& & & \{D_i\}[t_1,t_n] & & \\ & & \{D_i\}[t_1, t_4] \ar@{.>}[ur] & .... & \{D_i\}[t_{n-3}, t_n]\ar@{.>}[ul] & & & &\\
& \{D_i\}[t_1,t_2]\ar@{.>}[ur] \ar@{.>}[r] & & \{D_i\}[t_3,t_4] \ar@{.>}[ul] \ar@{.>}[r] & & \{D_i\}[t_{n-1}, t_n] \ar@{.>}[ul] & & & & &\\
\{C_i\}(t_1)\ar[ur] & \ar[l]^{\rm{FCH}} \{C_i\}(t_2)\ar[u] ... & \ar@{.>}[l] \{C_i\}(t_3) \ar[ur] & \ar[l]^{\rm{FCH}} \{C_i\}(t_4) \ar[u] & & \{C_i\}(t_{n-1}) \ar@{.>}[l] \ar[u] & \ar[l]^{\rm{FCH}} \{C_i\}(t_n) \ar[ul] & & & &}$$
$~$
> *Fig. 5*: a) Interpreting sequential object tokens as representing the same persistent individual constructs an FCH to link them. The FCH is depicted as acting backwards in time as it is built from the new observation to the old one. Here $\rm{SC}_i(t_j)$ denotes Singular Category $i$ at $t_j$, $\rm{OT}_i(t_j)$ denotes Object Token $i$ at $t_j$, etc. $\rm{Id}$ = Identity, and $\rm{Inst}$ = Instance. b) Families of FCHs link sets $\{C_i \}(t_j)$ of object tokens instantiated at different times. The set $\{D_i \}[t_j,t_k]$ of abstracted singular category plus FCH pairs representing objects persistent from $t_j$ to $t_k$ can be viewed as a channel between sets of linked object tokens. c) A CCD representing an object history during a time interval $t_1 ... t_n$. The set $\{D_i \}[t_1,t_n]$ is a colimit cocone for sequences of object tokens consistent with persistence from $t_1$ to $t_n$.
The “essential” identifying properties of objects can change over time, though they cannot all change together without causing identification failure. In the case of human beings, for example, both (approximate) age and core personality characteristics are identifying properties; hence a child with an adult friend’s personality is not identified as one’s adult friend. Slow, asynchronous changes in the composition of singular categories and hence small departures from identity of the linking maps between them do not alter the structures of the above diagrams. Such changes do, however, render FCH construction more difficult.
Contexts, event files and episodic memories
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Objects are never encountered in complete isolation; even the most austere psychophysics experiments have a computer screen and the surrounding laboratory as a context. In real-life settings, objects are typically encountered in interacting groups. Object tokens have, therefore, lateral synchronic associations as well as the diachronic links implemented by FCHs. The event-file construct of @Hommel provides a “snapshot” of such associations over the few-second to few-minute timeframes intuitively regarded as single “events.” Event files capture interactions between objects as well as their significance and affective consequences for the observer. These kinds of information provide crucial input into the FCH construction processes that allow the objects participating in the event to be identified [@Eichenbaum07; @Zimmer; @Fields3].
Event files as defined by @Hommel are effectively tokens; each represents a discrete event that can be encoded and then retrieved as a discrete episodic memory. Single events are by definition localized in time and hence cannot be repeated; recalling an event and hence (partially and perhaps inaccurately) reconstructing an event file, in particular, occurs in a current context and itself constitutes a distinct event. The recognition of event *types* as such is not well characterized experimentally. It seems reasonable to expect, however, that events tokens (i.e. events files) and event types form classifications under the action of a satisfaction relation that maps tokens to types. We consider this further in §\[mereological\] below in the broader context of mereological complexity and reasoning.
Learning new categories and Cone-Cocone Diagrams {#CCCD}
------------------------------------------------
With high frequency in infancy and childhood but typically reduced frequency thereafter, humans encounter not just individual objects, but object types that they have never encountered before. Humans often learn to recognize such novelties from just one “training” encounter. Understanding how humans achieve such one-shot learning is a major challenge for cognitive neuroscience, just as replicating this ability is a major challenge for machine learning. Besides achieving efficient, preferably one-shot learning from exemplars, the problem has (at least) two additional components: recognizing novelty and switching from classification mode to learning mode. As @Oudeyer have emphasized, it is *learnable* novelty that must be recognized; otherwise precious resources are wasted on attempts to learn the unlearnable.
Consider a novel object that is easily classified as an instance of a familiar entry-level category: a novel cat, for example. The object is recognized as novel because its object token does not match any existing singular category, cannot be linking to any existing object token by a plausible FCH, or both. Interacting with the object over an extended period (several views, a few minutes) or encountering it again after a short delay allows certain of its properties to be identified as unchanging; in the case of a cat, these may include size, shape, color pattern, face and voice but not location or behavior. The (short) sequence of distinct object tokens recorded during such interactions serves, in other words, to associate some properties of the object into a provisional singular category. The principle of association here is, once again, persistence: each successive object token indicates the object as persistent, and the features encoded by object tokens in the sequence are similar enough to be treated as identical.
We can, in this case, consider the distinct feature instances encoded by the distinct object tokens in the sequence to be feature “tokens” and consider the object tokens themselves, which the criterion of persistence identifies as representing one individual object, as jointly defining a “type” that organizes those tokens. The construction employed in §\[object-file\] above can then be employed to construct a classification of these tokens into these types. This classification is the Chu-space dual of the classification of object tokens by singular categories shown in Fig. 5a.
The requirement of a familiar entry-level category can now be relaxed: suppose that what is encountered is not a novel cat, but an entirely novel animal, perhaps a pangolin or a platypus. In this case an entirely category must be learned. However novel the object encountered is, it must have *some* familiar features, e.g. its approximate size and shape, whether it has a face, perhaps some aspect of its behavior. Even very young infants can use features of these kinds to initiate classification and identify novelty [@Rakison10]. Placement in any familiar category allows the construction of a singular category as outlined above. Construction of a non-singular category – e.g. \[pangolin\] – merely requires abstraction, i.e. allowance of inexact matches.
The problem of maintaining a singular category across changes in essential features introduced in §\[object-token\] above can now be seen as a special case of category learning. A singular category is robust against feature changes if the FCHs linking its instances are strong enough that persistence at the object token level can induce persistence at the singular category level. The “flow of association” in this case is the reverse of that depicted in Fig. 5c; the properties composing the singular category are in this case the “tokens” that are held together by the persistent object history as a “type.”
Reversing the arrows in a CCD (e.g. Fig. 5c) yields a cone, the dual of a cocone. A system capable of both object history construction and its dual, category learning with singular category maintenance as a special case, is thus characterized by a *cone-cocone diagram* (CCCD); such a diagram can be represented by making all of the arrows in a CCD such as Fig. 5c double-headed. Continuing the notation used in Fig. 5c, we denote the corresponding CCCD by $\mathbf{Dg}_i[t_1, t_n]$.
A CCCD captures the simultaneous upward and downward flow of constraints that characterize human vision and, it is reasonable to suppose, other sensory modalities both functionally and neuro-architecturally [@Hochstein]. The duality expressed by a CCCD is the duality found between dorsal and ventral attention systems, or between high-precision expectations and high-precision inputs in an active inference system. It resolves the central paradox of familiarity: that familiarity can confer either high or low salience in a context-dependent way. The “switch” between these dual constraint flows appears to be implemented, in humans, by the amygdala - insula - cingulate axis at the core of the salience network (e.g. @Uddin15).
Local logics embedded in CCCDs {#cccd-logic-1}
------------------------------
Recall that any classification generates a natural local logic in accordance with Definition \[local-2\], and that @Barwise1 [Prop. 12.7] ensures that any local logic defined on a classification can be identified with the local logic generated by the classification (cf. Example \[building\]). These ideas can now be applied to the classifications defined above to characterize the categorization and identity maintenance processes in terms of the actions of local logics.
To begin, we can immediately apply the principle of Definition \[local-2\] to relating “instantaneous” object files (tokens) to short sequences of object files (types) indicating object permanence, to obtain a local logic ${\mathsf{Lg}}(\mathcal{C}_i)$ with regular theory ${\mathsf{Th}}(\mathcal{C}_i) = (\{ C_i \}, \vdash)$. This ${\mathsf{Th}}(\mathcal{C}_i) = (\{ C_i \}, \vdash)$ expresses the effective criteria for short-term object permanence and hence captures, albeit implicitly, an important part of the semantics of “object” for the system it describes. Likewise, in , we have a local logic ${\mathsf{Lg}}({\mathcal A}_i[t_1, t_n])$ with regular theory ${\mathsf{Th}}({\mathcal A}_i[t_1, t_n]) = (\{D_i\}[t_1, t_n], \vdash)$ (for each $i$) that captures the effective criteria for longer-term object permanence and hence additional components of the semantics of “object.” In both cases all tokens are normal (see Definition \[local-2\]). On recalling Definition \[logic-info\], we can take as a working principle that sequences such as in are logic infomorphisms satisfying the properties of @Barwise1 [12.3]. Accordingly, an underlying semantic structure is built into Fig. 5c, and hence to the ensuing CCCD diagram $\mathbf{Dg}_i[t_1, t_n]$.
Sequents serve purposes in this development. On the one hand, they are implicitly assumed in the preceding discussion (see also §\[local\] and §\[ontologies\]). On the other hand, we recall from §\[shannon\] that on relaxing the sequent relation $\vdash$ to a conditional probability (see ), we see how a sequence of logic infomorphisms may function as a chain of Bayesian inferences. This is consistent with the use of Bayesian methods reviewed in §\[bayesian\], and suggests that the diagrams $\mathbf{Dg}_i[t_1,t_n]$ may be considered as effective carriers of Bayesian inference through sequences of episodic memories, and idea that is extended in §\[mereological\] below.
As regards ontologies, we recall that both types $\{C_i\}$ and $\{D_i\}[t_1, t_n]$ are not strictly sets of (formal) concept symbols. We suggest that a weaker sense of ontology is obtainable on assuming the relations $\leq, ~\perp, |$ in Definition \[ontology-1\]. Generally, however, we can acknowledge the viewpoint of @Kalfoglou2 which sees the local logics themselves as characterizing ontologies. This derivative sense of ontology is useful for our purposes since ontological partitions into “entities” tend to induce “spatial” boundaries around conceptual and/or perceptual partitions @Smith1996. Such induced boundaries can be identified with coarse-grainings and hence induced geometries, as will be discussed in §\[mereotop\] below.
Parts and wholes: Using Chu spaces and information channels to represent mereological complexity {#mereological}
================================================================================================
The Formal Ontology introduced by @Husserl developed a theory of “parts” and “wholes” towards a foundation for mereological reasoning, a methodology that also has roots in the works of Aristotle, Brentano, Whitehead, and others (as reviewed and developed in @Casati1 [@Lando; @Les; @Simons; @Smith1996]). Formalizations of mereological reasoning (e.g. @Casati1 [@Smith1996]) have found wide application in geographic information systems (GIS) and formal ontologies. The implementation of mereological reasoning in humans is not, however, well understood; indeed we have been able to find only a single neuroimaging study explicitly comparing mereological and functional classifications [@muehlhaus14]. As mereological reasoning appears specifically to fail in the “weak central coherence” phenotype of autism spectrum conditions [@Happe06], understanding its implementation is potentially of clinical relevance.
Perceptual identification of mereologically-complex objects {#mcomplex-1}
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The macroscopic objects perceptible by humans are by definition mereologically complex: they have multiple perceptible parts, each of which has further parts, etc. Such objects can, moreover, be assembled into larger complexes, with perceptual scenes being ubiquitous, transient examples. Many such larger complexes are, however, not transient but rather meaningful, persistent objects in their own right. A fundamental challenge posed by human object perception is to understand what mereological complexes are perceived as “whole” objects, how object tokens representing such complexes are constructed, and how such object tokens are linked into persistent histories despite changes in the properties and even identities of the “parts” making up the complex.
A specific example of a mereological hierarchy is shown in Fig. 6. Individual human beings, such as author CF, are entry-level (EL) objects and hence are represented by EL object tokens. Human beings are inevitably members of larger complexes, including families, extended families, tribes, ethnic groups, nations, etc. The smaller instances of such complexes (e.g. human nuclear families) can be directly perceived; larger instances may not be perceptible but can be referred to using language, images, and abstract graphics. Hence object tokens can be constructed for such complexes. Object tokens representing “parts” such as CF are naturally linked to object tokens representing complexes, such as CF’s family, by “$\mathrm{part\_of}$” relations. Such relations similarly link parts of CF to CF. Entry-level objects appear to play a special role in such hierarchies; “$\mathrm{part\_of}$” links are transitive both above and below EL objects, but not across EL objects. A part of CF is not a part of CF’s family, just as a part of a car is not a part of a fleet of cars.
{width="20cm"}
> *Fig. 6*: Example of an object token (OT) hierarchy extending both above and below a mereologically-complex entry-level (EL) object, one of the present authors (CF). Each OT has an associated singular category (SC) specifying identifying static and behavioral features. These SCs are in turn associated with general categories, some of which are shown here. Solid arrows show typical “$\mathrm{part\_of}$”, “$\mathrm{has\_a}$” and “$\mathrm{is\_a}$” links. Dashed arrows show induced $\mathrm{part\_of}$ links; red “X” indicated the failure of “$\mathrm{part\_of}$” transitivity across the EL OT.
Hierarchies of tokens linked by “$\mathrm{part\_of}$” relations are commonplace in AI systems. The existence of such “$\mathrm{part\_of}$” hierarchies raises, however, the questions of what the “$\mathrm{part\_of}$” relation is, how it is established, and how it is maintained over time. The correspondence between object tokens and singular categories provides a partial answer: “$\mathrm{part\_of}$” relations between object tokens correspond to “$\mathrm{has\_a}$” relations between singular categories, which in turn correspond to “$\mathrm{has\_a}$” relations between general categories (Fig. 6). While the object token is the locus of learning for the first exemplars of EL objects encountered in infancy and childhood, once a general category has been learned the “$\mathrm{part\_of}$” links between new object tokens can be induced by inter-category “$\mathrm{has\_a}$” links. The mechanisms by which such link induction is implemented remain to be elucidated; we consider formal structures supporting this process below.
Mereological hierarchies as hierarchies of CCCDs {#mcomplex-2}
------------------------------------------------
Let us first consider the Chu space ${\mathsf{C}}= (C_{{\mathsf{o}}}, \Vdash_{{\mathsf{C}}}, C_{{\mathsf{a}}})$, where $C_{{\mathsf{o}}}$ and $C_{{\mathsf{a}}}$ are sets of object tokens and their corresponding singular categories and $\Vdash_{{\mathsf{C}}}$ is the “Identifies” relation in Fig. 6. Recall from §\[FCA-1\] the pair of maps $(\alpha,\omega)$ (there considered as a Galois connection) given by: $$\label{mereo-1}
\begin{aligned}
\alpha &: {\mathcal P}(C_{{\mathsf{o}}}) {\longrightarrow}{\mathcal P}(C_{{\mathsf{a}}}) ~\text{with}~ {\alpha}(x) = \{{\mathsf{a}}: \forall x \in X, ~x \Vdash_{{\mathsf{C}}} {\mathsf{a}}\} \\
\omega &: {\mathcal P}(C_{{\mathsf{a}}}) {\longrightarrow}{\mathcal P}(C_{{\mathsf{o}}}) ~\text{with}~ \omega(A) = \{x: \forall {\mathsf{a}}\in A,~ x \Vdash_{{\mathsf{C}}} {\mathsf{a}}\}.
\end{aligned}$$ Here $\alpha$ clearly maps an object token to its (unique) singular category and $\omega$ maps a singular category to its (unique) object token.
\[mereo-2\] $~$
- Suppose $X$ is a set of subsets consisting of *parts of objects*. Then we define $\omega \circ \alpha (X)$ to be the set of subsets of *whole parts of objects as obtained from $X$*.
- Suppose $Y$ is a set of subsets consisting of *parts of attributes*. Then we define $\alpha \circ \omega(Y)$ to be the set of subsets of *whole parts of attributes as obtained from $Y$*.
The usage “whole parts” is employed here to emphasize that “wholes” on one level may be “parts” at the level(s) above.
Likewise, for a given classification we have ${\mathcal A}= \langle \rm{Tok}({\mathcal A}), \rm{Typ}({\mathcal A}), \Vdash_{{\mathcal A}} \rangle$ , and for $a \in X \subseteq \rm{Typ}({\mathcal A}), b \in A \subseteq \rm{Tok}({\mathcal A})$, we have: $$\label{mereo-3}
\begin{aligned}
\alpha^* &: {\mathcal P}(\rm{Typ}({\mathcal A})) {\longrightarrow}{\mathcal P}(\rm{Tok}({\mathcal A})) ~\text{with}~ \alpha^*(x) = \{b: \forall a \in X, ~x \Vdash_{{\mathcal A}} b \} \\
\omega^* &: {\mathcal P}(\rm{Tok}({\mathcal A})) {\longrightarrow}{\mathcal P}(\rm{Typ}({\mathcal A})) ~\text{with}~ \omega^*(A) = \{a: \forall b \in A,~ a \Vdash_{{\mathcal A}} b \}.
\end{aligned}$$ Iterating these conditions allows us to move one rung at a time through the mereological hierarchy when incorporating information channels.
To see how this mereological hierarchy can be constructed, we first of all construct a (quasi-hierarchial) complex of CCCDs following §\[object-token\], §\[CCCD\] and §\[cccd-logic-1\]. Recalling , we may view the index $i$ as indicating a ‘level’ in a complex of CCCDs constructed from connected sequences. In such sequences, the time intervals will generally be distinct. Thus we commence with families of time dependent logic infomorphisms arising from such morphisms between different classifications as specified in : $$\label{cccd-1}
{\mathcal A}_i[t_{i_1},t_{i_n}] {\longrightarrow}{\mathcal B}_j[t_{j_1},t_{j_n}]$$ at ‘levels’ $i,j$ (possibly $j=i$), each respecting the ‘parts’ to ‘wholes’ condition of . Both classifications lead to their corresponding diagrams as in Fig. 5c, as explained in §\[object-token\], with an induced (logic) infomorphism $\mathbf{C}_i {\longrightarrow}\mathbf{C}_j$ between the cocone vertex classifications of the corresponding CCCDs derived from Fig. 5c (again, the time intervals are understood). Following the formulism of §\[CCCD\], we thus obtain induced (logic) infomorphisms: $$\label{cccd-2}
\begin{aligned}
\mathbf{Dg}_i[t_{i_1}, t_{i_n}] &{\longrightarrow}\mathbf{Dg}_j[t_{j_1}, t_{j_n}] \\
\mathbf{C}_i &\mapsto \mathbf{C}_j
\end{aligned}$$ Schematically, this leads to a typical quasi-hierarchial configuration as depicted in Fig. 7 below. Note that the assumption of taking logic infomorphisms provides an underlying semantic structure to the various mechanisms as discussed in §\[cognitive\] and §\[tt-flow\].
$$\xymatrix@C=4pc{... & ... & ... & ... & & \\& \mathbf{Dg}_j[t_{j_1}, t_{j_n}]\ar@{.>}[d] \ar@{.>}[dl] & & \ar@{.>}[d] & \\
\mathbf{Dg}_w[t_{w_1}, t_{w_n}] \ar@{.>}[u] & \mathbf{Dg}_k[t_{k_1}, t_{k_n}] \ar@{.>}[dl] \ar@{.>}[d] & \mathbf{Dg}_{\ell}[t_{\ell_1}, t_{\ell_n}] \ar@{.>}[ul] \ar@{.>}[u] & \mathbf{Dg}_p[t_{p_1}, t_{p_n}] \ar@{.>}[l]\ar@{.>}[d] \ar@{.>}[r] & \\
\mathbf{Dg}_v[t_{v_1}, t_{v_n}]\ar@{.>}[u] & & \mathbf{Dg}_m[t_{m_1}, t_{m_n}]\ar@{.>}[u] \ar@{.>}[dl] \ar@{.>}[ul] & \mathbf{Dg}_q[t_{q_1}, t_{q_n}] \ar@{.>}[dl] \ar@{.>}[l] \ar@{.>}[d] \ar@{.>}[r] & &\\
\ar@{.>}[u]& \ar@{.>}[ul] \ar@{.>}[dl] \mathbf{Dg}_s[t_{s_1}, t_{s_n}] & \ar@{.>}[l] \mathbf{Dg}_u[t_{u_1}, t_{u_n}]\ar@{.>}[dl]\ar@{.>}[u] & \mathbf{Dg}_r[t_{r_1}, t_{r_n}] \ar@{.>}[dl] \ar@{.>}[l] \ar@{.>}[d] \ar@{.>}[r] & \\
... & ... & ... & ...& &
}$$ $~$
> *Fig. 7*: A typical complex of interactive CCCDs corresponding to a mereological object-token hierarchy that is maintained over time.
Observe that the configurations depicted in Fig. 7 are not strictly hierarchial, even though the corresponding colimits are iterated. Why it is not strictly hierarchial is plain to see: a part can be a part of many wholes, and even of wholes at different levels (for instance, an employee can be part of a division, but also part of a company). As will be seen later, it is in this respect that a mereotopological complex differs from a standard category-theory hierarchy in the sense of e.g. @Baas2004 [@EV2007]. Here, in particular, because of the existence of co-planar complexes and bi-directional arrows (Fig. 8), it is not always the case that that a relevant object of level $n+1$, say, is the colimit of at least one diagram at level $n$. However, somewhat in line with @Baas2004, we may also interpret an “observer” as an entity that is external or internal to the system, or the system’s environment itself, through which “selection” induces the further levels of structure.
{width="20cm"}
> *Fig. 8*: A two-layer mereological network with “parts” at the lower level, and “complexes” at the upper level. Each level comprises both tokens and types. For example, the “complex” level could consist of EL types such as \[cat\], \[dog\], \[chair\], etc. Each of these has many tokens that are specific individuals. The “part” level can include visually-identifiable, but non-essential features of these types, such as “has four legs”, “has fur”, “walks with a gait $X$”, “is white with brown spots”, etc. Here again there are tokens: e.g. four particular legs, a particular pattern of white-with-brown spots, etc.
Lateral connections at each level indicate, e.g. the probable co-occurrence in a scene. Vertical arrows indicate mereological inclusions going upwards and top-down prediction from current “understanding” or “prior probabilities” going downwards. These do not necessarily select the same relations, as they often do not in real-life situations.
Cocones exist in both directions in this mereological structure: linked (time dependent) colimit cocones as underlying a typical CCCD, and CCCDs as linked by a network of logic infomorphims $ ... {\longrightarrow}\mathbf{C}_i {\longrightarrow}\mathbf{C}_{i+1} {\longrightarrow}\mathbf{C}_{i+2} {\longrightarrow}... $ between the cocone vertex classifications.
“Learning” in this system would consist of: 1) associating altogether new lower-and upper-level types into the network; 2) distinguishing new high-level individuals as clusters of low-level tokens; 3) Convergence toward better predictions (i.e. all vertical arrows are bidirectional). “Building” in this system is associating a bunch of upward arrows with an upper-level token. “Abstraction” would be the grouping of upper-level tokens into an upper-level type while preserving all arrows. Previous work demonstrating general models of ANNs (§\[mlp\]) shows that these processes are allowed in principle; different specific choices of algorithms for these processes would be expected to produce different hierarchical structures.
From mereology to mereotopology: Distinguishing objects by boundaries {#mereotop}
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The detection of edges and their extension into contours that segment an image into bounded, non-overlapping regions is one of the earliest stages of visual processing (reviewed by @wagemans1). What, however, distinguishes a two- or even three-dimensional array of bounded, non-overlapping pattern elements – e.g. an array of color or texture patches – from an array of bounded, non-overlapping objects? As discussed above, animacy, agency and independent manipulability are important indicators of bounded objecthood during infancy and early childhood when object categories are first being learned and populated with exemplars. What, however, are the inferences that enforce boundedness for objects, and how does the constraint of having a boundary affect the informational relations outlined above?
The key idea of mereotopology is that the parts of an object must be *inside* the object, i.e. contained within its boundary [@Casati1; @Smith1996]. This constraint is, clearly, more easily satisfied for boundaries that are (at least approximately) smooth and convex. As simplicity and hence resource efficiency appear to be general principles of perceptual system organization [@wagemans2], one can expect perceivers to “see” smooth, convex boundaries – e.g. convex hulls of geometrically more complex objects – more easily. Imposing smoothness and convexity – e.g. by constructing the convex hull of a geometrically more complex object – is a form of coarse-graining. We can, therefore, suggest that constructing an “exterior” boundary around a collection of parts that then serves as a boundary for the whole simply is a coarse-graining operation. In this case, the simplicial methods introduced in §\[simplicial\] immediately become relevant, and indeed provide a general method of constructing object boundaries from the bottom up in any mereological hierarchy representable in the CCCD form as in Fig. 7.
As noted earlier, a scene is a mereological complex; segmenting a scene by adding boundaries makes it a mereotopological complex. At the “top” of the mereotopological hierarchy, a whole scene can be considered a multilayer complex of simplicial complexes (i.e. identified “whole” objects) of simplices (identifiable “part” objects). Recall from §\[simpl-def\] that any such complex, at any level of the hierarchy, has associated barycentric coordinates and a natural metric. Distances *within* a simplicial complex at level $n$ of the hierarchy, however, can also be viewed as distances *between* its component simplices at level $n-1$. Boundaries, therefore, induce geometric relations between the bounded objects. In this sense, spatial relations can be viewed as “emergent” from the distinctions between objects, a view with striking similarities to recent proposals within fundamental physics (cf. @Fields17b).
Channels, inter-object boundaries and interactions
--------------------------------------------------
The spatial separation between objects induced by their boundaries – and hence by their distinguishability – generates a time-dependent exchange of information and hence an *interaction* as this term is traditionally understood. Again working from the top down in a mereotopological hierarchy of simplicial complexes indentified with local classifications, every channel between classifications at level $n$ can also be viewed as a channel between the corresponding “objects”, i.e. simplicial complexes. This channel corresponds to the boundary between the “objects” if they are adjacent, i.e. if the corresponding simplicial complexes share $(n-1)$-level faces. It is natural to think of the information transmitted along the channel as “encoded on” this boundary, i.e. as encoded holographically as this term is used in physics @Fields17b. If the objects are not adjacent, i.e. if the connecting channel is a composition at level $n$, the channel can be thought of as passing through a shared “environment” interposed between the objects. The components of the composed channel cannot in general be expected to be isomorphisms; hence the structure of this interposed environment affects the interaction between the objects.
Perceiving object motion requires tracking the identity of the “moving” object through time; hence it involves a temporal sequence of mereological hierarchies along the lines of Fig. 7. The structure of the top-level scene is different at each time increment; hence the metric relations between component simplicial complexes is time-dependent. Changing the relative positions of objects re-shapes their shared environment, in general altering the interaction between them. The distance and material, e.g. transparency or electrical permittivity, dependence of physical interactions can, therefore, be viewed as qualitatively “emergent” from the simplicial structure of classifications.
Conclusion
==========
Category theory provides a rich set of tools for investigating relationships between structures, and hence for representing information flow between structures. These methods have been applied widely in computer science, and are seeing increasing applications in physics. As we have reviewed in Part I above, applications of category theoretic methods in the cognitive sciences – mainly in the investigation of ontologies and ontology convergence – have mainly been carried out at a high level of abstraction. In Part II, we have begun the process of characterizing object perception in category-theoretic terms, particularly in terms of Chu spaces, classifications, simplicial complexes, and local logics. These formal methods provide a natural and intuitively-clear representation perception as a multi-stage process in which types at one level may serve as tokens at a higher level. The co-cone – cone duality captured in CCCDs is particularly useful as a representation of the bidirectional information flow employed by the Bayesian predictive coding systems that the brain appears to implement at multiple scales. The analysis we present in §\[tt-flow\] extends previous work on the representation of ANNs by making this duality explicit. It also makes explicit the essential role of inferences – here captured as infomorphisms – in tracking object identity through time.
Human beings, and presumably other animals with relative complex cognitive systems, employ both abstraction and mereological hierarchies to categorize objects. We show in §\[mereological\] that networks of time-indexed CCCDs provide a natural representation of the mutually-constraining relationship between these two categorization methods, particularly as they are employed in object-identity tracking. We then explore briefly the emergence of spatial relationships and interactions between objects from their description as simplicial complexes embedded in the larger simplicial complex that constitutes a perceptual scene. This top-down, emergence-based approach to mereotopology differs significantly from previous approaches that are geared to different priorities, and different applications [@Le; @Smith1996].
This initial foray into the categorical representation of cognitive processes raises a number of questions and illuminates several open problems. One of the deepest is whether the satisfaction relations $\Vdash$ operating between tokens and types at any of the processing levels considered here are well-defined. While it must be assumed that they are to develop models, and associations, it remains possible that “$\Vdash$” is token, type, context, or time dependent, as studies of the dependence of language on unspecified “background knowledge” [@Searle83] or of cognition generally on “embodiment” [@Anderson03; @Chemero13] suggest. Problems that require further work include the implications of the present results for the representation of ontologies and particularly for ontology convergence between agents that have encountered non- or only partially-overlapping collections of individual objects, the extent to which local logics define or constrain semantic relations between either tokens or types, the extent to which cognitively-significant differences in spatial scale can be captured by coarse-graining, and the question of why the geometry emergent from human visual perception should be three-dimensional. There also remains the open question of how cognition is affected when one or the other of the categorization systems breaks down, as appears to be the case with the mereological system in autism where a sense of context, and an overall gestalt of certain situations may be adversely impacted [@Happe06]. It is, finally, possible that the conceptual ground which we have covered could be further supplemented by certain related techniques of higher dimensional algebra [@Brown2011] and those of $n$-categories [@Leinster2004], topics which remain for further investigation.
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[^1]: A topological space $X$ is said to be a *$T_0$-space* if given distinct points of $X$, there is an open set of $X$ that contains one but not the other. $T_0$-spaces naturally give rise to a partial order defined on the set of points of $X$, where $x \leq y$, if for each open set $U \subseteq X$, $y \in U$ implies $x \in U$, and conversely.
[^2]: We adopt this natural definition of a simplicial homotopy as found in [@Friedman; @Goerss]. In [@Friedman] it is compared with the traditional, more technically oriented definition as seen in other textbooks on the subject.
[^3]: The nerve specifies, in effect, which simplices adjoin each other by “sharing an edge.” @Porter1 provides a number of illustrative examples providing an intuition leading to the discussion in @GP1.
[^4]: Recall that a *$\sigma$-algebra* over $\Omega$ is a set $\Sigma$ of subsets of $\Omega$, such that $\emptyset \in \Sigma, ~\Omega - e \in \Sigma$, for each $e \in \Sigma$, and $\bigcup E \in \Sigma$, for each countable set $E \subseteq \Sigma$. $\mu$ is a *probability measure* on $\Sigma$, if and only if it satisfies the Kolmogorov axioms: $\mu(\emptyset) = 0,~ \mu(\Omega -e) = 1 - \mu(e)$, and $\mu(\bigcup E) = \sum_{e \in E} \mu(e)$ if $E$ is countable, and $\mu(e_1 \cap e_2) = 0$, for all $e_1 \neq e_2 \in E$.
[^5]: The diagrams included in – are reproduced from [@Baianu2006; @Brown2003], with permission from R. Brown
|
---
abstract: 'We characterize the para-associative ternary quasigroups (flocks) applicable to knot theory, and show which of these structures are isomorphic. We enumerate them up to order 64. We note that the operation used in knot-theoretic flocks has its non-associative version in extra loops. We use a group action on the set of flock colorings to improve the cocycle invariant associated with the knot-theoretic flock (co)homology.'
address:
- '(M.N.) Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gda[ń]{}sk, 80-308 Gda[ń]{}sk, Poland'
- '(A.P., A.Z.) Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland'
author:
- Maciej Niebrzydowski
- Agata Pilitowska
- 'Anna Zamojska-Dzienio'
title: 'Knot-theoretic flocks'
---
Introduction and preliminary definitions
========================================
Knot-theoretic ternary quasigroups are algebraic structures suitable for colorings of regions in the knot diagrams. Their operations generalize the ternary relations of the form $d=ab^{-1}c$ from the Dehn presentation of the knot group, just like the quandle operations generalize the conjugation present in the Wirtinger relations. Knot-theoretic ternary quasigroups are introduced in full generality in [@Nie17], but see also [@Nie14] and [@NeNe17].
In this paper, we work with the sub-family of knot-theoretic ternary quasigroups introduced in [@Nie14]. They do not require orientation to produce coloring invariants of knots and knotted surfaces. We generalize the results of [@NPZ19], where we described the structure of knot-theoretic ternary groups. Replacing the ternary associativity condition with a similar condition of para-associativity, somewhat surprisingly leads to the structures based on nonabelian groups in place of the abelian ones. The idempotent case corresponds to homomorphisms from the knot group, and the non-idempotent case involves a central involution in a group. It is possible to define various group actions on the set of flock colorings, which are compatible with the Reidemeister moves. We use one of them to strenghten the flock cocycle invariant obtained from the (co)homology of ternary algebras introduced in [@Nie17]. We also note the non-associative version of the obtained flock operations using extra loops. Let us begin with the necessary definitions.
A [*ternary groupoid*]{} is a non-empty set $X$ equipped with a ternary operation $[\, ]\colon X^3\to X$. It is denoted by $(X,[\, ])$.
A ternary groupoid $(X,[\, ])$ is called a *ternary quasigroup* if for every $a,b,c\in X$ each of the following equations is uniquely solvable for $z\in X$: $$\begin{aligned}
&[zab]=c, \\
&[azb]=c, \label{eq:m}\\
&[abz]=c.\end{aligned}$$
We say that an operation $[\, ] \colon X^3 \to X$ is *associative* if for all $a,b,c,d,e\in X$ $$[[abc]de]=[a[bcd]e]=[ab[cde]].$$
An associative ternary quasigroup is called a *ternary group*; see [@Post] for a treatise on $n$-ary groups.
An operation $[\, ] \colon X^3 \to X$ is *para-associative* if for all $a,b,c,d,e\in X$ $$[[abc]de]=[a[dcb]e]=[ab[cde]].$$
Various categories of para-associative groupoids were studied by Wagner in [@Wag53]. See also [@HoLa17].
A para-associative ternary quasigroup is called a *flock*. The connection of flocks with affine geometry was investigated by Dudek in [@Dud99].
We say that a ternary groupoid $(X,[\, ])$ is *idempotent* if $[aaa]=a$ for all $a\in X$. Idempotent flocks were used, for example, in [@Cer43].
![A relation in the Dehn presentation can be realized using a para-associative operation $[xyz]=xy^{-1}z$.[]{data-label="Dehngen"}](Dehngenbr.eps){height="3"}
A good topological motivation for considering para-associativity comes from relations in the Dehn presentation of the knot group. Recall that in the Dehn presentation generators are assigned to the regions in the complement of a knot diagram $D$ on a plane, and relations correspond to the crossings and are as in Fig. \[Dehngen\](A). One of the generators, for example the one corresponding to the unbounded region, is set equal to identity. Geometrically, a generator can be viewed as a loop originating from a fixed point $P$ beneath the diagram, piercing a region to which it is assigned, and returning to $P$ through a region labeled with the identity element. See e.g. [@Kau83] for more details about Dehn presentation. Note that the fundamental group relations can be realized using a para-associative operation $[xyz]=xy^{-1}z$, see Fig. \[Dehngen\](B). We will show that para-associativity leaves a bit of room for generalizing this operation.
![The third Reidemeister move and the nesting conditions.[]{data-label="rad3br"}](rad3brno.eps){height="5"}
![Coloring of three-dimensional regions near a double point curve in a knotted surface diagram.[]{data-label="surfdehn"}](surfdehnfl.eps){height="3"}
The following two nesting conditions obtained from the coloring of regions in the third Reidemeister move were defined in [@Nie14], see Fig. \[rad3br\]: $$\forall_{a,b,c,d\in X} \quad [ab[bcd]]=[a[abc][[abc]cd]], \tag{LN}$$ $$\forall_{a,b,c,d\in X} \quad [[abc]cd]=[[ab[bcd]][bcd]d]. \tag{RN}$$
By adding the adjective *knot-theoretic* when talking about a ternary groupoid of some sort, we mean that the said groupoid satisfies the conditions LN and RN. Thus we consider, for example, *knot-theoretic ternary quasigroups*, *knot-theoretic ternary groups*, and *knot-theoretic flocks*.
Colorings of a knot or a knotted surface diagram $D$ with elements of a knot-theoretic flock $(X,[\, ])$ are defined in a simple way, as in Fig. \[Dehngen\](B) and Fig. \[surfdehn\]. More specifically, they are functions $\mathcal{C}\colon Reg(D)\to X$, from the set of regions in the complement of $D$, such that a color $x$ near a crossing (resp. double-point curve) is expressed as $x=[yzw]$, where $y$, $z$, $w$ are the remaining three colors taken in a cyclic order, and the regions colored by $x$ and $y$ are separated by an over-arc (resp. over-sheet). For Yoshikawa diagrams, it is required that near a marker the colors are assigned in an $a$, $b$, $a$, $b$ fashion, that is, opposite regions receive the same color. The number of such colorings is an invariant under the applicable moves (Reidemeister, Roseman or Yoshikawa moves), see [@Nie17; @KimNel] for more details.
In a ternary quasigroup $(X,[\, ])$, for every element $a\in X$, the unique solution of the equation $[aza]=a$ is called the *skew element to* $a$ and is denoted by $\bar{a}$.
An operation $[\, ] \colon X^3 \to X$ is *semi-commutative* if for all $a,b,c\in X$ $$[abc]=[cba].$$ It follows that semi-commutative flocks are ternary groups.
In [@NPZ19], we obtained a precise characterization of knot-theoretic ternary groups, and applied it to the theory of flat links on possibly non-orientable surfaces. One of the main results of [@NPZ19] is as follows:
\[mainchartgr\] Each knot-theoretic ternary group $(A,[\, ])$ is determined by an abelian group $(A,+)$ and an element $a\in A$ which is either zero (in the idempotent case) or has order two in $(A,+)$. For every $x,y,z\in A$ $$[xyz]=x-y+z+a\quad {\rm and}\quad \bar{x}=x+a.$$
In particular, we obtained a connection with Takasaki quandles (including dihedral quandles): $$\label{Takasaki}
[xy\bar{x}]=x-y+\bar{x}+a=x-y+x+a+a=2x-y.$$
We are now ready to consider the para-associative case.
The structure of knot-theoretic flocks
======================================
In a para-associative groupoid $(X,[\, ])$ an element $a\in X$ is called *special* if $[aax]=[xaa]$ for all $x\in X$.
In [@Dud99] the following theorem, analogous to the Gluskin-Hossz[ú]{} theorem ([@Hos63; @DudGla; @Glu65; @Post]), was proven.
\[HGflock\] A para-associative groupoid $(X,[\, ])$ with a special element is a flock if and only if there exists a binary group $(X,\cdot)$ and its anti-automorphism $\theta$ such that $\theta^2(x)=x$, and $$[xyz]=x\cdot\theta(y)\cdot z\cdot b$$ for all $x$, $y$, $z\in X$, where $b$ is a central element of $(X,\cdot)$ such that $\theta(b)=b$.
First, we will show that knot-theoretic flocks satisfy the requirement of existence of a special element from the above theorem.
For any elements $x$, $y$, $z$ of a flock $(X,[\, ])$ $$\begin{aligned}
&[y\bar{x}x]=[yx\bar{x}]=[x\bar{x}y]=[\bar{x}xy]=y,\\
&\bar{\bar{x}}=x,\\
&\overline{[xyz]}=[\bar{x}\,\bar{y}\,\bar{z}].\end{aligned}$$
\[special\] Let $(X,[\, ])$ be a knot-theoretic flock. Then every element $x\in X$ is special.
From para-associativity and the condition RN, for any $a$, $b$, $c$, $d\in X$ we have $$[d[bbc]a]=[[dcb]ba]=[[dc[cba]][cba]a]=[d[[cba][cba]c]a].$$ From the uniqueness of solution of equation (\[eq:m\]) it follows that $$[bbc]=[[cba][cba]c].$$ Substituting $\bar{b}$ for $c$, we get $$b=[bb\bar{b}]=[[\bar{b}ba][\bar{b}ba]\bar{b}]=[aa\bar{b}],$$ and after substituting $\bar{b}$ for $b$, we obtain $\bar{b}=[aab]$. Similarly, using the condition LN and para-associativity: $$[a[cbb]d]=[ab[bcd]]=[a[abc][[abc]cd]]=[a[c[abc][abc]]d].$$ It follows that $$[cbb]=[c[abc][abc]],$$ and after substituting $\bar{b}$ for $c$, we get $$b=[\bar{b}bb]=[\bar{b}[ab\bar{b}][ab\bar{b}]]=[\bar{b}aa],$$ which is equivalent to $\bar{b}=[baa]$. To summarize, for any $a$, $b\in X$, $$[aab]=\bar{b}=[baa], \label{skewfl}$$ that is, every element is special.
Note that if $\bar{x}=x$ for all $x\in X$, then such knot-theoretic flocks are examples of heaps ([@Wag53; @HoLa17]), i.e. para-associative ternary groupoids $(X,[\, ])$ satisfying $[aax]=x=[xaa]$ for all $a$, $x\in X$.
A flock $(X,[\,])$ in which the equation (\[skewfl\]) is satisfied for any $x$, $y\in X$ is knot-theoretic.
Using $[yxx]=\bar{y}$ and para-associativity, we obtain the condition LN as follows: $$[ab[bcd]]=[a[cbb]d]=[a\bar{c}d]=[a[c[abc][abc]]d]=[a[abc][[abc]cd]].$$
The equation $[xxy]=\bar{y}$, for $x$, $y\in X$, yields the condition RN: $$[[abc]cd]=[a[ccb]d]=[a\bar{b}d]=[a[[bcd][bcd]b]d]=[[ab[bcd]][bcd]d].$$
Now we obtain a characterization of knot-theoretic flocks.
\[struct\] A ternary groupoid $(X,[\, ])$ is a knot-theoretic flock if and only if there exists a binary group $(X,\cdot)$ such that for any $x$, $y$, $z\in X$ $$\label{knfl}
[xyz]=x\cdot y^{-1}\cdot z\cdot b,$$ where $b$ is either the identity $e$ of the group $(X,\cdot)$, or is a central element of order two in $(X,\cdot)$.
From Theorem \[HGflock\] and Lemma \[special\] $$[xyz]=x\cdot\theta(y)\cdot z\cdot b,$$ where $\theta$ is an anti-automorphism such that $\theta(b)=b$, and $b$ is in the center of the group $(X,\cdot)$. From (\[skewfl\]) we have $$\bar{e}=[eee]=e\cdot \theta(e)\cdot e\cdot b=b.$$ Then, for any $y\in X$, $$b=\bar{e}=[yye]=y\cdot \theta(y)\cdot e\cdot b.$$ Thus, $e=y\cdot\theta(y)$, that is, $\theta(y)=y^{-1}$. Since $\theta(b)=b$, it follows that $b^2=e$. Note that, for any $x\in X$, $$\bar{x}=[xee]=x\cdot e^{-1}\cdot e\cdot b=x\cdot b.$$ Now suppose that a ternary groupoid $(X,[\, ])$ has an operation of the form (\[knfl\]). Then it is a ternary quasigroup and satisfies the para-associative condition: $$\begin{aligned}
&[[xyz]vw]=(x\cdot y^{-1}\cdot z\cdot b)\cdot v^{-1}\cdot w\cdot b=\\
&[x[vzy]w]=x\cdot(v\cdot z^{-1}\cdot y\cdot b)^{-1}\cdot w\cdot b=\\
&[xy[zvw]]=x\cdot y^{-1}\cdot (z\cdot v^{-1}\cdot w\cdot b)\cdot b,\end{aligned}$$ since $b$ is in the center of the group $(X,\cdot)$ and of order one or two. The nesting conditions LN and RN are also easy to check.
Note that one can obtain the core group operation (see e.g. [@Wada] for details) in a knot-theoretic flock by taking $$[xy\bar{x}]=x\cdot y^{-1}\cdot \bar{x}\cdot b=x\cdot y^{-1}\cdot x\cdot b\cdot b
=x\cdot y^{-1}\cdot x.$$
By Theorem \[struct\] each knot-theoretic flock $(X,[\,])$ is defined by a group $(X,\cdot)$ and an element $b\in Z(X)$ of order one or two. In this case, we write $(X,[\,])=\mathcal{F}((X,\cdot),b)$ and call the group $(X,\cdot)$ *associated* to the knot-theoretic flock $(X,[\,])$. The next result shows the relationship between isomorphic knot-theoretic flocks and their associated groups and central elements.
\[isomorphism\] Let $(X_1,[\, ]_1)=\mathcal{F}((X_1,\cdot),b_1)$ and $(X_2,[\, ]_2)=\mathcal{F}((X_2,\ast),b_2)$ be two knot-theoretic flocks. Then knot-theoretic flocks $(X_1,[\, ]_1)$ and $(X_2,[\, ]_2)$ are isomorphic if and only if there exists a group isomorphism $f\colon (X_1,\cdot)\to (X_2,\ast)$ such that $b_2=f(b_1)$.
The implication “$\Leftarrow$" directly follows by [@Dud99 Proposition 4.10].
To prove the converse let $h\colon (X_1,[\, ]_1)\to (X_2,[\, ]_2)$ be a flock isomorphism. This means that for $x,y\in X_1$ we have $$\begin{aligned}
\label{eq:ktfiso}
&h(x\cdot y)=h(x\cdot b_1^{-1}\cdot y \cdot b_1)= h([xb_1y]_1)=[h(x)h(b_1)h(y)]_2=h(x)\ast (h(b_1))^{-1}\ast h(y)\ast b_2.\end{aligned}$$ (The first equality holds by centrality of the element $b_1$.)
Let us define the mapping $$f\colon X_1\to X_2, \quad x\mapsto h(x)\ast (h(b_1))^{-1}\ast b_2.$$ Clearly, $f(b_1)=h(b_1)\ast (h(b_1))^{-1}\ast b_2=b_2$. We will show that $f$ is a group isomorphism. Since $h\colon X_1\to X_2$ is a bijection, then $f$ is a bijection, too. Further, by and centrality of the element $b_2\in X_2$ we immediately obtain for $x,y\in X_1$: $$\begin{aligned}
&f(x\cdot y)=h(x\cdot y)\ast (h(b_1))^{-1}\ast b_2\stackrel{ \eqref{eq:ktfiso}}=h(x)\ast (h(b_1))^{-1}\ast h(y)\ast b_2\ast (h(b_1))^{-1}\ast b_2=\\
&h(x)\ast (h(b_1))^{-1}\ast b_2\ast h(y)\ast (h(b_1))^{-1}\ast b_2=f(x)\ast f(y),\end{aligned}$$ which finishes the proof.
Using Theorem \[isomorphism\] and GAP [@GAP2019] (in particular, the Small Groups library), we were able to enumerate the knot-theoretic flocks obtained from non-abelian groups, up to order 64. Table \[count\_all\_nonab\] lists the orders for which such flocks exist. For the ones obtained from abelian groups (that is, for knot-theoretic ternary groups), see [@NPZ19].
$$\begin{array}{|r|cccccccccccccccccc|}\hline
n &6 &8 &10 &12 &14 &16 &18 &20 &21 &22 &24 &26 &27 &28 &30 &32 &34 &36\\\hline
\text{all}&1 &4 &1 &5 &1 &23 &3 &5 &1 &1 &24 &1 &2 &4 &3 &127 &1 &16\\
\text{idempotent}&1 &2 &1 &3 &1 &9 &3 &3 &1 &1 &12 &1 &2 &2 &3 &44 &1 &10\\\hline
\end{array}$$
$$\begin{array}{|r|cccccccccccccccccc|}\hline
n &38 &39 &40 &42 &44 &46 &48 &50 &52 &54 &55 &56 &57 &58 &60 &62 &63 &64\\\hline
\text{all} &1 &1 &23 &6 &4 &1 &112 &3 &5 &14 &1 &20 &1 &1 &17 &1 &2 &886\\
\text{idempotent} &1 &1 &11 &5 &2 &1 &47 &3 &3 &12 &1 &10 &1 &1 &11 &1 &2 &256\\\hline
\end{array}$$
A generalization to extra loops
-------------------------------
A [*binary quasigroup*]{} is a groupoid $(Q,*)$ such that the equation $x*y=z$ has a unique solution in $Q$ whenever two of the three elements $x$, $y$, $z$ of $Q$ are specified. A [*loop*]{} $(L,*)$ is a quasigroup with an identity element $e$ such that $x*e=x=e*x$, for all $x\in L$. See, for example, [@Pfl90] for an introduction to the theory of loops and binary quasigroups. An [*extra loop*]{} is a loop $(L,*)$ satisfying one of the following equivalent conditions: $$\begin{aligned}
&1.\ (x*(y*z))*y=(x*y)*(z*y),\\
&2.\ (y*z)*(y*x)=y*((z*y)*x),\\
&3.\ ((x*y)*z)*x=x*(y*(z*x)),\end{aligned}$$ for all $x$, $y$, and $z\in L$. Any group is an extra loop. A classical example of an extra loop that is not a group is as follows. Let $(G,\cdot)$ be a group, and $M(G,2)$ be the set $G\times\{0,1\}$ equipped with the operation $(g,0)*(h,0)=(gh,0)$, $(g,0)*(h,1)=(hg,1)$, $(g,1)*(h,0)=(gh^{-1},1)$, and $(g,1)*(h,1)=(h^{-1}g,0)$. Then $(M(G,2),*)$ is a nonassociative Moufang loop if and only if $(G,\cdot)$ is nonabelian, and $(M(D_4,2),*)$, where $D_4$ is the dihedral group with eight elements, is an extra loop. The smallest nonassociative extra loops have 16 elements. The structure of extra loops was investigated, for example, in [@KiKu04]. In extra loops (and more generally in Moufang loops) the subloop generated by any two elements is a group. Elements have their inverses, satisfying the left and the right inverse properties: $x^{-1}*(x*y)=y$ and $(y*x)*x^{-1}=y$.
We will show that the operation (\[knfl\]) has its generalization in extra loops, but first we need a few more definitions. For elements $x$, $y$, $z$ of a loop $L$, their [*associator*]{} $(x,y,z)\in L$ is defined as the unique element satisfying the equation $$(x*y)*z=(x,y,z)*(x*(y*z)).$$ The [*nucleus*]{} $N(L)$ of $L$ consists of all elements $x\in L$ such that $$(x,y,z)=(y,x,z)=(y,z,x)=e,$$ for all $y$, $z\in L$. The [*center*]{} $Z(L)$ is the subloop $\{ x\in N(L): x*y=y*x,\ \textrm{for all}\ y\in L\}$.
Let $(L,*)$ be an extra loop. Then the operations $[xyz]_1=((x*y^{-1})*z)*k$ and $[xyz]_2=(x*(y^{-1}*z))*k$, where $k\in Z(L)$ is of order one or two, satisfy the conditions LN and RN. They can be used for defining knot/knotted surface coloring invariants via unoriented diagrams using the coloring scheme from Fig. \[Dehngen\](B) and Fig. \[surfdehn\].
The operation $[\, ]_1$ generalizes the operation $[xyz]=((x*y^{-1})*z)$ that was introduced in [@Nie14]. Note that since $k$ is in $Z(L)$, it can be moved throughout any word containing it. The operation $[\, ]_1$ is used an even number of times on the right and on the left hand sides of the axioms LN and RN, and therefore $k$ also appears an even number of times. Thus, since $k$ is of order at most two, it can be eradicated from these expressions, and the proof that $[\, ]_1$ satisfies LN and RN reduces to the proof contained in [@Nie14 Lemma 5.7].
Note that because of our coloring conventions, the operation $[\, ]_1$ has to satisfy the equalities $$[[abc]_1cb]_1=a,\ [c[abc]_1a]_1=b,\ \textrm{and}\ [ba[abc]_1]_1=c,$$ for all $a$, $b$, $c\in L$ (they are true for knot-theoretic flocks). Since $k$ appears twice on the left hand sides of these equations, it can be disregarded, and the equalities follow from the inverse properties of extra loops, as in [@Nie14]. The proofs for the operation $[xyz]_2$ are analogous.
Cocycle invariants and group actions on the set of colorings
============================================================
The (co)homology theory for algebras satisfying the nesting conditions LN and RN was developed in [@Nie17]. The operation (\[knfl\]) defining knot-theoretic flocks considerably simplifies the terms appearing in [@Nie17], and allows for an independent treatment.
Let $(X,[\,])=\mathcal{F}((X,\cdot),k)$ be a knot theoretic flock. We will suppress the symbol of the group operation in the expressions below. The chain groups $C_n(X)=Z\langle X^{n+2}\rangle$ are defined as the free abelian groups generated by $(n+2)$-tuples $(x_0,x_1,\ldots, x_n,x_{n+1})$ of elements of $X$, for $n\geq -1$, with $C_{-2}(X)=\mathbb{Z}$. The differential $\partial_n\colon C_n(X) \to C_{n-1}(X)$ takes the form: $$\begin{aligned}
&\partial_n(x_0,x_1,\ldots,x_n,x_{n+1})=(x_1,\ldots,x_n,x_{n+1})\\
&+\sum_{i=1}^n(-1)^i\{(x_0x_i^{-1}x_{i+1}k^i,x_1x_i^{-1}x_{i+1}k^{i-1},\ldots,
x_{i-1}x_i^{-1}x_{i+1}k,\hat{x}_i,x_{i+1},\ldots,x_n,x_{n+1})\\
&+(x_0,x_1,\ldots,x_{i-1},\hat{x}_i,x_{i-1}x_i^{-1}x_{i+1}k,x_{i-1}x_i^{-1}x_{i+2}k^2,\ldots,x_{i-1}x_i^{-1}x_{n+1}k^{n+1-i})\}\\
&+(-1)^{n+1}(x_0,x_1,\ldots,x_n),\end{aligned}$$ where $\hat{x}_i$ denotes a missing element. We also set $\partial_{-1}(x_0)=0$ and $\partial_0(x_0,x_1)=x_1-x_0$. There is a degenerate subcomplex $\{C^D,\partial\}$ in which $C_n^D(X)$ is the free abelian group generated by $(n+2)$-tuples $x=(x_0,x_1,\ldots, x_n,x_{n+1})$ of elements of $X$ containing a triple $a$, $b$, $ba^{-1}bk$ on three consecutive coordinates, for some $a$ and $b\in X$. For $n<1$, we take $C_n^D(X)=0$. The normalized homology yields knot and knotted surface invariants via cycles assigned to colorings of oriented diagrams, see [@Nie17] and some details below. The knot-theoretic flock cohomology is defined in a standard dual way, with the coboundary $\delta$ obtained via $(\delta f)(c)=f(\partial c)$. Thus, the 1-cocycles (used for link diagrams) are functions $f\colon X\times X\times X\to A$, where $A$ is an abelian group, satisfying two conditions for all $a$, $b$, $c$, $d\in X$: $$\begin{aligned}
(1)\quad &f(a,b,ba^{-1}bk)=0,\\
(2)\quad &f(b,c,d)-f(a,ab^{-1}ck,ab^{-1}d)-f(ab^{-1}ck,c,d)\\
&+f(a,b,bc^{-1}dk)+f(ac^{-1}d,bc^{-1}dk,d)-f(a,b,c)=0.\end{aligned}$$ 2-cocycles (used for knotted surface diagrams in $\mathbb{R}^3$) are functions $\phi\colon X\times X\times X\times X\to A$, satisfying for all $a$, $b$, $c$, $d$, $e\in X$: $$\begin{aligned}
(1)\quad &\phi(a,b,ba^{-1}bk,c)=\phi(c,a,b,ba^{-1}bk)=0,\\
(2)\quad &\phi(b,c,d,e)-\phi(a,ab^{-1}ck,ab^{-1}d,ab^{-1}ek)
-\phi(ab^{-1}ck,c,d,e)+\phi(a,b,bc^{-1}dk,bc^{-1}e)\\
&+\phi(ac^{-1}d,bc^{-1}dk,d,e)-\phi(a,b,c,cd^{-1}ek)
-\phi(ad^{-1}ek,bd^{-1}e,cd^{-1}ek,e)+\phi(a,b,c,d)=0.\end{aligned}$$ The cocycle invariants using knot-theoretic flocks are defined in analogy to the construction in [@CJKLS03]. A cocycle invariant can be viewed as a multiset consisting of evaluations of a given cocycle on the cycles assigned to all colorings of a given diagram. Here we give a more detailed explanation for links, directing the reader to [@Nie17] for a description of cocycle invariants for knotted surfaces.
![The chains assigned to flock-colored crossings.[]{data-label="flockcycles"}](flockcycles.eps){height="3.5"}
Let $D$ be an oriented link diagram, and let $\mathcal{C}\colon Reg(D)\to X$ be its coloring with a knot-theoretic flock $(X,[\,])=\mathcal{F}((X,\cdot),k)$, where $Reg(D)$ denotes the set of regions in the complement of $D$. We denote the $\pm 1$ sign of the crossing $cr$ by $\epsilon(cr)$, its source region (i.e., the region with all the co-orientation arrows of $cr$ pointing out of it) by $r_s$, and the target region (all co-orientation arrows point into it) by $r_t$. The region of $cr$ separated from $r_s$ by an under-arc will be denoted by $r_m$. Then $$c(\mathcal{C})=\sum_{cr\in D} \epsilon(cr)(\mathcal{C}(r_s),\mathcal{C}(r_m),\mathcal{C}(r_t))$$ is a cycle in the first homology (with $\mathbb{Z}$ coefficients) of $(X,[\,])$. See Fig. \[flockcycles\], which shows the chains assigned to crossings. Now let $\{\mathcal{C}_1,\ldots,\mathcal{C}_n\}$ be the set of all colorings of $D$ with $(X,[\,])$, and let $\phi$ be a 1-cocycle from the cohomology of $(X,[\,])$, with values in the abelian group $A$. Then the cocycle invariant $\Psi(D,\phi)$ is defined as the multiset $$\{\phi(c(\mathcal{C}_1)),\ldots, \phi(c(\mathcal{C}_n))\}.$$ It is an invariant of Reidemeister moves, so we can write $\Psi(L,\phi)$ instead of $\Psi(D,\phi)$, where $D$ is a diagram for a link $L$.
Let $X$ be a permutation group with 12 elements numbered as follows: 1. (),\
2. (1,2,3,5)(4,10,7,12)(6,11,9,8), 3. (1,3)(2,5)(4,7)(6,9)(8,11)(10,12), 4. (1,4,8)(2,6,10)(3,7,11)(5,9,12),\
5. (1,5,3,2)(4,12,7,10)(6,8,9,11), 6. (1,6,3,9)(2,7,5,4)(8,10,11,12), 7. (1,7,8,3,4,11)(2,9,10,5,6,12),\
8. (1,8,4)(2,10,6)(3,11,7)(5,12,9), 9. (1,9,3,6)(2,4,5,7)(8,12,11,10), 10. (1,10,3,12)(2,11,5,8)(4,6,7,9),\
11. (1,11,4,3,8,7)(2,12,6,5,10,9), 12. (1,12,3,10)(2,8,5,11)(4,9,7,6).\
Note that $k=(1,3)(2,5)(4,7)(6,9)(8,11)(10,12)$ is of order two, and belongs to the center of $X$. From now on, we will write the numbers assigned to these elements instead of the elements themselves. The twelve tables below show an example of a 1-cocycle $\Phi$ for the knot theoretic flock $(X,[\,])=\mathcal{F}((X,\cdot),k)$ with values in $\mathbb{Z}_3$, generated with the help of GAP. The tables include the values of $\Phi$ on the triples $(i,j,l)$ with $i$, $j$, $l\in X$. The value of $\Phi(i,j,l)$ can be found in the $i$-th table, in the intersection of the $j$-th row and the $l$-th column. For example, $\Phi(1,2,3)=2$ and $\Phi(10,3,6)=1$. The cocycle invariant $\Psi(L,\Phi)$ is quite effective in distinguishing links. It has 27 distinct values on the 48-element set of nontrivial links with two components, that have up to eight crossings in the minimal braid form in the table from [@Git04]. It distinguishes all of them from the trivial link with two components, for which the value of the cocycle invariant is $12^3=1728$. The number of colorings can distinguish only 5 classes among these links. The values of the cocycle invariant in Table \[cocvalues\] are given as polynomials. For example $768 + 408t + 552t^2$ calculated for the closure of the braid $\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_1\sigma_3\sigma_2^{-1}\sigma_3$ means that out of 1728 cycles assigned to the colorings of this link, $\Phi$ has value 0 on 768 of them, value 1 on 408 cycles, and value 2 on 552 cycles. That is, the values of the cocycle are encoded in the powers of $t$.
$$i=1\hspace{0.5cm}
\begin{array}{|cccccccccccc|}
\hline
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 2 & 1 \\
0 & 2 & 2 & 0 & 0 & 2 & 0 & 1 & 0 & 2 & 0 & 1 \\
2 & 1 & 0 & 1 & 2 & 0 & 1 & 1 & 2 & 2 & 2 & 2 \\
0 & 1 & 2 & 2 & 2 & 0 & 2 & 2 & 0 & 1 & 0 & 1 \\
0 & 0 & 2 & 2 & 2 & 0 & 2 & 1 & 0 & 0 & 0 & 1 \\
0 & 2 & 1 & 1 & 2 & 0 & 1 & 2 & 0 & 1 & 1 & 2 \\
0 & 1 & 2 & 0 & 0 & 2 & 2 & 0 & 1 & 1 & 0 & 0 \\
1 & 2 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 1 & 1 & 0 \\
0 & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 1 & 0 \\
0 & 2 & 1 & 2 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 0 & 2 & 1 & 2 \\
\hline
\end{array}
\hspace{0.7cm} i=2\hspace{0.5cm}
\begin{array}{|cccccccccccc|}
\hline
1 & 0 & 2 & 0 & 0 & 0 & 2 & 1 & 0 & 2 & 0 & 2 \\
2 & 2 & 1 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 \\
2 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 2 \\
2 & 0 & 1 & 0 & 2 & 0 & 0 & 2 & 0 & 2 & 1 & 0 \\
0 & 0 & 2 & 1 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 2 \\
2 & 2 & 1 & 1 & 0 & 0 & 0 & 2 & 1 & 2 & 1 & 0 \\
2 & 0 & 0 & 2 & 2 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 2 & 0 & 0 & 2 & 1 & 2 & 0 & 2 \\
2 & 1 & 2 & 2 & 1 & 0 & 2 & 2 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 & 2 & 2 & 0 & 1 & 0 & 1 & 0 & 1 \\
2 & 0 & 1 & 0 & 2 & 0 & 2 & 1 & 1 & 2 & 0 & 0 \\
0 & 2 & 2 & 1 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 1 \\
\hline
\end{array}$$ $$i=3\hspace{0.5cm}
\begin{array}{|cccccccccccc|}
\hline
0 & 0 & 1 & 2 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 2 \\
2 & 1 & 0 & 1 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 2 & 2 & 1 & 2 & 2 & 0 & 0 & 2 & 1 & 1 & 0 \\
2 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 2 & 0 & 0 & 0 & 1 & 0 \\
2 & 0 & 0 & 1 & 1 & 2 & 0 & 0 & 2 & 2 & 1 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 2 \\
0 & 1 & 0 & 1 & 2 & 2 & 1 & 2 & 1 & 0 & 2 & 0 \\
0 & 2 & 2 & 0 & 1 & 0 & 0 & 0 & 2 & 0 & 2 & 2 \\
1 & 1 & 0 & 2 & 0 & 2 & 0 & 2 & 0 & 1 & 2 & 1 \\
\hline
\end{array}
\hspace{0.7cm} i=4\hspace{0.5cm}
\begin{array}{|cccccccccccc|}
\hline
1 & 2 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 2 & 0 & 1 \\
1 & 2 & 2 & 0 & 2 & 0 & 1 & 2 & 0 & 0 & 1 & 2 \\
0 & 0 & 2 & 1 & 1 & 2 & 0 & 0 & 0 & 2 & 0 & 1 \\
0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 2 & 0 & 0 & 2 & 2 & 1 & 1 & 2 & 1 & 0 \\
2 & 1 & 1 & 2 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
2 & 2 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & 2 & 0 & 1 \\
1 & 1 & 2 & 0 & 1 & 0 & 2 & 2 & 1 & 1 & 2 & 1 \\
2 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 0 & 1 \\
0 & 1 & 0 & 1 & 1 & 2 & 2 & 0 & 1 & 1 & 0 & 1 \\
2 & 0 & 0 & 0 & 1 & 2 & 1 & 1 & 0 & 0 & 2 & 0 \\
\hline
\end{array}$$ $$i=5\hspace{0.5cm}
\begin{array}{|cccccccccccc|}
\hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 2 & 2 & 0 \\
2 & 1 & 1 & 2 & 0 & 2 & 1 & 0 & 1 & 1 & 0 & 0 \\
2 & 2 & 0 & 2 & 0 & 1 & 2 & 1 & 0 & 2 & 0 & 2 \\
1 & 0 & 1 & 2 & 1 & 0 & 2 & 0 & 1 & 1 & 2 & 0 \\
2 & 2 & 0 & 2 & 0 & 2 & 2 & 2 & 2 & 0 & 0 & 2 \\
0 & 2 & 1 & 1 & 0 & 2 & 1 & 0 & 1 & 0 & 2 & 2 \\
1 & 0 & 2 & 2 & 0 & 1 & 0 & 2 & 0 & 1 & 2 & 1 \\
2 & 2 & 2 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 & 2 & 1 & 2 & 0 \\
0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\
0 & 0 & 0 & 2 & 1 & 0 & 0 & 1 & 0 & 2 & 0 & 0 \\
\hline
\end{array}
\hspace{0.7cm} i=6\hspace{0.5cm}
\begin{array}{|cccccccccccc|}
\hline
1 & 2 & 1 & 0 & 1 & 0 & 0 & 2 & 1 & 2 & 2 & 0 \\
0 & 0 & 0 & 1 & 2 & 0 & 1 & 0 & 2 & 1 & 0 & 0 \\
2 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 1 & 2 & 1 \\
1 & 0 & 0 & 2 & 0 & 0 & 2 & 2 & 1 & 2 & 0 & 1 \\
0 & 2 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 1 & 0 & 0 \\
2 & 1 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 2 & 0 & 1 \\
0 & 2 & 2 & 2 & 0 & 0 & 2 & 2 & 0 & 0 & 2 & 2 \\
1 & 1 & 0 & 2 & 1 & 0 & 1 & 1 & 0 & 2 & 0 & 1 \\
0 & 0 & 0 & 2 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 2 \\
0 & 1 & 2 & 0 & 0 & 1 & 0 & 2 & 0 & 2 & 0 & 2 \\
1 & 2 & 2 & 0 & 1 & 0 & 1 & 0 & 1 & 2 & 2 & 1 \\
2 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 2 & 1 & 1 & 1 \\
\hline
\end{array}$$ $$i=7\hspace{0.5cm}
\begin{array}{|cccccccccccc|}
\hline
0 & 2 & 2 & 0 & 0 & 2 & 0 & 0 & 1 & 1 & 2 & 1 \\
2 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 \\
2 & 2 & 0 & 1 & 2 & 0 & 1 & 0 & 0 & 2 & 0 & 1 \\
0 & 1 & 0 & 0 & 2 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\
2 & 0 & 0 & 0 & 1 & 1 & 0 & 2 & 2 & 2 & 0 & 1 \\
0 & 0 & 1 & 2 & 1 & 1 & 0 & 2 & 2 & 1 & 1 & 1 \\
1 & 1 & 1 & 0 & 0 & 1 & 2 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 1 & 2 & 0 & 1 & 0 & 2 & 2 & 2 & 2 & 0 \\
0 & 2 & 0 & 2 & 1 & 0 & 0 & 2 & 0 & 1 & 1 & 0 \\
2 & 2 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 2 \\
0 & 0 & 1 & 1 & 2 & 1 & 1 & 2 & 1 & 2 & 2 & 1 \\
0 & 0 & 0 & 0 & 1 & 2 & 0 & 2 & 0 & 0 & 2 & 2 \\
\hline
\end{array}
\hspace{0.7cm} i=8\hspace{0.5cm}
\begin{array}{|cccccccccccc|}
\hline
2 & 0 & 2 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 0 \\
2 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 2 & 0 \\
2 & 1 & 0 & 2 & 1 & 1 & 0 & 1 & 0 & 1 & 2 & 1 \\
2 & 1 & 0 & 0 & 2 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 2 & 1 & 0 & 0 & 2 & 1 & 0 & 0 & 1 & 2 & 1 \\
2 & 2 & 2 & 0 & 1 & 2 & 0 & 0 & 2 & 2 & 2 & 1 \\
0 & 1 & 0 & 2 & 0 & 0 & 2 & 0 & 1 & 2 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 1 \\
0 & 2 & 2 & 0 & 2 & 0 & 2 & 0 & 2 & 0 & 2 & 2 \\
2 & 1 & 2 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 1 & 0 & 1 \\
0 & 2 & 2 & 0 & 1 & 2 & 1 & 0 & 0 & 2 & 1 & 0 \\
\hline
\end{array}$$ $$i=9\hspace{0.55cm}
\begin{array}{|cccccccccccc|}
\hline
1 & 1 & 0 & 0 & 0 & 2 & 1 & 2 & 0 & 2 & 2 & 1 \\
1 & 2 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 \\
2 & 0 & 0 & 1 & 2 & 2 & 1 & 2 & 0 & 2 & 0 & 2 \\
0 & 1 & 2 & 1 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 0 \\
0 & 0 & 0 & 2 & 1 & 2 & 0 & 1 & 0 & 0 & 0 & 2 \\
2 & 1 & 1 & 2 & 2 & 0 & 1 & 2 & 0 & 2 & 0 & 2 \\
0 & 2 & 1 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 1 \\
2 & 0 & 2 & 0 & 0 & 2 & 0 & 1 & 0 & 2 & 1 & 2 \\
1 & 0 & 1 & 0 & 0 & 0 & 1 & 2 & 2 & 0 & 0 & 1 \\
2 & 0 & 2 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 0 & 2 \\
2 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\
\hline
\end{array}
\hspace{0.7cm} i=10\hspace{0.35cm}
\begin{array}{|cccccccccccc|}
\hline
1 & 1 & 0 & 2 & 2 & 2 & 0 & 1 & 1 & 0 & 0 & 1 \\
0 & 2 & 2 & 1 & 1 & 2 & 2 & 2 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 2 & 0 & 1 & 2 & 0 & 2 & 0 & 1 & 0 \\
1 & 1 & 0 & 1 & 2 & 1 & 2 & 1 & 1 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 1 & 1 & 1 \\
2 & 0 & 2 & 2 & 2 & 0 & 0 & 2 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 2 \\
0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 \\
1 & 0 & 0 & 2 & 2 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 0 & 2 & 1 & 0 & 2 & 0 & 0 & 2 & 0 & 2 & 2 \\
0 & 1 & 0 & 0 & 2 & 1 & 0 & 2 & 2 & 1 & 1 & 0 \\
\hline
\end{array}$$ $$i=11\hspace{0.45cm}
\begin{array}{|cccccccccccc|}
\hline
0 & 2 & 0 & 0 & 0 & 1 & 2 & 0 & 1 & 2 & 2 & 2 \\
2 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 2 \\
0 & 1 & 0 & 0 & 1 & 1 & 0 & 2 & 2 & 0 & 0 & 0 \\
0 & 2 & 2 & 1 & 2 & 1 & 1 & 0 & 2 & 1 & 0 & 2 \\
2 & 0 & 1 & 0 & 0 & 2 & 2 & 0 & 0 & 2 & 0 & 1 \\
0 & 1 & 2 & 1 & 2 & 2 & 0 & 2 & 0 & 0 & 2 & 2 \\
0 & 1 & 0 & 0 & 1 & 2 & 0 & 0 & 0 & 2 & 1 & 0 \\
0 & 2 & 1 & 2 & 2 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
2 & 1 & 2 & 2 & 0 & 1 & 0 & 1 & 0 & 2 & 0 & 0 \\
0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 2 & 1 & 2 & 0 & 0 & 1 & 0 & 0 \\
\hline
\end{array}
\hspace{0.7cm} i=12\hspace{0.35cm}
\begin{array}{|cccccccccccc|}
\hline
1 & 1 & 0 & 0 & 0 & 1 & 2 & 1 & 2 & 2 & 1 & 0 \\
2 & 0 & 1 & 2 & 2 & 0 & 1 & 1 & 2 & 2 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 2 & 2 & 0 \\
0 & 2 & 1 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 2 & 0 \\
0 & 2 & 0 & 0 & 2 & 0 & 2 & 0 & 1 & 1 & 1 & 0 \\
2 & 0 & 0 & 0 & 2 & 1 & 1 & 1 & 0 & 1 & 2 & 0 \\
2 & 0 & 1 & 1 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 2 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 0 \\
2 & 0 & 1 & 1 & 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\
\hline
\end{array}$$
------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$480 + 264t + 120t^2$ $\sigma_1\sigma_1,\ \sigma_1\sigma_1\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1},\ \sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_3\sigma_2^{
-1}\sigma_3$
$480 + 408t + 552t^2$ $\sigma_1\sigma_1\sigma_1\sigma_1,\ \sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_3^{-1}\sigma_2\sigma_4\sigma_3^{-1}\sigma_4$
$768 + 120t + 552t^2$ $\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2,\ \sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_1\sigma_2^{-1}\sigma_2^{-1},\sigma_1\sigma_1\sigma_1\sigma_1\sigma_
2\sigma_2\sigma_1^{-1}\sigma_2\sigma_2$,
$\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_3\sigma_2^{-1}\sigma_3\sigma_3$
$864 + 576t$ $\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}$
$1152$ $\sigma_1\sigma_1\sigma_1\sigma_1\sigma_1\sigma_1,\ \sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_3\sigma_2^{-1}\sigma_3$
$912 + 408t + 408t^2$ $\sigma_1\sigma_2^{-1}\sigma_1\sigma_3\sigma_2^{-1}\sigma_3,\ \sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_2$
$864$ $\sigma_1\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2,\ \sigma_1\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_3\sigma_2^{-1}\sigma_3,\ \sigma_1\sigma_1\sigma_1\sigma_
2\sigma_1^{-1}\sigma_2\sigma_2\sigma_3\sigma_2^{-1}\sigma_3$,
$\sigma_1\sigma_1\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_2,\ \sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_
3^{-1}\sigma_2\sigma_4\sigma_3^{-1}\sigma_4$
$480 + 120t + 264t^2$ $\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_3^{-1}\sigma_2\sigma_3^{-1}$
$1152 + 288t$ $\sigma_1\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_1\sigma_2^{-1}$
$624 + 408t + 120t^2$ $\sigma_1\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_2^{-1},\ \sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_3^{-1}\sigma_2\sigma_3^{
-1},\ \sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_3\sigma_2^{-1}\sigma_3\sigma_3$,
$\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_2^{-1}\sigma_3\sigma_2^{-1}\sigma_3$
$624 + 264t + 552t^2$ $\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_3\sigma_2\sigma_2\sigma_3,\ \sigma_1\sigma_2^{-1}\sigma_1\sigma_3\sigma_2^{-1}\sigma_2^{-1}\sigma_2^{-1}\sigma_3$
$1440 + 288t$ $\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}$
$864 + 576t$ $\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_3^{-1}\sigma_2\sigma_3^{-1}$
$768 + 408t + 264t^2$ $\sigma_1\sigma_1\sigma_1\sigma_2^{-1}\sigma_1^{-1}\sigma_1^{-1}\sigma_2^{-1}$
$768 + 264t + 408t^2$ $\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1\sigma_1\sigma_2,\ \sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_1\sigma_3\sigma_2^{-1}\sigma_3,\ \sigma_1\sigma_1\sigma_
1\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2,$
$\sigma_1\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_3\sigma_2^{-1}\sigma_3$
$1152 + 288t$ $\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_1^{-1}\sigma_2,\ \sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_3\sigma_2^{-1}\sigma_2^{-1}\sigma_3$
$480 + 552t + 408t^2$ $\sigma_1\sigma_1\sigma_1\sigma_1\sigma_1\sigma_1\sigma_1\sigma_1$
$768 + 552t + 120t^2$ $\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_3\sigma_2^{-1}\sigma_3\sigma_4\sigma_3^{-1}\sigma_4$
$624 + 408t + 408t^2$ $\sigma_1\sigma_2^{-1}\sigma_1\sigma_3\sigma_2\sigma_2\sigma_4^{-1}\sigma_3\sigma_4^{-1}$
$1440$ $\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_3\sigma_2^{-1}\sigma_2^{-1}\sigma_3$
$864 + 288t + 288t^2$ $\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_1\sigma_3\sigma_2^{-1}\sigma_3,\ \sigma_1\sigma_1\sigma_2^{-1}\sigma_1^{-1}\sigma_1^{-1}\sigma_3\sigma_2^{
-1}\sigma_3$
$1632 + 1128t + 696t^2$ $\sigma_1\sigma_2^{-1}\sigma_3\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_3\sigma_2^{-1}$
$1776 + 552t + 1128t^2$ $\sigma_1\sigma_1\sigma_2\sigma_3^{-1}\sigma_2\sigma_1^{-1}\sigma_2\sigma_3\sigma_3\sigma_2$
$624 + 696t + 408t^2$ $\sigma_1\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_2\sigma_2$
$912 + 696t + 120t^2$ $\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_2\sigma_2\sigma_3\sigma_2^{-1}\sigma_3$
$912 + 264t + 552t^2$ $\sigma_1\sigma_1\sigma_2\sigma_1\sigma_1\sigma_3^{-1}\sigma_2\sigma_3^{-1}$
$768 + 408t + 552t^2$ $\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_1\sigma_3\sigma_2^{-1}\sigma_3$
------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: The values of the cocycle $\Phi$ on the 48 links with two components in the minimal braid form with up to 8 crossings.[]{data-label="cocvalues"}
Let groups $G$ and $G'$ act on sets $A$ and $B$, respectively. Then the actions are said to be [*equivalent*]{} if there is an isomorphism $\beta\colon G\to G'$ and a bijective map $\alpha\colon A\to B$ such that, for all $g\in G$ and $a\in A$, $$\label{equivalentact}
\alpha(a{\mathchar"5E}g)=\alpha(a){\mathchar"5E}\beta(g),$$ where ${\mathchar"5E}$ denotes the appropriate group action. This defines an equivalence relation on group actions; see e.g. [@Rose] for more material on this topic. We will now consider group actions on the set of colorings, but for this notion to be useful, it has to take into account the Reidemeister (or Roseman, etc.) moves. We write the following definition on a more general level of knot-theoretic ternary quasigroups (see [@Nie17] for the corresponding colorings, generalizing the flock colorings).
\[compatible\] Let $\mathfrak{C}(D,(X,[\, ]))$ denote the set of colorings of a knot diagram $D$ with a knot-theoretic ternary quasigroup $(X,[\, ])$. A Reidemeister move changing $D$ into a diagram $D'$ results in local changes of colorings, yielding a bijection $\alpha\colon\mathfrak{C}(D,(X,[\, ]))\to\mathfrak{C}(D',(X,[\, ]))$. Suppose that there is a group action $G\times\mathfrak{C}(D,(X,[\, ]))\to\mathfrak{C}(D,(X,[\, ]))$. We say that this action is [*compatible with the Reidemeister moves*]{}, if for any such move, and $D'$ and $\alpha$ as above, there is an isomorphism $\beta\colon G\to G'$, with $G'$ acting on $\mathfrak{C}(D',(X,[\, ]))$, so that the actions, $\alpha$, and $\beta$, satisfy the equation (\[equivalentact\]).
![The group of central colorings acting on colorings.[]{data-label="action1"}](action1b.eps){height="4.5"}
\[directmult\] Let $\mathfrak{C}(D,\mathcal{F}((Z(X),\cdot),e))$ be the set of flock colorings of a diagram $D$ using elements of the center $Z(X)$ of the group $X$, with $e$ being the identity element of the group $X$. Then it forms a group acting on the set of colorings $\mathfrak{C}(D,\mathcal{F}((X,\cdot),k))$, for any central involution $k$ of $X$. The action is compatible with the Reidemeister moves.
The colorings from $\mathfrak{C}(D,\mathcal{F}((Z(X),\cdot),e))$ form a group with the operation of region-wise multiplication of colors. The identity element is the coloring in which all the regions are labeled by $e$. The inverse of a coloring $\mathcal{C}\in \mathfrak{C}(D,\mathcal{F}((Z(X),\cdot),e))$ is the coloring $\mathcal{C}^{-1}$ such that $\mathcal{C}^{-1}(r)=(\mathcal{C}(r))^{-1}$, for any region $r$ of $D$. We also note that taking a product of two colorings with elements from the center, but with $k\neq e$, gives a coloring with $k=e$. If $D$ and $D'$ differ by a Reidemeister move, then the corresponding groups of central colorings are isomorphic. There is an action of the group $\mathfrak{C}(D,\mathcal{F}((Z(X),\cdot),e))$ on the set of colorings $\mathfrak{C}(D,\mathcal{F}((X,\cdot),k))$ given by the region-wise multiplication of colors. Centrality of colors of the acting coloring ensures that the result is again a flock coloring, for any central involution $k\in X$ (see Fig. \[action1\]). It is not difficult to see that this action is compatible with the Reidemeister moves: if $\alpha$ is as in Definition \[compatible\], then we can take $\beta$ to be the restriction of $\alpha$ to $\mathfrak{C}(D,\mathcal{F}((Z(X),\cdot),e))$.
![A group $X$ acting on colorings by conjugation.[]{data-label="action2"}](action2b.eps){height="3.5"}
\[actionbyconjugation\] Another example of an action of a group on the set of colorings that is compatible with the Reidemeister moves is given by the group $X$ (and its subgroups) acting on $\mathfrak{C}(D,\mathcal{F}((X,\cdot),k))$ by conjugation. More specifically, there is a mapping $g\times\mathcal{C}\mapsto\mathcal{C}{\mathchar"5E}g$ defined by $\mathcal{C}{\mathchar"5E}g(r)=g\mathcal{C}(r)g^{-1}$ for all regions $r$ of the diagram $D$. Centrality of $k$ is important, as can be seen in Fig. \[action2\]. Definition \[compatible\] is satisfied, as we can take $\beta$ to be the identity isomorphism. Then $\alpha(\mathcal{C}{\mathchar"5E}g)=(\alpha(\mathcal{C})){\mathchar"5E}g$.
We can generalize the action from Example \[actionbyconjugation\]. Let $H$ and $S$ be subgroups of $X$. Then the direct product $H\times S$ acts on $\mathfrak{C}(D,\mathcal{F}((X,\cdot),k))$ via $\mathcal{C}{\mathchar"5E}(h,s)(r)=h\mathcal{C}(r)s^{-1}$, for $(h,s)\in H\times S$.
Now we incorporate the actions on colorings compatible with the Reidemeister moves into cocycle invariants.
\[cocyinvrefi\] Let $(X, [\, ])$ be a knot-theoretic ternary quasigroup, and $D$ be an oriented link diagram. Suppose that $G\times \mathfrak{C}(D,(X,[\, ]))\to\mathfrak{C}(D,(X,[\, ]))$ is a group action compatible with the Reidemeister moves, and that $$\mathcal{O}_1=\{\mathcal{C}_1,\ldots,\mathcal{C}_{n_1}\},
\mathcal{O}_2=\{\mathcal{C}_{n_1+1},\ldots,\mathcal{C}_{n_2}\},\ldots,
\mathcal{O}_s=\{\mathcal{C}_{n_{s-1}+1},\ldots,\mathcal{C}_{n_s}\}$$ are the orbits of this action. Let $\phi$ be a cocycle from the first cohomology of $(X,[\, ])$ with values in an abelian group $A$. Then the multiset of multisets $$\{\{\phi(c(\mathcal{C}_1)),\ldots,\phi(c(\mathcal{C}_{n_1}))\},
\{\phi(c(\mathcal{C}_{n_1+1})),\ldots,\phi(c(\mathcal{C}_{n_2}))\},\ldots,
\{\phi(c(\mathcal{C}_{n_{s-1}+1})),\ldots,\phi(c(\mathcal{C}_{n_s}))\}\}$$ is a refinement of the cocycle invariant, that is not changed by the Reidemeister moves.
Let $\alpha\colon\mathfrak{C}(D,(X,[\, ]))\to\mathfrak{C}(D',(X,[\, ]))$, $\beta$ and $G'$ be as in Definition \[compatible\]. We have: $\mathcal{O}_i=\{\mathcal{C}_{n_{i-1}+1},\ldots,\mathcal{C}_{n_i}\}$ is an orbit of the action of $G$ if and only if $\alpha(\mathcal{O}_i)=\{\alpha(\mathcal{C}_{n_{i-1}+1}),\ldots,\alpha(\mathcal{C}_{n_i})\}$ is an orbit of the action of $G'$. Indeed: $$\mathcal{C}_i{\mathchar"5E}g=\mathcal{C}_j \iff
\alpha(\mathcal{C}_i{\mathchar"5E}g)=\alpha(\mathcal{C}_j) \iff
\alpha(\mathcal{C}_i){\mathchar"5E}\beta(g)=\alpha(\mathcal{C}_j).$$ In [@Nie17] we proved that the homology class of a cycle assigned to a knot-theoretic ternary quasigroup coloring is not changed by the Reidemeister moves. It follows that $$c(\mathcal{C}{\mathchar"5E}g)\sim c(\alpha(\mathcal{C}{\mathchar"5E}g))
,\ \textrm{and}\ \phi(c(\mathcal{C}{\mathchar"5E}g))=\phi(c(\alpha(\mathcal{C}{\mathchar"5E}g))),$$ for all $\mathcal{C}\in\mathfrak{C}(D,(X,[\, ]))$ and $g\in G$, what ends the proof.
We can use Lemma \[cocyinvrefi\], and the action from Example \[actionbyconjugation\], to improve the cocycle invariant $\Psi(L,\Phi)$ on the set of links from Table \[cocvalues\]. More specifically, we consider the action by conjugation with the three-element subgroup generated by $(1,4,8)(2,6,10)(3,7,11)(5,9,12)$. For the closures of the braids $\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1\sigma_1\sigma_2$, $\sigma_1\sigma_1\sigma_2^{-1}\sigma_1\sigma_1\sigma_3\sigma_2^{-1}\sigma_3$, $\sigma_1\sigma_1\sigma_1\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2$, and $\sigma_1\sigma_1\sigma_1\sigma_1\sigma_2\sigma_1^{-1}\sigma_2\sigma_3\sigma_2^{-1}\sigma_3$, the value of the cocycle invariant $\Psi(L,\Phi)$ is $768 + 264t + 408t^2$. In particular, there are 1440 colorings for each of these links. Also, in each case the set of colorings splits into 216 one-element orbits and 408 three-element orbits. This also divides the sets of cycles corresponding to the colorings. The values of the cocycle $\Phi$ on these groupings of cycles allow us to separate the four braids into two classes. We will write the results as polynomials in the brackets with multiplicities, with a multiplicity giving the number of orbits with the same value of the polynomial (which describes the values of $\Phi$ on a given subset of cycles). For the first two braids we have: $$\{ 132 [ 1 ], 212 [ 3 ], 60 [ t ], 68 [3t],
24 [ t^2 ], 128 [3t^2] \},$$ where, for example, $128 [3t^2]$ means that there are 128 three-element orbits such that $\Phi$ has value 2 on each coloring in the orbit. For the third and the fourth braid the results are: $$\{ 132 [ 1 ], 212 [ 3 ], 24 [t], 80 [3t],
60 [t^2], 116 [3t^2] \}.$$ Thus, some additional links are distinguished.
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abstract: 'We prove a splitting theorem for complete gradient Ricci soliton with nonnegative curvature and establish a rigidity theorem for codimension one complete shrinking gradient Ricci soliton in $\mathbb R^{n+1}$ with nonnegative Ricci curvature.'
address:
- 'Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada'
- 'Department of Mathematics, University of Oregon, Eugene, OR 97403, USA'
- 'Department of Mathematics, Nanjing University, Nanjing, China, 210093 and Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada'
author:
- 'Pengfei Guan, Peng Lu and Yiyan Xu'
title: 'A rigidity theorem for codimension one shrinking gradient Ricci solitons in $\mathbb R^{n+1}$'
---
[^1]
Introduction
============
A complete Riemannian metric $g$ on a smooth manifold $M^n$ is called a gradient Ricci soliton (GRS) (Hamilton [@Hamilton93], Perelman [@Perelman02]) if there exists a smooth function $f$ on $M$ such that $$\label{GraShrRicciSolitonEqu1}
\operatorname{Ric} + \operatorname{Hess}f=\frac{\lambda}{2}g,$$ where $\lambda\in \mathbb{R}$. Below we assume that $\lambda=1,~0,~ \hbox{or}~ -1$; these cases correspond to the GRS of shrinking, steady, or expanding type, respectively.
The classification of GRS, especially the noncompact shrinking GRS, has been a subject of interest to many people. By the work of Perelman [@Perelman03], Ni and Wallach [@NW08] and Cao, Chen, and Zhu [@CCZ08], any 3-dimensional complete noncompact nonflat shrinking GRS must be the round cylinder $S^2\times \mathbb{R}$ or its $\mathbb{Z}_2$ quotient. Naber [@Naber10] proved that four dimensional complete noncompact shrinking GRS with bounded nonnegative curvature operator are finite quotients of generalized cylinders $S^2 \times \mathbb{R}^2$ or $S^3 \times \mathbb{R}$. Note that there are several rigidity results for higher dimensional complete noncompact shrinking GRS under various geometric assumptions [@NW08; @Zhang09; @PW10; @CWZ11; @MS13; @Cai13].
On the other hand, Feldman, Ilmanen and Knopf [@FIK03] constructed $U(n)-$invariant shrinking Kähler GRS on the holomorphic line bundles $\mathcal{O}(-k)$ , $1\leq k\leq n$, over $P^{n-1}$, $n\geq 2$. Their examples are cone-like at infinity, and have Euclidean volume growth, positive scalar curvature and quadratic curvature decay. However the Ricci curvature of these examples changes signs, more precisley the Ricci curvature is negative along the vertical (fiber) direction and positive along horizontal direction. We do not know whether there is any nontrivial (the universal cover does not split) example of complete noncompact nonflat shrinking GRS with nonnegative Ricci curvature.
The constant rank theorem is a powerful tool in the study of convex properties of solutions of nonlinear differential equations [@CF85; @SWYY; @BG09]. In this paper we first establish a constant rank theorem for Ricci tensor and for curvature operator of GRS (shrinking, steady, or expanding) and the corresponding splitting property of the GRS in section \[SplitThmPf\].
\[SplGSRSThm1\] Let $(M^n,g,f)$ be a GRS satisfying .
1. \[SplStrRicThm\] If $g$ has nonnegative sectional curvature, then the rank of Ricci curvature is constant. Thus, either Ricci curvature is strictly positive or the universal covering $(\widetilde{M},\tilde{g})=(N,h)\times \mathbb{R}^{n-k}$ splits isometrically and $(N,h)$ has strictly positive Ricci curvature;
2. \[SplStrCurOpeThm\] If $g$ has nonnegative curvature operator, then the rank of curvature operator is constant. Thus, either the curvature operator is strictly positive or the universal covering $(\widetilde{M},\tilde{g})=(N,h)\times \mathbb{R}^{n-k}$ splits isometrically and $(N,h)$ has strictly positive curvature operator.
Note that since Ricci solitons satisfy Ricci flow, the splitting theorem can be obtained from the maximum principle for tensors in the parabolic setting. The maximum principle of this type was first proved by Hamilton for compact manifolds [@Hamilton86], we also refer [@Ni04] for the corresponding result for complete noncompact manifolds under certain growth condition on tensors.
.1cm With the help of Theorem \[SplGSRSThm1\], to classify shrinking GRS with nonnegative curvature operator, one only needs to consider GRS with positive curvature operator. Since compact GRS with positive curvature operator must be of constant curvature, therefore, to prove the rigidity of the complete shrinking GRS with nonnegative curvature operator one only needs to rule out the noncompact shrinking GRS with positive curvature operator. Note that in the Kähler case, Ni [@NL05] proved that a complete shrinking GRS with positive bisectional curvature must be compact; therefore the GRS is isometric to complex projective space $\mathbb{P}^m$ by the Mori-Siu-Yau theorem.
In the second part of this paper, we consider codimension one shrinking GRS $(M^n,g)$ isometrically embedded in $\mathbb R^{n+1}$. If it has nonnegative Ricci curvature, then it has nonnegative curvature operator. By the classical convexity theorem of Sacksteder-van Heijenoort, $M$ is a convex hypersurface. The main result of this paper is the following.
\[GSRSHypCyl\] A complete codimension one shrinking GRS isometrically embedded in $\mathbb R^{n+1}$ with positive Ricci curvature must be compact. As a consequence, if $(M^n,g, f)$ is a complete shrinking GRS isometrically embedded in $R^{n+1}$ with nonnegative Ricci curvature, then $(M,g)$ is a generalized cylinder $S^{k}\times \mathbb{R}^{n-k},
\, 2 \leq k\leq n$.
The main ingredients of our proof are the estimate of the mean curvature and the eigenvalue estimate of a generalized Cheng-Yau operator associated to shrinking GRS. These results will be proved in section \[MeaCurGroEstSec\] and section \[StrGenCyl\], respectively.\
**Acknowledgement**: P.L. is partially supported by a Simons grant. The work was done when Y.X. was supported by CRC Postdoctoral Fellowship at McGill University.
Preliminaries of Gradient Ricci Soliton
=======================================
We collect some well known identities for gradient GRS below, they can be found in [@Hamilton93]. For a GRS satisfying , let $\{e_i\}$ is an orthonormal basis of $TM$, the Ricci curvature satisfies the following formula: $$\begin{gathered}
\nabla R= 2 \operatorname{Ric}(\nabla f), \label{RicSolIde1}\\
\Delta_f R_{ij}= 2\lambda R_{ij}-2\overset{\circ}{R}(\operatorname{Ric})_{ij}, \label{LapRicSolEqu}\end{gathered}$$ where $$\begin{aligned}
\Delta_f =\nabla_{ii}^2 - \nabla_i f \nabla_i \quad\text{ and }\quad
\overset{\circ}{R}(\operatorname{Ric})_{ij}=\sum_kR(e_i,\operatorname{Ric}(e_k),e_j,e_k).\end{aligned}$$
Moreover, after identifying $\Lambda^2T_xM$ with $\mathfrak{so}(T_xM)$, the curvature operator is symmetric endomorphism $\mathfrak{R}\in S^2(\mathfrak{so}(T_xM) )$, $$\mathfrak{R}_{\alpha\beta}=R_{ijkl}\phi_\alpha^{ij}\phi_\beta^{kl},\quad\phi_\alpha=\phi_\alpha^{ij}e_i\wedge e_j\in\Lambda^2T_xM .$$ For a GRS satisfying , the curvature operator satisfies the following formula: $$\label{LapCurOpeSolEqu}
\Delta_f\mathfrak{R}_{\alpha\beta}=2\lambda\mathfrak{R}_{\alpha\beta}-\mathfrak{R}^2_{\alpha\beta}-
\mathfrak{R}^\sharp_{\alpha\beta},$$ where $$\label{PosQuaCurOpe1}
\begin{split}
\langle\mathfrak{R}^\sharp(\phi_\alpha),\phi_\alpha\rangle&=\langle{\rm ad}\circ(\mathfrak{R}\wedge \mathfrak{R})\circ {\rm ad}^*(\phi_\alpha),\phi_\alpha\rangle \\
&=\sum_{\beta,\gamma}\langle[\mathfrak{R}(\phi_\beta),\mathfrak{R}(\phi_\gamma)],\phi_\alpha
\rangle\langle[\phi_\beta,\phi_\gamma],\phi_\alpha\rangle,
\end{split}$$ here ${\rm ad}: \Lambda^2(\mathfrak{so}(T_xM))\rightarrow \mathfrak{so}(T_xM), \phi\wedge\varphi\mapsto{\rm ad}(\phi\wedge\varphi)=[\phi,\varphi]$ is the adjoint representation.
The following identities involving the curvature and potential function are satisfied for shrinking GRS [@Hamilton93]\[IdeCGSRS1\], $$\begin{gathered}
R+\Delta f= \frac{n}{2}, \label{RicSolIde0} \\
R+|\nabla f|^2-f=C_0(=0)\label{RicSolIde2}\end{gathered}$$ for some constant $C_0$ (By adding a constant to $f$, we assume $C_0=0$ below).
The behavior of the potential function plays an important role in understanding the structure of shrinking GRS [@Perelman03; @NW08; @CCZ08]. The following estimates are due to [@Perelman02; @CZ10; @HM11].
\[ProPotGSRS\] Let $(M^n,g,f)$ be complete non-compact shrinking GRS satisfying . Let $x_0\in M$ be the point such that $f(x_0) =\min_{x \in M} f(x)$. Then the potential function $f$ satisfies the estimates $$\begin{gathered}
\frac{1}{4}(r(x)-5n)_+^2\leq f(x)\leq\frac{1}{4}(r(x)+\sqrt{2n})^2,\label{PotQuaGroPotEst1} \\
|\nabla f|\leq \frac{1}{2} (r(x)+2\sqrt{f(x_0)}),\label{PotGraGroPotEst1}\end{gathered}$$ where $r(x)=d(x,x_0)$ is the distance function and $a_+:= \max\{a, 0\}$. Consequently, $$\label{FunExpGrwFin1}
\int_M|u|e^{-f}d\mu <+\infty$$ for any continuous function $u$ on $M$ satisfying $|u(x)|\leq Ae^{\alpha r^{2}(x)}$ where $0\leq \alpha<\frac{1}{4}$ and $A>0$. In particular, the weighted volume of $M$ $\int_Me^{-f}d\mu$ is finite.
Chen [@Chen09] proved that the scalar curvature of complete ancient solution of Ricci flow is always nonnegative. For complete non-flat shrinking GRS $(M^n,g,f)$ the asymptotic estimates for the potential function $f$ also controls the curvature growth rates. In particular, and imply that the scalar curvature grows at most quadratically, $$\label{PloGroScaCur1}
0\leq R(x)\leq \frac{1}{4}(r(x)+\sqrt{2n})^2.$$ When $(M,g,f)$ is assumed to have nonnegative Ricci survature, by we have $$\langle \nabla R, \nabla f\rangle= 2 \operatorname{Ric}(\nabla f,\nabla f )\geq0,$$ thus scalar curvature $R$ is increasing along the gradient flow of potential $f$. It is established by Ni [@NL05 Proposition 1.1] that there exists a $\delta_0 =\delta(M) \in (0, 1)$ such that $$\label{ScaStrPosBou1}
R\geq \delta_0>0.$$ Combining $|\operatorname{Ric}|^2\leq R^2$ with , we conclude that the gradient of scalar curvature grows at most polynomial fast, $$\label{PloGroGraScaCur1}
|\nabla R|^2=|2 \operatorname{Ric}(\nabla f)|^2\leq 4R^2|\nabla f|^2.$$ By , , , together with the Bochner identity, $$\begin{split}
\Delta R&=\Delta(f-|\nabla f|^2)\\
&=\Delta f-2\big(|\nabla^2f|^2+\langle\nabla\Delta f, \nabla f\rangle+ \operatorname{Ric}(\nabla f, \nabla f)\big)\\
&= \frac{n}{2}-R -2|\operatorname{Ric}-\frac{1}{2}g|^2-2\langle\nabla (\frac{n}{2}-R), \nabla f\rangle-2
\operatorname{Ric}(\nabla f, \nabla f)\\
&=\frac{n}{2}-R-2|\operatorname{Ric}-\frac{1}{2}g|^2+2 \operatorname{Ric}(\nabla f, \nabla f).
\end{split}$$ Hence, $$\label{PolyGrowLapSca}
\begin{split}
|\Delta R| &\leq \frac{n}{2}+|R|+2|\operatorname{Ric}-\frac{1}{2}g|^2+2|\operatorname{Ric}||\nabla f|^2\\
&\leq n+2R^2+2R|\nabla f|^2.
\end{split}$$
We will need the following classification result for compact GRS which follows from the works of Hamilton [@Hamilton82; @Hamilton86] (dimensions three and four) and Böhm and Wilking [@BW08] (dimensions $\geq 5$).
\[GSRSPosCOCla\] A compact GRS with positive curvature operator must be a space form.
Splitting Theorem of Gradient Ricci Soliton {#SplitThmPf}
===========================================
In this section, we establish the constant rank Theorem \[SplGSRSThm1\] for GRS with nonnegative curvature via strong maximum principle. We will show that the distribution of the null space of the Ricci tensor is of constant dimension and is invariant under parallel translation. That would yield a splitting theorem for GRS. Similar conclusion also holds for the curvature operator.
Ricci curvature and constant ranking Theorem \[SplGSRSThm1\] \[SplStrRicThm\] {#ProSplThmRic1}
-----------------------------------------------------------------------------
Let $A=(a_{ij})_{n \times n}$ be a symmetric matrix. Define $$\label{DefSigk}
\det(I+tA)=\sum_{l=0}^n \sigma_l (A)t^l.$$ Note that $\sigma_l (A)$ is a smooth function of variables $a_{ij}$. When $A= \operatorname{diag}[\lambda_1,\cdots,\lambda_n]$ is a diagonal matrix, then $\sigma_l (A)$ is the $l$-th elementary polynomial of $\lambda_1,\cdots,\lambda_n$.
If $A$ is any $n\times n$ symmetric matrix, we denote $$\label{DerSigEqu1}
\sigma_l^{ij}(A) := \frac{\partial\sigma_l(A)}{\partial a_{ij}}, \quad
\sigma_l^{ij,kl}(A) := \frac{\partial^2\sigma_l(A)}{\partial a_{ij}\partial a_{kl}}.$$ In particular, we have $$\sigma_1^{ij} (A)=\delta_{ij},\quad \sigma_2^{ij} (A)=(\sum_{k=1}^na_{kk})\delta_{ij}-a_{ij}.$$
We also denote by $(A|i)$ the $(n-1) \times(n-1)$ matrix obtained from $A$ by deleting the $i$-th row and $i$-th column, and by $(A|ij)$ the $(n-2) \times(n-2)$ matrix obtained from $A$ by deleting the $i$, $j$-th row and $i$, $j$-th column.
The following two propositions (See Proposition \[DerSigRan1\] and Proposition \[DerSigRan2\] below) are well known (e.g., see [@BG09]).
\[DerSigRan1\]If $A$ is a diagonal matrix. For any $l, i, j$ we have $$\sigma_l^{ij} (A) =\left\{
\begin{array}{ll}
\sigma_{l-1}(A|i), & \hbox{if $i=j$;} \\
0, & \hbox{otherwise.}
\end{array}
\right.$$ and $$\sigma_l^{ij,kl} (A)=\left\{
\begin{array}{ll}
\sigma_{l-2}(A|ik), & \hbox{if $i=j$, $k=l$, $i\neq k$;} \\
-\sigma_{l-2}(A|ij), & \hbox{if $i=l$, $j=k$, $i\neq j$;} \\
0, & \hbox{otherwise.}
\end{array}
\right.$$
Let $(M^n,g,f)$ be a GRS with nonnegative Ricci curvature as in Theorem \[SplGSRSThm1\]. In this case, we take $A=Ric$. We may assume that $ r := \min_{x\in M} \operatorname{rank} \operatorname{Ric}(x) <n$; otherwise we have $\operatorname{Ric}>0$. Let $x_0 \in M$ be a point such that $\operatorname{rank} \operatorname{Ric}(x_0)=r$. Pick a small open neighborhood $\mathcal{O}$ of $x_0$. We define function $\phi$ on $\mathcal{O}$ by $$\phi (x) =\sigma_{r+1}(\operatorname{Ric} (x)).$$ To prove Theorem \[SplGSRSThm1\] \[SplStrRicThm\], we first show that there is a positive constant $C$ independent of $\phi$ such that on $\mathcal{O}$ $$\Delta\phi(x) \leq C(\phi(x)+ |\nabla \phi (x) |).$$
In the following we shall use the notations used in [@BG09]. For any $x\in\mathcal{O}$, let $\lambda_1(x)\leq
\lambda_2(x)\leq \cdots \leq \lambda_n(x)$ be the eigenvalues of $\operatorname{Ric}(x)$. There is a positive constant $C_0>0$ depending $\mathcal{O}$, such that $\lambda_1(x)\leq \lambda_2(x)\leq \cdots \lambda_{n-r}(x) \leq \frac{C_0}{10^{100n}}$ and $C_0\leq \lambda_{n-r+1}(x)\leq \lambda_{n-r+2}(x)\leq \cdots \leq \lambda_n(x)$ for all $x\in \mathcal{O}$. Let $G=\{n-r+1,n-r+2,\cdots,n\}$ and $B=\{1,\cdots,n-r\}$ be the “good" and “bad" sets of indices for eigenvalues of $\operatorname{Ric}$, respectively. Define diagonal matrix $\Lambda_G=\operatorname{diag}[0, \cdots,0, \lambda_{n-r+1},\lambda_{n-r+2},
\cdots,\lambda_{n}]$ and $\Lambda_B=\operatorname{diag}[\lambda_{1},\cdots,\lambda_{n-r},0, \cdots, 0]$. Use notation $h=O(k)$ if $|h(x)|\leq Ck(x)$ for $x\in \mathcal{O}$ with some positive constant $C$ under control. In particular, $\lambda_i=O(\phi)$ for all $i\in B$, and $$(\sum_{i\in B}\lambda_i)\sigma_{r}(\Lambda_G)=O(\phi). \label{ZeroOrdEqu1}$$
Based on Proposition \[DerSigRan1\], with the notation as above, we have
\[DerSigRan2\] Let $A=Ric$ as above. Then we have that on $\mathcal{O}$ $$\frac{\partial \sigma_{r+1}(A)}{\partial a_{ij}}=\left\{
\begin{array}{ll}
\sigma_{r}(\Lambda_G)+O(\phi), & \hbox{if $i=j\in B$;} \\
O(\phi), & \hbox{otherwise.}
\end{array}
\right.$$ and $$\frac{\partial^2\sigma_{r+1}(A) }{\partial a_{ij}\partial a_{kl}}=\left\{
\begin{array}{ll}
\sigma_{r-1}(\Lambda_G|i)+O(\phi)=\frac{1}{\lambda_i}\sigma_{r}(\Lambda_G)+O(\phi), &
\hbox{if $i=j\in G,k=l\in B$;} \\
\sigma_{r-1}(\Lambda_G|k)+O(\phi)=\frac{1}{\lambda_k}\sigma_{r}(\Lambda_G)+O(\phi), &
\hbox{if $i=j\in B,k=l\in G$;} \\
\sigma_{r-1}(\Lambda_G)+O(\phi), & \hbox{if $i=j\in B,k=l\in B,i\neq k$;} \\
-\sigma_{r-1}(\Lambda_G|i)+O(\phi)= -\frac{1}{\lambda_i}\sigma_{r}(\Lambda_G)+O(\phi),
& \hbox{if $i=l\in G,j=k\in B$;} \\
-\sigma_{r-1}(\Lambda_G|j)+O(\phi)= -\frac{1}{\lambda_j}\sigma_{r}(\Lambda_G)+O(\phi),
& \hbox{if $i=l\in B,j=k\in G$;} \\
-\sigma_{r-1}(\Lambda_G)+O(\phi), & \hbox{if $i=l\in B,j=k\in B,i\neq j$;} \\
0, & \hbox{otherwise.}
\end{array}
\right.$$
From Proposition \[DerSigRan2\], we compute the first derivative $$\begin{gathered}
\phi_\alpha=\sigma_{r+1}^{ij}R_{ij,\alpha}= \sigma_{r}(\Lambda_G)\sum_{i\in B}R_{ii,\alpha}+O(\phi),
\label{FirOrdEqu1}\end{gathered}$$ and the second derivative $$\begin{aligned}
\label{SecOrdEqu1}
\phi_{\alpha\beta}&=&\sigma_{r+1}^{ij}R_{ij,\alpha\beta}+\sigma_{r+1}^{ij,kl}R_{ij,\alpha}R_{kl,\beta}\nonumber\\
&= &\sigma_{r}(\Lambda_G)\sum_{i\in B}R_{ii,\alpha\beta} + \sum_{i\in G}\sum_{k\in B}\frac{1}{\lambda_i}\sigma_{r}(\Lambda_G)R_{ii,\alpha}R_{kk,\beta} + \sum_{i\in B}\sum_{k\in G }\frac{1}{\lambda_k}\sigma_{r}(\Lambda_G)R_{ii,\alpha}R_{kk,\beta}\nonumber\\
&&+ \sum_{i,k\in B}\sigma_{r-1}(\Lambda_G)R_{ii,\alpha}R_{kk,\beta}- 2 \sum_{i\in G}\sum_{j\in B}\frac{1}{\lambda_i}\sigma_{r}(\Lambda_G)R_{ij,\alpha}R_{ji,\beta}\nonumber\\
&&-\sum_{i,j\in B}\sigma_{r-1}(\Lambda_G)R_{ij,\alpha}R_{ji,\beta}+O(\phi).
\end{aligned}$$ Take trace of and using , we get $$\label{SigRanDifIne1}
\begin{split}
\Delta\phi & =\sigma_{r}(\Lambda_G)\sum_{i\in B}\Delta R_{ii} +2\left ( \sum_{i \in G}
\frac{1}{\lambda_i}R_{ii,\alpha} \right ) \phi_\alpha - 2 \sigma_{r}(\Lambda_G)\sum_{i\in
G}\sum_{j\in B}\frac{1}{\lambda_i}|\nabla R_{ij}|^2\\
&\quad + \frac{\sigma_{r-1}(\Lambda_G)}{ \sigma_{r}^2(\Lambda_G)}
\phi_\alpha^2 -\sigma_{r-1}
(\Lambda_G) \sum_{i, j\in B}|\nabla R_{ij}|^2+O(\phi)\\
& =\sigma_{r}(\Lambda_G)\sum_{i\in B}\Delta R_{ii} - 2 \sigma_{r}(\Lambda_G)\sum_{i\in
G}\sum_{j\in B}\frac{1}{\lambda_i}|\nabla R_{ij}|^2-\sigma_{r-1}
(\Lambda_G) \sum_{i, j\in B}|\nabla R_{ij}|^2\\
&\quad+O(\phi)+O(|\nabla\phi|).
\end{split}$$ By identity , $$\label{SigRanDifIne2}
\begin{split}
\Delta\phi
=& \sigma_{r}(\Lambda_G)\sum_{i\in B}(\nabla_{\nabla f}R_{ii}+2\lambda R_{ii}
-2\overset{\circ}{R}(\operatorname{Ric})_{ii})\\
&- 2 \sigma_{r}(\Lambda_G)\sum_{i\in G}\sum_{j\in B}\frac{1}{\lambda_i}|\nabla R_{ij}|^2- \sigma_{r-1}(
\Lambda_G)\sum_{i,j\in B}|\nabla R_{ij}|^2+O(\phi)+O(|\nabla\phi|).
\end{split}$$
To deal with the first term in the righthand side of , by and we have $$\label{LapRic1Ord1}
\sigma_{r}(\Lambda_G)\sum_{i\in B}(\nabla_{\nabla f}R_{ii}+2\lambda R_{ii})=
O(\phi)+O(|\nabla\phi|).$$ By the assumption of nonnegative sectional curvature [^2], $$\label{LapRicQuaCur1}
\overset{\circ}{R}(\operatorname{Ric})_{ii}=\sum_kR(e_i, \operatorname{Ric}(e_k),e_i,e_k)
\geq \lambda_1\sum_kR(e_i,e_k,e_i,e_k)\geq 0.$$ Combine , , and , $$\label{SigRanDifIne3}
\begin{split}
\Delta\phi
&\leq C(\phi+ |\nabla\phi|) - 2 \sigma_{r}(\Lambda_G)\sum_{i\in G}\sum_{j\in B}\frac{1}{
\lambda_i}|\nabla R_{ij}|^2- \sigma_{r-1}( \Lambda_G)\sum_{i,j\in B}|\nabla R_{ij}|^2.
\end{split}$$
Hence we have proved $$\Delta\phi\leq C(\phi+|\nabla\phi|).$$ Since $\phi \geq 0$ on $\mathcal{O}$ and $\phi(x_0)=0$, it follows from the strong maximum principle that $ \phi\equiv 0$ on $\mathcal{O}$. We conclude that $\phi\equiv 0$ in $M$, i.e. $\operatorname{rank}
\operatorname{Ric} \equiv r$.
Next we consider the null space of Ricci curvature $\operatorname{null} \operatorname{Ric}$. It follows from that $$2 \sigma_{r}(\Lambda_G)\sum_{i\in G}\sum_{j\in B}\frac{1}{\lambda_i}|\nabla R_{ij}|^2+ \sigma_{r-1}(
\Lambda_G)\sum_{i,j\in B}|\nabla R_{ij}|^2\equiv 0.$$ Hence for any $v\in \operatorname{null} \operatorname{Ric}$, $\nabla \operatorname{Ric}(v)=0$. On the other hand, for any section $v \in \operatorname{null} \operatorname{Ric}$ and for any index $k$ we have $$0=\nabla_{k}(R_{ij}v^i)=(\nabla_kR_{ij})v^j +R_{ij}\nabla_kv^i,$$ thus $R_{ij}\nabla_kv^i = -(\nabla_k R_{ij})v^j=0$. This shows that $\nabla_k v \in \operatorname{null} \operatorname{Ric}$ and that $\operatorname{null}
\operatorname{Ric}$ is invariant under parallel translation.
Finally we show that the universal covering space of the GRS $(M,g)$ splits. Since the distribution $\operatorname{null} \operatorname{Ric}$ is invariant under parallel translation, $\operatorname{null} \operatorname{Ric}$ is involutive. Let $(\operatorname{null} \operatorname{Ric})^\perp$ be the distribution that generated by orthogonal complements of $\operatorname{null} \operatorname{Ric}$. For any sections $X, Y\in(\operatorname{null}
\operatorname{Ric})^\perp$, $V\in \operatorname{null}\operatorname{Ric}$, then $$g([X,Y],V)=g(\nabla_XY-\nabla_YX,V)=-g(Y,\nabla_XV)+g(X,\nabla_YV)=0.$$ Thus the distribution $(\operatorname{null} \operatorname{Ric})^\perp$ is also involutive. The classical deRham splitting theorem (see [@Besse87 Theorem 10.43]) implies that $(M,g)$ locally splits.
Now consider the the universal covering space $(\widetilde{M},\tilde{g})$. We denote by $L$ the leaf of the integral manifold of $\operatorname{null} \operatorname{Ric}$, then $L$ is Ricci flat. By equation , on every leaf, ${\operatorname{Hess}}f=\frac{1}{2} g $. Consequently, $L$ is isometric to $\mathbb{R}^{n-r}$. Hence $(\widetilde{M},\tilde{g})=(N,h)\times
\mathbb{R}^{n-r}$ split isometrically along the null space of Ricci curvature, where $(N,h)$ has strictly positive Ricci curvature. We have finished the proof of Theorem \[SplGSRSThm1\] \[SplStrRicThm\].
Curvature operator and constant rank Theorem \[SplGSRSThm1\](II)
----------------------------------------------------------------
Similarly, we can establish the constant rank theorem Theorem \[SplGSRSThm1\] \[SplStrCurOpeThm\] for curvature operators. We may assume that $r:= \min_{x\in M}{\rm rank\,}\mathfrak{R}(x)< \frac{n(n-1)}
{2}$. There is a point $x_0\in M$ such that ${\rm rank\,}\mathfrak{R}(x_0) =r$. Pick an orthonormal basis $\{ e_i \}$ around $x_0$. Let $\{\varphi_\alpha=\varphi_\alpha^{ij}e_i\wedge e_j\}$ be the eigenvectors of curvature operator $\mathfrak{R}$, i.e. $\mathfrak{R}(\varphi_\alpha)=\lambda_\alpha\varphi_\alpha$. Define $\phi=\sigma_{r+1}(\mathfrak{R})$.
Below we adopt notations similar to the ones used in section \[ProSplThmRic1\]. Using Proposition \[DerSigRan2\] and equation and by a computation similar to the derivation of (\[SigRanDifIne2\]) we get $$\label{SigCRRanDifIne1}
\begin{split}
\Delta\phi& \leq C(\phi+ |\nabla \phi|)+ \sigma_{r}(\Lambda_G)\sum_{\alpha\in B} \left (\nabla_{\nabla f}\mathfrak{R}_{
\alpha\alpha}+ 2\lambda \mathfrak{R}_{\alpha\alpha}-\mathfrak{R}^2_{\alpha\alpha} - \mathfrak{R}^\sharp_{
\alpha\alpha} \right )\\
&\quad -2 \sigma_{r}(\Lambda_G)\sum_{\alpha\in G}\sum_{\beta\in B}\frac{1}{\lambda_\alpha}|\nabla \mathfrak{R}_{\alpha\beta}|^2- \sigma_{r-1}(\Lambda_G)\sum_{\alpha, \beta\in B}|\nabla \mathfrak{R}_{
\alpha\beta}|^2 \\
&\leq C(\phi+ |\nabla \phi|) - 2 \sigma_{r}(\Lambda_G)\sum_{\alpha\in G}\sum_{\beta\in B}\frac{1}{\lambda_\alpha}
|\nabla \mathfrak{R}_{\alpha\beta}|^2- \sigma_{r-1}(\Lambda_G)\sum_{\alpha, \beta\in B}
|\nabla \mathfrak{R}_{\alpha\beta}|^2\\
& \leq C(\phi+ |\nabla \phi|),
\end{split}$$ in some small neighborhood $\mathcal{O}$ of $x_0$. To get the second inequality above we have used the following $$\begin{aligned}
\langle\mathfrak{R}^\sharp(\varphi_\alpha), \varphi_\alpha \rangle = \sum_{\alpha, \beta}\lambda_\beta\lambda_\gamma\langle[\varphi_\beta,\varphi_\gamma],\varphi_\alpha\rangle^2
\geq 0,\end{aligned}$$ which follows from and the assumption of the nonnegative curvature operator.
Since $\phi\geq 0$, and $\phi(x_0)=0$, by applying the strong maximum principle to we get $\phi\equiv 0$. We conclude that curvature operator $\mathfrak{R}$ has constant rank.
By a similar proof as for the Ricci curvature case the null space of $\mathfrak{R}$ is invariant under parallel transilation. Moreover, it follows from (\[LapCurOpeSolEqu\]) that $\operatorname{null}\mathfrak{R} \subset
\operatorname{null}\mathfrak{R}^\sharp$. By we have $ \langle[\varphi_\beta,
\varphi_\gamma], \phi\rangle=0$ for $\phi\in \operatorname{null} \mathfrak{R}$ and for any $\beta\neq\gamma$ with $\lambda_\beta>0$ and $\lambda_\gamma>0$. Since $\mathfrak{R}$ is a self-adjoint operator, $$\phi\in \operatorname{image} \mathfrak{R}\Leftrightarrow \langle \phi, \varphi_\alpha\rangle=0,
\quad \forall~ \alpha {~\rm with~}\lambda_\alpha=0.$$ For any section $\phi, \omega \in \operatorname{image} \mathfrak{R}$, we have $$\langle[\phi,\omega],\psi\rangle=\sum_{\beta,\gamma} \langle\phi,\varphi_\beta\rangle \langle \omega,\varphi_\gamma\rangle\langle[\varphi_\beta,\varphi_\gamma],\psi \rangle
=0, \quad \forall \psi\in \operatorname{null} \mathfrak{R},$$ hence $[\phi,\omega]\in \operatorname{image} \mathfrak{R}$. This implies that the image of $\mathfrak{R}$ is a Lie subalgebra. Ambrose-Singer theorem ensures that the Lie algebra $\mathfrak{hol}(M,g)$ of Holonomy group is reduced to a lower dimension, so by deRham splitting theorem (see [@Besse87 Theorem 10.43]) the universal covering space $(\widetilde{M},\tilde{g})$ is a Riemannian product. Since one of the product factor is flat from our construction, $(\widetilde{M},\tilde{g})=(N,h)\times \mathbb{R}^{n-m}$, $\frac{m(m-1)}{2}=r$, splits isometrically, where $(N,h)$ has strictly positive curvature operator. The proof of Theorem \[SplGSRSThm1\] is completed.
Mean Curvature Growth Estimate {#MeaCurGroEstSec}
==============================
In this section, we establish the following a priori interior estimate of the mean curvature for a convex hypersurface in $\mathbb R^{n+1}$. As a consequence, we can control the mean curvature growth for embedded codimension one GRS in $\mathbb R^{n+1}$, see and .
Let $X: M^n \rightarrow \mathbb{R}^{n+1}$ be a hypersurface with induced metric $g$ and (outer) unit normal $\nu$. Let $\{e_1,\cdots,e_n\}$ be a local orthonormal frame filed on $M$, then $$\label{FirDerPosNorHyp}
X_{ij} = -h_{ij}\nu, \quad 1\leq i, j\leq n,$$ where $h=(h_{ij})$ is the second fundamental form. Let $\sigma_k=\sigma_k(h)$ be the $k$-th elementary symmetric function of the eigenvalues of $h$. In particular, $H=\sigma_1(h)$ and $R=2\sigma_2(h)$ are the mean curvature and the scalar curvature respectively. If the scalar curvature of $M$ is positive, we take the unit normal $\nu$ such that $h$ lies in Garding’s $\Gamma_2$-cone. In particular, the differential operator $$\label{EllipticSig2Ope}
\square_h:=\sigma_2^{ij}(h)\nabla_{ij}^2$$ is elliptic, where $\sigma_2^{ij}(h)$ is defined in .
\[IntEstMeaCurConLem\] Let $X : (M^n, g)\rightarrow \mathbb{R}^{n+1}$ be a convex hypersurface with positive scalar curvature. If there exists a unit constant vector $a$ such that $\langle X, a\rangle$ is a nonegative proper function, then we have the interior estimate $$\label{IntEstMeaCurCon}
H(x)\leq C(n)\sup_{\{y \, | \,\langle X(y), a\rangle\leq 2\langle X(x),a\rangle \}}
(1+R^2(y)+ \frac{1}{R(y)}
+ \frac{1}{R^2(y)}|\nabla R|^2 (y)+ \frac{1}{R(y)}|\Delta R|(y) ).$$
We note that on a shrinking GRS, one can split out lines and reduce the GRS to be a convex hypersurface such that there exists automatically a vector $a$ such that $\langle X, a\rangle$ is a nonnegative proper function, see .
First of all, the following identity is well known (see, for example, (2.11) in [@CY77]) and will be used to prove the theorem above.
Let $X: M^n \rightarrow \mathbb{R}^{n+1}$ be a hypersurface with the second fundamental form $h$, then $$\label{EllEquMeaCur1}
\square \sigma_1:=\sigma_2^{ij}\sigma_{1,ij}=\Delta\sigma_2+|\nabla h|^2-|\nabla \sigma_1|^2+2\sigma_2|h|^2-(\sigma_1\sigma_2-3\sigma_3)\sigma_1,$$ where $( \sigma_2^{ij}):= (\frac{\partial \sigma_2}{\partial h_{ij}})$ is defined in .
To prove Theorem \[IntEstMeaCurConLem\], let $\phi(x)= r -\langle X(x), a\rangle$ be a cut off function with $r \geq 1$, we will apply second derivative test to the auxiliary function $$\phi^2(x)\sigma_1(x)$$ in the domain $\Omega_{r}: = \{x \in M | \, \langle X(x),
a \rangle\leq r \}$ to estimate $\sigma_1(x)$.
We may assume that $\phi^2 \sigma_1$ achieves its maximum at an interior point $\bar{x} \in \Omega_r$. Let $0\leq \lambda_1(x)\leq \lambda_2(x)\leq \cdots \leq \lambda_n(x)$ be the principle curvature of $M$ at $x \in M$. Moreover, in a neighborhood of $\bar{x}$, we choose a local orthonormal frame $\{ e_i \}$ such that $ h_{ij}(\bar{x})=\lambda_i(\bar{x})\delta_{ij}$.
.1cm We consider three cases.
1. : $\lambda_n(\bar{x})\leq \max\{n^2, 100n\}$. Then $\sigma_1(\bar{x})\leq C(n)$, and thus $$\label{IntEstMeaCurCaseI}
\phi^2(\bar{x})\sigma_1(\bar{x})\leq C(n) r^2.$$
2. : $\lambda_n(\bar{x})> \max\{n^2, 100n\}$ and $\lambda_{n-1}(\bar{x})\geq \lambda_n^{-\frac{1}{2}}
(\bar{x})$, then the scalar curvature $R(\bar{x})=\sum_{i\neq j}\lambda_i\lambda_j\geq \lambda_n\lambda_{n-1} \geq \lambda_n^{\frac{1}{2}}(\bar{x})$ and $\sigma_1(\bar{x})=\sum_{i=1}^n\lambda_i\leq n\lambda_n\leq nR^2(\bar{x})$. Hence $$\label{IntEstMeaCurCaseII}
\phi^2(\bar{x})\sigma_1(\bar{x})\leq nR^2(\bar{x}) r^2 .$$
3. \[MeaEstCaseIII\] : $\lambda_n(\bar{x})> \max\{n^2, 100n\}$ and $\lambda_{n-1}(\bar{x}) < \lambda_n^{-\frac{1}{2}}
(\bar{x})$. Then $\lambda_i(\bar{x}), i\neq n$ is much smaller than $\lambda_n(\bar{x})$. In this case, we have $$\label{MeaMaxEigCom1}
\lambda_i<\frac{1}{n^3}\lambda_n,\, \, i \neq n,\quad \hbox{and} \quad \lambda_n< \sigma_1=\sum_{i=1}^n\lambda_i<(1+\frac{1}{n^2})\lambda_n.$$
To estimate $\phi^2(\bar{x})\sigma_1(\bar{x})$ from above, we are left to consider case \[MeaEstCaseIII\]. Note that the function $$\zeta:=\ln\big(\phi^2\sigma_1\big)=2\ln\phi+\ln \sigma_1$$ on $\Omega_{r}$ also achieve the maximum at $\bar{x}$. Apply the first and second derivative test, we get that at $\bar{x}$ $$\label{GraAuxZer3}
0=\zeta_i(\bar{x})=2\frac{\phi_i}{\phi}+\frac{\sigma_{1,i}}{\sigma_1}, \quad \forall~ i=1, \cdots,n.$$ and the matrix $$\label{HesAuxCon3}
\begin{split}
0&\geq \zeta_{ij}(\bar{x})\\
&=2\frac{\phi_{ij}}{\phi}-2\frac{\phi_i\phi_j}{\phi^2}+\frac{\sigma_{1,ij}}{\sigma_1}
-\frac{\sigma_{1,i}\sigma_{1,j}}{\sigma_1^2}\\
&=2\frac{\phi_{ij}}{\phi}+\frac{\sigma_{1,ij}}{\sigma_1}
-\frac{3}{2}\frac{\sigma_{1,i}\sigma_{1,j}}{\sigma_1^2},
\end{split}$$ where we used in the last step.
Note that the positive scalar curvature on $M$ implies that the operator $\sigma_2^{ij}\nabla_{ij}^2$ is elliptic, see . Take the contraction of with $\sigma_2^{ij}$ and use , we get that at $\bar{x}$ $$\label{ConAulMeaEqu1}
\begin{split}
0&\geq \sigma_2^{ij}\zeta_{ij} \\
&=2\frac{\sigma_2^{ij}\phi_{ij}}{\phi}+\frac{\Delta\sigma_2+|\nabla h|^2-|\nabla \sigma_1|^2+2\sigma_2|h|^2-(\sigma_1\sigma_2-3\sigma_3)\sigma_1}{\sigma_1}
- \frac{3}{2} \frac{\sigma_2^{ij}\sigma_{1,i}\sigma_{1,j}}{\sigma_1^2}.
\end{split}$$ Below we will deal with the three terms in the righthand side of separately. All the related calculations are at point $\bar{x}$.
For the first term, from , we have $$\label{SquPsoConaFunEqu1}
\sigma_2^{ij}\phi_{ij}=-\sigma_2^{ij}\langle X_{ij},a\rangle=2\sigma_2\langle \nu, a\rangle.$$
To deal with the second term, note that $$\label{GraSFFMeaDiff1}
\begin{split}
|\nabla h|^2-|\nabla \sigma_1|^2&=\sum_{i,j,k} h_{ij,k}^2-\sum_{i,j,k} h_{ii,k}h_{jj,k}\\
&=\sum_k\sum_{i\neq j} h_{ij,k}^2-\sum_k\sum_{i\neq j} h_{ii,k}h_{jj,k}\\
&=2\sum_{i\neq j} h_{ii,j}^2+\sum_{ i\neq j\neq k} h_{ij,k}^2-\sum_k\sum_{i\neq j} h_{ii,k}h_{jj,k},
\end{split}$$ where we have used the Codazzi equation $h_{ij,k}=h_{ik,j}$ to get the last equality. The term $2\sum_{i\neq j} h_{ii,j}^2$ in will play a crucial role in our estimate. The term $\sum_{ i\neq j\neq k } h_{ij,k}^2$ can be discarded. However we need to control the negative term $-\sum_k\sum_{i\neq j} h_{ii,k}h_{jj,k}$. In fact, by G[å]{}rding’s theory on hyperbolic polynomial (see [@GLL Lemma 3.2] or [@GRW Lemma 2.2]), we have $$\label{GraBelGraSca1}
-\sum_{i\neq j}h_{ii,k}h_{jj,k}\geq -\frac{1}{2}\frac{ |\sigma_{2,k}| ^2}{\sigma_2},
~\forall~ k=1,\cdots,n.$$ Therefore $$\label{GraSFFMeaDiff2}
|\nabla h|^2-|\nabla \sigma_1|^2\geq 2\sum_{i\neq j} h_{ii,j}^2-\frac{1}{2}\frac{|\nabla\sigma_2|^2}{\sigma_2}.$$
We now handle the third term in . Since the principle curvatures $\lambda_i, \, i \neq n$, are very small compared with $\lambda_n$, thus $\sigma_2^{nn}=\sum_{i=1}^{n-1}\lambda_i \leq \sigma_1$ is also very small. We may use to substitute the partial derivative of mean curvature along the $i=n$ direction, which in turn is bounded. We compute that for any $\epsilon>0$ $$\label{ConGraTermEst1}
\begin{split}
\frac{\sigma_2^{ij}\sigma_{1,i}\sigma_{1,j}}{\sigma_1^2}
&=\frac{\sigma_2^{nn}\sigma_{1,n}^2+\sum_{i=1}^{n-1}\sigma_2^{ii}\sigma_{1,i}^2}{\sigma_1^2} \\
& \leq 4 \frac{\sigma_2^{nn}\phi_{n}^2}{\phi^2}+\sigma_1\sum_{i=1}^{n-1}\Big(\frac{\sigma_{1,i}}{\sigma_1}
\Big)^2 \\
& = 4 \frac{\sigma_2^{nn} \phi_{n}^2}{\phi^2} + \frac{\sum_{i=1}^{n-1} \big ( h_{nn,i}+\sum_{
j=1}^{n-1} h_{jj,i}\big )^2} {\sigma_1} \\
& \leq 4 \frac{\sigma_2^{nn} \phi_{n}^2}{\phi^2} + (1+\epsilon)\frac{\sum_{i=1}^{n-1} h_{nn,i}^2}{\sigma_1}
+(1+\frac{4} {\epsilon})\frac{\sum_{i=1}^{n-1} \big(\sum_{j=1}^{n-1}h_{jj,i}\big)^2}{\sigma_1}. \\
\end{split}$$
To deal with the last term in , we consider the scalar curvature $\sigma_2$. For $i\leq n-1$, $$\label{GradScaCurGraSFF}
\sigma_{2,i}=\sum_{j=1}^n\sigma_2^{jj}h_{jj,i}=\sigma_2^{nn}h_{nn,i}+\sum_{j=1}^{n-1}(\sigma_1
-\lambda_j)h_{jj,i},$$ where we have used Proposition \[DerSigRan1\] to get $\sigma_2^{jj} = \sigma_1
-\lambda_j$ at $\bar{x}$. As a consequence, we get $$\label{GraQuaTerEst1}
\begin{split}
\Big ( \sum_{j=1}^{n-1}h_{jj,i}\Big )^2&=\Big ( \frac{\sigma_{2,i}}{\sigma_1}-\frac{\sigma_2^{nn}}{\sigma_1}
h_{nn,i}+\sum_{j=1}^{n-1}\frac{\lambda_j}{\sigma_1}h_{jj,i}\Big )^2\\
&\leq 3\Big(\frac{(\sigma_{2,i})^2}{\sigma_1^2}+\frac{(\sigma_2^{nn})^2}{\sigma_1^2}h_{nn,i}^2+(n-1)
\sum_{j=1}^{n-1}\frac{\lambda_j^2}{\sigma_1^2}h_{jj,i}^2\Big)\\
&\leq 3\Big(\frac{(\sigma_{2,i})^2}{\sigma_1^2}+\frac{(\sigma_2^{nn})^2}{\sigma_1^2}h_{nn,i}^2
+\frac{2n}{\sigma_1^3}\sum_{j=1}^{n-1}h_{jj,i}^2 \Big),\\
\end{split}$$ where we have used $\lambda_j^2\leq \lambda_n^{-1}< 2\sigma_1^{-1}$ for $j\leq n-1$ to get the last inequality (see ).
Use equation again, we get that for $i \leq n-1$ $$\label{GraQuaTerEst3}
h_{ii,i}=\frac{\sigma_{2,i}}{\sigma_2^{ii}}-\frac{\sigma_2^{nn}}{\sigma_2^{ii}}h_{nn,i} -\sum_{j=1, j
\neq i}^{n-1} \frac{\sigma_2^{jj}}{\sigma_2^{ii}}h_{jj,i}.$$ It follows from that $\sigma_1\geq \sigma_2^{ii}=\sigma_1-\lambda_i\geq \frac{n-1}{n}
\sigma_1$ for $i\leq n-1$. Then $$\label{GraQuaTerEst4}
\begin{split}
h_{ii,i}^2 &\leq 3\Big(\frac{\big(\sigma_{2,i}\big)^2}{(\sigma_2^{ii})^2}+\frac{(\sigma_2^{nn})^2}{(\sigma_2^{ii}
)^2}h_{nn,i}^2+(n-2)\sum_{j=1, j\neq i}^{n-1} \Big(\frac{\sigma_2^{jj}}{\sigma_2^{ii}}\Big)^2h_{jj,i}^2\Big)\\
& \leq 3\Big(\frac{n}{n-1}\Big)^2 \Big(\frac{\big(\sigma_{2,i}\big)^2}{\sigma_1^2}+\frac{(\sigma_2^{nn}
)^2}{\sigma_1^2}h_{nn,i}^2 + (n-2) \sum_{j=1, j \neq i}^{n-1}h_{jj,i}^2 \Big)\\
&\leq 6 \Big(\frac{\big(\sigma_{2,i}\big)^2}{\sigma_1^2}+\frac{(\sigma_2^{nn})^2}{\sigma_1^2}
h_{nn,i}^2+n\sum_{j=1, j \neq i}^{n-1} h_{jj,i}^2\Big).
\end{split}$$
Combine and and use $\sigma_1 >n^2$, we have $$\label{GraQuaTerEst5}
\begin{split}
\Big ( \sum_{j=1}^{n-1}h_{jj,i}\Big )^2
&\leq 3\Big[\frac{(\sigma_{2,i})^2}{\sigma_1^2}+\frac{(\sigma_2^{nn})^2}{\sigma_1^2}h_{nn,i}^2+
\frac{2n}{\sigma_1^3}\Big(\sum_{j=1, j\neq i}^{n-1}h_{jj,i}^2 \\
&\quad +6\big ( \frac{\big(\sigma_{2,i}\big)^2}{\sigma_1^2}+\frac{(\sigma_2^{nn})^2}{\sigma_1^2}h_{nn,i}
^2+n\sum_{j=1, j\neq i}^{n-1}h_{jj,i}^2\big )\Big)\Big]\\
&\leq 21\Big[\frac{(\sigma_{2,i})^2}{\sigma_1^2}+\frac{(\sigma_2^{nn})^2}{\sigma_1^2}h_{nn,i}^2+\frac{2n^2}
{\sigma_1^3}\sum_{j=1, j\neq i}^{n-1} h_{jj,i}^2\Big)\Big].
\end{split}$$
Plug into , we have $$\label{ConGraTermEstSma2}
\begin{split}
\frac{\sigma_2^{ij} \sigma_{1,i} \sigma_{1,j}}{\sigma_1^2}
&\leq 4 \frac{\sigma_2^{nn} \phi_{n}^2}{\phi^2} + (1+\epsilon)\frac{\sum_{i=1}^{n-1} h_{nn,i}^2}{\sigma_1} \\
& \quad +(1+\frac{4}{\epsilon})\frac{21}{\sigma_1}\Big[\frac{ \sum_{i=1}^{n-1} (\sigma_{2,i})^2}{\sigma_1^2}
+\frac{(\sigma_2^{nn})^2}{\sigma_1^2} \sum_{i=1}^{n-1} h_{nn,i}^2+\frac{2n^2}{\sigma_1^3} \sum_{i=1}^{n-1}
\sum_{j=1,j \neq i}^{n-1} h_{jj,i}^2\Big)\Big]\\
&\leq 4 \frac{\sigma_2^{nn} \phi_{n}^2}{\phi^2} + (1+\delta)\frac{1}{\sigma_1}\sum_{i=1}^{n-1}
\sum_{j=1, j\neq i}^{n}h_{jj,i}^2+21(1+\frac{4}{ \epsilon})\frac{ \sum_{i=1}^{n-1} (\sigma_{2,i})^2}{\sigma_1^3},
\end{split}$$ where $$\label{DefDeltaEpl1}
\delta: =\epsilon+21(1+\frac{4}{\epsilon})
\big(\frac{\sigma_2^{nn}}{\sigma_1}\big)^2+21(1+\frac{4}{\epsilon})\frac{2n^2}{\sigma_1^3}.$$
Put , and into , we get $$\label{ConAulMeaEqu2}
\begin{split}
0 &\geq \frac{4\sigma_2\langle\nu,a\rangle}{\phi}+\frac{\Delta\sigma_2}{\sigma_1}+
\frac{1}{\sigma_1}\big(2\sum_{i\neq j} h_{ii,j}^2-\frac{1}{2}\frac{|\nabla\sigma_2|^2}{\sigma_2}\big)
+\Big( \frac{2 |h|^2}{\sigma_1^2}-1\Big)\sigma_2\sigma_1\\
&\quad +3\sigma_3- \frac{3}{2}\Big(\frac{4\sigma_2^{nn}\phi_{n}^2}{\phi^2}+(1+\delta)\frac{1}{\sigma_1}
\sum_{i \neq j}h_{ii,j}^2+21(1+\frac{4}{\epsilon})\frac{|\nabla\sigma_{2}|^2
}{\sigma_1^3}\Big)\\
&\geq \frac{4 \sigma_2\langle\nu,a\rangle}{\phi}+\frac{\Delta\sigma_2}{\sigma_1}
+\Big(2- \frac{3}{2} (1+\delta)\Big) \frac{1}{\sigma_1} \sum_{i \neq j }h_{ii,j}^2-\frac{|\nabla\sigma_2|^2}{2\sigma_2\sigma_1} \\
&\quad +\Big(\frac{2|h|^2}{\sigma_1^2} -1\Big)\sigma_2\sigma_1 - \frac{6\sigma_2^{nn}\langle e_n,a
\rangle^2}{\phi^2}- \frac{63}{2}(1+\frac{4}{\epsilon})\frac{|\nabla\sigma_{2}|^2}{2\sigma_1\sigma_2},
\end{split}$$ where we have used the Newton-MacLaurin inequality $\sigma_1^2\geq\frac{2n}{n-1}\sigma_2\geq 2\sigma_2$.
By \[MeaEstCaseIII\] assumption and we have $$\label{StrPosCoeMea1}
\frac{2|h|^2}{\sigma_1^2} -1 \geq \frac{2 \lambda_n^2}{\sigma_1^2} -1 \geq \frac{2n^2}{n^2+1} -1 \geq \frac{1}{2}.$$ Again by we have $$\sigma_2^{nn}=\sum_{i=1}^{n-1}\lambda_i<n\lambda_n^{-\frac{1}{2}}<\frac{2n}{\sqrt{\sigma_1}}.$$ Take $\epsilon=\frac{1}{10}$ in and use $\sigma_1\geq 100 n$, we get $$\label{NeaCalGraSFF1}
\delta \leq \frac{1}{10}+1000 \frac{4n^2}{\sigma_1^3}+1000\frac{2n^2}{\sigma_1^3} <\frac{1}{3}.$$
Hence it follows from , and that in Case III at point $\bar{x}$ $$\label{MeaCurIntEst2}
\begin{split}
(\phi^2\sigma_1)&\leq \frac{2}{\sigma_2}\Big[-4\sigma_2\langle \nu, a\rangle\phi+ \frac{12n}
{\sqrt{\sigma_1}} \langle e_n, a\rangle^2 +\Big(-\frac{\Delta\sigma_2}{\sigma_1}+1000\frac{|\nabla\sigma_2|^2}{\sigma_2\sigma_1}
\Big)\phi^2\Big].
\end{split}$$
Combine the three cases , and all together and use $H(x)=\sigma_1(x)$ and $R(x) =2\sigma_2(x)$, we have proved the interior estimate of the mean curvature $$\sup_{x \in \Omega_{r}} \phi^2(x) H(x) \leq C(n) \sup_{y \in \Omega_{r}} (1+R^2(y)+ \frac{1}{R(y)}
+ \frac{1}{R^2(y)}|\nabla R|^2 (y)+ \frac{1}{R(y)}|\Delta R|(y) ) r^2. \label{eq add 1}$$
Given any $x \in M$, we take $r =2\langle X(x),a\rangle$, then $\phi(x)= \langle X(x),a\rangle$. Hence by , we obtain . Theorem \[IntEstMeaCurConLem\] is proved.
The Structure of Generalized Cylinder {#StrGenCyl}
=====================================
In this section, we prove Theorem \[GSRSHypCyl\]. First we define an operator which generalizes Cheng-Yau’s self-adjoint operator associated to a given Codazzi tensor [@CY77].
\[SlfAdjDifOpe\] Let $(M^n, g)$ be a Riemannian manifold and let $f$ be a smooth function on $M$. Let $\psi=\sum_{ij}\psi_{ij} \omega^i \omega^j$ be a symmetric $(2,0)$-tensor on $M$. Then the operator $$\square_{\psi,f} :=\psi_{ij} \nabla^2_{ij}-\psi_{ij} f_i \nabla_j$$ is self-adjoint with respect the weighted measure $e^{-f}d \mu$ if and only if the divergence of $\psi$ $\sum_j\psi_{ij,j}=0$ for all $i$. Here $\nabla$ is the Levi-Civita connection and $d \mu$ is the volume element associated with $g=\sum_i\omega_i^2$.
In particular, given a symmetric Codazzi tensor $h= h_{ij}\omega^i\omega^j$ with $h_{ij,k}= h_{ik,j}$, define $\sigma_2(h)$, $\sigma_2^{ij}(h)$ as in and , then the tensor $(\psi(h))_{ij} :=\big(\sum_k h_{kk}\big) \delta_{ij}-
h_{ij}=\sigma_2^{ij}(h)$ is divergence free and the operator $$\label{GenCYOpeDef1}
\begin{split}
\square_{\psi(h),f}
&=\sigma_2^{ij}(h) \nabla^2_{ij}-\sigma_2^{ij}(h)f_i \nabla_j,
\end{split}$$ is a self-adjoint operator with respect to the weighted measure $e^{-f}d\mu$.
For any compact supported $C^2$ functions $u$ and $v$ on $M$, by Stoke’s theorem, $$\begin{split} \int_{M} \square_{\psi,f} u \cdot v e^{-f}d\mu
&=\int_{M} (\psi_{ij}u_{i}ve^{-f})_j d\mu-\int_{M} \psi_{ij}u_i
v_je^{-f} d\mu
-\int_{M} \psi_{ij,j}u_ive^{-f} d\mu \\
& = -\int_{M}\psi_{ij}u_iv_je^{-f} d\mu.
\end{split}$$ Therefore the operator $\square_{\psi, f}$ is self-adjoint with respect to the measure $e^{-f}d\mu$.
On a GRS satisfying , there is another natural intrinsic self-adjoint operator $\square_{\operatorname{Ric}}=R_{ij}\nabla_{ij}^2 $ with respect the weighted measure $e^{-f}d\mu$. We believe that this operator should be useful.
We need the following proposition (analogue of [@CY77 Proposition 2]).
\[PinTypIneSqu1\] Let $(M^n,g)$ be a compact manifold with boundary and let $f$ be a smooth function on $M$. Suppose $\psi=\sum_{ij}\psi_{ij} \omega^i \omega^j$ is a semipositive symmetric $(2,0)$-tensor which is divergence free. Then the (possibly degenerate) elliptic operator $\square_{\psi, f}$ has the following property. For any $C^2$ positive function $u$ and any non-negative $C^2$ function $v$ satisfying $v|_{\partial M}=0$, we have $$\label{FirEigSqu1}
\Big(-\int_M v\square_{\psi, f} ve^{-f} d \mu \Big) \Big(\int_Mv^2e^{-
f} d\mu \Big)^{-1}\geq \inf_M\frac{-\square_{\psi, f} u}{u}.$$
We only need to prove assuming $\square_{\psi, f}$ is non-degenerate elliptic, as one may replace $\square_{\psi, f}$ by $\square_{\psi, f}+\epsilon \Delta_{f}$ and let $\epsilon\rightarrow 0$.
Let $\lambda$ be the first (positive) eigenvalue and $v_\lambda$ be the first eigenfunction of $\square_{\psi, f}$ over $M$ with the zero boundary condition. Then it is well-known that the left hand of is always not less than $\lambda$ and $v_\lambda$ is positive in the interior of $M$.
Consider the function $\frac{v_\lambda}{u}$. At the interior point where $\frac{v_\lambda}{u}$ attains its maximum, we have $$\begin{aligned}
& 0= \nabla \frac{v_\lambda}{u}=\frac{u \nabla v_\lambda-v_\lambda \nabla
u}{u^2}, \label{GraEigFunQuo1} \\
& 0\geq \nabla^2\frac{v_\lambda}{u}=\frac{u \nabla^2 v_\lambda-v_\lambda \nabla^2
u}{u^2}. \label{HessEigFunQuo1}\end{aligned}$$ Hence $$\begin{split}
\lambda&=\frac{-\square_{\psi,f} v_\lambda}{v_\lambda}=-\frac{\psi_{ij}\nabla^2_{ij} v_\lambda}{v_\lambda}+\frac{\psi_{ij}f_i \nabla_jv_\lambda}{v_\lambda}\\
&\geq-\frac{\psi_{ij} \nabla^2_{ij} u}{u}+\frac{\psi_{ij} f_i \nabla
_ju}{u}= \frac{-\square_{\psi, f}u}{u}.
\end{split}$$ This verifies .
Let $X: M^n \rightarrow \mathbb{R}^{n+1}$ be a hypersurface with induced metric $g$ and (outer) unit normal $\nu$. Let $h$ be the second fundamental form and let $H$ be the mean curvature. Given a smooth function $f$ on $M$, we define two operators using a local orthonormal frame $\{e_1,\cdots,e_n\}$ $$\square_h := \sigma_2^{ij}(h) \nabla_{ij}^2 \quad \text{ and } \quad
\square_{h,f}: = \square_h - \sigma_2^{ij}(h) f_i \nabla_j,$$ where $\sigma_2(h)$, $\sigma_2^{ij}(h)$ are defined in and . From Proposition , the operator $\square_{h,f}$ is a self-adjoint operator with respect to the weighted measure $e^{-f}d\mu$.
We compute $\square_{h,f}\nu$. Note that $$\label{DerPosNorHyp1}
X_{ij} = -h_{ij}\nu, \quad \nu_i = h_{il}e_l, \quad
\nu_{ij}= h_{ij,l}e_l-h^2_{ij}\nu, \qquad 1\leq i, j\leq n.$$ From , we have $$\begin{aligned}
%\square_h X&=&(H\delta_{ij}-h_{ij})(-h_{ij}\nu) =-R \nu, \label{SquPsoFunEqu0}\\
\square_{h,f} \nu &=&\sigma_2^{ij}(h)(h_{ij,l}e_l-h^2_{ij}\nu)-\sigma_2^{ij}(h)f_ih_{jl})e_l\nonumber\\
&=&\frac{1}{2}\nabla R-(\sigma_1(h)\sigma_2(h)-3\sigma_3(h))\nu- \operatorname{Ric}(\nabla f). \label{SquNorVecEqu1}\end{aligned}$$
For a GRS satisfying , we have the following equation analog to the one considered by Cheng-Yau in [@CY77].
\[SquNorVecEquGSRSPro\] Let $(M^n,g,f)$ be an isometrically embedded hypersurface in $\mathbb{R}^{n+1}$ with a GRS structure . Then $$\label{SquNorVecEquGSRS1}
\square_{h,f}\nu = -(\sigma_1(h)\sigma_2(h)-3\sigma_3(h))\nu.$$
The proposition follows from and .
Now we can prove Theorem \[GSRSHypCyl\]. Since $(M,g,f)$ be a gradient shrinking Ricci soliton with nonnegative Ricci curvature, then by the maximum principle, either $(M,g,f)$ is flat or the scalar curvature is strictly positive (see ). Suppose $(M,g)$ is an Euclidean hypersurface with nonnegative Ricci curvature and positive scalar curvature, then the second fundamental form $h$ is semipositive. It follows that $M$ is a convex hypersurface and the curvature operator is semipositive.
With the splitting Theorem \[SplGSRSThm1\], we can assume that the universal covering $(\widetilde{M},\tilde{g})=(N,h)\times \mathbb{R}^{n-k}$ split isometrically and $(N,h)$ has strictly positive Ricci curvature. However, it is not illuminating to see that whether $(N,h)$ or one of its quotient admit an isometric embedding in $\mathbb{R}^{k+1}$. Alternatively, since $(M,g)$ is Euclidean hypersurface, we can also establish an splitting theorem easily in an extrinsic way.
First of all, if $M$ is noncompact convex Euclidean hypersurface, then the Gauss image must lies in a closed hemisphere of $S^n$; moreover, if $(M,g)$ has positive section curvature, then Gauss image is an open convex subset of $S^n$ [@Wu74]. Therefore, there is a unit vector $a\in S^n$ so that $\langle \nu, a\rangle\geq 0$ on $M$. If $\langle \nu, a\rangle=0$ at one point, then we claim that $\langle \nu, a\rangle$ is identically zero. From we get $$\square_{h,f}\langle \nu, a\rangle=-(\sigma_1(h)\sigma_2(h)-3\sigma_3(h))\langle \nu, a\rangle.$$ Note that the positive scalar curvature on $M$ implies that the operator $\square_{h,f}$ is elliptic, see . Hence by applying the maximum principle we conclude that either $\langle \nu, a\rangle$ is everywhere positive or $\langle \nu, a\rangle\equiv 0$. In the later case, since $a$ is constant tangent vector and therefore parallel, we can split out one line globally along $a$. Continue by induction, we prove that $M^n=N\times \mathbb{R}^{n-k}$ for some $2\leq k\leq n$, where $N$ does not contain any straight lines.
We note that on a codimension one shrinking GRS isometrically embedded in $\mathbb{R}^{n+1}$, we have the equation $$\sigma_2(h)=\frac{1}{2}R=\frac{1}{2}(f-|\nabla f|^2).$$ In the same way as did for the Ricci curvature in Theorem \[SplGSRSThm1\], we can deduce the constant rank theorem for the second fundamental form, and therefore the splitting theorem.
In the following we will show by contradiction argument that $N$ must be compact. With the splitting structure, let us assume $M(=N)$ is noncompact and there exists a vector $a$ such that $\langle\nu, a\rangle>0$. In this case, $M$ is essentially a graph along $-a$, the set $$\label{CompactSet}
\Omega_{r}=\{x \in M |\langle X(x),
-a\rangle\leq r \}$$ is compact for all $r>0$ and $\langle X(x),
-a\rangle$ is asymptotic to the geodesic distance of $M$, see [@Wu74; @CY77].
Combine Proposition \[PinTypIneSqu1\] and Proposition \[SquNorVecEquGSRSPro\], we have that for any compact region $\Omega \subset M$ and for any nonnegative $C^2$ function $u$ with $u\big|_{\partial \Omega}=0$ $$\label{FirEigNorEqu}
\begin{split}
\min_{\Omega}\Big(\sigma_1\sigma_2-3\sigma_3\Big)&= \min_\Omega \Big(\frac{-\square_{h,f} \langle\nu, a\rangle}{\langle\nu, a\rangle}\Big)\\
&\leq \Big(-\int_\Omega u\square_{h,f}ue^{-f}\Big)\Big(\int_\Omega u^2e^{-f}\Big)^{-1}\\
&=\Big(\int_\Omega\sigma_2^{ij}u_iu_je^{-f}\Big)\Big(\int_\Omega u^2e^{-f}\Big)^{-1}.\\
\end{split}$$
We apply to $u(x)= r
-\langle X(x), -a\rangle$ on $\Omega_{r}$ and get $$\label{FirEigPosVec}
\begin{split}
\Big(&\int_{\Omega_{r}}\sigma_2^{ij}u_iu_je^{-f}\Big)\Big(\int_{\Omega_{r}} u^2e^{-f}\Big)^{-1}\\
&=\Big(\int_{\Omega_{r}}(H\delta_{ij}-h_{ij})\langle e_i,a\rangle\langle e_j,a\rangle e^{-f}\Big)\Big(\int_{\Omega_{r}}(r-\langle X,-a\rangle)^2e^{-f}\Big)^{-1}\\
&\leq \Big(\int_{\Omega_{r}}He^{-f}\Big)\Big(\frac{r^2}{4}\int_{\Omega_{\frac{r}
{2}}}e^{-f}\Big)^{-1}\\
&= 4r^{-2}\Big(\int_{\Omega_{r}}He^{-f}\Big)\Big(\int_{\Omega_{\frac{r}{2}}}e^{-f}\Big)^{-1}.
\end{split}$$
Since $(M,g)$ is a shrinking GRS with positive Ricci curvature, combine the fact that the scalar curvature have a strictly lower bound with the scalar curvature growth estimate , and , implies that $$\label{meaIntWeiMea1}
H(x)\le C(n)(1+r(x))^4,$$ where we used the fact that $\langle X(x), -a \rangle$ is asymptotic to the intrinsic geodesic distance function $r(x)$ of $M$. In particular, the mean curvature is integrable with respect to the weighted measure $e^{-f}d\mu$, $$\label{MeaIntWeiMea1}
\int_M H e^{-f}d \mu<\infty.$$ Combine with , and , we have $$\label{NonComAloCyl1}
\min_{\Omega_{r}}\Big(\sigma_1\sigma_2-3\sigma_3\Big)\leq C(n)r^{-2}.$$ Let $r$ go to infinity, this implies that $$\label{NonComAsyCyl1}
\inf_{M}\Big(\sigma_1\sigma_2-3\sigma_3\Big)=0.$$ On the other hand, since $h$ is in the so-called G[å]{}rding cone $\Gamma_2$, by Newton-MacLaurin inequality we have $$\sigma_1\sigma_2-3\sigma_3\geq C(n)\sigma_2^{\frac{3}{2}}>\delta,$$ where the uniform positive lower bound of scalar curvature is ensured by . This contradicts with , and consequently, $M$ must be compact.
Since $M$ is compact hypersurface, then Ricci tensor has full rank at the elliptic point, i.e. there exist $x_0\in M$ such that ${\rm Ric}(x_0)>0$. With Theorem \[SplGSRSThm1\], we conclude that Ricci tensor has constant rank; therefore the Ricci curvature and also the curvature operator are positive everywhere. Finally, by Theorem \[GSRSPosCOCla\], the compact shrinking GRS with positive curvature operator has to be the round sphere or its metric quotient. Since $M$ is an Euclidean hypersurface, then it must be a round sphere. Now the proof of Theorem \[GSRSHypCyl\] is complete.
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[^1]: Research of P.G. is supported in part by NSERC Discovery Grant.
[^2]: This is the only place that we need nonnegative sectional curvature.
|
---
abstract: |
We construct the magnetic dual of QCD with one adjoint Weyl fermion. The dual is a consistent solution of the ’t Hooft anomaly matching conditions, allows for flavor decoupling and remarkably constitutes the first nonsupersymmetric dual valid for [*any*]{} number of colors. The dual allows to bound the anomalous dimension of the Dirac fermion mass operator to be less than one in the conformal window.\
[*Preprint: CP$^3$-Origins-2011-01*]{}
author:
- 'Matin [Mojaza]{}$^{\color{rossoCP3}{\varheartsuit}}$'
- 'Marco [Nardecchia]{}$^{\color{rossoCP3}{\varheartsuit}}$'
- 'Claudio [Pica]{}$^{\color{rossoCP3}{\varheartsuit}}$'
- 'Francesco [Sannino]{}$^{\color{rossoCP3}{\varheartsuit}}$'
title: |
\
Dual of QCD with One Adjoint Fermion
---
Introduction
============
One of the most fascinating possibilities is that generic asymptotically free gauge theories have magnetic duals. In fact, in the late nineties, in a series of ground breaking papers Seiberg [@Seiberg:1994bz; @Seiberg:1994pq] provided strong support for the existence of a consistent picture of such a duality within a supersymmetric framework. Supersymmetry is, however, quite special and the existence of such a duality does not automatically imply the existence of nonsupersymmetric duals. One of the most relevant results put forward by Seiberg has been the identification of the boundary of the conformal window for supersymmetric QCD as function of the number of flavors and colors. The dual theories proposed by Seiberg pass a set of mathematical consistency relations known as ’t Hooft anomaly matching conditions [@Hooft]. Another important tool has been the knowledge of the all-orders supersymmetric beta function [@Novikov:1983uc; @Shifman:1986zi; @Jones:1983ip]. Recently we provided several analytic predictions for the conformal window of nonsupersymmetric gauge theories using different approaches [@Sannino:2004qp; @Dietrich:2006cm; @Ryttov:2007cx; @Sannino:2009za; @Pica:2010mt; @Pica:2010xq; @Ryttov:2010iz; @Mojaza:2010cm; @Sannino:2009aw; @Ryttov:2009yw].
We initiated in [@Sannino:2009qc] the exploration of the possible existence of a QCD nonsupersymmetric gauge dual providing a consistent picture of the phase diagram as function of number of colors and flavors. Arguably the existence of a possible dual of a generic nonsupersymmetric asymptotically free gauge theory able to reproduce its infrared dynamics must match the ’t Hooft anomaly conditions [@Hooft]. We have exhibited several solutions of these conditions for QCD in [@Sannino:2009qc]. An earlier exploration already appeared in [@Terning:1997xy]. In [@Sannino:2009me] we have also analyzed theories with fermions transforming according to higher dimensional representations. Some of these theories have been used to construct sensible extensions of the standard model of particle interactions of technicolor type passing precision data and known as Minimal Walking Technicolor models [@Sannino:2004qp; @Dietrich:2006cm]. Other interesting studies of technicolor dynamics making use of higher dimensional representations appeared in [@Christensen:2005cb]. These are the known extension of technicolor type possessing the smallest intrinsic $S$ parameter [@Peskin:1990zt; @Peskin:1991sw; @Kennedy:1990ib; @Altarelli:1990zd] while being able to display simultaneously (near) conformal behavior before adding the backreaction due to traditional type extended technicolor interactions (ETC) [@Eichten:1979ah]. For recent analysis of the relevant properties of the $S$ parameter, using also duality arguments we refer to [@Sannino:2010ca; @Sannino:2010fh; @DiChiara:2010xb]. As for the issue of the effects of the ETC interactions on the technicolor dynamics and associated conformal window, it has recently been argued in [@Fukano:2010yv] that before adding the ETC interactions the technicolor theory, in isolation, should already be conformal for the combined model to be phenomenologically viable. A deeper understanding of the gauge dynamics of (near) conformal gauge theories is therefore needed making the study of gauge duals very relevant. For example, the [*magnetic*]{} dual allows to predict, in principle, the critical number of flavors below which the electric theory looses large distance conformality. It has also been show in [@Sannino:2010fh] that it is possible to use gauge dualities to compute the $S$ parameter in the nonperturbative regime of the electric theory.
Here we put the idea of nonsupersymmetric gauge duality on a much firmer ground by showing that for certain scalarless gauge theories with a spectrum similar to the one of QCD the gauge dual passes a large number of consistency checks.
The theory we choose is QCD with $N_f$ Dirac flavors and one adjoint Weyl fermion. A relevant feature of this theory is that it possesses the same global symmetry of super QCD despite the fact that squarks are absent. This means that there are four extra anomaly constraints not present in the case of ordinary QCD, moreover we will show that the dual can be constructed for any number of colors greater than two.
The magnetic dual is a new gauge theory featuring magnetic quarks and a Weyl adjoint fermion, new gauge singlet fermions which can be identified as states composite of the electric variables, as well as scalar states needed to mediate the interactions between the magnetic quarks and the gauge singlet fermions. The new scalars allow for a consistent flavor decoupling which was an important consistency check in the case of supersymmetry.
We will show that the dual allows to bound the anomalous dimension of the Dirac fermion mass operator to be less than one in the conformal window, and also estimate the critical number of flavors below which large distance conformality is lost in the electric variables.
The Electric Theory: QCD with one Adjoint Weyl Fermion
======================================================
The electric theory is constituted by a scalarless $SU(N)$ gauge theory with $N_f$ Dirac fermions and $N$ larger than two, as in QCD, but with an extra Weyl fermion transforming according to the adjoint representation of the gauge group. The quantum global symmetry is: $$SU_L(N_f) \times SU_R(N_f) \times U_V(1) \times U_{AF}(1)\ .$$ At the classical level there is one more $U_A(1)$ symmetry destroyed by quantum corrections due to the Adler-Bell-Jackiw anomaly. Therefore of the three independent $U(1)$ symmetries only two survive, a vector like $U_V(1)$ and an axial-like anomaly free (AF) one indicated with $U_{AF}(1)$. The spectrum of the theory and the global transformations are summarized in table \[Electric\].
$$\begin{array}{|c| c | c c c c | } \hline
{\rm Fields} & \left[ SU(N) \right] & SU_L(N_f) &SU_R(N_f) & U_V(1)&U_{AF}(1) \\ \hline \hline
\lambda &{\rm Adj} & 1 &1 &~~0& ~~1 \\
Q &{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}&{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}&1&~~1 & -\frac{N}{N_f} \\
\widetilde{Q} & \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}&1 & \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}& -1 & -\frac{N}{N_f} \\
G_{\mu}&{\rm Adj} &1&1 &~~0 & ~~0\\
\hline \end{array}$$
The global anomalies are:
$$\begin{aligned}
SU_{L/R}(N_f)^3 = \pm N \ , \quad
U_{V}(1) \, SU_{L/R}(N_f)^2 = \pm \frac{N}{2} \ , \quad
U_{AF}(1) \, SU_{L/R}(N_f)^2 = - \frac{N^2}{2N_f} \ ,\label{eq:An1}\\
U_{AF}(1)^3 = N^2 - 1 - 2\frac{N^4}{N_f^2} \ , \quad
U_{AF}(1)\, U_{V}(1)^2 = - 2N^2 \ , \quad
[{\rm Gravity}]^2U_{AF}(1) = -(N^2 + 1) \label{eq:An2} \end{aligned}$$
The first two anomalies are the same as in QCD and they are associated to the triangle diagrams featuring three $SU(N_f)$ generators (either all right or all left) at the vertices , or two $SU(N_f)$ generators (all right or all left) and one $U_V(1)$ charge.
Solutions of the ’t Hooft anomaly conditions for any number of colors
=====================================================================
We seek solutions of the anomaly matching conditions in the conformal window of the electric theory. This means that we consider a sufficiently large number of Dirac flavors so that the electric coupling constant freezes at large distances.
Following Seiberg we assume the dual to be a new $SU(X)$ gauge theory with global symmetry group $SU_L(N_f)\times SU_R(N_f) \times U_V(1) \times U_{AF}(1)$ featuring [*magnetic*]{} quarks ${q}$ and $\widetilde{q}$ transforming in the fundamental representation of $SU(X)$ together with a magnetic Weyl fermion $\lambda_{m}$ in the adjoint representation. We also introduce a minimal set of gauge singlet fermionic particles which can be viewed as composite states in terms of the electric variables but are considered elementary in the dual description. We limit to gauge singlet fermionic states which can be interpreted as made by three electric fields for any number of colors [^1]. The idea behind this choice is that composite states made by more fields could be constructed also in the dual theory from the new elementary fields.
We summarize the dual spectrum in table \[QCDAdual\]. The dual gauge group can be different from the electric one since the only physical quantities are gauge singlets. On the other hand the global symmetries must be the original ones[^2] given that can be physically probed.
We wish to find solutions to the ’t Hooft anomaly conditions valid for any number of colors.
$$\begin{array}{|c|c|c|c c c c|c|} \hline
{\rm Fields} & \text{Composite eq.} &\left[ SU(X) \right] & SU_L(N_f) &SU_R(N_f) & U_V(1)& U_{AF}(1) & \# ~{\rm of~copies} \\ \hline
\hline
\lambda_m & - & {\rm Adj} & 1 & 1 & 0 & z' & 1 \\
q & - &{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}&\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}&1&~~y & - \frac{X}{N_f} z &1 \\
\widetilde{q} & - & \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}&1 & {{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}& -y& - \frac{X}{N_f} z &1 \\
M & Q \lambda \widetilde{Q} & 1 & {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}& \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} & 0 & 1-2\frac{N}{N_f} & \ell_{M} \\
\widetilde{M} & \overline{Q \widetilde{Q}}\lambda & 1 & \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} &{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}& 0 & 1+2\frac{N}{N_f} & \ell_{\widetilde{M}} \\
\Lambda_S & \overline{Q \lambda} Q \, , \overline{ \widetilde{Q} \lambda} \widetilde{Q} \, , \overline{\lambda \lambda } \lambda & 1 & 1 & 1 & 0 & -1 & \ell_{\Lambda_S} \\
\Lambda_L & \overline{Q \lambda} Q & 1 & {\rm Adj} & 1 & 0 & -1 & \ell_{\Lambda_L} \\
\Lambda_R & \overline{\widetilde{Q} \lambda} \widetilde{Q} & 1 & 1 & {\rm Adj} & 0 & -1 & \ell_{\Lambda_R} \\
\Lambda_G & \lambda GG& 1 & 1 & 1 & 0 & 1 & \ell_{\Lambda_G}\\
\Lambda & \lambda \lambda \lambda & 1 & 1 & 1 & 0 & 3 & \ell_{\Lambda}\\
\hline \end{array}$$
$z^{\prime}$, $z$ and $y$ are arbitrary Abelian charges for the magnetic quarks and adjont fermion while the ones of the $SU(X)$ magnetic singlets are derived from the electric constituents. From table \[QCDAdual\] we write below all the relevant anomalies for the dual theory which we require to match the electric ones given in eqs. and $$\begin{aligned}
[SU(X)]^2 U_{AF}(1) = &X z' + \frac{1}{2}\left(- \frac{X}{N_f} z\right) N_f \times 2 = X (z'-z)
:= 0\\
SU_{L/R} (N_f)^3 = &\pm [ N_f ( \ell_{M} - \ell_{\widetilde{M}} ) -X]
:= \pm N\\
SU_{L/R} (N_f)^2 U_V(1) =& \pm \frac{1}{2} y X
:= \pm \frac{N}{2} \\
SU_{L/R} (N_f)^2 U_{AF}(1) = &\frac{1}{2} \left(- \frac{X}{N_f} z\right) X + \frac{1}{2} \ell_M N_f (1-2 \frac{N}{N_f}) + \frac{1}{2} \ell_{\widetilde{M}} N_f (1+2 \frac{N}{N_f}) - \ell_{\Lambda_{L/R}} N_f \nonumber \\
&:= - \frac{N^2}{2 N_f} \end{aligned}$$ $$\begin{aligned}
U_V(1)^2 U_{AF}(1) = &2 \times y^2 \left(- \frac{X}{N_f} z\right) X N_f
:= - 2 N^2\\
U_{AF}(1)^3 = &(X^2-1){z'}^{3} + X N_f\left(- \frac{X}{N_f} z\right)^3 \times 2 + \ell_M N_f^2 (1-2 \frac{N}{N_f})^3 + \ell_{\widetilde{M}} N_f^2 (1 + 2\frac{N}{N_f})^3 \nonumber \\
&+ \ell_{\Lambda_L} (N_f^2-1) (-1)^3 + \ell_{\Lambda_R} (N_f^2-1) (-1)^3 + \ell_{\Lambda_G} + (-1)^3 \ell_{\Lambda_S} + 3^3 \ell_\Lambda \nonumber \\
& := N^2 - 1 - 2 \frac{N^4}{N_f^2}\\
[{\rm Gravity}]^2 U_{AF}(1) = &(X^2-1)z' + X N_f\left(- \frac{X}{N_f} z\right) \times 2 + \ell_M N_f^2 (1-2 \frac{N}{N_f}) + \ell_{\widetilde{M}} N_f^2 (1 + 2\frac{N}{N_f}) \nonumber \\
&- \ell_{\Lambda_L} (N_f^2-1) - \ell_{\Lambda_R} (N_f^2-1)+ \ell_{\Lambda_G} - \ell_{\Lambda_S} + 3 \ell_\Lambda
:= -N^2 - 1 \ .\end{aligned}$$
The electric $U_{AF}(1)$, anomaly free condition, i.e. $[SU(X)]^2 U_{AF}(1) = 0$ provides the first nontrivial constraint, i.e. $z'=z$.
Consider now the ’t Hooft anomaly matching conditions from $SU_{L}(N_f)^2 U_V(1)$ and the $U_V(1)^2 U_{AF}(1)$: $$\frac{Xy}{2}=\frac{N}{2} \ , \qquad
X^2y^2z=N^2 \ .
$$ which can be simultaneously solved for $$z=1 \qquad {\rm and} \qquad y=N/X \ .$$ The difference between the $SU_{L}(N_f)^2 U_{AF}(1)$ and $SU_{R}(N_f)^2 U_{AF}(1)$ anomalies forces .
At this point we remain with 7 unknowns $\left(X,\ell_{\Lambda_M},\ell_{\Lambda_{\widetilde{M}}},\ell_{\Lambda_L},\ell_{\Lambda_S},\ell_{\Lambda_G},\ell_{\Lambda} \right)$ and 4 independent anomaly matching conditions. We can use the 4 equations to solve for $\ell_{\Lambda_M},\ell_{\Lambda_{\widetilde{M}}},\ell_{\Lambda_L}$ and $\ell_{\Lambda}$. Notice that the system is linear in these 4 unknowns. Obviously we expect that the solution is not unique and we can parametrize the set of all the solutions using combinations of the remaining three unknowns: $X,\ell_{\Lambda_S},\ell_{\Lambda_G}$. Before showing the result we already know that the solution of the system can depend only on $X$ and linearly on $z_D \equiv \ell_{\Lambda_G}-\ell_{\Lambda_S}$. The solution depends only on $z_D$ because $\Lambda_S$ and $\Lambda_{G}$ form vector like pairs. The sum $\ell_{\Lambda_G}+\ell_{\Lambda_S}$ is not constrained by the ’t Hooft anomaly matching conditions.
Here is the solution: $$\begin{aligned}
\ell_\Lambda &= \frac{ \left( N+X \right)\left( 3N-X \right) \left( N+ N_f+X \right) \left( X +N- N_f \right) -6 z_D N_f^{2} {N}^{2}} {6 {N_f}^{2} \left( 3{N}^{2}-2 \right) } \label{eq:lL}\\
\ell_{\Lambda_L} &= - \frac{ \left( N+X \right)\left( 3N-X \right) \left( N+ N_f+X \right) \left( X +N- N_f \right) -4 z_D N_f^{2}} {4 {N_f}^{2} \left( 3{N}^{2}-2 \right) } \label{eq:lLL}\\
\ell_{M} &= \frac{ \left( N+X \right)\left( 3 N^2 + 3 N N_f - 2 N X - 4 + X^2 - N_f X \right) \left( X +N + N_f \right) +4 z_D N_f^{2}} {4 {N_f}^{2} \left( 3{N}^{2}-2 \right) }\label{eq:lLM}\\
\ell_{\widetilde{M}} &= \frac{ \left( N+X \right)\left( 3 N^2 - 3 N N_f - 2 N X - 4 + X^2 + N_f X \right) \left( X +N - N_f \right) +4 z_D N_f^{2}} {4 {N_f}^{2} \left( 3{N}^{2}-2 \right) }
$$ The four equations are further constraint by being nonnegative integers. The problem is then reduced to finding combinations of $X$ and $z_D$ fulfilling this requirement.
In this way we have completely characterized *all* the possible solutions of the ’t Hooft anomaly matching in a dual theory with matter content presented in Table \[QCDAdual\]. We notice that one can reproduce a Seiberg-like solution for $\ell_M =1$ and the remaining indices set to zero.
As we shall show, the above equations can be rewritten in a more transparent way. The following linear combination of the first two equations and gives: $$\begin{aligned}
z_D = -3 \ell_\Lambda -2\ell_{\Lambda_L} \ .\label{eqzd}\end{aligned}$$ Therefore, imposing positivity of the multiplicities, we can only have solutions for which $z_D \leq 0$. Furthermore, we can also form another linear combination independent on $z_D$: $$\begin{aligned}
\ell_\Lambda + N^2\ell_{\Lambda_L} = \frac{ \left( N+X \right)\left( X-3N \right) \left(X+N+ N_f \right) \left( X +N- N_f \right)} {12 {N_f}^{2} } \ . \label{twolambda}\end{aligned}$$ It is more convenient to work with the sum and difference of $\ell_{M}$ and $\ell_{\widetilde{M}}$: $$\begin{aligned}
\ell_M-\ell_{\widetilde{M}}&=\frac{N+X}{N_f}\label{eq:3N} = d\\
\ell_M+\ell_{\widetilde{M}}&= d^2 + 2 \ell_{\Lambda_L} ,\end{aligned}$$ where we have defined $d$ in the first equation. $d$ must both be a positive integer implying $X \geq N_f - N$. The positivity condition of eq. further requires the inequality $X\geq 3N$ when $X >N_f - N$.
In summary, the original set of ’t Hooft anomaly matching conditions can be neatly rewritten as: $$\begin{aligned}
\ell_M-\ell_{\widetilde{M}}&=& d \label{eq:3N} \\
\ell_M+\ell_{\widetilde{M}}&=& d^2 + 2 \ell_{\Lambda_L} \\
\ell_\Lambda &=& \frac{(d-1) \ d \ (d+1) (d \ N_f - 4 N ) N_f}{12} - N^2 \ell_{\Lambda_L} \label{eq:f}\\
z_D &=& -3 \ell_\Lambda -2\ell_{\Lambda_L} \ .\end{aligned}$$ In this form, the constraint on the $\ell$’s being positive integers is always fulfilled and thus all solutions are given in terms of the *generic parameters* $d$ and $\ell_{\Lambda_L}$. One immediate consequence is that the dual gauge group $SU(X)$ is $SU(d \ N_f - N)$ with $d$ a positive integer. As we shall show in the Appendix \[involutionApp\] this structure of the gauge group admits an involution of the duality operation, meaning that if one dualizes again it is possible to recover the $SU(N)$ group. This, however, is only a necessary condition for the complete involution of the duality operation to hold exactly, meaning that one has still to demonstrate that, at the fixed point the dual of the dual is actually the electric theory.
The first, and most relevant solution, is obtained for $d=1$ and it mimics Seiberg’s solution for the dual of super QCD. We have, in fact, that: $$d = 1 \quad \Rightarrow \quad X = N_f - N \ , \quad \ell_{M} = 1 \ , \quad
\ell_\Lambda = \ell_{\Lambda_L} = \ell_{\widetilde{M}} = z_D = 0 \ .$$ No other solutions exist for $d=1$, i.e. $\ell_{\Lambda_L}$ vanishes. The dual gauge group is therefore $SU(N_f - N)$.
We now show that the minimal value of $-z_D = \ell_{\Lambda_S} - \ell_{\Lambda_G}= 2 \ell_{\Lambda_L} + 3 \ell_{\Lambda}$ different from zero is $5$. Solutions for which $-z_D = 2 \ , 3 \ , 4$ require either $\ell_\Lambda$ or $\ell_{\Lambda_L}$ to be zero, but one finds from , that: $$\begin{aligned}
\text{for} &\quad \ell_\Lambda = 0 \quad \Rightarrow \quad min(\ell_{\Lambda_L}) = 3 \\
\text{for} &\quad \ell_{\Lambda_L} = 0 \quad \Rightarrow \quad min(\ell_{\Lambda}) = 5\\
\text{thus} &\quad min(-z_D) = 5 \quad \text{with} \quad \ell_\Lambda = \ell_{\Lambda_L}=1.\end{aligned}$$ For any $d$ greater than one for which $\ell_{\Lambda_L}$ and $\ell_\Lambda$ assume values lower than the ones provided above there is no solution to the ’t Hooft anomaly matching conditions. This can be shown by first noting that by setting one of the two $\ell$s to zero in the minimal value of the other $\ell$ is always obtained by setting $d=2$. The second step is to set $\ell_\Lambda$ to zero and therefore obtain: $$\left( \frac{N_f}{N} - 2\right) \frac{N_f}{N} = \ell_{\Lambda_L} \ .$$ It is clear that the minimum $\ell_{\Lambda_L}$ must be three for this equation to hold. When setting instead $\ell_{\Lambda_L}=0$ we obtain: $$\ell_\Lambda = \left(N_f - 2 N \right)N_f \ .$$ This equation, given that the $\ell$s are positive integers, requires $N_f > 2N$ and therefore the minimum $N_f$ must be: $$N_f = 2 N+ 1 \ ,$$ which means: $$\ell_\Lambda= 2N +1 \ ,$$ and therefore $\ell_\Lambda > 5$. If both $\ell_\Lambda$ and $\ell_{\Lambda_L}$ are equal to zero one *either* recovers the $d=1$ or one can saturate the inequality $X\geq 3N$ and find the following solutions: $$\text{for} \quad d >1 \quad \text{and} \quad \ell_\Lambda = \ell_{\Lambda_L} = z_D =0 \quad \Rightarrow \quad
X = 3 N \ ,\quad
d = 4 \frac{N}{N_f} \ , \quad
\ell_M = \frac{d^2+d}{2} \ , \quad
\ell_{\widetilde{M}} = \frac{d^2 - d}{2}$$ These solutions are, however, unnatural since $X$ is $N_f$ independent and furthermore $d$ is by definition restricted to be an integer. A solution of this type implies that in the conformal window it is impossible to change the number of flavors in the electric theory and remain with an integer $d$ for the same number of colors assuming one had started with a number of flavors for which $d$ was an integer. Therefore, we do not consider these solutions to represent viable magnetic duals. Solutions with $z_D \leq -5$ are certainly non-minimal strongly indicating that the $d=1$ solution is the relevant one.
Requiring the electric theory to be asymptotically free imposes further general constraints. We start with recalling the first coefficient of the beta function for the electric theory which is: $$\begin{aligned}
\beta_0^e = 3 N - \frac{2}{3} N_f \ .\end{aligned}$$ For the theory to be asymptotically free, one imposes $\beta_0^e \geq 0$, leading to the constraint on $N_f$: $$\begin{aligned}
N_f \leq \frac{9}{2} N \ .\end{aligned}$$ For $\ell_\Lambda \geq 0$ from eq. we find the condition: $$\frac{(d-1) \ d \ (d+1) (d \ N_f - 4 N ) N_f}{12} \geq N^2 \ell_{\Lambda_L} \ ,$$ leading to the constraint, already noted earlier: $$\text{for } d\neq 1 \ , \quad d\geq 4 \frac{N}{N_f} \ .$$ We can set another constraint on $\alpha = {N_f}/{N}$ rewriting the inequality above as: $$\alpha^2 - \frac{4}{d}\alpha - \frac{12 \ell_{\Lambda_L}}{(d^2-1)d^2} \geq 0,$$ which leads to $$\alpha = \frac{N_f}{N} \geq 2 \frac{d^2 -1 + \sqrt{(d^2-1)(d^2 -1 + 3 \ell_{\Lambda_L})}}{(d^2-1)d} \ .$$ Combining the two constraints on $N_f/N$ we obtain: $$2 \frac{d^2 -1 + \sqrt{(d^2-1)(d^2 -1 + 3 \ell_{\Lambda_L})}}{(d^2-1)d} \leq \frac{N_f}{N} \leq \frac{9}{2}\quad
\text{with} \quad d \geq 4 \frac{N}{N_f} \ .$$
Expressing instead the bound on $\ell_{\Lambda_L}$, we see that it is bounded from above, hence constraining the number of possible solutions to be finite for a fixed value of $d$: $$\ell_{\Lambda_L} \leq \frac{1}{12} d \ (d^2-1)(d \frac{N_f}{N}-4) \frac{N_f}{N}.$$
The most natural solution is the one with the lowest value assumed by all indices which corresponds to $d=1$ and therefore has associated gauge group $SU(N_f - N)$. We will, therefore, concentrate on this theory below and summarize here the fermionic spectrum which resembles the supersymmetric version of the theory.
$$\begin{array}{|c|c|c c c c|} \hline
{\rm Fields} &\left[ SU(N_f-N) \right] & SU_L(N_f) &SU_R(N_f) & U_V(1)& U_{AF}(1) \\ \hline
\hline
\lambda_m & {\rm Adj} & 1 & 1 & 0 & 1 \\
q &{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}&\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}&1&~~\frac{N}{N_f - N} & - \frac{N_f - N }{N_f} \\
\widetilde{q}& \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}&1 & {{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}& -\frac{N}{N_f - N}& - \frac{N_f - N}{N_f} \\
M & 1 & {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}& \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} & 0 & 1-2\frac{N}{N_f} \\
\hline \end{array}$$
Decoupling of flavors
=====================
Following Seiberg a consistent dual description requires that decoupling of a flavor in the electric theory corresponds to decoupling of a flavor in the magnetic theory. The resulting magnetic theory is still the dual of the electric theory with one less flavor. For nonsupersymmetric theories one can still expect a similar phenomenon to occur, if no phase transition takes place as we increase the mass of the specific flavor we wish to decouple. In fact, for nonsupersymmetric theories, this idea is similar to require the [*mass persistent*]{} condition used by Preskill and Weinberg [@Preskill:1981sr] according to which one can still use the ’t Hooft anomaly matching conditions for theories with one extra massless flavor to constrain the solutions with one less flavor if, when giving mass to the extra flavor, no phase transition occurs. We now provide a time-honored example of the use of the [*mass persistent*]{} condition. For example if one starts with three flavors QCD the ’t Hooft anomaly conditions cannot be satisfied by massless baryonic states and therefore chiral symmetry must break. However for two flavors one can find a solution and therefore one cannot decide if chiral symmetry breaks unless the two-flavor case is embedded in the three flavor case. Using the three flavors anomaly conditions to infer that chiral symmetry must break for the two flavors case requires that as we take the [*strange*]{} quark mass large compared to the intrinsic scale of the theory no phase transition occurs apart from the explicit breaking of the flavor symmetry. A counter example is QCD at nonzero baryonic matter density for which such a phase transition is expected to occur and therefore two-flavors QCD can be realized without the breaking of the flavor symmetry [@Sannino:2000kg]. A more detailed analysis of the validity of the ’t Hooft anomaly conditions at nonzero matter density appeared in [@Hsu:2000by].
To investigate the decoupling of each flavor one needs to introduce bosonic degrees of freedom. These are not constrained by anomaly matching conditions but are kept massless by the requirement that the magnetic and electric theories must display large distance conformality. Interactions among the mesonic degrees of freedom and the fermions in the dual theory cannot be neglected in the regime when the dynamics is strong. For this, we need to add Yukawa terms in the dual Lagrangian. Here we investigate the case $d=1$ when decoupling a flavor in the electric theory. The diagram below shows how the non-abelian global and gauge symmetries of the electric and magnetic theories change upon decoupling of a Dirac flavor which is indicated by a down arrow for both theories: $$\begin{aligned}
\begin{array}{ccc}
{Electric} & dualizing & Magnetic \\[2mm]
[SU(N)] \times SU_L(N_f) \times SU_R (N_f) &\longrightarrow & [SU(N_f - N)] \times SU_L(N_f) \times SU_R(N_f) \\[2mm]
\Downarrow & & \Downarrow \\[2mm]
[SU(N)] \times SU_L(N_f-1)\times SU_R(N_f-1) & \longrightarrow & [SU(N_f-1-N)] \times SU_L(N_f-1)\times SU_R(N_f-1) \nonumber
\end{array}\end{aligned}$$ The abelian symmetries $U_V(1) \times U_{AF}(1)$ remain intact. It is clear from the diagram above that to ensures duality the decoupling of a flavor in the magnetic theory must also entail a breaking of the dual gauge symmetry. The dual theory is vector-like and therefore the Vafa-Witten theorem [@Vafa:1983tf] forbids the spontaneous breaking of the magnetic gauge group. We are, then, forced to introduce colored scalar fields that break the symmetry through a Higgs-mechanism.
*How do we introduce the correct scalar spectrum and associated Yukawa terms for the magnetic dual?*
We start by identifying the part of the magnetic spectrum which must acquire a mass term when adding an explicit mass term in the electric theory for the $N_f$-th flavor. We therefore consider the following group decomposition: $$\small
\begin{array}{| c | c | c l |} \hline
& \{ SU(X) \mid SU_L(N_f) , SU_R(N_f) \} & \{ SU(X-1) \mid SU_L(N_f-1) , SU_R(N_f-1) \} & \\ \hline
\hline
\lambda_m & \{ \text{Adj} \mid 1 , 1\} & \{ \text{Adj} \mid 1, 1\}\oplus \{ {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\mid 1, 1\} \oplus \{ \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \mid 1, 1\} \oplus \{ 1 \mid 1, 1\} & = \lambda_m \oplus \lambda'_{m,1} \oplus \lambda'_{m,2} \oplus \lambda_m^S\\[2mm]
q & \{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\mid \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}, 1 \} & \{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\mid \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}, 1 \} \oplus \{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\mid 1, 1 \} \oplus \{1 \mid \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}, 1 \} \oplus \{1 \mid 1, 1 \} & = q \oplus q'_1 \oplus q'_2 \oplus q^S \\[2mm]
\widetilde{q} & \{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \mid 1, {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\} & \{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \mid 1, {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\} \oplus \{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \mid 1, 1 \} \oplus \{1 \mid 1, {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\} \oplus \{1 \mid 1, 1 \} & = \widetilde{q} \oplus \widetilde{q}'_1 \oplus \widetilde{q}'_2 \oplus \widetilde{q}^S\\[2mm]
M & \{ 1 \mid {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}, \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \} & \{ 1 \mid {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}, \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \} \oplus \{1 \mid {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}, 1 \} \oplus \{1 \mid 1, \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \} \oplus \{1 \mid 1, 1 \} & = M \oplus M'_1 \oplus M'_2 \oplus M^S\\[1mm]
\hline
\end{array}$$ which clearly indicates that the flavor decoupling in the electric theory must lead to the generation of mass terms for the following states $$\lambda'_{m,1}, ~~\lambda'_{m,2},~~ q'_1, ~~q'_2,~~ \widetilde{q}'_1, ~~\widetilde{q}'_2, ~~M'_1,~~ M'_2,~~
\lambda_m^S, ~~q^S,~~ \widetilde{q}^S, ~~M^S\ ,$$ for the duality to be consistent. We will introduce mass terms through a Higg-mechanism via scalar fields acquiring the appropriate vacuum expectation values (vev) in order to induce the correct mass terms. To identify the scalar spectrum we consider Yukawa interactions such as $\phi M q$, with $\phi$ a new scalar field which should generate a mass term, upon $\phi$ condensation, of the type $q'_2 M'_1$ and $q^S M^S$. Note that the fermionic singlet states also receive mass terms via the same couplings. To provide a complete decoupling of all the non-singlet states the following unique set of quantum numbers for the complex scalars is needed: $$\begin{aligned}
\lambda_m \widetilde{q} &= \{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \mid 1, {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}, -y, \frac{N}{N_f} \} \quad \longrightarrow \phi = \{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\mid 1, \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}, y, -\frac{N}{N_f} \} \\
\lambda_m q &= \{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\mid \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}, 1, y, \frac{N}{N_f} \} \quad ~~~\longrightarrow \widetilde{\phi} = \{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \mid {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}, 1, -y , -\frac{N}{N_f} \} \\
M q & = \{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\mid 1, \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}, y, -\frac{N}{N_f} \} \quad \longrightarrow \phi_3 = \{ \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \mid 1, {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}, - y, \frac{N}{N_f} \} \sim \phi^*\\
M \widetilde{q} & = \{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}} \mid {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}, 1, -y, -\frac{N}{N_f} \} \quad \longrightarrow \phi_4 = \{ {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}\mid \overline{ {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}, 1 , y, \frac{N}{N_f} \} \sim \widetilde{\phi}^*\end{aligned}$$ Where on the left we showed the fermionic bilinear we wish to generate and on the right the needed corresponding complex scalar.
The similarity of the two last scalar fields with the conjugate of the first two fields is at the level of quantum numbers only. As a minimalistic approach, however, we choose them to be pairwise equivalent, and thus reduce the number of degrees of freedom. The Yukawa terms giving mass to the non-singlet states are thus: $$\mathcal{L}_Y = \widetilde{y}_\lambda \phi \lambda_m \widetilde{q} + y_\lambda \widetilde{\phi} \lambda_m q + y_M \phi^* M q + \widetilde{y}_M \widetilde{\phi}^* M \widetilde{q} + {\rm h.c.} \ .$$
These terms automatically give masses to the singlet states as well. However, each singlet receives contribution from two terms. We now must ensure that there will be no combination of singlet states with vanishing mass.
Denote the vev’s of the $\phi$’s with: $$\langle \phi \rangle = \phi_{X}^{\overline{N_f}} = v\\ \ , \qquad
\langle \widetilde{\phi} \rangle = \widetilde{\phi}^{X}_{N_f} = \widetilde{v} \ ,$$ with $X$ and $N_f$ the particular dual color and flavor indices along which we align the scalar condensates.
When the scalar fields assume these vev’s, the Yukawas reduce to: $$\begin{aligned}
\mathcal{L}_Y =& {\widetilde}{y}_\lambda v \left( \lambda'_{m,1}{\widetilde}{q}'_1 + \lambda_m^S {\widetilde}{q}^S \right)
+y_\lambda {\widetilde}{v} \left( \lambda'_{m,2}q_1 + \lambda_m^S q^S \right) +
y_M v^* \left( M'_1q_2 + M^S q^S \right) + {\widetilde}{y}_M {\widetilde}{v}^* \left( M'_2 {\widetilde}{q}'_2 + M^S {\widetilde}{q}^S \right) + {\rm h.c.} \ .\end{aligned}$$ Thus, the singlet mass-matrix is $$\begin{aligned}
\mathcal{M}_S =
\begin{array}{c} \lambda_m^S \\[1mm] q^S \\[1mm] {\widetilde}{q}^S \\[1mm] M^S \end{array}
\begin{pmatrix}
0 & y_\lambda {\widetilde}{v} & {\widetilde}{y}_\lambda v & 0 \\[1mm]
y_\lambda {\widetilde}{v} & 0 & 0 & y_M v^* \\[1mm]
{\widetilde}{y}_\lambda v & 0 & 0 & {\widetilde}{y}_M {\widetilde}{v}^* \\[1mm]
0 & y_M v^* & {\widetilde}{y}_M {\widetilde}{v}^* & 0
\end{pmatrix},\end{aligned}$$ with $$\det \mathcal{M}_S = \left(y_\lambda {\widetilde}{y}_M \mid {\widetilde}{v} \mid^2 - {\widetilde}{y}_\lambda y_M \mid v \mid^2 \right )^2.$$ All symmetries are correctly broken if and only if $$y_\lambda {\widetilde}{y}_M \mid {\widetilde}{v} \mid^2 \neq {\widetilde}{y}_\lambda y_M \mid v \mid^2 \ ,$$ and furthermore all the unwanted singlets correctly decouple.
We are still missing an essential ingredient, i.e. the possibility to communicate to the magnetic theory the introduction of a mass term in the electric theory for some of the flavors. The scalar vevs are expected to be induced by the electric quark masses. The solution to this problem is suggested by inspecting the electric mass term which reads: $${\rm Tr}\left[m Q\widetilde{Q}\right] + {\rm h.c.} \equiv {\rm Tr}\left[m {\Phi_{Q\widetilde{Q} }} \right] + {\rm h.c.} \ ,$$ with $m$ the explicit mass matrix. If we wish to decouple a single flavor then the mass matrix has the only entry $m \delta_{N_f}^{\bar{N}^f}$ with zero for the remaining ones. We have also introduced the complex scalar $\Phi_{Q\widetilde{Q} }$ with the following quantum numbers with respect to the electric gauge group: $$\Phi_{Q\widetilde{Q} }= \{1 \mid {\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}, \overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}, 0, -2\frac{N}{N_f} \} \sim Q\widetilde{Q} \ .$$ This state resembles the supersymmetric auxiliary complex scalar field of $M$ and should be identified with the standard QCD meson. Adding $\Phi_{Q\widetilde{Q} }$ as part of the dual spectrum allows us to introduce the following interaction in the dual theory: $${\cal L}_{\Phi_{Q\widetilde{Q} }} = \frac{y_{\Phi_{Q\widetilde{Q} }}}{\mu}\, \widetilde{\phi}^* \Phi_{Q\widetilde{Q} }{\phi}^* + {\rm h.c.} \ .
\label{inducedmass}$$ This operator is the nonsupersmmetric Lagrangian equivalent of Seiberg’s superpotential term. The ultraviolet scale $\mu$ serves to adjust the physical dimensions since $\phi$ and $\widetilde{\phi}$ are canonically normalized elementary magnetic fields while $\Phi_{Q\widetilde{Q} }$ has dimension three being an electric field made by $Q\widetilde{Q}$. It was shown in [@Sannino:2008pz; @Sannino:2008nv] that due to the presence of a nonzero mass term $\Phi_{Q\widetilde{Q} }$ acquires a vev in a theory with large distance conformality with the following explicit dependence on the mass and anomalous dimension $\gamma = - \partial \ln m / \partial \ln \Lambda$: $$\begin{aligned}
\langle {\Phi_{Q\widetilde{Q} }}^{\bar{N}_f}_{N_f} \rangle &\propto& -m \mu^2 \ , \qquad ~~~~~~~~~~~~~~~~~~~~~0 <\gamma < 1 \ , \label{BZm} \\
\langle {\Phi_{Q\widetilde{Q} }}^{\bar{N}_f}_{N_f} \rangle &\propto & -m \mu^2 \log \frac{\mu^2}{{|\langle {\Phi_{Q\widetilde{Q} }} \rangle|}^{2/3} }\ , ~~~~~~ \gamma \rightarrow 1 \ , \label{SDm} \\
\langle {\Phi_{Q\widetilde{Q} }}^{\bar{N}_f}_{N_f} \rangle &\propto & -m^{\frac{3-\gamma} {1+\gamma}}
\mu^{\frac{4\gamma} {1+\gamma}}\ , ~~~~~~~~~~~~~~~~~~~~~~1<\gamma \leq 2 \ .
\label{UBm}
\end{aligned}$$ An interesting reanalysis of the ultraviolet versus infrared dominated components of the vevs above which, however, does not modify these results can be found in [@DelDebbio:2010ze]. This shows that for any physically acceptable value of the anomalous dimension the relevant scalar degrees of freedom of the magnetic theory also acquire a mass term, decouple and are expected to develop a vev able to induce masses for the remaining states. In the magnetic theory the dual of $\Phi_{Q\widetilde{Q} }$ is denoted by $\Phi_{m}$. The operator , involving $\phi$, $\widetilde{\phi}$ and $\Phi_m$, should be at most a marginal one at the infrared fixed point for the theory to display large distance conformality. Therefore the physical mass dimension of $\Phi_{m}$ is two and $\Phi_{Q\widetilde{Q} } = \mu \Phi_{m}$. This implies the fundamental result that the anomalous dimension at the fixed point of the electric quark bilinear cannot exceed one!
Since the anomalous dimension does not exceed one at the interacting fixed point, $\Phi_m$ cannot be interpreted as an elementary field here. This is completely analogous to Seiberg’s case. This is so since $\Phi_m$ precisely maps into the auxiliary field of the mesonic chiral superfield in superQCD and therefore cannot propagate. To be able to integrate the $\Phi_m$ field out at the interacting fixed point one needs to add an associated quadratic term leading to the Lagrangian: $${\cal L}_{\Phi_{Q\widetilde{Q} }} + \frac{\Phi_{Q\widetilde{Q} }^{\ast}\Phi_{Q\widetilde{Q} }}{\mu^2} = \left( {y_{\Phi_{Q\widetilde{Q} }}}\, \widetilde{\phi}^* \Phi_{ m}{\phi}^* + {\rm h.c.} \right) + \Phi_m^{\ast}\Phi_m\ .
\label{njl}$$ Integrating out the auxiliary field $\Phi_m$ leads to the following quartic Lagrangian for the $\phi$ fields: $${\cal L}_{\phi^4} =
- y_{\Phi_{Q\widetilde{Q}}}^2 \,
{\widetilde{\phi}^{*r}}_{c1} \,
{\widetilde{\phi}}^{c2}_r \,
\phi_{c2}^{l} \,
{\phi^{* c1}}_l \ ,$$ with $c_i$ the dual color indices and $r$ and $l$ the right and left flavor indices respectively.
Adding an explicit quark mass in the electric theory corresponds to adding the following operator: $${\rm Tr} \left[ m \Phi_{Q\widetilde{Q}}\right] + {\rm h.c.} = \mu{\rm Tr} \left[ m \Phi_{m}\right] + {\rm h.c.}$$ in the magnetic theory which induces a vauum expectation value for the scalar field $\Phi_{Q\widetilde{Q}} = - m \mu^2$ in perfect agreement with the field theoretical result shown in .
We note that the second term in , in the electric variables, is the Nambu Jona-Lasinio [@Nambu:1961tp; @Nambu:1961fr] (NJL) four-fermion operator. In the dual variables it means that one can view the ordinary fermionic condensate, at the fixed point, as a composite state of two elementary magnetic scalars. The duality picture offers the first simple explanation of why the anomalous dimension of the fermion condensate does not exceed one at the boundary of the conformal window.
Finally we comment on the fact that requiring the electric and the magnetic theory to be both asymptotically free one deduces the following range of possible values of $N_f$: $$\frac{3}{2}N \leq N_f \leq \frac{9}{2} N \ .$$ It is natural to identify this range of values of $N_f$ with the actual extension of the conformal window
Conclusion
==========
We constructed possible magnetic duals of QCD with one adjoint Weyl fermion by classifying [*all*]{} the solutions of the ’t Hooft anomaly matching conditions of the type shown in the table \[QCDAdual\]. We assumed the number of flavors to be sufficiently large for the electric and magnetic theory to develop an infrared fixed point. The ’t Hooft anomaly conditions constrained the fermionic spectrum and led to a dual gauge group of the type $SU(dN_f - N)$ with $d$ a positive integer. We have shown that the case of $d=1$ leads to the minimal amount of fermionic matter needed to saturate the anomalies and moreover any other choice of $d$ does not allow to move in the flavor space without simultaneously change the number of colors of the electric theory. These results strongly suggest that $SU(N_f - N)$ constitute the obvious candidate for the dual gauge group.
Imposing consistent flavor decoupling allowed to determine the spectrum of the scalars and the Yukawa sector of the dual theory.
An important result is that we provided a consistent picture for the existence of the first nonsupesymmetric dual valid for [*any*]{} number of colors.
We have also shown that the anomalous dimension of the electric fermion mass operator can never exceed one at the infrared fixed point for the dual theory to be consistent. This is a remarkable result showing that one can obtain important non-perturbative bounds on the anomalous dimension of vector-like nonsupersymmetric gauge theories using duality arguments.
Our dual theory can be already tested with todays first principle lattice techniques. In fact, given that our theory resembles super QCD but without the fundamental scalars, in the electric theory, establishing the existence of duality in our model is the first step towards checking Seiberg’s duality on the lattice.
The Involution Theorem {#involutionApp}
======================
What happens if we dualize again the magnetic theory? The simplest possibility is that one recovers the electric theory. In fact one can imagine more general situations but for the time being we will assume this to be the case. The duality transformation is therefore a mathematical [*involution*]{}. This condition leads to an interesting and general theorem on the gauge structure of any dual gauge group.
Consider an electric $SU(N)$ gauge theory with $N_f$ flavors possessing a magnetic dual constituted of an $SU(X)$ gauge theory with also $N_f$ flavors. The same number of flavors insures that the global symmetries, encoding the physically relevant information of the theory, match in both theories.
The duality transformation from the electric to the magnetic theory is an [*involution*]{} if a second duality transformation acting on the magnetic theory gives back the original electric $SU(N)$ theory.
Denoting the duality transformation with an arrow, we summarize the involution as follows: $$\begin{aligned}
SU(N) \longrightarrow SU(X) \longrightarrow SU(N)\end{aligned}$$
We are now ready to enunciate the following theorem:
In order for the $SU(X)$ gauge group to respect [*involution*]{} we must have: $$X = P(N_f) - N \ ,$$ where $P(N_f)$ denotes any integer valued polynomial in $N_f$.
We start by noting that the involution property requires $X$ to be linear in $N$. This is so since, if the duality transformation requires the gauge group $X$ to depend on a generic power $p$ of $N$, i.e. $X \sim N^p$, then a second duality transformation on the magnetic gauge group leads to a second gauge group depending on $(N^{p})^p$. The involution condition requires $p$ to be one.
Furthermore, $X$ can still depend on any continuous and integer valued function of $N_f$ via the polynomial $P(N_f) = \sum_{p=0}^{n} \alpha_p N_f^p$.
Combining the two requirements we have that $X = P(N_f) - \beta N$, for some integer $\beta$. Imposing [*involution*]{} one last time we require: $$\begin{aligned}
N = P(N_f) - \beta X \rightarrow P(N_f) - \beta \left (P(N_f) - \beta N \right ) = \left (1-\beta \right ) P(N_f) + \beta^2 N.\end{aligned}$$ Only $\beta = 1$ solves this equation non-trivially (i.e. $\beta=-1$ and $P(N_f) = 0$ is the trivial solution), thus proofing the theorem.
For the present work, the theorem guarantees that *any* solution of the ’t Hoof anomaly matching conditions above respects the involution condition since $X = d \ N_f - N$.
Note that the [*involution theorem*]{} corresponds to the minimum requirement for the duality transformation to be an [*involution*]{}, even before taking into account the specific spectrum of the dual theory.
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[^1]: Note that for $N=3$ ordinary baryons are also made by three states, however since we will find consistent solutions of the ’t Hooft anomaly matching conditions for any $N$ we have not included them in the spectrum.
[^2]: There might be exceptions such as possible enhanced global symmetries but it is not the case here.
|
---
abstract: 'Differential equations have arithmetic analogues [@book] in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the “linear" ones. The equations themselves were introduced in a previous paper [@adel2]. In the present paper we deal with the solutions of these equations as well as with the Galois groups attached to the solutions.'
address: |
University of New Mexico\
Albuquerque, NM 87131
author:
- Alexandru Buium and Taylor Dupuy
title: |
Arithmetic differential equations on $GL_n$, III\
Galois groups
---
ø § i[\_1]{}
\[section\]
\[section\] \[theorem\][Corollary]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Definition]{} \[theorem\][Remark]{} \[theorem\][**Example**]{}
Introduction, main definitions, and main results
================================================
In a series of papers beginning with [@char] an arithmetic analogue of differential equations was introduced in which derivations are replaced by Fermat quotient operators. Cf. [@book] for an overview. It is then natural to ask for an arithmetic analogue of linear differential equations. Classically a linear differential equation has the form $$\label{classical}
\frac{d}{dz} U=A\cdot U$$ where $A$ is, say, a matrix of meromorphic functions on a domain in the complex plane ${\mathbb C}$ with complex variable $z$, and $U$ is an invertible matrix of unknown meromorphic functions (on a smaller domain). A basic object attached to \[classical\] is its differential Galois group which is an algebraic subgroup of $GL_n({\mathbb C})$. This concept is classical, going back to Picard and Vessiot. A modern version of the theory was developed by Kolchin [@kolchin] in the framework of differential algebra. In the present paper we ask for arithmetic analogues, in the spirit of [@char; @book], of all of these concepts. The beginnings of such a theory were sketched in [@adel2], where a concept of arithmetic linear differential equation on an algebraic group was introduced; the present paper deals with the solutions of these equations, and especially with the Galois groups attached to these solutions. Our paper is, in principle, a sequel to [@adel1; @adel2] but it is entirely independent of these papers. Indeed very little of the theory in [@char; @book] will be needed here and everything that will be needed will be reviewed in this Introduction. Our main purpose here will be to attach a Galois group to each given solution of a given linear arithmetic differential equation and to study some basic properties of this group; morally the Galois groups of such equations should (and in some sense will) appear as subgroups of “$GL_n({\mathbb F}_1^a)$" where ${\mathbb F}_1^a$ is the “algebraic closure of the field with one element"; cf. [@borgerf1] for this interpretation.
Main definitions
----------------
We denote by $R$ the unique complete discrete valuation ring with maximal ideal generated by an odd prime $p$ and with residue field $k=R/pR={\mathbb F}_p^a$, the algebraic closure of ${\mathbb F}_p$. So $R$ can be identified with the ring $W(k)$ of $p$-typical vectors on $k$. We denote by $\phi:R\ra R$ the unique ring homomorphism lifting the $p$-power Frobenius on the residue field $k$ and we denote by $\d:R\ra R$ the map $\d a=\frac{\phi(a)-a^p}{p}$. We morally view $\d$ as an arithmetic analogue of a derivation [@char; @book]. We denote by $R^{\d}$ the monoid of constants $\{\lambda\in R;\d \lambda=0\}$; so $R^{\d}$ consists of $0$ and all roots of unity in $R$. Recall that the reduction mod $p$ map $R^{\d}\ra k$ is an isomorphism of monoids. Also we denote by $K$ the fraction field of $R$. As usual we denote by ${\mathfrak g}{\mathfrak l}_n(A)$ the ring of $n\times n$ matrices with coefficients in a ring $A$ and we denote by $GL_n(A)$ the group of invertible elements of that ring. If $A=R$ we will often write $$G:=GL_n:=GL_n(R),\ \ {\mathfrak g}:={\mathfrak g}{\mathfrak l}_n:={\mathfrak g}{\mathfrak l}_n(R).$$ More generally for a smooth scheme $X$ over $R$ we will often write $X$ instead of $X(R)$. If $u=(u_{ij})\in {\mathfrak g}{\mathfrak l}_n(A)$ then we set $\phi(u)=(\phi(u_{ij}))$, $\d u=(\d u_{ij})$, $u^{(p)}=(u^p_{ij})$; hence $\phi(u)=u^{(p)}+p\d u$. In what follows we fix a matrix $\Delta(x)\in {\mathfrak g}{\mathfrak l}_n(A)$ with entries in the ring $A=\cO(GL_n)\h=R[x,\det(x)^{-1}]\h$ where $x$ is an $n\times n$ matrix of indeterminates and $\h$ means $p$-adic completion. (This matrix is usually canonically associated to the problem at hand and is uniquely determined by natural symmetry conditions that come with the problem; see [@adel2]. We will not be concerned with explaining these conditions here but rather we will concentrate on the abstract case when $\Delta$ is arbitrary or on specific Examples, cf. \[GLn\], \[SLn\], \[SO(q)\] below). Set $\Phi(x)=x^{(p)}+p\Delta(x)$. Moreover for $\alpha\in {\mathfrak g}{\mathfrak l}_n={\mathfrak g}{\mathfrak l}_n(R)$ set $$\Delta^{\alpha}(x)=\alpha \cdot \Phi(x)+\Delta(x)=\alpha x^{(p)}+(1+p\alpha)\Delta(x).$$ By a [*$\Delta$-linear equation*]{} we will then understand an equation of the form $$\label{typically}
\d u =\Delta^{\alpha}(u)$$ where $u\in G=GL_n=GL_n(R)$; $u$ is a referred to as a [*solution*]{} to the equation \[typical\] and the set $G^{\alpha}$ of all $u\in GL_n$ such that \[typically\] holds is referred to as the [*solution set*]{} of \[typically\]. If we set $\epsilon=1+p\alpha$ and $\Phi^{\alpha}(x)=\epsilon \cdot \Phi(x)$ then \[typically\] is equivalent to $$\label{dodoly}
\phi(u)=\Phi^{\alpha}(u).$$ This concept of linearity is always relative to a given $\Delta$. (If $\Delta$ has been fixed and is clear from the context $\Delta$-linear equations are also referred to as $\d$-linear equations [@adel2].) Note, by the way, that there is a natural concept of equivalence on ${\mathfrak g}{\mathfrak l}_n(A)$ which lies in the background of our discussion; two matrices $\Delta_1$ and $\Delta_2$ in ${\mathfrak g}{\mathfrak l}_n(A)$ are equivalent if and only if there exists $\alpha\in {\mathfrak g}{\mathfrak l}_n(R)$ such that $\Delta_1=\Delta_2^{\alpha}$. We have that $\d u=\Delta_1(u)$ is $\Delta_2$-linear if and only if $\Delta_1$ and $\Delta_2$ are equivalent.
A function ${\mathcal I}\in R[x,\det(x)^{-1}]\h$ will be called a [*prime integral*]{} for the $\Delta$-linear equation \[typically\] if for any solution $u$ of \[typically\] we have $$\d({\mathcal I}(u))=0.$$ (Intuitively ${\mathcal I}$ is “constant" along the solutions of \[typically\].) More generally an $m$-tuple of functions ${\mathcal I}\in (R[x,\det(x)^{-1}]\h)^m$ is called a prime integral of our equation if each of the components of ${\mathcal I}$ is a prime integral.
The basic examples we have in mind are those in [@adel2] and are going to be reviewed below; they are related to the classical groups and for their basic properties we refer to [@adel2]. For the purpose of the present article we will not need to review these properties.
\[GLn\] We say that $\Delta$ is of type $GL_n$ if $\Delta=0$. So in this case $\Phi(x)=x^{(p)}$ and \[typically\] and \[dodoly\] become $$\label{typical}
\d u =\alpha \cdot u^{(p)}$$ and $$\label{dodo}
\phi(u)=\epsilon\cdot u^{(p)}$$ respectively. It is worth noting that \[dodo\] is [*not*]{} an instance of a linear difference equation in the sense of [@SVdP]. Indeed a linear difference equation for $\phi$ has the form $$\label{difference}
\phi(u)=\epsilon \cdot u$$ rather than the form \[dodo\].
\[SLn\] We say that $\Delta$ is of type $SL_n$ if $$\Delta(x)=\frac{\lambda(x)-1}{p}\cdot x^{(p)}$$ where $p\not| n$ and $$\label{deflambda}
\lambda(x):=\left( \frac{\det(x^{(p)})}{\det(x)^p}\right)^{-1/n}.$$ Here the $-1/n$ power is computed using the usual series $(1+pt)^a\in \bZ_p[[t]]$ for $a\in \bZ_p$. In this case $\Phi(x)=\lambda(x) \cdot x^{(p)}$ and the equations \[typically\] and \[dodoly\] become $$\label{typical2}
\d u= \left(\lambda(u) \cdot \alpha+\frac{\lambda(u)-1}{p}\right) \cdot u^{(p)}$$ and $$\label{dodo2}
\phi(u)=\lambda(u)\cdot \epsilon \cdot u^{(p)}$$ respectively. Note that, in this case, $\Phi(u)\in SL_n$ for any $u\in SL_n$. In this context, following [@adel2], it is useful to introduce the $\d$-Lie algebra ${\mathfrak s}{\mathfrak l}_{n,\d}$ of $SL_n$ as being the set of all $\alpha\in {\mathfrak g}{\mathfrak l}_n$ such that $1+p\alpha\in SL_n$, in other words $${\mathfrak s}{\mathfrak l}_{n,\d}=\{\alpha\in {\mathfrak g}{\mathfrak l}_n; tr(\alpha)+...+p^{n-1}\det(\alpha)=0\}.$$ This is not a subgroup of $({\mathfrak g}{\mathfrak l}_n,+)$ where $+$ is the usual addition of matrices but rather a subgroup of $({\mathfrak g}{\mathfrak l}_n,+_{\d})$ where $a+_{\d}b:=a+b+pab$; the latter group is the group of $R$-points of a group in the category of $p$-adic formal schemes; cf. [@adel2].) This is in analogy with the Lie algebra ${\mathfrak s}{\mathfrak l}_n$ of $SL_n$ which is given by $${\mathfrak s}{\mathfrak l}_n=\{\alpha\in {\mathfrak g}{\mathfrak l}_n;tr(\alpha)=0\}.$$ Note also that if $\alpha\in {\mathfrak s}{\mathfrak l}_{n,\d}$ then ${\mathcal I}(x):=\det(x)$ is a prime integral for the $\Delta$-linear equation $\d u=\Delta^{\alpha}(u)$. Indeed if $u$ is a solution if this equation and $\epsilon=1+p\alpha$ then $$\begin{array}{rcl}
\phi(\det(u)) & = & \det(\phi(u))\\
\ & = & \det\left(\lambda(u)\cdot \epsilon \cdot u^{(p)} \right)\\
\ & = & \lambda(u)^n \cdot \det(\epsilon) \cdot \det(u^{(p)})\\
\ & = & \det(u)^p,
\end{array}$$ hence $\d(\det(u))=0$.
\[SO(q)\] Let $q\in GL_n$ be defined as $$\label{scorpion3}
\left(\begin{array}{cl} 0 & 1_r\\-1_r & 0\end{array}\right),\ \
\left(
\begin{array}{ll} 0 & 1_r\\1_r & 0\end{array}\right),\ \
\left( \begin{array}{lll} 1 & 0 & 0\\
0 & 0 & 1_r\\
0 & 1_r & 0\end{array}\right),\ $$ where $n=2r, 2r, 2r+1$ respectively. Let $SO(q)\subset SL_n$ be the identity component of the subgroup defined by the equations $x^t q x=q$; for $q$ as above $SO(q)$ is denoted by $Sp_{2r}, SO_{2r}, SO_{2r+1}$ respectively. We say that $\Delta$ is of type $SO(q)$ if $$\Delta(x)=x^{(p)}\cdot \frac{1}{p}(\Lambda(x)-1),$$ where $$\Lambda(x)=(((x^{(p)})^tqx^{(p)})^{-1} (x^tqx)^{(p)})^{1/2}.$$ Here, again, the $1/2$ power is computed using the usual series $(1+pT)^a\in {\mathfrak g}{\mathfrak l}_n(\bZ_p[[T]])$ for $a\in \bZ_p$, $T=(t_{ij})$. In this case we have $\Phi(x)=x^{(p)}\cdot \Lambda(x)$. Recall from [@adel2] that $\Phi(x)^tq\Phi(x)=(x^tqx)^{(p)}$. Note also that, in this case, $\Phi(u)\in SO(q)$ for any $u\in SO(q)$; cf. [@adel2]. In this context, following [@adel2], it is useful to introduce the $\d$-Lie algebra ${\mathfrak s}{\mathfrak o}(q)_{\d}$ of $SO(q)$ as being the set of all $\alpha\in {\mathfrak g}{\mathfrak l}_n$ such that $1+p\alpha\in SO(q)$, in other words $${\mathfrak s}{\mathfrak o}(q)_{\d}=\{\alpha\in {\mathfrak g}{\mathfrak l}_n; \alpha^t q+q\alpha+p\alpha^tq\alpha=0\}.$$ This is, again, a subgroup of $({\mathfrak g}{\mathfrak l}_n,+_{\d})$; and this is, again, in analogy with the Lie algebra ${\mathfrak s}{\mathfrak o}(q)$ of $SO(q)$ which is given by $${\mathfrak s}{\mathfrak o}(q)=\{\alpha\in {\mathfrak g}{\mathfrak l}_n;\alpha^t q+q\alpha=0\}.$$ Note also that if $\alpha\in {\mathfrak s}{\mathfrak o}(q)_{\d}$ then ${\mathcal I}(x):=x^tqx$ is a prime integral for the $\Delta$-linear equation $\d u=\Delta^{\alpha}(u)$. Indeed, if $u$ is a solution of this equation and $\epsilon=1+p\alpha$ then, using the identity $\Phi(x)^t q \Phi(x)=(x^t qx)^{(p)}$, we get $$\begin{array}{rcl}
\phi(u^tqu) & = & \phi(u)^t q \phi(u)\\
\ & = & \Phi(u)^t \epsilon^t q \epsilon \Phi(u)\\
\ & = & \Phi(u)^t q \Phi(u)\\
\ & = & (u^tqu)^{(p)},
\end{array}$$ which implies $\d(u^tqu)=0$.
Main results
------------
One has an existence and uniqueness result for our equations \[typically\]; cf. Propositions \[exu\], \[mama\], \[tata\], and Remark \[eggs\] in the body of the paper:
\[drink\] Let $u_0\in GL_n$ and $\alpha\in {\mathfrak g}{\mathfrak l}_n$ and let $\Delta$ be arbitrary. Then the following hold:
1\) There is a unique $u\in GL_n$ satisfying \[typically\] such that $u\equiv u_0$ mod $p$.
2\) If $\Delta$, $u_0$, and $\alpha$ have entries in a complete valuation subring $\cO$ of $R$ then $u$ also has entries in $\cO$.
3\) If $u_0\in SL_n$, $\alpha\in {\mathfrak s}{\mathfrak l}_{n,\d}$, and $\Delta$ is of type $SL_n$ then $u\in SL_n$.
4\) If $u_0\in SO(q)$, $\alpha\in {\mathfrak s}{\mathfrak o}(q)_{\d}$, and $\Delta$ is of type $SO(q)$ then $u\in SO(q)$.
5\) If $u_0$ and $\alpha$ have entries in a valuation $\d$-subring $\cO$ of $R$ with finite residue field and either $\Delta$ is of type $GL_n$ (i.e. $\Delta=0$) or $\Delta$ is of type $SL_n$ and $u\in SL_n$ then $u$ has entries in a $\d$-subring of $R$ which is generically finite over $\cO$.
Here by a $\d$-subring $\cO$ of $R$ we understand a subring with $\d \cO\subset \cO$. By a valuation subring of $R$ we mean the intersection of $R$ with a subfield of the field of fractions $K$ of $R$. Also an extension of integral domains is called generically finite if the induced extension between fraction fields is finite.
The above theorem allows us to introduce the first steps in a $\d$-Galois theory attached to $\Delta$-linear equations \[typical\]. In particular we will attach $\d$-Galois groups to such equations and prove results about their form in “generic" cases. Here are some details. Start with a $\d$-subring $\cO\subset R$, let $\alpha\in {\mathfrak g}{\mathfrak l}_n(\cO)$ and let $u\in GL_n(R)$ be a solution of \[typically\]. Consider the subring $\cO\{u\}$ of $R$ generated by $\cO$ and $u,\d u, \d^2 u,...$; so $\d \cO\subset \cO$. Consider the group $Aut_{\d}(\cO\{u\}/\cO)$ of all $\cO$-algebra automorphisms $\sigma$ of $\cO\{u\}$ such that $\sigma\circ \d=\d\circ \sigma$ on $\cO\{u\}$. Consider furthermore the subgroup $\tilde{G}_{u/\cO}$ of $Aut_{\d}(\cO\{u\}/\cO)$ consisting of all $\sigma\in Aut_{\d}(\cO\{u\}/\cO)$ such that $u^{-1}\cdot \sigma(u)\in GL_n(\cO)$. There is natural map, which is an injective group homomorphism, $$\label{torpedo}
\tilde{G}_{u/\cO}\ra GL_n(\cO)$$ sending any $\sigma$ into $c_{\sigma}:=u^{-1}\cdot \sigma(u)$. Finally define the $\d$-[*Galois group*]{} of $u/\cO$ as the image $G_{u/\cO}$ of \[torpedo\]. In particular $G_{u/\cO}\simeq \tilde{G}_{u/\cO}$. Our next task is to “compute/bound" $\d$-Galois groups. We begin with $\Delta$ of type $SL_n$ and $SO(q)$:
\[shesleeps\]
1\) Assume $\Delta$ is of type $SL_n$ and let $\alpha\in {\mathfrak g}{\mathfrak l}_n(\cO)$, $u\in G^{\alpha}$. Then for any $c\in G_{u/\cO}$ we have $\d(\det(c))=0$.
2\) Assume $\Delta$ is of type $SO(q)$ and let $\alpha\in {\mathfrak g}{\mathfrak l}_n(\cO)$, $u\in SO(q)\cap G^{\alpha}$. Then for any $c\in G_{u/\cO}$ we have $\d(c^t q c)=0$.
Cf. Propositions \[forget\] plus \[clock\].
Theorem \[shesleeps\] shows that if $\Delta$ is of type $SL_n$ or $SO(q)$ the $\d$-Galois group $G_{u/\cO}$ is “close to being contained" in $SL_n$ and $SO(q)$ respectively (provided $u$ is in these groups respectively). Indeed $G_{u/\cO}$ being contained in $SL_n$ (respectively $SO(q)$) means $\det(c)=1$ (respectively $\det(c)=1$ and $c^tqc=q$) for $c\in G_{u/\cO}$. Theorem \[shesleeps\], however, merely guarantees that $\d(\det(c))=0$ or $\d(c^tqc)=0$, which is a “slightly" weaker property.
Next we consider the case $\Delta$ is of type $GL_n$ (i.e. $\Delta=0$). To state our result below we let $W\subset G$ be the Weyl group of all matrices obtained from the identity matrix by permuting its columns. Let $T\subset G$ be the maximal torus of diagonal matrices with entries in $R$ and consider the normalizer $N=WT=TW$ of $T$ in $G$. We denote by $1\in G$ the identity matrix. Also consider the subset (not a subgroup!) $G^{\d}$ of $G$ consisting of all elements of $G$ with entries in the monoid of constants $R^{\d}$. Let $N^{\d}=N\cap G^{\d}$ and $T^{\d}=T\cap G^{\d}$. Then $N^{\d}$ and $T^{\d}$ are subgroups (not just subsets!) of $G$. Also $N^{\d}=WT^{\d}=T^{\d}W$. We also use below the notation $K^a$ for the algebraic closure of the fraction field $K$ of $R$; the Zariski closed sets of $GL_n(K^a)$ are then viewed as (possibly reducible) varieties over $K^a$. A subgroup of $GL_n(K^a)$ is called diagonalizable if it is conjugate in $GL_n(K^a)$ to a subgroup of the group of diagonal matrices. The next result illustrates some “generic" features of our $\d$-Galois groups in case $\Delta=0$; assertion 1) shows that the $\d$-Galois group is generically “not too large". Assertion 2) shows that the $\d$-Galois groups are generically “as large as possible". As we shall see presently, the meaning of the word [*generic*]{} is different in each of the $2$ situations: in situation 1) [*generic*]{} means [*outside a Zariski closed set*]{}; in situation 2) [*generic*]{} means [*outside a set of the first category*]{} (in the sense of Baire category).
\[food\] Assume $\Delta(x)=0$.
1\) There exists a Zariski closed subset $\Omega\subset GL_n(K^a)$ not containing $1$ such that for any $u\in GL_n(R) \backslash \Omega$ the following holds. Let $\alpha=\d u \cdot (u^{(p)})^{-1}$ and let $\cO$ be a valuation $\d$-subring of $R$ containing $\alpha$. Then $G_{u/\cO}$ contains a normal subgroup of finite index which is diagonalizable.
2\) There exists a subset $\Omega$ of the first category in the metric space $$X=\{u\in GL_n(R);u\equiv 1\ \ \text{mod}\ \ p\}$$ such that for any $u \in X\backslash \Omega$ the following holds. Let $\alpha=\d u\cdot (u^{(p)})^{-1}$. Then there exists a valuation $\d$-subring $\cO$ of $R$ containing $R^{\d}$ such that $\alpha\in {\mathfrak g}{\mathfrak l}_n(\cO)$ and such that $G_{u/\cO}=N^{\d}$.
Cf. Propositions \[dimless\], \[transcendence\], in the body of the paper.
The groups $W$ and $N^{\d}$ should be morally viewed as “incarnations" of “$GL_n({\mathbb F}_1)$" and “$GL_n({\mathbb F}_1^a)$" where “${\mathbb F}_1$" and “${\mathbb F}_1^a$" are the “field with element" and “its algebraic closure" respectively; cf. [@borgerf1]. This suggests that the $\d$-Galois theory we are proposing here should be viewed as a Galois theory over “${\mathbb F}_1$". By the way Theorem \[food\] suggests the following question: [*Is the $\d$-Galois group $G_{u/\cO}$ [*always*]{} a subgroup of $N$?*]{} The answer to this turns out to be negative in general (cf. Example \[counterexample\]) but something close to an affirmative answer may still be true.
We end with a couple of remarks comparing the theory above with some familiar situations.
\[expostuff\] It is worth comparing Equation \[dodo\] with the familiar linear equations in analysis in the case $n=1$; in case $n=1$ Equation \[dodo\] is, of course, $$\label{dododo}
\phi(u)=\epsilon \cdot u^p$$ where $\epsilon=1+p\alpha$, $\alpha\in R$, $u\in R^{\times}$. This equation can be solved as follows. Write $\epsilon=\exp(p\beta)$, where $\exp:pR\ra 1+pR$ is the group isomorphism given by the $p$-dic exponential and $\beta\in R$. Then the set of solutions to \[dododo\] consists of all $u\in R^{\times}$ of the form $$\label{research}
u=\zeta \cdot \exp\left( \sum_{n=1}^{\infty} p^n \phi^{-n}(\beta)\right)$$ where $\zeta\in R^{\times}$, $\d \zeta=0$. On the other hand consider the group homomorphism $\psi:R^{\times}\ra R$ defined by $$\label{psimap}
u\mapsto \psi(u)=\frac{1}{p}\log\left(\frac{\phi(u)}{u^p}\right)=\sum_{n=1}^{\infty}(-1)^{n-1}
\frac{p^{n-1}}{n}\left(\frac{\d u}{u^p}\right)^n$$ where $\log$ is the $p$-adic logarithm. Then Equation \[dododo\] is equivalent to the equation $$\label{teach}
\psi(u)=\beta$$ Now the homomorphism $\psi$ above should be viewed as an analogue of the logarithmic derivative map ${\mathcal M}(D)^{\times}\ra {\mathcal M}(D)$, $$u\mapsto u'/u,$$ where ${\mathcal M}(D)$ is the field of meromorphic functions on a disk $D\subset {\mathbb C}$, say, and $u'=\frac{du}{dz}$, where $z$ is a complex variable. So the analogue, in analysis, of Equation \[teach\] is the equation $$\label{dududu}
\frac{u'}{u}=\beta,$$ where $\beta \in {\mathcal M}(D)$. For $\beta$ holomorphic in $D$ the solutions to Equation \[dududu\] are of the form $$\label{junior}
u=c\cdot \exp\left(\int \beta dz\right)$$ where $\exp$ is the complex exponential and $c\in {\mathbb C}$. Hence the elements \[research\] in $R^{\times}$ should be viewed as arithmetic analogues of the functions \[junior\] in ${\mathcal M}(D)$.
It is worth comparing the $\Delta$-linear equations \[typical\] with Lang’s framework in [@lang]. Indeed in [@lang] Lang considers the map $$\label{characteristic}
GL_n(k)\ra GL_n(k),\ \ a\mapsto a^{(p)}\cdot a^{-1},$$ where $k$ is an algebraically closed field of characteristic $p$. This is a non-abelian cocycle for the adjoint action of $GL_n(k)$ on itself. A natural lift of \[characteristic\] to characteristic zero is the map $$\label{tv}
GL_n(R)\ra GL_n(R),\ \ a\mapsto \phi(a)\cdot a^{-1}.$$ The fiber of \[tv\] over $\alpha\in {\mathfrak g}{\mathfrak l}_n(R)$ consists of the solutions $u\in GL_n(R)$ to the [*linear difference equation*]{} \[difference\] which, as already noted, is quite different from the equation \[dodo\]. By the way the equation \[difference\] can be studied in at least two ways leading to two rather different theories: one way is from the viewpoint of difference algebra [@SVdP]; the other way is from the $\d$-arithmetic viewpoint [@book]. The $\d$-arithmetic viewpoint on equations \[difference\] tends to lead to profinite groups; our $\d$-arithmetic study of the equations \[dodo\] will lead to torsion groups (hence to inductive, rather than projective, limits of finite groups). This makes the $\d$-arithmetic study of equations \[dodo\] and the $\d$-arithmetic study of equations \[difference\] quite different in nature. Neverthless there are cases (such as that of abelian varieties [@char]) where one encounters combinations of profinite and torsion groups; so it is conceivable that the $\d$-arithmetic theories of \[dodo\] and \[difference\] can be unified.
On the other hand \[characteristic\] has another natural lift to characteristic zero which is $$\label{coveringg}
GL_n(R)\ra GL_n(R),\ \ a\mapsto a^{(p)}\cdot a^{-1}.$$ (This map is not induced by an endomorphism of the scheme $GL_n$ but rather by an endomorphism of the $p$-adic completion of $GL_n$.) Composing this with inversion $b\mapsto b^{-1}$ one gets a map $$\label{covering}
GL_n(R)\ra GL_n(R),\ \ a\mapsto a\cdot (a^{(p)})^{-1}.$$ Note now that the set of solutions to any of the equations \[dodo\] is a fiber of the map $$\label{coverin}
GL_n(R)\ra GL_n(R),\ \ a\mapsto \phi(a)\cdot (a^{(p)})^{-1}$$ But \[covering\] and \[coverin\] induce by restriction the same map $GL_n(\bZ_p)\ra GL_n(\bZ_p)$. This connection points towards a link between the arithmetic of usual coverings such as \[coveringg\] and the “$\d$-Galois theory" of $\Delta$-linear equations such as \[dodo\]. Also, in some sense, our paradigm here can be viewed as a lift to characteristic zero, in the framework of “$\d$-geometry", of Lang’s characteristic $p$ algebro-geometric paradigm.
The paper is organized as follows. In section 2 we provide the proof of Theorem \[drink\]. In section 3 we amplify our definitions and foundational discussion and we prove, in particular, Theorems \[shesleeps\] and \[food\].
Acknowledgement
---------------
The authors are indebted to P. Cartier for inspiring discussions. Also the first author would like to acknowledge partial support from the Hausdorff Institute of Mathematics in Bonn, from the NSF through grant DMS 0852591, from the Simons Foundation (award 311773), and from the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PN-II-ID-PCE-2012-4-0201.
Existence, uniqueness, and rationality of solutions
===================================================
The following proposition is an existence and uniqueness result for solutions of $\Delta$-linear equations. In the Propositions below $\Delta(x)$ is arbitrary unless otherwise stated and, as usual, $\Phi(x)=x^{(p)}+p\Delta(x)$.
\[exu\] Let $u_0\in GL_n(R)$, and $\alpha \in {\mathfrak g}{\mathfrak l}_n(R)$. Then the $\Delta$-linear equation $\d u=\alpha \cdot \Phi(u)+ \Delta(u)$ has a unique solution $u\in GL_n(R)$ such that $u\equiv u_0$ mod $p$.
[*Proof*]{}. Recall that the equation above is equivalent to $\phi(u)=\epsilon \cdot \Phi(u)$ where $\epsilon=1+p\alpha$. To check the uniqueness of the solution assume $\phi(u)=\epsilon \cdot \Phi(u)$ and $\phi(v)=\epsilon \cdot \Phi(v)$ with $u,v \in GL_n(R)$, $u\equiv v$ mod $p$. Then we prove by induction that $u\equiv v$ mod $p^n$. Indeed if the latter is the case then $u^{(p)}\equiv v^{(p)}$ mod $p^{n+1}$ and $\Delta(u)\equiv \Delta(v)$ mod $p^{n+1}$ hence $\Phi(u)\equiv \Phi(v)$ mod $p^{n+1}$. Hence $\phi(u)\equiv \phi(v)$ mod $p^{n+1}$. Hence $u\equiv v$ mod $p^{n+1}$.
To check the existence of a solution $u$ such that $u\equiv u_0$ mod $p$ we define a sequence of matrices $u_n\in GL_n(R)$ by the formula $$u_{n+1}=\phi^{-1}(\epsilon \cdot \Phi(u_n)),\ \ n\geq 0.$$ We claim that for all $n\geq 0$ we have $$\phi(u_n)\equiv \epsilon \cdot \Phi(u_n)\ \ \ \text{mod}\ \ p^{n+1}.$$ Assuming the claim we get $u_{n+1}\equiv u_n$ mod $p^{n+1}$ hence $u_n$ converges $p$-adically to some $u\in GL_n(R)$. Also $\phi(u)=\epsilon \cdot \Phi(u)$ which ends our proof. We are left with checking the claim. We proceed by induction. The case $n=0$ is clear. Assume now $\phi(u_n) \equiv \epsilon \cdot \Phi(u_n)$ mod $p^{n+1}$. Hence $$\phi^{-1}(\epsilon \cdot \Phi(u_n))\equiv u_n\ \ mod\ \ p^{n+1},$$ hence $$\Phi(\phi^{-1}(\epsilon \cdot \Phi(u_n)))\equiv \Phi(u_n)\ \ mod\ \ p^{n+2}.$$ Hence $$\begin{array}{rcl}
\epsilon\cdot \Phi(u_{n+1}) & = & \epsilon\cdot \Phi(\phi^{-1}(\phi(u_{n+1})))\\
\ & \ & \ \\
\ & = & \epsilon \cdot \Phi(\phi^{-1}(\epsilon\cdot \Phi(u_n)))\\
\ & \ & \ \\
\ & \equiv & \epsilon \cdot \Phi(u_n)\ \ \ \text{mod}\ \ \ p^{n+2}\\
\ & \ & \ \\
\ & = & \phi(u_{n+1}),\end{array}$$ and the induction step follows.
If, in Proposition \[exu\], $\Delta=0$, $n=1$, and $u_0\equiv \zeta$ mod $p$ where $\zeta\in R$ is a root of unity, the solution $u$ has a closed form: $$u=\zeta \cdot \epsilon_{-1} \cdot \epsilon_{-2}^p\cdot \epsilon_{-3}^{p^2}\cdot ...\ \ \text{(convergent product)}$$ where $\epsilon_i=\phi^i(\epsilon)$ for $i\in \bZ$. This computation implies the formula in Remark \[expostuff\].
\[eggs\] If in Proposition \[exu\] we have $\Delta$ of type $SL_n$, $u_0\in SL_n(R)$, and $\alpha\in {\mathfrak s}{\mathfrak l}_{n,\d}$ then $u\in SL_n(R)$. Indeed this follows because $\Phi(a)\in SL_n(R)$ and $\phi^{-1}(a)\in SL_n(R)$ for all $a\in SL_n(R)$; hence if $u_n$ is as in the proof of that Proposition then $u_n\in SL_n(R)$. Similarly if $\Delta$ is of type $SO(q)$, $u_0\in SO(q)$, and $\alpha\in {\mathfrak s}{\mathfrak o}(q)_{\d}$ then $u\in SO(q)$. The above proves assertions 3 and 4 in Theorem \[drink\]. Alternatively these assertions can be deduced as follows. Let ${\mathcal I}(x)$ be $\det(x)$ or $x^tqx$ respectively and let $u$ be such that $\d u=\Delta^{\alpha}(u)$ with $u\equiv u_0$ mod $p$ with either $u_0\in SL_n$, $\alpha\in {\mathfrak s}{\mathfrak l}_{n,\d}$ or $u_0\in SO(q)$, $\alpha\in {\mathfrak s}{\mathfrak o}(q)_{\d}$ respectively. By the discussion in Examples \[SLn\] and \[SO(q)\] we have $\d({\mathcal I}(u))=0$ hence ${\mathcal I}(u)$ is either $0$ or a root of unity in $R$. On the other hand we have ${\mathcal I}(u)\equiv {\mathcal I}(u_0)$ mod $p$ hence, since ${\mathcal I}(u_0)=0$, we have ${\mathcal I}(u)\equiv 0$ mod $p$. Since ${\mathcal I}(u)$ is either $0$ or a root of unity we conclude it must be $0$, hence $u\in SL_n$ or $u\in SO(q)$ respectively.
In notation of Propositon \[exu\] the natural reduction map $G^{\alpha}\ra GL_n(k)$ is a bijection. So each solution set $G^{\alpha}$ has a natural structure of group; but of course with this structure $G^{\alpha}$ is not a subgroup of $GL_n(R)$.
Let us address the question of “rationality" of solutions of $\Delta$-linear equations.
Let $\cO\subset R$ be a subring. Recall that $\cO$ is called a $\d$-subring if $\d\cO\subset \cO$. Also we say $\cO$ is a a valuation subring of $R$ if $\cO$ is the intersection of $R$ with a subfield of $K$. Any valuation subring of $R$ is a discrete valuation ring with maximal ideal generated by $p$. Note that if $\cO$ is a valuation subring which is complete then either $\cO=R$ or there exists $\nu\geq 1$ such that $\cO=R^{\phi^{\nu}}$, the fixed ring of $\phi^{\nu}$; in particular such an $\cO$ is automatically a $\d$-subring. An extension $\cO\subset \cO'$ of subrings of $R$ will be called generically finite if the extension of their fraction fields is finite; if in addition $\cO$ is a valuation subring then $\cO'$ is a localization of a finite extension of $\cO$; if, in addition $\cO$ is complete then any generically finite extension of $\cO$ in $R$ is finite.
\[mama\] Assume $\cO$ is a complete valuation subring of $R$ (hence also a $\d$-subring). If in Proposition \[exu\] we have $$\Delta\in {\mathfrak g}{\mathfrak l}_n(\cO[x,\det(x)]\h),\ \ u_0\in GL_n(\cO),\ \ \alpha\in {\mathfrak g}{\mathfrak l}_n(\cO)$$ then $u\in GL_n(\cO)$.
[*Proof*]{}. Let $\cO=R^{\phi^{\nu}}$. Then $\phi^{\nu}(u_0)=u_0$ and $\phi^{\nu}(\alpha)=\alpha$ hence $\phi^{\nu}(\epsilon)=\epsilon$, where $\epsilon=1+p\alpha$. Also $\phi^{\nu}(\Delta(a))=\Delta(\phi^{\nu}(a))$, and hence $\phi^{\nu}(\Phi(a))=\Phi(\phi^{\nu}(a))$, for all $a\in GL_n(R)$. Since $\phi(u)=\epsilon \cdot \Phi(u)$ and $u\equiv u_0$ mod $p$ it follows that $$\phi^{\nu+1}(u)=\phi^{\nu}(\epsilon) (\phi^{\nu}(\Phi(u)))=\epsilon \cdot \Phi( (\phi^{\nu}(u)))$$ and $\phi^{\nu}(u)\equiv \phi^{\nu}(u_0)\equiv u_0$ mod $p$. By the uniqueness in Proposition \[exu\] it follows that $\phi^{\nu}(u)=u$ hence $u\in GL_n(\cO)$.
\[tata\] Assume $\cO$ is a valuation $\d$-subring of $R$ with finite residue field. Assume in Proposition \[exu\] that one of the following holds:
1\) $\Delta$ is of type $GL_n$ (i.e. $\Delta=0$), $u_0\in GL_n(\cO)$, and $\alpha\in {\mathfrak g}{\mathfrak l}_n(\cO)$.
2\) $\Delta$ is of type $SL_n$, $u_0\in SL_n(\cO)$, and $\alpha\in {\mathfrak s}{\mathfrak l}_{n,\d}\cap {\mathfrak g}{\mathfrak l}_n(\cO)$.
Then there exists a generically finite extension of $\d$-subrings $\cO\subset \cO'$ of $R$ such that $u\in GL_n(\cO')$.
[*Proof*]{}. Assume we are in case 2; case 1 is similar (and indeed slightly easier).
By Proposition \[mama\] if $\widehat{\cO}$ is the completion of $\cO$ then $u\in GL_n(\widehat{\cO})$ hence there exists $\nu\geq 0$ such that $\phi^{\nu+1}(u)=u$. Let $N=n^2$ and identify the points of ${\mathbb A}^N$ with $n\times n$ matrices. Let $$\lambda_{\nu}(u)=\phi^{\nu}(\lambda(u))\cdot \phi^{\nu-1}(\lambda(u))^p\cdot ...\cdot
\lambda(u)^{p^{\nu}}.$$ Using $\phi(u)=\lambda(u) \cdot \epsilon \cdot u^{(p)}$, and setting $\epsilon_j=\phi^j(\epsilon)$, we get $$\label{marga}
u=\phi^{\nu+1}(u)=\lambda_{\nu}(u)\cdot \varphi(u),$$ where $\varphi:{\mathbb A}^{N}\ra {\mathbb A}^{N}$ is the morphism of schemes over $\cO$ defined on points by $$\varphi(v)=\epsilon_{\nu}(\epsilon_{\nu-1}(\epsilon_{\nu-2}(...(\epsilon v^{(p)})^{(p)})^{(p)}...)^{(p)}.$$ Let $K^a$ be an algebraic closure of $K$, let $F$ be the fraction field of $\cO$, and let $F^a$ be the algebraic closure of $F$ in $K^a$. Note that $\varphi:{\mathbb A}^N(K^a)\ra {\mathbb A}^N(K^a)$ is obtained by composing maps $\eta\mapsto \epsilon_j \eta$ with copies of the map $\eta\mapsto \eta^{(p)}$; both these types of maps are given by homogeneous polynomials (of degree $1$ and $p$ respectively) and have the property that the pre-image of $0$ is $0$. Hence $\varphi$ is given by $$\varphi(\eta)=(\Phi_1(\eta),...,\Phi_N(\eta))$$ where $\Phi_1,...,\Phi_N\in F[x_1,...,x_N]$ are homogeneous polynomials of degree $p^{\nu+1}>1$ and $\varphi^{-1}(0)=\{0\}$; hence $\Phi_1,...,\Phi_N$ have no common zero in ${\mathbb A}^N(K^a)$ except at the origin. Consider an extra variable $x_0$ and consider the projective variety $V\subset {\mathbb P}^{N}$ defined by the equations $$\label{cheese}
\Phi_j(x_1,...,x_N)-x_0^{p^{\nu+1}-1}x_j=0.$$ Clearly the intersection of $V$ with the hyperplane $x_0=0$ is empty. So $V$ has dimension zero hence $V(K^a)$ is finite. Since $V$ is defined over $F$ we have $V(K^a)=V(F^a)$. By equation \[marga\] the point $$(\lambda_{\nu}(u)^{-1/(p^{\nu+1}-1)}:u)\in {\mathbb P}^{N}(K)$$ belongs to $V(K)$ hence it belongs to $V(F^a)$. (Here the $1/(p^{\nu+1}-1)$-power is computed, again, using the series $(1+pt)^a\in \bZ_p[[t]]$ for $a\in \bZ_p^{\times}$). It follows that $$\label{ham}
\lambda_{\nu}(u)^{1/(p^{\nu+1}-1)}\cdot u\in {\mathbb A}^{N}(F^a)={\mathfrak g}{\mathfrak l}_n(F^a)$$ hence $$\det(\lambda_{\nu}(u)^{1/(p^{\nu+1}-1)}\cdot u)\in F^a.$$ Since, by Remark \[eggs\], $\det(u)=1$ we get $(\lambda_{\nu}(u)^{1/(p^{\nu+1}-1)})^n\in F^a$ hence $$\lambda_{\nu}(u)^{1/(p^{\nu+1}-1)}\in F^a.$$ By \[ham\] again we get $u\in GL_n(F^a)$ which ends the proof.
Note that the Propositions in this section imply Theorem \[drink\] in the Introduction. The consideration of the variety cut out by equations \[cheese\] is a trick from [@FS] and is an indication of an interesting link between the paradigm of the present paper and the arithmetic of dynamical systems on projective space.
$\d$-Galois groups
==================
Recall that $\d$-Galois groups were defined in the Introduction. We will review here the notation involved and some related concepts. Then we will prove a series of Propositions amounting to Theorem \[food\].
As usual we often denote by $G$ the group $GL_n(R)$ and by ${\mathfrak g}{\mathfrak l}_n$ the Lie algebra ${\mathfrak g}{\mathfrak l}_n(R)$. Let $\Delta(x)\in {\mathfrak g}{\mathfrak l}_n(R[x,\det(x)^{-1}]\h)$, $x$ an $n\times n$ matrix of indeterminates, and let $\Phi(x)=x^{(p)}+p\Delta(x)$. Let $\alpha \in {\mathfrak g}{\mathfrak l}_n$, $\Delta^{\alpha}(x)=\alpha\cdot \Phi(x)+\Delta(x)$, and consider the $\Delta$-linear equation $$\label{tree1}
\d u=\Delta^{\alpha}(u).$$ Recall that if $\Phi^{\alpha}(x)=\epsilon\cdot \Phi(x)$, $\epsilon=1+p\alpha$, then this equation is equivalent to the equation $$\label{tree2}
\phi(u)=\Phi^{\alpha}(u).$$ Let $G^{\alpha}$ be the set of solutions to Equation \[tree1\], let $u\in G^{\alpha}$ be a fixed solution, let $\Phi_u(x)=\Phi(u)^{-1}\Phi(ux)$, $\Delta_u(x)=\frac{1}{p}(\Phi_u(v)-v^{(p)})$, and let $G_u$ be the set of solutions $v\in G$ to the $\Delta_u$-linear equation $$\label{tree3}
\d v=\Delta_u(v),$$ equivalently to the equation $$\label{cry}
\phi(v)=\Phi_u(v).$$ Note that $$uG_u\subset G^{\alpha}.$$ Indeed if $c\in G_u$ we have $$\phi(uc)=\phi(u)\cdot \phi(c)=\epsilon \cdot \Phi(u)\cdot \phi(c)=\epsilon\cdot \Phi(uc)$$ so $uc\in G^{\alpha}$.
Let now $\cO$ be a $\d$-subring of $R$. Assume $\alpha\in {\mathfrak g}{\mathfrak l}_n(\cO)$ and let $u\in GL_n(R)$ be a solution of Equation \[tree1\]. Recall from the Introduction the group $Aut_{\d}(\cO\{u\}/\cO)$ of all $\cO$-algebra automorphisms $\sigma$ of $\cO\{u\}$ such that $\sigma\circ \d=\d\circ \sigma$ on $\cO\{u\}$, its subgroup $\tilde{G}_{u/\cO}$ and the injective group homomorphism $\tilde{G}_{u/\cO}\ra GL_n(\cO)$ sending any $\sigma$ into $c_{\sigma}:=u^{-1}\cdot \sigma(u)$. Then the $\d$-Galois group $G_{u/\cO}$ was defined as the image $G_{u/\cO}$ of the homomorphism $\tilde{G}_{u/\cO}\ra GL_n(\cO)$.
In the special cases of interest to us the $\d$-Galois group has a “$\d$-theoretic description/bound" which we now discuss. Let $x',x'',...$ be new matrices of indeterminates and consider the polynomial ring $\cO\{x\}:=\cO[x,x',x'',...]$. There is a unique ring endomorphism $\phi$ of $\cO\{x\}$ whose restriction to $\cO$ is $\phi$ and such that $\phi(x)=x^{(p)}+px'$, $\phi(x')=(x')^{(p)}+px''$, etc. Define the map $\d:\cO\{x\}\ra \cO\{x\}$ by $\d f=p^{-1}(\phi(f)-f^p)$. We let $I_{u/\cO}$ be the kernel of the unique $\cO$-algebra map $\cO\{x\}\ra R$, sending $x\mapsto u$, $x'\ra \d u$, $x''\mapsto \d^2 u$, etc. (the ideal of $\d$-algebraic relations among the entries of $u$); note that $\cO\{u\}$ is then the image of the map $\cO\{x\}\ra R$ above. We let $\Sigma_{u/\cO}$ be the subgroup of $GL_n(\cO)$ consisting of all matrices $c$ such that the $\cO$-automorphism $\sigma_c:\cO\{x\}\ra \cO\{x\}$ defined by $\sigma_c(x)=xc$, $\sigma(x')=\d(xc)$, $\sigma(x'')=\d^2(xc)$, etc. satisfies $\sigma_c(I_{u/\cO})=I_{u/\cO}$. Similarly let $I^0_{u/\cO}$ be the kernel of the map $\cO[x]\ra \cO[u]$, $x\mapsto u$, and let $\Sigma^0_{u/\cO}$ be the subgroup of $GL_n(\cO)$ consisting of all matrices $c$ such that the $\cO$-automorphism $\sigma^0_c:\cO[x]\ra \cO[x]$ defined by $\sigma_c(x)=xc$, satisfies $\sigma^0_c(I^0_{u/\cO})=I^0_{u/\cO}$.
Here is the “$\d$-theoretic description/bound" of the $\d$-Galois group in our cases of interest:
\[forget\]
1\) $G_{u/\cO}=\Sigma_{u/\cO}$.
2\) If $\Delta(x)$ is of type $GL_n$, $SL_n$ or $SO(q)$, we have $G_{u/\cO}\subset G_u$.
3\) If $\Delta(x)$ is of type $GL_n$ we have $G_{u/\cO}=\Sigma^0_{u/\cO}\cap G_u$.
[*Proof*]{}. To check assertion 1 let, first, $c\in G_{u/\cO}$, so there exists $\sigma\in \tilde{G}_{u/\cO}$ with $\sigma u=c u$. Then we claim that $c\in \Sigma_{u/\cO}$. Indeed this follows from the commutativity of the diagram $$\label{redd}\begin{array}{ccc}
\cO\{x\} & \stackrel{\sigma_c}{\ra} & \cO\{x\}\\
\downarrow & \ & \downarrow\\
\cO\{u\} & \stackrel{\sigma}{\ra} & \cO\{u\}
\end{array}$$ Conversely, if $c\in \Sigma_{u/\cO}$ then $\sigma_c:\cO\{x\}\ra \cO\{x\}$ obviously induces an automorphism $\sigma:\cO\{u\}\ra \cO\{u\}$ commuting with $\d$ and sending $u$ into $uc$ so $c\in G_{u/\cO}$.
To check assertions 2 and 3 we need a preliminary discussion in which we assume that $\Delta(x)$ is of type $GL_n$, $SL_n$, or $SO(q)$.
Let us start with an $\cO$-algebra automorphism $\sigma$ of $\cO\{u\}$ such that $c:=u^{-1}\cdot \sigma(u)\in GL_n(\cO)$ and let $\epsilon=1+p\alpha$. We may (uniquely) extend $\sigma$ to an automorphism of $S^{-1}\cO\{u\}$ where $S$ is the multiplicative system consisting of all elements of $\cO\{u\}$ of the form $\det(u)^m+pf$ where $m\in \bZ_{\geq 0}$, $f\in \cO\{u\}$. We claim that $$\label{freeze}
\sigma(\Phi(u))=\Phi(\sigma(u)).$$ (The left hand side makes sense because $\Phi(u)=\epsilon^{-1}\cdot \phi(u)$ has entries in $S^{-1}\cO\{u\}$.) This is clear if $\Delta$ is of type $GL_n$ because in this case $\Phi(x)$ has polynomial entries. Let us check \[freeze\] in case $\Delta$ is type $SO(q)$; the case of $SL_n$ is similar. Indeed, since $$\label{corridor}
\Lambda(u)=(u^{(p)})^{-1}\Phi(u)=(u^{(p)})^{-1}\epsilon^{-1}\phi(u)
=(u^{(p)})^{-1}(1+p\alpha)^{-1}(u^{(p)}+p\d u)$$ it follows that $\Lambda(u)$ has entries in $S^{-1}\cO\{u\}$ so $$\sigma(\Phi(u))=\sigma(u^{(p)}\cdot \Lambda(u))=\sigma(u^{(p)})\cdot \sigma(\Lambda(u))=(uc)^{(p)}\cdot \sigma(\Lambda(u)),$$ $$\Phi(\sigma u)=(uc)^{(p)}\cdot \Lambda(uc).$$ So it is enough to check that $\sigma(\Lambda(u))=\Lambda(uc)$. Since both matrices in the latter equality are $\equiv 1$ mod $p$ in $GL_n(R)$ (for the first one use \[corridor\]) it is enough to check that their squares are equal. But, since $M(x):=\Lambda(x)^2$ has entries rational functions of $x$, we get: $$\sigma(\Lambda(u))^2=\sigma(\Lambda(u)^2)=\sigma(M(u))=M(\sigma u)=
M(uc)=\Lambda(uc)^2,$$ which concludes the proof of \[freeze\]. Using \[freeze\] in equation \[calculuss\] below we get $$\label{calculus}
\phi(\sigma(u))=\phi(uc)=\phi(u)\cdot \phi(c)=\epsilon \cdot \Phi(u) \cdot \phi(c),$$ $$\label{calculuss}
\sigma(\phi(u))=\sigma(\epsilon \cdot \Phi(u))=\epsilon \cdot \sigma(\Phi(u))
=\epsilon \cdot \Phi(\sigma(u))=\epsilon\cdot \Phi(uc).$$
To check assertion 2 let $c\in G_{u/\cO}$ and let us prove that $c\in G_u$. Let $\sigma\in \tilde{G}_{u/\cO}$, $\sigma u=uc$. Since $\sigma\circ \d=\d\circ \sigma$ on $\cO\{u\}$ it follows that $\sigma\circ \phi=\phi\circ \sigma$ on $\cO\{u\}$ so, by \[calculus\] and \[calculuss\], $\Phi(uc)=\Phi(u)\cdot \phi(c)$ hence $c\in G_u$.
To check assertion 3, assume $\Delta=0$ (hence $\cO\{u\}=\cO[u]$). Let, first, $c\in G_{u/\cO}$ and let us prove that $c\in \Sigma^0_{u/\cO}\cap G_u$. By assertion 2 we already know that $c\in G_u$. Also $c\in \Sigma^0_{u/\cO}$ by the commutativity of the diagram $$\label{reddd}\begin{array}{ccc}
\cO[x] & \stackrel{\sigma^0_c}{\ra} & \cO[x]\\
\downarrow & \ & \downarrow\\
\cO[u] & \stackrel{\sigma}{\ra} & \cO[u]
\end{array}$$ Conversely let $c\in \Sigma^0_{u/\cO}\cap G_u$ and let us prove that $c\in G_{u/\cO}$. Indeed since $\sigma_c^0(I^0_{u/\cO})=I^0_{u/\cO}$, it follows that $\sigma^0_c:\cO[x]\ra \cO[x]$ induces an automorphism $\sigma:\cO[u]\ra \cO[u]$ with $\sigma(u)=uc$. On the other hand since $c\in G_u$ we have $\Phi(uc)=\Phi(u)\cdot \phi(c)$ hence, by \[calculus\] and \[calculuss\], $\phi(\sigma(u))=\sigma(\phi(u))$. It follows that $\sigma\circ \phi=\phi\circ \sigma$ on $\cO[u]$ hence $\sigma\circ \d=\d\circ \sigma$ on $\cO[u]$. So $\sigma\in \tilde{G}_{u/\cO}$, hence $c\in G_{u/\cO}$ and we are done.
For our discussion below we recall from the Introduction that we denote by $T, W, N$ the torus of diagonal matrices in $G$, the Weyl group of permutation matrices in $G$ and the normalizer of $T$ in $G$ respectively; so $N=TW=WT$. Also if $G^{\d}=\{a\in G;\d a=0\}$ we set $T^{\d}=T\cap G^{\d}$, $N^{\d}=N\cap G^{\d}=T^{\d}W=WT^{\d}$; $G^{\d}$ is a subset of $G$ while $T^{\d}$ and $N^{\d}$ are subgroups of $G$.
We say that $\Phi$ is right compatible with $N$ if $\Phi(ac)=\Phi(a)\cdot c^{(p)}$ for all $a\in G$ and all $c\in N$.
If $\Delta$ is if type $GL_n, SL_n, SO(q)$ then $\Phi$ is right compatible with $N$. By the way if $\Delta$ is of type $GL_n$ (i.e. in case $\Delta=0$) right compatibility of $\Phi(x)=x^{(p)}$ with $N$ simply means that $(ac)^{(p)}=a^{(p)}c^{(p)}$ for $a\in G$ and $c\in N$.
\[rightt\] If $\Phi$ is right compatible with $N$ then $N^{\d}\subset G_u$.
[*Proof*]{}. Trivial.
\[fuego\] Assume $\Delta=0$ and set $N_{u/\cO}=N^{\d}\cap \Sigma^0_{u/\cO}$.
1\) Assume the entries of one of the rows of $u$ are algebraically independent over $\cO$. Then $G_{u/\cO}\subset N^{\d}$ hence $$G_{u/\cO}=N_{u/\cO}.$$
2\) Assume the entries of $u$ are algebraically independent over $\cO$; then $$G_{u/\cO}=N^{\d}\cap GL_n(\cO).$$
3\) Assume $\sigma$ is an $\cO$-automorphism of $\cO[u]$ such that $\sigma(u)=uc$ with $c\in GL_n(\cO)\cap G_u$. Then $c\in G_{u/\cO}$.
4\) Assume $n=1$. Then $G_{u/\cO}\subset N^{\d}=G^{\d}$.
5\) We have an equality $$\bigcap_{u\in G}G_u=N^{\d}.$$
[*Proof*]{}. To prove 1 let $c\in G_{u/\cO}$, hence $c\in G_u$, i.e. $(uc)^{(p)}=u^{(p)}\phi(c)$. If $c=(c_{ij})$ then for all $m$ and $j$ $$\sum_{i=1}^n u_{mi}^p \phi(c_{ij})=(\sum_{i=1}^n u_{mi}c_{ij})^p.$$ Let $m$ be such that $u_{m1},...,u_{mn}$ are algebraically independent over $\cO$. Identifying the coefficients of the monomials in $u_{m1},...,u_{mn}$ in the latter equality we get that for each $j$ there exists an index $\tau(j)$ such that $c_{ij}=0$ for all $i\neq \tau(j)$ and such that $c_{\tau(j)j}^p=\phi(c_{\tau(j)j})$. Since $c$ is non-singular we must have that $\tau$ is a permutation and $c\in N^{\d}$.
To prove assertion 2 note that $G_{u/\cO}\subset N^{\d}\cap GL_n(\cO)$ by assertion 1. Also $N^{\d}\subset G_u$ by Lemma \[rightt\] and, since $\cO[x]\ra \cO[u]$ is a isomorphism, we also have $\Sigma^0_{u/\cO}=GL_n(\cO)$; hence, using Proposition \[forget\], $N^{\d}\cap GL_n(\cO)\subset G_u\cap \Sigma^0_{u/\cO}=G_{u/\cO}$.
To prove assertion 3 let $\sigma^0_c:\cO[x]\ra \cO[x]$ be the unique $\cO$-algebra homomorphism such that $\sigma^0_c(x)=xc$. Then $\sigma^0_c(I^0_{u/\cO})=I^0_{u/\cO}$ by the commutativity of the diagram \[reddd\]; hence $c\in \Sigma^0_{u/\cO}$, hence $c\in G_{u/\cO}$.
To prove assertion 4 let $c\in G_{u/\cO}$; then $uc\in G^{\alpha}$ hence $\phi(u)\phi(c)=\epsilon u^p c^p$ where $\epsilon=1+p\alpha$. Since $\phi(u)=\epsilon u^p$ we get $\phi(c)=c^p$ hence $c \in G^{\d}=N^{\d}$.
To prove 5 note that the inclusion $\supset$ follows from Lemma \[rightt\]. To prove the inclusion $\subset$ let $c$ be in the intersection. Since $R$ is uncountable one can find $u$ with entries algebraically independent over the ring generated by the entries of $c, \d c, \d^2 c,...$. Then one concludes that $c\in N^{\d}$ by using the same argument as in the proof of assertion 1.
\[clock\]
1\) Assume $\Delta$ is of type $SL_n$ and let $u\in GL_n$. Then ${\mathcal I}(x)=\det(x)$ is a prime integral for the $\Delta_u$-linear equation $\d v=\Delta_u(v)$; in other words for any $v\in G_u$ we have $\d(\det(v))=0$.
2\) Assume $\Delta$ is of type $SO(q)$ and let $u\in SO(q)$. Then ${\mathcal I}(x)=x^tqx$ is a prime integral for the $\Delta_u$-linear equation $\d v=\Delta_u(v)$; in other words for any $v\in G_u$ we have $\d(v^t q v)=0$.
[*Proof*]{}. To check 1) note that since $v\in G_u$ we have $$\lambda(uv)\cdot (uv)^{(p)}=\lambda(u) \cdot u^{(p)}\cdot \phi(v).$$ Taking determinants we get $$\lambda(uv)^n\cdot \det((uv)^{(p)})=\lambda(u)^n\cdot \det(u^{(p)})\cdot \det(\phi(v)).$$ Taking into account the definition of $\lambda(x)$ we get $$(\det(uv))^p=\det(u)^p\cdot \det(\phi(v))$$ hence $\det(v)^p=\det(\phi(v))=\phi(\det(v))$ which implies $\d(\det(v))=0$.
To check 2) note that by Equation \[cry\] we have $$\phi(v)=\Phi(u)^{-1}\Phi(uv).$$ On the other hand recall that we have an identity $\Phi(x)^tq\Phi(x)=(x^t qx)^{(p)}$. We get that $$\Phi(u)^tq\Phi(u)=(u^tqu)^{(p)}=q^{(p)}=q,$$ hence $$(\Phi(u)^t)^{-1}q\Phi(u)^{-1}=q,$$ hence $$\begin{array}{rcl}
\phi(v^tqv) & = & \phi(v)^t q \phi(v)\\
\ & \ & \ \\
\ & = & \Phi(uv)^t (\Phi(u)^t)^{-1} q \Phi(u)^{-1} \Phi(uv)\\
\ & \ & \ \\
\ & = & \Phi(uv)^t q \Phi(uv)\\
\ & \ & \ \\
\ & = & (v^t u^t q uv)^{(p)}\\
\ & \ & \ \\
\ & = & (v^tqv)^{(p)},
\end{array}$$ which implies that $\d(v^tqv)=0$.
Our next task will be to compute/bound the $\d$-Galois group $G_{u/\cO}$ in case $\Delta=0$. One of the morals will be that this group tends to be contained in $N$; but this is not always the case as shown in the following:
\[counterexample\] Let $\cO=\bZ_{(p)}$, $n=2$, and assume $p\equiv 1$ mod $3$. Consider the matrices $$u= (
[cc]{} 1 &\
1& \^2
), c= (
[cc]{} 1 & $-1$\
0 & $-1$
), uc= (
[cc]{} 1 & \^2\
1 &
),$$ where $\zeta\in \bZ_p\subset R$ is a cubic root of unity. Note that $\det u=\zeta^2-\zeta\not\equiv 0$ mod $p$ so $u, c, uc\in GL_2(R)$. Then $u$ is a solution to the $\Delta$-linear equation equation $$\d u=0,$$ where $\Delta=0$. We will show that $G_{u/\cO}\not\subset N$. Indeed $u, c, uc \in G^{\d}\backslash N$ and $u^{(p)}=u$, $c^{(p)}=c$, $(uc)^{(p)}=uc$ so $c\in G_u$. Also we have $\cO[u]= \bZ_{(p)}[\zeta]$ and the unique non-trivial automorphism $\sigma$ of $\bZ_{(p)}[\zeta]$ sending $\sigma(\zeta)=\zeta^2$ satisfies $\sigma(u)=uc$. By assertion 3 in Lemma \[fuego\] we have $c\in G_{u/\cO}$; so in particular $G_{u/\cO}\not\subset N$, and our claim is proved. By the way in this case $G_{u/\cO}=\langle c \rangle$ is cyclic of order $2$.
\[sapun\] Assume $\Delta=0$ and $\cO$ is a valuation $\d$-subring of $R$ with finite residue field. Then $G_{u/\cO}$ is a finite group.
[*Proof*]{}. Let $u_0\in G^{\d}$ be the unique element such that $u\equiv u_0$ mod $p$. Let $F$ be the field of fractions of $\cO$, let $F'$ be the field generated by $F$ and the roots of unity appearing as entries in $u_0$, and let $\cO'=R\cap F'$. Then $\cO'$ is a valuation $\d$-subring of $R$ generically finite over $\cO$ and $u_0\in GL_n(\cO')$. In particular $\cO'$ has a finite residue field. Since $\alpha\in GL_n(\cO')$, by Proposition \[tata\], we get $u\in GL_n(\cO'')$ for some generically finite extension $\cO''$ of $\cO'$. Then, by the equality $\cO\{u\}=\cO[u]$, $G_{u/\cO}$ is finite.
In what follows we view $R$ as a complete metric space with respect to the $p$-adic metric. So we can talk about open balls in $R$. Any open ball has the form $X=b+p^NR$ for some $b\in R$ and $N\in \bZ_{\geq 0}$; any such $X$ is also closed and is, again, a complete metric space with respect to the induced metric. Now recall that a subset of a metric space is called [*of the first category*]{} if it is a countable union of subsets each of which has the property that its closure has an empty interior. By the Baire-Hausdorff theorem [@Y], p. 11, any subset of the first category in a non-empty complete metric space $X$ is different from $X$. This applies then to any open ball $X$ in $R$.
\[transcendence\] Assume $\Delta=0$. There exists a subset $\Omega$ of the first category in the metric space $$X=\{u\in GL_n(R);u\equiv 1\ \ \text{mod}\ \ p\}$$ such that for any $u \in X\backslash \Omega$ the following holds. Let $\alpha=\d u \cdot (u^{(p)})^{-1}$. Then there exists a valuation $\d$-subring $\cO$ of $R$ containing $R^{\d}$ such that $\alpha\in {\mathfrak g}{\mathfrak l}_n(\cO)$ and such that $G_{u/\cO}=N^{\d}$.
\[vanishing\] Let $x, x',..., x^{(r)}$ are a $m$-tuples of indeterminates and let $f\in R[x,x',...,x^{(r)}]\h$. Assume the map $f_*:R^m\ra R$ defined by $$f_*(a)=f(a, \d a, ..., \d^m a)$$ vanishes on a product of open balls. Then f vanishes on the whole of $R^m$.
[*Proof*]{}. By [@char], Remark 1.6, $f=0$ if and only if $f_*=0$. So it is enough to show that for any $b_j\in R$, $1\leq j\leq m$, the $R$-algebra homomorphism $$R[x,x',...,x^{(r)}]\h \ra R[x,x',...,x^{(r)}]\h,\ \ \ x_j^{(i)}\mapsto \d^i(b_j+p^Nx_j),$$ is injective. To check this we may assume $b_j=0$ for all $j$. But then the assertion follows from the fact that $$R[x,x',...,x^{(r)}]\h \subset K[[x,x',...,x^{(r)}]]=K[[x,\phi(x),...,\phi^r(x)]]$$ and from the fact that the endomorphism of $K[[x,\phi(x),...,\phi^r(x)]]$ defined by $\phi^i(x)\mapsto p^N\phi^i(x)$ is injective.
\[Independent\] Let $E$ be a countable subfield of $K$ and let $X_1,...,X_m\subset R$ be open balls. Then one can find a subset $\Omega$ of the first category in the metric space $X=X_1\times...\times X_m$ such that for all $u=(u_1,...,u_m)\in X\backslash \Omega$ the family $$(\d^i u_j)_{i\geq 0, 1\leq j\leq m}$$ is algebraically independent over $E$.
[*Proof*]{}. Let ${\mathcal F}=E[x,x',x'',...]$ be the polynomial ring where each of $x, x', x'',...$ is an $m$-tuple of indeterminates . Hence ${\mathcal F}$ is countable. Then for each $f\in {\mathcal F}$ with $f\neq 0$ set $$X_f:=\{u\in X;f(u,\d u,\d^2 u,...)=0\}.$$ Now we claim that each $X_f$ is closed in the metric space $X$ and has empty interior; indeed $X_f$ is the zero locus in $X$ of $f_*:R^m\ra R$ and our claim follows from Lemma \[vanishing\]. The present Lemma follows now by taking $$\Omega= \bigcup_{0\neq f\in {\mathcal F}}X_f.$$
[*Proof of Proposition \[transcendence\]*]{}. Let $E$ be the subfield of $K$ generated over ${\mathbb Q}$ by all the roots of unity in $K$; i.e. $E={\mathbb Q}(R^{\d})$. Now $X$ in the Proposition is a product of balls so by Lemma \[Independent\] there exists a subset of the first category $\Omega\subset X$ such that for all $u\in X\backslash \Omega$ the family $(\d^r u_{ij})_{r\geq0, 1\leq i,j\leq n}$ is algebraically independent over $E$. Let $\epsilon=\phi(u)\cdot (u^{(p)})^{-1}$, $\alpha=(\epsilon-1)/p$ and consider the fields $$F_s=E(\d^r \alpha_{ij}; 0\leq r\leq s,\ 1\leq i,j\leq n)=E(\phi^r (\epsilon_{ij}); 0\leq r\leq s,\ 1\leq i,j\leq n)$$ and $F=\cup_{s} F_s$. Let $\cO$ be a valuation $\d$-subring of $R\cap F$ containing $R^{\d}$ and the entries of $\alpha$ (e.g. one can take the “maximal“ choice” $\cO=R\cap F$). Note that for $s\geq 1$ we have equalities of fields $$\label{fields}
E(\d^r u_{ij}; 0\leq r\leq s,\ 1\leq i,j\leq n)=F_{s-1}(u_{ij}; 1\leq i,j\leq n).$$ Now the field in the left hand side of the \[fields\] has transcendence degree $(s+1)n^2$ over $E$. Since $F_{s-1}$ has transcendence degree at most $sn^2$ over $E$ it follows from \[fields\] that $(u_{ij})_{ij}$ are algebraically independent over $F_{s-1}$. Since this is true for all $s$ it follows that $(u_{ij})_{ij}$ are algebraically independent over $F$. By assertion 2 in Lemma \[fuego\], $G_{u/\cO}=N^{\d}$.
The next Proposition shows that the $\d$-Galois group cannot be “too large" at least if we take our data in a Zariski open set of the set of all data. In the statement below by a Zariski $K$-closed set in $GL_n(R)$ we understand the intersection of $GL_n(R)$ with a Zariski $K$-closed set of $GL_n(K^a)$; in other words a $K$-closed set of $GL_n(R)$ is the zero set in $GL_n(R)$ of a collection of polynomials with coefficients in $K$ in $n^2$ variables. A subgroup $\Gamma$ of $GL_n(R)$ is called diagonalizable if there exists $g\in GL_n(K^a)$ such that $g^{-1}\Gamma g$ consists of diagonal matrices.
\[dimless\] There exists a Zariski $K$-closed set $\Omega$ in $G=GL_n(R)$ not containing $1$ such that for any $u\in G\backslash \Omega$ the following holds. Let $\alpha=\d u \cdot (u^{(p)})^{-1}$ and let $\cO$ be a valuation $\d$-subring of $R$ containing the entries of $\alpha$. Then $G_{u/\cO}$ contains a normal subgroup of finite index which is diagonalizable.
In order to prove Proposition \[dimless\] we need a series of Lemmas: \[Z\], \[later\], \[gee\]. In the discussion below (pertaining to these Lemmas only!) it is convenient to temporarily change some of the notation used so far. Indeed we let ${\mathcal C}$ be an uncountable algebraically closed field of characteristic zero (such as $K^a$ or ${\mathbb C}$) and all schemes will be schemes over ${\mathcal C}$. By a variety we will understand a reduced (not necessarily irreducible) scheme of finite type over ${\mathcal C}$. We use the same letter $X$ to denote a variety $X$ over ${\mathcal C}$ and its set $X({\mathcal C})$ of ${\mathcal C}$-points. In particular we denote by $G$ the group scheme $GL_n$ over ${\mathcal C}$ and also the “abstract" group $GL_n({\mathcal C})$; we denote by $T$ the group scheme of diagonal matrices over ${\mathcal C}$ and also the “abstract" group $T({\mathcal C})$ of diagonal matrices with entries in ${\mathcal C}$. If $X$ is a variety and $x\in X$ is a point we always understand $x$ is a ${\mathcal C}$-point and we denote by $\dim_x X$ the maximum of the dimensions of the irreducible components of $X$ passing through $x$. Also, in what follows, we let $p$ be any integer $\geq 2$ (not necessarily prime).
\[Z\] Let $X\subset G$ be the Zariski closed subset consisting of all $v\in G$ satisfying the following properties:
1\) $(v^m)^{(p)}=(v^{(p)})^m$ for all $m\geq 0$,
2\) $(v^m)^{(p)}(v^{-m})^{(p)}=1$ for all $m\geq 0$.
Then $X$ has exactly one irreducible component passing through $1$ and that component is $T$.
The equalities 1) and 2) are viewed as equalities in ${\mathfrak g}={\mathfrak g}{\mathfrak l}_n({\mathcal C})$; note however that, by 1) and 2), for any $v\in X$ we have that $(v^m)^{(p)} \in G$ for all $m\in \bZ$ and hence 1) holds for all $m\in \bZ$ as an equality in $G$.
The set $X$ contains the group $N=WT=TW$ generated by the Weyl group $W$ and the group $T$ of diagonal matrices with entries in ${\mathcal C}$. It is not clear whether $X$ actually coincides with the group $N$.
\[sauce\] Let ${\mathbb X}$ be the closed subscheme of $G$ defined by the equations 1) and 2) in the statement of Lemma \[Z\]; hence the variety ${\mathbb X}_{red}$ coincides with $X$. It is interesting to note that tangent space of ${\mathbb X}$ at $1$ is the whole of the tangent space of $G$ i.e. the Lie algebra $L(G)$ of $G$; indeed, equations 1) and 2) are easily seen to hold when $v$ is replaced by $1+\epsilon \xi$, where $\epsilon^2=0$ and $\xi$ is an arbitrary element of ${\mathfrak g}{\mathfrak l}_n({\mathcal C})$. In particular ${\mathbb X}$ is not reduced.
[*Proof of Lemma \[Z\]*]{}. Let $v\in X$, let $\langle v \rangle \subset G$ be the group generated by $v$, let $H_v \subset G$ be the Zariski closure of $\langle v \rangle$ in $G$ (which is an algebraic subgroup of $G$, cf. [@hum], p. 54), and let $H^{\circ}_v$ be the identity component of $H_v$. Clearly $H_v$ is commutative.
[*Claim.*]{} For all $v\in X$ we have $H_v^{\circ}\subset T$.
To check the claim note first that $\langle v \rangle \subset X$ hence $H_v\subset X$. Denote by $\Phi:G\ra {\mathfrak g}$ the map $\Phi(v)=v^{(p)}$. Clearly we have $\Phi(v^r v^s)=\Phi(v^r)\Phi(v^s)$ for all $r,s\in \bZ$ hence we have $\Phi(gh)=\Phi(g)\Phi(h)$ for all $g,h \in H_v$. Let $\varphi:H_v\ra {\mathfrak g}$ be the restriction of $\Phi$; then the regular map $\varphi$ takes values in $G$ and is a group homomorphism hence its image $H'_v:=\varphi(H_v)\subset G$ is a subgroup which is constructible. Hence $H'_v$ is a closed subgroup of $G$ (cf. [@hum], p. 54) and hence $\varphi:H_v\ra H'_v$ is an algebraic group homomorphism. Consider the commutative diagram of (possibly reducible) varieties $$\begin{array}{rcl}
H_v & \subset & G\\
\varphi \downarrow & \ & \downarrow \Phi\\
H'_v & \subset & {\mathfrak g}\end{array}$$ and the induced tangent maps between the corresponding tangent spaces at the identity $$\begin{array}{rcl}
L(H_v) & \subset & L(G)\\
d_1\varphi \downarrow & \ & \downarrow d_1 \Phi\\
L(H'_v) & \subset & L(G)\end{array}$$ (Here $L(\ )$ denotes the Lie algebra functor. The linear map $d_1\Phi$ is not a Lie algebra map. The map $d_1\varphi$, on the other hand, is, of course, a Lie algebra map because its source and target are abelian.) One can compute $d_1\Phi$ explicitly: letting $v=1+\epsilon \xi\in GL_n({\mathcal C}[\epsilon])$, $\epsilon^2=0$, we have $$\Phi(v)=(1+\epsilon \xi)^{(p)}=diag(1+\epsilon p\xi_{11},...,1+\epsilon p\xi_{nn}).$$ Hence the image of $d_1\Phi$ is contained in the Lie algebra $L(T)$ of the torus $T$. Since $d_1\varphi$ is surjective (because we are in characteristic zero) it follows that $L(H'_v)\subset L(T)$. Hence the identity component $(H'_v)^{\circ}$ of $H'_v$ is contained in $T$. Now, clearly, $\Phi^{-1}(T)=T$. Hence $H_v^{\circ}\subset \Phi^{-1}((H'_v)^{\circ})\subset \Phi^{-1}(T)=T$ and our claim is proved.
For any subtorus $S\subset T$ let us denote by $C(S)$ the centralizer of $S$ in $G$; moreover, for any integer $e\geq 1$ denote by $S^{1/e}$ the set of all $v\in G$ such that $v^e\in S$. By the above Claim and by the commutativity of $H_v$ it follows that for any $v\in X$ we have that $H_v^{\circ}$ is a subtorus of $T$ and there exists $e\geq 1$ such that $v\in C(H_v^{\circ})\cap H_v^{1/e}$. In particular we have $$X=\bigcup_{S,e} (C(S)\cap S^{1/e}\cap X)$$ where $S$ runs through the (countable!) set of subtori of $T$ and $e$ runs through the set of positive integers. Since ${\mathcal C}$ is uncountable no irreducible variety over ${\mathcal C}$ is a countable union of proper closed subvarieties; in particular, applying this to the irreducible components of $X$ it follows that there exists $e\geq 1$ and finitely many subtori $S_1,...,S_q\subset T$ such that $$\label{deco}
X=\bigcup_{i=1}^q (C(S_i)\cap S_i^{1/e}\cap X).$$ To conclude the proof of the Lemma we assume (as we always can) that ${\mathcal C}={\mathbb C}$. Let $V$ be an irreducible component of $X$ passing through $1$. We will prove that $V=T$ and this will end the proof. Assume $V\neq T$ and seek a contradiction. Since $V\neq T$ it follows that $V\not\subset T$ hence $V\backslash T$ is Zariski open in $V$ hence dense in $V$ in the complex topology. So there exists a sequence $x_n\ra 1$ (in the complex topology) with $x_n\in X\backslash T$. By \[deco\] and by replacing $x_n$ with a subsequence we may assume $x_n\in C(S_i)\cap S_i^{1/e}\cap X$ for some $i$. Let $[x_n]\in C(S_i)/S_i$ be the class of $x_n$ and choose an embedding $\rho: C(S_i)/S_i\ra GL_{\nu}({\mathcal C})$ for some $\nu$. Then $\rho([x_n])\ra 1$ hence the eigenvalues of $\rho([x_n])$ tend to $1$. But $[x_n]^e=1$, hence $\rho([x_n])^e=1$, for all $n$. So the eigenvalues of $\rho([x_n])$ are $e$-th roots of unity so they form a discrete set. We get that for $n$ sufficiently big the eigenvalues of $\rho([x_n])$ are equal to $1$. But a matrix of finite order with all eigenvalues equal to $1$ must be the identity. Hence $\rho([x_n])=1$ hence $[x_n]=1$ hence $x_n\in S_i\subset T$ for some $n$, a contradiction. This ends the proof of the Lemma.
The next lemma is completely standard; we just include it for convenience.
\[later\] Let $\pi:Z\ra Y$ be a morphism of varieties over ${\mathcal C}$ and assume $\sigma:Y\ra Z$ is a section of $\pi$. Assume $Y$ is irreducible and for $y\in Y$ consider the variety $\pi^{-1}(y)$. Let $y_0\in Y$ and assume the point $\sigma(y_0)$ is a connected component of $\pi^{-1}(y_0)$. Then there exists a Zariski open set $U\subset Y$ containing $y_0$ such that for all $y\in U$ the point $\sigma(y)$ is a connected component of $\pi^{-1}(y)$.
[*Proof*]{}. This is a standard consequence of the semicontuinty theorem for the local dimension of fibers. Indeed let $Z^1,...,Z^m$ be the irreducible components of $Z$, let $S=\sigma(Y)$ and assume $\sigma(y_0)\in Z^i$ for $1\leq i\leq r$ and $\sigma(y_0)\not\in Z^j$ for $r< j\leq m$. Let $U_0=\pi(S\backslash \bigcup_{j>r}Z^j)$. Also let $Y^i\subset Y$ be the closure of $\pi(Z^i)$ and let $\pi_i:Z^i\ra Y^i$ for $i\leq r$ be induced by $\pi$. By the semicontinuity theorem in [@hum], p.33, for $i\leq r$, there exist closed sets $T^i\subset Z^i$ not containing $\sigma(y_0)$ such that $$\label{fragrance}
\dim_x \pi_i^{-1}(\pi(x))\leq \dim_{\sigma(y_0)} \pi_i^{-1}(y_0)\ \ \text{for all}\ \ x\in Z^i\backslash T^i.$$ Consider the closed set $T:=T^1\cup...\cup T^r\cup Z^{r+1}\cup...\cup Z^m$ in $Z$ and the open subset $U=\pi(S\backslash T)=Y\backslash \pi(S\cap T)$ of $Y$. Then $y_0\in U$. Let $y\in U$ and let $F$ be an irreducible component of $\pi^{-1}(y)$ passing through $\sigma(y)$. Then $F\not\subset Z^j$ for $j>r$ (because if one assumes the contrary then $\sigma(y)\in S\cap Z^j\subset S\cap T$ hence $y\in \pi(S\cap T)$, a contradiction). So $F\subset Z^i$ for some $i\leq r$ and hence $F\subset \pi_i^{-1}(y)$. Since $y\not\in \pi(S\cap T)$ we have $\sigma(y)\not\in T$ hence $\sigma(y) \not\in T^i$; on the other hand $\sigma(y)\in F\subset Z^i$, hence $\sigma(y)\in Z^i\backslash T^i$. So by \[fragrance\] we get $$\dim_{\sigma(y)}F\leq \dim_{\sigma(y)}\pi_i^{-1}(y)\leq \dim_{\sigma(y_0)}\pi_i^{-1}(y_0)\leq \dim_{\sigma(y_0)}\pi^{-1}(y_0)=0.$$ So $\dim_{\sigma(y)}F=0$ hence $F=\{\sigma(y)\}$ and we are done.
\[gee\] Let $Y$ be the Zariski open set of $G=GL_n({\mathcal C})$ consisting of all $u\in G$ such that $u^{(p)}$ is invertible. Let $\Psi:Y\times G\ra {\mathfrak g}$ be the morphism defined by $$\Psi(u,v) =(u^{(p)})^{-1}(uv)^{(p)}.$$ For each $u\in Y$ let $X_u \subset G$ be the Zariski closed set consisting of all $v\in G$ such that
1\) $\Psi(u,v^m)=\Psi(u,v)^m$ for all $m\geq 0$,
2\) $\Psi(u,v^m) \Psi(u,v^{-m})=1$ for all $m \geq 0$.
Then there exists a Zariski open set $U\subset Y$ containing $1$ with the property that for any $u\in U$ and for any connected closed subgroup $S\subset G$ contained in $X_u$ we have that $S$ is a torus.
[*Proof*]{}. Let $Z\subset Y\times G$ be the closed set defined by the equations 1) and 2) together with the equation $(v-1)^n=0$. Note that this latter equation is equivalent to asking that $v$ be unipotent. Let $\pi:Z\ra Y$, $\pi(u,v)=u$, and let $pr_G:Y\times G\ra G$ be the second projection. Then $pr_G(\pi^{-1}(u))$ coincides with the set of unipotent matrices in $X_u$. Also note that $X_1$ coincides with $X$ in Lemma \[Z\]. Now, by Lemma \[Z\], there is exactly one irreducible component of $X_1$ passing through $1$ and that component is a torus so it does not contain unipotent matrices with the exception of $1$ itself. In particular $1$ is a connected component of $pr_G(\pi^{-1}(1))$. Now $\pi$ has a section $\sigma:Y\ra Z$, $\sigma(u)=(u,1)$. By Lemma \[later\] there exists a Zariski open set $U$ of $Y$ containing $1$ such that for all $u\in U$ we have that $(u,1)$ is a connected component of $\pi^{-1}(u)$. So $1$ is a connected component of the set of unipotent matrices in $X_u$. Now let $S\subset G$ be a closed connected subgroup contained in $X_u$. Then $1$ is a connected component of the set of unipotent matrices in $S$. This implies that $S$ contains no unipotent matrix except $1$ (because any unipotent matrix $\neq 1$ is contained in a subgroup isomorphic to the additive group). So the unipotent radical of $S$ is trivial, hence a torus by [@hum], p. 161.
Exactly as in Remark \[sauce\], if ${\mathbb X}_u$ is the subscheme of $G$ defined by equations 1) and 2) in Lemma \[gee\] then $({\mathbb X}_u)_{red}=X_u$ and the tangent space to ${\mathbb X}_u$ at $1$ is, again, the whole of the Lie algebra $L(G)={\mathfrak g}{\mathfrak l}_n({\mathcal C})$.
[*Proof of Proposition \[dimless\]*]{}. Consider the situation and notation in Lemma \[gee\] with ${\mathcal C}=K^a$. Choose a polynomial $F\in K^a[x]$ such that $$1\in D(F):=\{v\in GL_n(K^a);F(v)\neq 0\}\subset U.$$ Replacing $F$ by the product of its conjugates over $K$ we may assume $F\in K[x]$ and hence that $F\in R[x]$. Now let $u\in D(F)\cap GL_n(R)$, $\alpha=\d u \cdot (u^{(p)})^{-1}$, and let $\cO\subset R$ be a valuation $\d$-subring containing the entries of $\alpha$. Let $\overline{G_{u/\cO}}$ be the Zariski closure of $G_{u/\cO}$ in $GL_n(K^a)$. We want to show that the connected component $\overline{G_{u/\cO}}^{\circ}$ of $\overline{G_{u/\cO}}$ is a torus in $GL_n(K^a)$. Note that $u^{(p)}$ is invertible so $u\in Y$. Let $c\in G_{u/\cO}$ hence $c^m\in G_{u/\cO}\subset G_u$ for all $m\in \bZ$. Hence $(uc^m)^{(p)}=u^{(p)}\phi(c^m)$, hence $\Psi(u,c^m)=\phi(c^m)$. We claim that $c\in X_u$; indeed for $m\geq 0$ we have $$\Psi(u,c^m) = \phi(c^m)=\phi(c)^m= \Psi(u,c)^m$$ and also $$\Psi(u,c^m)\Psi(u,c^{-m}) = \phi(c^m)\phi(c^{-m})=\phi(1)=1.$$ Since $c$ was arbitrary in $G_{u/\cO}$ we conclude that $G_{u/\cO}\subset X_u$ hence $\overline{G_{u/\cO}}\subset X_u$. By Lemma \[gee\], $\overline{G_{u/\cO}}^{\circ}$ is a torus. Then clearly $$G_{u/\cO}\cap \overline{G_{u/\cO}}^{\circ}$$ is a normal subgroup of finite index in $G_{u/\cO}$ which is diagonalizable.
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abstract: 'A coexistent phase of spin polarization and color superconductivity in high-density QCD is investigated using a self-consistent mean-field method at zero temperature. The axial-vector self-energy stemming from the Fock exchange term of the one-gluon-exchange interaction has a central role to cause spin polarization. The magnitude of spin polarization is determined by the coupled Schwinger-Dyson equations with a superconducting gap function. As a significant feature, the Fermi surface is deformed by the axial-vector self-energy and then rotation symmetry is spontaneously broken down. The gap function results in being anisotropic in the momentum space in accordance with the deformation. As a result of numerical calculations, it is found that spin polarization barely conflicts with color superconductivity, but almost coexists with it.'
address: |
${^a}$[*Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan* ]{}\
${^b}$[*College of Bioresouce Sciences, Nihon University, Fujisawa, 252-8510, Japan*]{}\
${^c}$[*Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan*]{}\
${^d}$[*Institute for Quantum Energy, Nihon University*]{}\
${^e}$[*Department of Physics, Kyoto University, Kyoto 606-8502, Japan*]{}
author:
- 'E.Nakano$^a$, T.Maruyama$^{b, c, d}$ and T.Tatsumi$^e$'
title: '**Spin Polarization and Color Superconductivity in Quark Matter**'
---
=-2.0cm = 1.5em
Introduction
============
Recently much interest is given for high-density QCD, especially for quark Cooper-pair condensation phenomena at high-density quark matter (called as color superconductivity (CSC)), in connection with, e.g., physics of heavy ion collisions and neutron stars [@CSC1; @BL; @CSC2]. Its mechanism is similar to the BCS theory for the electron-phonon system [@BCS], in which the attractive interaction of electrons is provided by phonon exchange and causes the Cooper instability near the Fermi surface. As for quark matter, the quark-quark interaction is mediated by colored gluons, and is often approximated by some effective interactions, e.g., the one-gluon-exchange (OGE) or the instanton-induced interaction, both of which give rise to the attractive quark-quark interaction in the color anti-symmetric ${\bf 3^*}$ channel. CSC leads to spontaneous symmetry breaking of color $SU(3)$ into $SU(2)$ as a result of condensation of quark Cooper pairs [@BL; @CSC2].
In this paper we would like to address another phenomenon expected in quark matter: spin polarization or ferromagnetism of quark matter. We examine the possibility of the spin-polarized phase with CSC in quark matter. As far as we know, interplay between the color superconducting phase and other phases characterized by the non-vanishing mean fields of the spinor bilinears $\langle \bar{\psi} \Gamma \psi \rangle$ has not been explored except for the case of chiral symmetry breaking [@CSC3]. Our main concern here is to investigate the possibility of the quark Cooper instability under the axial-vector mean-field, $\langle \bar{\psi}\gamma^\mu\gamma_5\psi \rangle$ which is responsible for spin polarization of quark matter. It would be worth mentioning in this context that ferromagnetism (or spin polarization) and superconductivity are fundamental concepts in condensed matter physics, and their coexistent phase has been discussed for a long time [@MagSup1]. As a recent progress, a superconducting phase have been discovered in ferromagnetic materials and many efforts have been made to understand the coexisting mechanism [@MagSup2].
Besides being interesting in its own right, the coexistence problem may be related to some physical phenomena. Recently, a new type of neutron stars, called as “magnetars”, with a super strong magnetic field of $\sim O(10^{15}$G) has been discovered [@MAG1; @MAG2]. they may raise an interesting question for the origin of the magnetic field in compact stars, since its strength is too large to regard it as a successor from progenitor stars, unlike canonical neutron stars [@MAG3]. Since hadronic matter spreads over inside neutron stars beyond the nuclear density ($\rho_0$$\sim$$0.16$${\rm fm^{-3}}$), it should be interesting to consider the microscopic origin of the magnetic field in magnetars. In this context, a possibility of ferromagnetism in quark matter due to the OGE interaction has been suggested by one of the authors (T.T.) within a variational framework [@Tatsu]; a competition between the kinetic and the Fock exchange energies gives rise to spin polarization, similarly to Bloch’s idea for itinerant electrons. Salient features of spin polarization in the relativistic system are also discussed in Ref. [@Tatsu]. Thus, it might be also interesting to examine the possibility of the spin-polarized phase with CSC in quark matter, in connection with magnetars.
We investigate spin polarization in the color superconducting phase by a self-consistent framework, in which quark Cooper pairs are formed under the axial-vector mean-field. We shall see that this phenomenon is a manifestation of spontaneous breaking of both color $SU(3)$ and rotation symmetries.
We adopt here the OGE interaction as an effective quark-quark interaction. Since the Fermi momentum is very large at high density, asymptotic freedom of QCD implies that the interaction between quarks is very weak [@perry]. So it may be reasonable to think that the OGE interaction has a dominant contribution for the quark-quark interaction. In the framework of relativistic mean-field theories, the axial-vector and tensor mean-fields, which stem from the Fock exchange terms, $\langle \bar{\psi} \gamma_5 \gamma_\mu \psi \rangle$ and $\langle \bar{\psi} \sigma_{\mu \nu} \psi \rangle$, may have a central role to split the degenerate single-particle energies of the two spin states, and then leads to spin polarization, e.g., see [@MaruTatsu] for discussion in nuclear matter. As for quark matter, several types of the color singlet mean-fields appear after the Fierz transformation in the Fock exchange terms, but we retain only the axial-vector mean-field as the origin of spin polarization, because the OGE interaction by no means holds the tensor mean-field due to chiral symmetry in QCD, unlike nuclear matter [@MaruTatsu]. Presence of the axial-vector mean-field deforms the quark Fermi seas according to their spin degrees of freedom, and thereby the gap function should be no more isotropic in the momentum space. We assume here an anisotropic gap function $\Delta$ on the Fermi surface by a physical consideration and solve the coupled Schwinger-Dyson equations self-consistently by way of the Nambu formalism to find the axial-vector mean-field $U_A$ and the superconducting gap function $\Delta$. Thus we discuss the interplay between spin polarization and superconductivity in quark matter.
In Section 2 we give a framework to deal with the present subject. The explicit structure of the anisotropic gap function $\Delta$ in the color, flavor, and Dirac spaces is carefully discussed there and in the Appendix B and Appendix C. Numerical results about $U_A$ and $\Delta$ are given in Section 3, where phase diagram of spin polarization and color superconductivity is given in the mass-baryon number density plane. Section 4 is devoted to summary and concluding remarks.
Formalism
=========
In this section we present our formalism to treat CSC and spin polarization. We consider quark matter with flavor $SU(2)$ and color $SU(3)$ symmetries, and assume that the interaction action is described by the OGE interaction as $$\begin{aligned}
I_{int}=-g^2\frac{1}{2}\int{\rm d^4}x \int{\rm d^4}y
\left[\bar{\psi}(x)\gamma^\mu \frac{\lambda_a}{2} \psi(x)\right]
D_{\mu \nu}(x,y)
\left[\bar{\psi}(y)\gamma^\nu \frac{\lambda_a}{2} \psi(y)\right], \end{aligned}$$ where $\psi$ is the quark field, $D_{\mu \nu}(x,y)$ is the gauge boson (gluon) propagator, and $\lambda_a=1,2,\cdots,8$ are the $SU(3)$ Gell-Mann matrices. Using the Nambu formalism [@BL; @Nambu] the effective action is given within the mean-field approximation as $$I_{MF}=\frac{1}{2} \int \frac{{\rm d}^4 p}{(2 \pi)^4}
\left( \begin{array}{l}
\bar{\psi}(p) \\
\bar{\psi}_c(p) \\
\end{array} \right)^T
G^{-1}(p)
\left( \begin{array}{l}
\psi(p) \\
\psi_c(p) \\
\end{array} \right) \\
\label{mfield}$$ with the inverse quark Green function $$G^{-1}(p)=\left( \begin{array}{cc}
\sla{p}-m+\sla{\mu}+V(p) &
\gamma_0 \Delta^\dagger(p) \gamma_0 \\
\Delta(p) &
\sla{p}-m-\sla{\mu}+\overline{V}(p) \\
\end{array} \right),
\label{fullg}$$ where $\sla{\mu}=\gamma_0 \mu$ with the chemical potential $\mu$. $V$ is a self-energy and $\Delta$ is the gap function for the quark Cooper pair; both terms $V$ and $\Delta$ should be provided by the Fock exchange terms of the OGE interaction. We define here $\psi_c(k)$ and $\overline{V}$ as $$\begin{aligned}
\psi_c(k) &=& C \bar{\psi}^T(-k),
\\
\overline{V} &\equiv& C V^T C^{-1} \end{aligned}$$ with the charge conjugation matrix $C$ which is explicitly given by $i\gamma_2 \gamma_0$ in Dirac representation.
The Green function $G(p)$ can be written straightforwardly from eq.(\[fullg\]) as $$G(p)=\left( \begin{array}{cc}
G_{11}(p) & G_{12}(p) \\
G_{21}(p) & G_{22}(p) \\
\end{array} \right) \label{fullg2} \\$$ with $$\begin{aligned}
G_{11}(p) &=& \left[ \sla{p}-m+\sla{\mu}+V(p)
-\gamma_0 \Delta(p)^\dagger \gamma_0
\left( \sla{p}-m-\sla{\mu}+\overline{V}(p) \right)^{-1}
\Delta(p) \right]^{-1}
\label{G11i} \\
G_{21}(p) &=& -\left( \sla{p}-m-\sla{\mu}+\overline{V}(p) \right)^{-1}
\Delta(p) G_{11}(p). \label{G21i}\end{aligned}$$ Following Nambu’s argument [@Nambu], we impose the self-consistency condition to obtain the Hartree-Fock ground state such that the self-energy by the residual interaction, $\Sigma_{Res.}$, vanishes: $$\Sigma_{Res.}=\Sigma_{M.F.}-\Sigma_{Int.}=0, \label{Res}$$ where $\Sigma_{M.F.}$ is defined by $$\begin{aligned}
\Sigma_{M.F.}(k) &=& G_0^{-1}(k)-G^{-1}(k) =-\left(
\begin{array}{cc}
V(k) & \gamma_0 \Delta^{\dagger}(k)
\gamma_0 \\
\Delta(k) & \overline{V}(k) \\
\end{array} \right)\\
\mbox{with} ~~~
G_0(p)&=&\left[\begin{array}{cc}
( \sla{p}-m+\sla{\mu} )^{-1} & 0 \\
0 & (\sla{p}-m-\sla{\mu} )^{-1} \\
\end{array}\right], \end{aligned}$$ and $\Sigma_{Int.}$ is given by the use of the OGE interaction. Within the first-order approximation in $g^2$, $\Sigma_{Int.}$ renders $$\begin{aligned}
\Sigma_{Int.}(k)&=&g^2 \int \frac{{\rm d}^4 p}{i(2 \pi)^4}
D^{a b}(k-p) \hat{\Gamma}_a G(p) \hat{\Gamma}_b \label{self11} \\
\hat{\Gamma}_a &\equiv& \left( \begin{array}{cc}
\gamma^\mu \frac{\lambda_\alpha}{2} & 0 \\
0 &
C\left(\gamma^\mu \frac{\lambda_\alpha}{2}\right)^TC^{-1} \\
\end{array} \right)
= \left( \begin{array}{cc}
\gamma^\mu \frac{\lambda_\alpha}{2} & 0 \\
0 & -\gamma^\mu \frac{\lambda^T_\alpha}{2} \\
\end{array} \right),
\end{aligned}$$ which is nothing else but the Fock exchange energy by the OGE interaction. Using eqs. (\[Res\]) - (\[self11\]), we obtain the self-consistent equation for $V(k)$ by the use of the diagonal component of the full Green function (\[G11i\]): $$\begin{aligned}
-V(k)=(-ig)^2 \int \frac{{\rm d}^4p}{i(2\pi)^4} [-iD^{\mu \nu}(k-p)]
\gamma_\mu \frac{\lambda_\alpha}{2} [-iG_{11}(p)]
\gamma_\nu \frac{\lambda_\alpha}{2}
\label{self1}.
\end{aligned}$$ The gap equation is also obtained from the off-diagonal component as $$-\Delta(k)=(-ig)^2 \int \frac{{\rm d}^4p}{i(2 \pi)^4} [-iD^{\mu \nu}(k-p)]
\gamma_\mu \frac{-(\lambda_\alpha)^T}{2}
[-iG_{21}(p) ]
\gamma_\nu \frac{\lambda_\alpha}{2}.
\label{gap1}$$ In the following sections, we present explicit forms of $V(p)$ and $\Delta(p)$ and then solve their coupled equations (\[self1\]) and (\[gap1\]).
Fermion propagator under the axial-vector self-energy
-----------------------------------------------------
We, hereafter, take the static approximation for the gauge-boson propagator as $$\begin{aligned}
D_{\mu \nu}(q) \approx -\frac{g_{\mu \nu}}{{{\mbox{\boldmath $q$}}}^2+M^2} \label{cpl}\end{aligned}$$ where $M$ is an effective gauge boson mass originated from the Debye screening $M^2 \sim N_f g^2 \mu^2/(2\pi^2)$ [@LeBe].
Since typical momentum transfer $|{\bf q}|$ at high density is of the order of the chemical potential, we may further introduce the zero-range approximation [@Ripka] for the propagator as $$\begin{aligned}
D_{\mu \nu}(q) \approx -\frac{g_{\mu \nu}}{Q^2+M^2},
\label{prop-zero}\end{aligned}$$ with a typical momentum scale $Q$ of $O(\mu)$. This approximation corresponds to the Stoner model [@Yoshi], which is popular in solid-state physics, and stands on the same concept of the NJL model [@NJL] as well.
To proceed, we assume, without loss of generality, that total spin expectation value is oriented to the negative $z$-direction in the spin-polarized phase which is caused by the finite axial-vector mean-field along the $z$-axis [^1] . As shown in Ref. [@MaruTatsu], rotation symmetry is spontaneously broken down in this phase while axial symmetry around the $z$-axis is preserved. Then two Fermi seas of the different spin states are deformed accordingly.
Applying the Fierz transformation for the Fock exchange energy term (14) we can see that there appear the color-singlet scalar, pseudoscalar, vector and axial-vector self-energies (Appendix D). In general we must take into account these self-energies in $V$, $V=U_s+i\gamma_5 U_{ps}+\gamma_\mu U_v^\mu+\gamma_\mu\gamma_5U_{av}^\mu$ with the mean-fields $U_\alpha$. Here we introduce an ansatz: the Femri distribution holds the reflection symmetry with respect to the $p_x - p_y$ plane, and only the mean-field parts $U_s$, $U^0_v$ and $U^3_{av}$ are retained in $V$. Later we will see that the self-consistent solution is obtained with the zero-range approximation (\[prop-zero\]) under this ansatz.
In this paper, furthermore, we disregard the scalar mean-field $U_s$ and the time component of the vector mean-field $U_v^0$ for simplicity since they are irrelevant for the spin degree of freedom; $U_v^0$ has only a role to shift the total energy ot the chemical potential, and may not affect any other physical properties. On the other hand, $U_s$ may significantly influence the spin-polarization properties through changing the quark effective mass. Instead of introducing the scalar mean-field explicitly, however, we treat the quark mass as a variable parameter, and discuss its effect in the next section.
According to the above assumptions and considerations the self-energy $V$ in eq.(\[fullg\]) renders $$V = \gamma_3 \gamma_5 U_A, ~~~U_A\equiv U_{av}^3 ,$$ with the axial-vector mean-field $U_A$. Then the diagonal component of the Green function $G_{11}(p)$ is written as $$G_{11}(p)=\left[ G_A^{-1}-
\gamma_0 \Delta^\dagger \gamma_0 \tilde{G}_A \Delta \right]^{-1}$$ with $$\begin{aligned}
G_A^{-1}(p) &=& \sla{p}-m+\sla{\mu}-\gamma_5 \gamma_3 U_A, \\
\tilde{G}_A^{-1}(p) &=& \sla{p}-m-\sla{\mu}-\overline{\gamma_5 \gamma_3} U_A,\end{aligned}$$ where $\overline{\gamma_5 \gamma_3}=\gamma_5 \gamma_3$ and $G_A(p)$ is the Green function with the axial-vector mean-field $U_A$ which is determined self-consistently by way of eq. (\[self1\]).
Before constructing the gap function $\Delta$, we first find the single-particle spectra and their eigenspinors in the absence of $\Delta$, which is achieved by diagonalization of the operator $G_A^{-1}$. In the usual case of no spin polarization this procedure gives nothing but the free energy spectra and plane waves. Then we choose a gap structure on the basis of a physical consideration as in the usual BCS theory.
From the condition that $\det G_A^{-1}(p_0)|_{\mu=0} =0$ one can obtain four single-particle energies $\epsilon_\pm$ (positive energies) and $-\epsilon_\pm$ (negative energies), which are given as $$\begin{aligned}
&& \epsilon_{\pm}({{\mbox{\boldmath $p$}}}) =
\sqrt{{{\mbox{\boldmath $p$}}}^2 + U_A^2 + m^2 \pm
2 U_A \sqrt{m^2 + p_z^2 }},
\label{eig}\end{aligned}$$ where the sign factor $\pm 1$ being in front of $U_A$ indicates the energy splitting between different spin states due to the presence of the axial-vector self-energy, which corresponds to the [*exchange splitting*]{} in the non-relativistic electron system [@Yoshi]. In the following, we call the “spin”-up (-down) states for the states $\pm \epsilon_+$ ($\pm \epsilon_-$). Eq. (\[eig\]) also shows that each Fermi sea for the “spin”-up (-down) state should undergo a deformation and lose rotation symmetry, once $U_A$ is finite. This is a genuine relativistic effect [@MaruTatsu]; actually the exchange splitting never produces deformation of the Fermi sea in the non-relativistic ferromagnetism, e.g. in the Stoner model [@Yoshi].
Here, it would be interesting to see the peculiarities of the quark Fermi seas in the presence of the axial-vector self-energy. In Fig. [\[FS\]]{} we sketch the profile of the Fermi seas projected onto the $p_z-p_t$ plane ($p_t = \sqrt{p_x^2 + p_y^2 }$) for the cases of (a) $U_A < m$, (b) $U_A > m$ and (c) $m=0$. As is already mentioned these seas still hold the axial symmetry around the $z$-axis and the reflection symmetry with respect to the $p_x-p_y$ plane. The region surrounded by the outer line show the Fermi sea of “spin”-down quarks, and the shaded region is that of “spin”-up quarks.
=13.5cm =6.8cm
We can see in Fig. [\[FS\]]{}a that the Fermi seas for the “spin”-down and “spin”-up states are deformed in the prolate and oblate shapes, respectively, where the minimum of the single-particle energy still resides at the origin ${{\mbox{\boldmath $p$}}}=0$. When $U_A > m$ as shown in Fig. [\[FS\]]{}b, there appear two minima at the points $(p_t, p_z) = (0, \pm \sqrt{U_A^2 - m^2})$ for the “spin”-down quark. Hence in the massless limit, $m \rightarrow 0$, the Fermi sea is described by two identical spheres with radii $\mu$ in the momentum space, which are centered at the points $(p_t, p_z) = (0, \pm U_A)$ (see Fig. [\[FS\]]{}c).
In what follows we use subscript ‘$n$’$(=1,2,3,4)$ for notational convenience as $\epsilon_n$ which means $\{\epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4 \}=\{\epsilon_-,\epsilon_+,-\epsilon_-,-\epsilon_+\}$. We define the spinor $\phi_n({{\mbox{\boldmath $p$}}})$ that satisfies the equation $G_A^{-1}(p_0=\epsilon_n-\mu)\phi_n({{\mbox{\boldmath $p$}}})=0$, which corresponds to the eigenspinor with the single-particle energy $\epsilon_n$ in the absence of the quark Cooper pairing. The spinor $\phi_n({{\mbox{\boldmath $p$}}})$ is explicitly given as $$\phi_n({{\mbox{\boldmath $p$}}})={\cal N}_n \left(
\begin{array}{c}
(\epsilon_n -(-1)^n \beta_p -U_A) (p_x-{\rm i} p_y) p_z \\
-((-1)^n \beta_p +m ) p_t^2\\
\left[ -((-1)^n \beta_p +m)(\epsilon_n-m-U_A)+p_z^2 \right](p_x-{\rm i} p_y)\\
p_t^2 p_z
\end{array} \right), \label{spinor}
$$ where ${\cal N}_n$$=\sqrt{[\beta_p-(-1)^n m]\left[ \epsilon_n+U_A+(-1)^n \beta_p \right]/
(\epsilon_n \beta_p)}/(2p_t^2 p_z)$ and $\beta_p \equiv \sqrt{p_z^2+m^2}$. It is to be noted that the spinors $\phi_n$ do not return to the eigenspinors of spin operator $\sigma_z$ even when $U_A \rightarrow 0$, but become mixtures of them, see Appendix A. Introducing the projection operator $\Lambda_n=\phi_n \phi_n^\dagger$ with properties $\Lambda_{m} \Lambda_n =\Lambda_n \delta_{m n}$ and $\sum_n \Lambda_n = {\bf 1}$, we can recast $G_A(p)$ in the spectral representation into $$\begin{aligned}
G_A &=& \sum_n \frac{\Lambda_n}{p_0-\epsilon_n+\mu} \gamma_0 \\
G_A^{-1} &=& \sum_n (p_0-\epsilon_n+\mu) \gamma_0 \Lambda_n.\end{aligned}$$
Gap structure
-------------
In this subsection, we give the explicit form of the gap function $\Delta$ in the Dirac, color and flavor spaces, and then calculate the diagonal component of the full Green function $G_{11}(p)$ in eq. (\[G11i\]), provided that only the axial-vector self-energy is taken for $V(p)$ in eq. (\[self1\]). In general various types of the gap structures are possible in the Dirac, color and flavor spaces; they depend on the form of interaction and the quark mass [@BL; @QMass], especially on the strange quark mass [@SMass]. Here we suppose a simple gap structure from a physical consideration, disregarding the finite mass effect.
Using the the spinor $\phi_n({{\mbox{\boldmath $p$}}})$ we assume that the gap function $\Delta$ in eq. (\[gap1\]) has a following form in the color and flavor spaces: $$\begin{aligned}
\Delta({{\mbox{\boldmath $p$}}})=\sum_n \tilde{\Delta}_n({{\mbox{\boldmath $p$}}}) B_n({{\mbox{\boldmath $p$}}}) \label{aa}\end{aligned}$$ with the operator $B_n({{\mbox{\boldmath $p$}}})$, $$B_n({{\mbox{\boldmath $p$}}})=\gamma_0 \phi_{-n}({{\mbox{\boldmath $p$}}}) \phi_{n}^\dagger({{\mbox{\boldmath $p$}}}) \label{Bn}.$$ where the subscript ‘$-n$’($=-1,-2,-3,-4$) indicates that the single-particle energy in the spinor is replaced by that of opposite sign, $\epsilon_{-n} \equiv -\epsilon_n$, without change of “spin”.
One can easily see what kind of quark pairs the gap function $\Delta$ (\[aa\]) represents. Utilizing the property, $\phi^T_{-n'}(-{{\mbox{\boldmath $p$}}}) C\gamma_0 \phi_{n}({{\mbox{\boldmath $p$}}})$$\propto$$\delta_{n' n}$, one can find for the general spinor $\psi({{\mbox{\boldmath $p$}}})=\sum_n a_n({{\mbox{\boldmath $p$}}}) \phi_{n}({{\mbox{\boldmath $p$}}})$ with arbitrary coefficients $a_n$, $$\begin{aligned}
\bar{\psi}_{c} B_n \psi =
\psi^T(-p) C \gamma_0 \phi_{-n} \phi^\dagger_{n} \psi(p) \propto
a_{n}(-{{\mbox{\boldmath $p$}}})a_{n}({{\mbox{\boldmath $p$}}}).\end{aligned}$$ This equation clearly shows that two quarks included in the Cooper pairing have opposite momenta to each other and belong to the same energy eigenstate as illustrated in Fig. \[FS\]b.
Now we should note that the antisymmetric nature of the fermion self-energy imposes a constraint on the gap function [@BL; @PiRi], $$\begin{aligned}
C \Delta({{\mbox{\boldmath $p$}}}) C^{-1}=\Delta^T({-{\mbox{\boldmath $p$}}}).\end{aligned}$$ Since $B_n$ satisfies the relation $C B_n({{\mbox{\boldmath $p$}}}) C^{-1} = B_n^T({- {\mbox{\boldmath $p$}}})$, $\tilde{\Delta}_n({{\mbox{\boldmath $p$}}})$ must be a symmetric matrix in the spaces of internal degrees of freedom. Taking into account the property that the most attractive channel of the OGE interaction is the color antisymmetric ${\bf 3^*}$ one, it must be the flavor singlet state. Thus we can choose the form of the gap function as $$\left[\tilde{\Delta}_n({{\mbox{\boldmath $p$}}})\right]_{\alpha \beta, ~i j}=
\epsilon_{\alpha \beta 3} \epsilon_{i j} \Delta_n({{\mbox{\boldmath $p$}}}), \label{2SC}$$ where $(\alpha \beta)$ and $(i j)$ are indices in three-color and two-flavor spaces, respectively. The form of gap function (\[2SC\]) in the color and flavor spaces is familiar for two-flavor CSC [@BL; @CSC2].
Using the properties of $\Lambda_n({\mbox{\boldmath $p$}})$ and $B_n({\mbox{\boldmath $p$}})$, we then obtain an explicit form of $G_{11}(p)$ as $$\begin{aligned}
[G_{11}(p)]_{\alpha \beta, i j}
&=& \left\{ \sum_n \left[ (p_0+\mu-\epsilon_n) -
\frac{\Delta_n^\dagger \Delta_n}{p_0+\epsilon_n-\mu} \right] \gamma_0
\Lambda_n \right\}^{-1}_{\alpha \beta, i j} \nonumber \\
&=& \sum_n \frac{p_0-\mu+\epsilon_n}{
p_0^2-(\epsilon_n-\mu)^2-\frac{1}{2}{\rm Tr}[\Delta_n^\dagger \Delta_n]
(1-\delta_{3 \alpha})+i\eta}
\Lambda_n \gamma_0 \, \delta_{\alpha \beta}\, \delta_{i j}
\label{G11} \end{aligned}$$ with $$\Delta_n^\dagger \Delta_n =
{\rm diag} \left( |\Delta_n|^2, \ |\Delta_n|^2, \ 0 \right)\ \hbox{in the
color space},$$ where $\eta$ is a positive infinitesimal. The quasiparticle energies are obtained by looking for the poles of $G_{11}(p)$: $$\begin{aligned}
E_{n}({{\mbox{\boldmath $p$}}})&&=\left\{
\begin{array}{ll}
\sqrt{(\epsilon_n({{\mbox{\boldmath $p$}}})-\mu)^2+|\Delta_n({{\mbox{\boldmath $p$}}})|^2} & \mbox{for color 1, 2} \\
\sqrt{(\epsilon_n({{\mbox{\boldmath $p$}}})-\mu)^2} & \mbox{for color 3}
\end{array}
\right.
\label{qusiE}\end{aligned}$$ The quark number density $\rho_q$ is also given as $$\begin{aligned}
\rho_q &\equiv&
-i \int \frac{{\rm d}^4p}{(2\pi)^4}
{\rm Tr} \left[(G_{11}(p)- G_{11}(p)|_{\mu=0}) \gamma_0\right]
\label{BN} \\
& = &
N_f \sum_{n=1,2} \int \frac{{\rm d}^3 p}{(2\pi)^3}
\left[ \theta(\mu-\epsilon_{n}) + 2 v_{n}^2({{\mbox{\boldmath $p$}}})
- 2 \left(1-v_{-n}^2({{\mbox{\boldmath $p$}}})\right) \right]
\label{BN2}\end{aligned}$$ with $$v^2_n({{\mbox{\boldmath $p$}}})=\frac{1}{2}\left(1-\frac{\epsilon_n({{\mbox{\boldmath $p$}}})-\mu}{E_n({{\mbox{\boldmath $p$}}})} \right),
\label{cof}$$ where the first two terms in eq. (\[BN2\]) show the quark contributions, while the last term the anti-quark contribution; $v_{n}^2({{\mbox{\boldmath $p$}}})$ is the occupation probability of the quark pairs with momentum ${{\mbox{\boldmath $p$}}}$ and represents diffuseness of the momentum distribution.
Similarly we can know the self-consistent solutions satisfy our ansatz about the mean-fields in $V$. From the above solutions we can easily obtaine that ${\rm Tr}[G_{11}(p) i \gamma_5]=0$, ${\rm Tr}[G_{11}(p)\gamma_i]\propto p_i$, ${\rm Tr}[G_{11}(p)\gamma_5\gamma_0]\propto p_z$ and ${\rm Tr}[G_{11}(p)\gamma_5\gamma_{1,2}]\propto p_x,p_y$ . Hence the pseudoscalar mean-field $U_{ps}$, the space-component of vector mean-field $U_v^i$, the axial-vector mean-fields $U_{av}^0$ and $U_{av}^{1,2}$ are vanished after the integration over angles.
Equation for the superconducting gap function
---------------------------------------------
Using eq. (\[G11\]), the off-diagonal component of the full Green function $G(p)$, given in eq. (\[G21i\]), can be represented in the similar way as $$\begin{aligned}
G_{21}(p) =-\sum_n \frac{\gamma_0 B_n \gamma_0}
{p_0^2-(\epsilon_n-\mu)^2-|\Delta_n|^2+i\eta}
\Delta_n \lambda_2 \tau_2, \label{G21} \end{aligned}$$ where $\tau_2$ is the Pauli matrix in the two-flavor space. Substituting eq. (\[G21\]) into the gap equation (\[gap1\]) and using the identity $\sum_{a=1}^8 (\lambda_a)^T \lambda_2 \lambda_a=-8/3\, \lambda_2$, we obtain $$\sum_{n'} B_{n'}({{\mbox{\boldmath $k$}}}) \Delta_{n'}({{\mbox{\boldmath $k$}}}) =
-i\frac{2}{3} g^2 \int \frac{{\rm d}^4 p}{(2\pi)^4}
D_{\mu\nu}{(k-p)} \sum_n
\left[ \frac{\gamma^\mu \gamma_0 B_n({{\mbox{\boldmath $p$}}}) \gamma_0 \gamma^\nu }
{p_0^2-(\epsilon_n-\mu)^2-|\Delta_n|^2+i\eta} \right]
\Delta_n({{\mbox{\boldmath $p$}}}), \label{gap1i}$$ where the factor $2/3$ is simply the Fierz coefficient for the color and flavor degrees of freedom (Appendix D). Furthermore multiplying both sides of eq. (\[gap1i\]) by $B_{n'}^\dagger({{\mbox{\boldmath $k$}}})$ and taking trace with respect to the Dirac indices, the coupled equations for the gap functions $\Delta_n$ are obtained after $p_0$ integration, $$\begin{aligned}
\Delta_{n'}({{\mbox{\boldmath $k$}}})=-\frac{2}{3} g^2 \int \frac{{\rm d}^3 p}{(2\pi)^3}
D_{\mu\nu}{(k-p)}
\sum_n T_{n' n}^{\mu \nu}({{\mbox{\boldmath $k$}}},{{\mbox{\boldmath $p$}}})
\frac{\Delta_n({{\mbox{\boldmath $p$}}})}{2 E_n({{\mbox{\boldmath $p$}}})} \label{GAP1j} \end{aligned}$$ where the function $T_{n' n}^{\mu \nu}({{\mbox{\boldmath $k$}}}, {{\mbox{\boldmath $p$}}})$ is defined as $$\begin{aligned}
T_{n' n}^{\mu \nu}({{\mbox{\boldmath $k$}}},{{\mbox{\boldmath $p$}}}) &\equiv& {\rm Tr}
\left[ B_{n'}^\dagger({{\mbox{\boldmath $k$}}}) \gamma^\mu \gamma_0
B_n({{\mbox{\boldmath $p$}}}) \gamma_0 \gamma^\nu \right] =
(\bar{\phi}_{-n'}({{\mbox{\boldmath $k$}}}) \gamma^\mu \phi_{-n}({{\mbox{\boldmath $p$}}}))
(\bar{\phi}_{n}({{\mbox{\boldmath $p$}}}) \gamma^\nu \phi_{n'}({{\mbox{\boldmath $k$}}})) \label{MN},\end{aligned}$$ a decomposition of $B_n({{\mbox{\boldmath $p$}}})$ in terms of gamma matrices and its properties are given in Appendix B.
Here we take the zero-range approximation in eq. (\[prop-zero\]). In terms of the polar coordinates ${{\mbox{\boldmath $p$}}}=\{p,\theta_p,\phi_p\}$, we can consider that the gap function $\Delta_n({{\mbox{\boldmath $p$}}})$ does not depend on the horizontal angle $\phi_p$ due to axial-symmetry around the $p_z$-axis. Thus we can explicitly perform the integration with respect to the angle $\phi_p$ in the gap equation (\[GAP1j\]): $$\begin{aligned}
\Delta_{n'}(k,\theta_k)=\frac{2}{3} \tilde{g}^2
\int \frac{{\rm d}p\, {\rm d}\theta_p}{(2\pi)^2} p^2 \sin\theta_p
\sum_n T_{n' n}(k,\theta_k,p,\theta_p)
\frac{\Delta_n(p,\theta_p)}{2 E_n(p,\theta_p)} \label{GAP1} \end{aligned}$$ with the effective coupling constant $\tilde{g}\equiv g/\sqrt{Q^2+M^2}$. As seen from the above equation, each of the gap functions couples with others by the function $T_{n' n}(k,\theta_k,p,\theta_p)$ defined as $$T_{n' n}(k,\theta_k,p,\theta_p) \equiv
\int \frac{{\rm d} \phi_p}{2\pi} g_{\mu \nu}T_{n' n}^{\mu \nu}({{\mbox{\boldmath $k$}}},{{\mbox{\boldmath $p$}}})
= \frac{k_t p_t}{2|\epsilon_{n'}({{\mbox{\boldmath $k$}}})| |\epsilon_n({{\mbox{\boldmath $p$}}})|}
\left[ (-1)^{n'+n} \frac{2 m^2+k_z p_z}{\beta_p \beta_k}+1 \right],
\label{Tss1}$$ where $p_t$$\equiv$$p\sin\theta_p$ and $p_z$$\equiv$$p\cos\theta_p$ and the same for $k_t$ and $k_z$. The term proportional to $p_z$ in eq. (\[Tss1\]) will disappear after the integration over $\theta_p$.
Equation for the axial-vector mean-field $U_A$
----------------------------------------------
Using eqs. (\[qusiE\]) and (\[cof\]), $G_{11}(p)$ is recasted in the form, $$\begin{aligned}
&&[G_{11}(p)]_{\alpha \beta, ~i j}
=\left[\sum_n \left(\frac{1-v_n^2({{\mbox{\boldmath $p$}}})}{p_0-E_n+i\eta}+
\frac{v_n^2({{\mbox{\boldmath $p$}}})}{p_0+E_n-i\eta} \right)
{\rm e}^{ip_0 \eta} \Lambda_n({{\mbox{\boldmath $p$}}}) \gamma_0 \right]
\delta_{\alpha \beta} \delta_{i j}. \end{aligned}$$ Substituting the above equation into eq. (\[self1\]), and integrating with respect to $p_0$, we obtain the self-consistent equation for $U_A$ in the zero-range approximation: $$\begin{aligned}
U_A
&=& -\frac{2}{9} \frac{N_f}{2} \tilde{g}^2 \int \frac{{\rm d}^3 p}{(2\pi)^3}
\sum_n \left[\theta(\mu-\epsilon_n({\mbox{\boldmath $p$}})) + 2 v_n^2({{\mbox{\boldmath $p$}}})\right] S_n({{\mbox{\boldmath $p$}}}),
\label{UA1} \end{aligned}$$ where the factor $-2/9$ stems from the Fierz coefficient of the color-singlet axial-vector channel of the OGE interaction (Appendix D), and $S_n({{\mbox{\boldmath $p$}}})$ is the expectation value of the spin operator, $\sigma_z$$\equiv$$-\gamma_0 \gamma_5 \gamma_3$, with respect to the spinor $\phi_n({{\mbox{\boldmath $p$}}})$: $$\begin{aligned}
S_n({{\mbox{\boldmath $p$}}}) \equiv {\rm Tr} ( \gamma_5 \gamma_3 \Lambda_n({{\mbox{\boldmath $p$}}}) \gamma_0 ) =
\phi^\dagger_n({{\mbox{\boldmath $p$}}}) (-\sigma_z) \phi_n({{\mbox{\boldmath $p$}}})=
\frac{U_A +(-1)^n \beta_p}{\epsilon_n({\mbox{\boldmath $p$}})}. \label{spin1}\end{aligned}$$ Thus $U_A$ is related to the expectation value of $\sigma_z$ summing over the state with momentum ${{\mbox{\boldmath $p$}}}$. An effect of the Cooper pairing enters into eq. (\[UA1\]) through the function $v_n({{\mbox{\boldmath $p$}}})$.
Weak coupling approximation
---------------------------
In this subsection we consider a high-density limit, which means the weak coupling limit due to asymptotic freedom of QCD, and then disregards the anti-quark pairing and contributions from the negative-energy sea (the Dirac sea) in eq. (\[GAP1\]). Actually it costs more energy to form the anti-quark pairing than the quark pairing for a large chemical potential. Taking the approximation, we also disregard the contribution from anti-quarks to calculate the quark number density in eq. (\[BN2\]) and the axial-vector mean-field eq. (\[UA1\]) consistently. [^2] In the following calculations we define gap functions of the quark pairing by subscript $\pm$ whcih corresponds to the “spin”-up (-down) of positive-energy states as $\Delta_- \equiv \Delta_1$ and $\Delta_+ \equiv \Delta_2$. The other symbols with the subscript $\pm$ have the same meaning, e.g., $\phi_\mp \equiv \phi_{1,2}$.
In addition, we assume that only quarks near the Fermi surface form the Cooper pairs, and thereby replace the gap function by an approximated form, $$\Delta_\pm ({{\mbox{\boldmath $p$}}}) \rightarrow \Delta_\pm({{\mbox{\boldmath $p$}}}) \theta(\delta-|\epsilon_\pm-\mu|),$$ where $\delta$ is a cut-off parameter around the Fermi surface. The function $\theta(\delta-|\epsilon_\pm({\mbox{\boldmath $p$}})-\mu|)$ is also regarded as a form factor to regularize the integration in the gap equation [@PiRi]. The step-function form factor mimics the asymptotic freedom; inner particles in Fermi sea costs large kinetic energy to create the pairing and takes large momentum transfer which indicates that coupling of this inner-process is small. There, however, might be more realistic form factors for finite density QCD, which are smoother functions of momentum and $\mu$ than ours, we think that they makes little change on qualitative results of the CSC and spin polarization. There are models with other form factors or cut-off functions [@CSC2; @CSC3].
Looking at the structure of the gap equation (\[GAP1\]) with (\[Tss1\]), one can find that the gap function is exactly parametrized as (Appendix C) $$\begin{aligned}
\Delta_\pm({{\mbox{\boldmath $p$}}})&=&\frac{p_t}{\epsilon_\pm({\mbox{\boldmath $p$}})}
\left( \mp\frac{m}{\beta_p} R + F \right) \nonumber\\
&\equiv&\frac{p_t}{\epsilon_\pm({\mbox{\boldmath $p$}})}\hat\Delta_\pm({\mbox{\boldmath $p$}}) \label{para}, \end{aligned}$$ where $R$ and $F$ are some constants and represent the antisymmetric and symmetric combinations of the gap functions; $R=\beta_p/m(\hat\Delta_--\hat\Delta_+)=\beta_p/(p_tm)(\epsilon_-\Delta_- -\epsilon_+\Delta_+)$ and $F=\hat\Delta_-+\hat\Delta_+=1/p_t(\epsilon_-\Delta_- +\epsilon_+\Delta_+)$. Their magnitudes are determined by the coupled equations; $$\begin{aligned}
F&=& \frac{2}{3}\tilde{g}^2
\int \frac{{\rm d}p\, {\rm d}\theta_p}{(2\pi)^2} p^2 \sin\theta_p
\frac{1}{4}\left[Q_+({\mbox{\boldmath $p$}}) (F - \frac{m}{\beta_p} R)
+Q_-({\mbox{\boldmath $p$}}) (F + \frac{m}{\beta_p} R) \right]
\label{eqF} \\
R&=& \frac{2}{3}\tilde{g}^2
\int \frac{{\rm d}p\, {\rm d}\theta_p}{(2\pi)^2} p^2 \sin\theta_p
\frac{m}{2\beta_p}\left[-Q_+({\mbox{\boldmath $p$}}) (F - \frac{m}{\beta_p} R)
+Q_-({\mbox{\boldmath $p$}}) (F + \frac{m}{\beta_p} R) \right]
\label{eqR}, \\
&&\mbox{where}~~~ Q_{\pm}({\mbox{\boldmath $p$}})=
\frac{p_t^2}{\epsilon_\pm({\mbox{\boldmath $p$}})^2 E_\pm({\mbox{\boldmath $p$}})}
\theta(\delta-|\epsilon_\pm({\mbox{\boldmath $p$}}) -\mu|).
\nonumber\end{aligned}$$ We can obviously see that $R\rightarrow 0$ as $m\rightarrow 0$.
Here we examine the polar-angle dependence of the anisotropic gap function at the Fermi surface $\Delta_\pm(p^F,\theta)$. The Fermi momentum $p^F(\theta)$ of each “spin” eigenstate is given as $$\begin{aligned}
p_t&=&p^F_\pm(\theta) \sin\theta, ~~p_z=p^F_\pm(\theta) \cos\theta ~~~\mbox{with}
\nonumber \\
p^F_\pm(\theta)&=&\left[ \mu^2-m^2+U_A^2 \cos(2 \theta)
\mp U_A \sqrt{4 \mu^2\cos^2\theta+4 m^2\sin^2\theta
-U_A^2\sin^2(2 \theta)} \right]^{1/2}, \end{aligned}$$ where the subscript $\pm$ corresponds to the “spin”-up (-down) state again. Substituting the above formula into the gap function (\[para\]), we get $$\Delta_\pm \left( p^F_\pm,\theta \right)=\frac{p^F_\pm(\theta) \sin \theta}{\mu}
\left[ \mp\frac{m}{\sqrt{m^2+\left(p^F_\pm(\theta) \cos \theta \right)^2}} R + F \right].$$ Note that this form exhibits a $P$- wave pairing nature: it is a genuine relativistic effect by the Dirac spinors (Appendix B). We show a schematic view of the above gap functions in Fig. \[Delta\].
As characteristic features, both the gap functions vanish at poles ($\theta=0,\pi$) and take maximal values near equator ($\theta=\pi/2$), keeping the relation, $\Delta_- \ge \Delta_+$ [^3]. Suppression of $\Delta_+$ and enhancement of $\Delta_-$ at $\theta=\pi/2$ for the case of $m\neq 0$ (Fig. \[Delta\]b) are originated from a finite value of $R$, while they vanish if quark is taken to be massless (Fig. \[Delta\]a). The anisotropic gap functions give rise to the different diffuseness in the momentum distribution of the two “spin” eigenstates, and thereby make some effects on spin polarization, unlike in the normal phase. The anisotropic diffuseness has two effects that it obscures the deformation in the momentum distribution due to their angle dependence and enlarges the difference of the state density between the two “spin” eigenstates through the relation $\Delta_- \ge \Delta_+$.
Results and Discussions
=======================
In this section we solve the coupled equations (\[UA1\]), (\[eqF\]) and (\[eqR\]) and investigate the effects of the superconducting gap on spin polarization. Before going to numerical calculations of $U_A$,$R$ and $F$, each of which is coupled with others by the self-consistent equations, it is instructive to see their parameter dependence by treating one of them as an input parameter. First we show $R$ and $F$ as functions of $U_A$ in Fig. \[ParaU\] where $\mu = 400, 450$ MeV and $\delta = 0.1 \mu$.
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$R$ starts from zero and almost linearly increases with $U_A$ (Fig. \[ParaU\]a), which is understood by seeing that $R$ is proportional to the difference, ${\hat\Delta}_--{\hat\Delta}_+$, due to finite $U_A$, see eq. (\[para\]) or Appendix C. Thus $R$ is induced by $U_A$ and closely coupled with it.
As for the behavior of $F$, it is barely affected by $U_A$ (slight decreasing with $U_A$ in the numerical value) (Fig. \[ParaU\]b). As seen from the dependence on $\mu$ the magnitude of $F$ is almost determined by the volume of the phase space in the gap equation, that is, by $\mu$ and $\delta$. This reflects the fact that $F$ is related to the sum, ${\hat\Delta}_++{\hat\Delta}_-$ (Appendix C). Thus we expect that $F$ increases with density when other parameters are fixed.
From the above results we have found that $F$ is not so much influenced by $U_A$. Next we examine the behavior of $U_A$ and $R$ when $F$ is treated to be an input parameter. In Fig. \[ParaF\] we show the parameter dependence of $U_A$ and $R$ on $F$, where $\mu=450$ MeV and we use three values of the cut-off parameter $\delta =0.05 \mu$, $0.1\mu$ and $0.15\mu$, and add the result of $U_A$ in the normal phase ($\delta=0$).
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Comparing the dependence of $U_A$ on $F$ (Fig. \[ParaF\]a) with that in the normal phase, we see a characteristic behavior for different values of $\delta$: there are regions where $U_A$ is larger than that in the normal phase for relatively small $F$, and this region seems to extend with $\delta$. On the other hand results from the self-consistent calculations show that $F$ becomes larger with $\delta$ so that its value corresponds to a region where $U_A$ is comparable with or slightly less than that in the normal phase, for any value of the chemical potential. This situation seems to be qualitatively unchanged, once the ratio of the effective coupling constants in the axial-vector channel $G_{axial}$ and the diquark channel $G_{diq}$ is kept, $G_{axial}:G_{diq}=2/9:2/3$, which comes from the Fierz transformation for color and flavor (Appendix D). However, if the coupling constant in each channel is taken independently, our results might be changed qualitatively.
Seeing the results for $R$ in Fig. \[ParaF\]b, we find that $R$ increases with $\delta$ due to the growth of the phase space and is hardly affected by $F$ except the region of small $F$ where $U_A$ varies rapidly as shown in Fig. \[ParaF\]a: it also shows that $R$ is closely related to $U_A$.
These parameter dependences also suggest that the regularization scheme for the gap equation, i.e.,the sharp momentum cut-off function, the form factor, etc., will give rise to a qualitative change to $U_A$. In the present cut-off function, $\theta(\delta-|\epsilon_\pm-\mu|)$, $U_A$ (spin polarization) coexists with CSC, except a slight competition, as will be shown later.
Self-consistent solutions
--------------------------
We demonstrate some self-consistent solutions here. Since we have little information to determine the values of the parameters $\tilde{g}$ and $\delta$ (there may be other more reasonable form factors than the present cut-off function), and our purpose is to figure out qualitative properties of spin polarization in the color superconducting phase, we mainly set in the following calculations them as $\tilde{g}=0.13$ MeV$^{-1}$ and $\delta=0.1\mu$, for example, which is not so far from the couplings in NJL-like models [@CSC3; @Ripka; @NJL].
We first examine spin polarization in the absence of CSC. In Fig. \[UA\] we show the the axial-vector mean-field $U_A$, with $\Delta_\pm $ being set to be zero, as a function of baryon number density $\rho_B (\equiv \rho_q /3)$ relative to the normal nuclear density $\rho_0=0.16$ fm$^{-3}$ for $m=14 \sim 25$ MeV (dashed lines).
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It is seen that the axial-vector mean-field (spin polarization) appears above a critical density and becomes larger as baryon number density gets higher. Moreover, the results for different values of the quark mass show that spin polarization grows more for the larger quark mass. This is because a large quark mass gives rise to much difference in the Fermi seas of two opposite “spin” states, which leads to growth of the exchange energy in the axial-vector channel.
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Next we solve the coupled equations (\[UA1\]), (\[eqF\]) and (\[eqR\]). Results for $U_A$, $R$ and $F$ are shown in Fig. \[UA\] (solid lines) and Fig. \[FR\], for values of the quark mass $m=14 \sim 25$ MeV. It is found again, by comparing these cases of the quark mass, that $U_A$ is very sensitive to the quark mass and increases with it as in the absence of CSC (Fig. \[UA\]). For the behavior of the gap functions, $R$ is induced by $U_A$ and both of $F$ and $R$ increase with $\rho_B$ due to the growth of the Fermi surface (Fig. \[FR\]). It is also seen that $F$ is not sensitive to the quark mass. To see the bulk behavior of pairing gap as a function of baryon number density, we also show, in Fig. \[Dpm\], their mean values with respect to the polar angle on the Fermi surface; $$\langle \Delta_\pm \rangle \equiv
\left( \int_0^\pi {\rm d}\theta \frac{\sin\theta}{2} \Delta_\pm^2 \right)^{1/2}.$$ The mean values $\langle \Delta_\pm \rangle$ begin to split with each other at a density where $U_A$ becomes finite. This reflects that $R$ is induced by $U_A$ and then has a negative (positive) contribution to $\Delta_+$ ($\Delta_-$).
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Here we would like to comment on the magnitude of $\langle \Delta_\pm \rangle$. These should be compared with the usual uniform gap function, and may look very large values of $O$(GeV) in our case. However these values would be largely reduced by taking a smooth form factor which models asymptotic freedom of QCD[@CSC3]; it further reduces the integral value in the gap equation, compared with our sharp cut-off function. In Fig. \[Spin\] we show the expectation value of the spin operator per quark, $\langle \sigma_z/N_q \rangle$, as a function of $\rho_B/\rho_0$ with and without the superconducting gap. The critical density becomes lower as the quark mass increases, and the peak positions of $\langle \sigma_z/N_q \rangle$ are located at relatively lower densities in each quark mass. The magnitude of $\langle \sigma_z/N_q \rangle$ is to be compared with $1$ for a free quark, because $|\psi_s^\dagger \sigma_z \psi_s/\psi_s^\dagger\psi_s|=1$ at the rest frame for the free spinor $\psi_s$. We arrange the results of three quark masses $m=14\sim 16$ MeV by $1$ MeV in Fig. \[Spin\]a to show a high sensitivity of spin polarization to the quark mass, which implies that the exchange energy from the attractive axial-vector interaction is strongly enhanced by the quark mass to produce the large axial-vector mean-field.
The exchange energy is also enhanced by larger chemical potential and the resulting axial-vector mean-field increases with it (see Fig.\[UA\]). But the spin expectation value per quark, which is relative to the axial-vector mean-field per quark ($\propto U_A/N_q$), has an upper limit since the increase of $N_q$ is far superior to that of $U_A$ for larger chemical potential, which gives rise to the peak positions in Fig. \[Spin\].
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The quark mass is very important in relation to the breaking of chiral symmetry in QCD. Models incorporating chiral dynamics have indicated that the dynamical mass becomes smaller as chiral symmetry is restored at a high density, while the current quark mass is small and explicitly breaks it [@HatsuKuni]. In our model, on the other hand, we treat the quark mass $m$ as a variable parameter so that we may simulate a change of the dynamical mass. In order to further examine the effect of the quark mass on spin polarization, we show the mass dependence at densities $\rho_B =$ 5$\rho_0$, $\rho_B =$ 10$\rho_0$ and $\rho_B =$ 15$\rho_0$ for the cases with and without the superconducting gap in Fig. \[SpinMass\].
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Spin polarization increases with the quark mass in all the three densities. In the figure we exhibit only the results for a narrow region of the mass parameter ($m=13 \sim 20$ MeV), while as for larger masses of $O(100 \mbox{MeV})$ (order of the strange quark mass) spin polarization monotonically increases without singular oscillations. Critical values of the quark mass at which spin polarization disappears become smaller as density increases in both cases.
In relation of $U_A$ to $m$ we can derive an exact result in the massless limit, $m \rightarrow 0$. In the normal phase where $\Delta =0$, eq. (\[UA1\]) becomes $$U_A
= -\frac{2}{9} \tilde{g}^2 \sum_{n=1,2} \int \frac{{\rm d}^3 p}{(2\pi)^3}
\, 3 \, \theta(\mu-\epsilon_n({{\mbox{\boldmath $p$}}}))
\frac{U_A +(-1)^n |p_z|}{\epsilon_n({{\mbox{\boldmath $p$}}})}
\label{uam0}$$ with $${\epsilon_\pm({{\mbox{\boldmath $p$}}})} = \sqrt{ \left(|p_z| \pm U_A \right)^2 + p_t^2 } .$$ The right-hand side of the above equation can be analytically integrated to give $$\begin{aligned}
U_A
&=& -\frac{2}{9} {\tilde g}^2 \frac{4 \pi}{(2\pi)^3}
3 \int^{\mu - U_A}_0 d p_z \int^{\sqrt{\mu^2 - (p_z + U_A)^2}}_0
d p_t p_t \frac{U_A + p_z}{\sqrt{ (p_z + U_A)^2 + p_t^2}}
\nonumber \\
&& -\frac{2}{9} {\tilde g}^2 \frac{4 \pi}{(2\pi)^3}
3 \int^{\mu + U_A}_0 d p_z \int^{\sqrt{\mu^2 - (p_z - U_A)^2}}_0
d p_t p_t \frac{U_A - p_z}{\sqrt{ (p_z - U_A)^2 + p_t^2}}
\nonumber \\
&=& 0
\label{uam01}\end{aligned}$$ Here we have assumed that $\mu > U_A$. In the massless limit, the Fermi sea is described by two complete spheres in the momentum space with radii $\mu$, whose centers are located at $(p_t, p_z) = (0, \pm U_A)$ (see Fig. \[FS\]c). The momentum distribution for quarks in the “spin”-down state occupies these two spheres, while the “spin”-up state does their overlap region (shaded). In the above integration for the “spin”-down quarks, the expectation value of the spin operator given by quarks with $0 \le p_z\le U_A$ is canceled with that of quarks with $U_A \le p_z \le 2 U_A$. The remaining contribution from the region, $2 U_A \le p_z \le \mu + U_A$, cancels with that by the “spin”-up quarks. Thus we can see that spin polarization disappears as $m \rightarrow 0$ in the absence of CSC.
This analytical result that $U_A \rightarrow 0$ as $m \rightarrow 0$ can be also understood as follows. The eigenstates of non-interacting massless fermions are classified by the definite helicity states: the positive energy state is right-handed (left-handed) with positive (negative) helicity, while the negative energy state those with negative (positive) helicity. This property is not spoiled by introducing the axial-vector mean-field, when we extend the meaning of helicity; the Dirac equations for the “left-” and “right-handed” positive-energy fermion fields $\psi_{L,R}$ are now given as $$\begin{aligned}
&&(p_0+{{\mbox{\boldmath $p$}}}\cdot \mbox{\boldmath $\sigma$}+U_A\sigma_3)\psi_L=0 \\
&&(p_0-{{\mbox{\boldmath $p$}}}\cdot \mbox{\boldmath $\sigma$}+U_A\sigma_3)\psi_R=0,\end{aligned}$$ which give the eigenvalues, $p_0=\sqrt{p_t^2+(p_z+U_A)^2}(\equiv \epsilon_L({{\mbox{\boldmath $p$}}}))$ for $\psi_L$ and $p_0=\sqrt{p_t^2+(p_z-U_A)^2}(\equiv \epsilon_R({{\mbox{\boldmath $p$}}}))$ for $\psi_R$. $\psi_L$ ($\psi_R$) is the eigenstate of generalized helicity $h=\mp 1$ projected onto the shifted momentum ${{\mbox{\boldmath $p$}}'}=\{ p_x, p_y, p_z \pm U_A \}$. If $\mu \neq 0$ they form the spherical Fermi seas, see Fig. \[FS\]c. Here it would be interesting to compare these eigenvalues with the limit form of $\epsilon_\pm({{\mbox{\boldmath $p$}}})$ in eq. (\[eig\]), $$\epsilon_\pm({{\mbox{\boldmath $p$}}}) \longrightarrow \sqrt{p_t^2+(|p_z| \pm U_A)^2} ~{\rm as}~m\rightarrow 0.$$ Then we can see the relations: $\epsilon_\pm({{\mbox{\boldmath $p$}}})=\epsilon_L({{\mbox{\boldmath $p$}}})\theta(\pm p_z)+\epsilon_R({{\mbox{\boldmath $p$}}})\theta(\mp p_z)$, which clearly show that the two Fermi seas of the eigenspinors (\[spinor\]) give the same Fermi seas of $\psi_{L,R}$ for a given chemical potential $\mu$. Thus we can take an alternative view of the Fermi seas in terms of the definite helicity states by rearranging the eigenspinor (\[spinor\]) properly in the massless limit. In each Fermi sea for $\psi_{L,R}$ the particle number with the definite $h$ becomes the same, and thereby the total spin-expectation value becomes vanished.
In the color superconducting phase, on the other hand, the situation is different because the momentum distribution becomes diffused due to the creation of the Cooper pairs near the Fermi surface. For $m \rightarrow 0$ and $\Delta_\pm \neq 0$, eq. (\[UA1\]) becomes $$U_A
= -\frac{2}{9} \tilde{g}^2 \sum_{n = 1, 2} \int \frac{{\rm d}^3 p}{(2\pi)^3}
2 v_n^2({{\mbox{\boldmath $p$}}})
\frac{U_A +(-1)^n |p_z|}{\epsilon_n({{\mbox{\boldmath $p$}}})} ,
\label{uas}$$ where $v_n^2({{\mbox{\boldmath $p$}}})$ indicates the diffused part of the momentum distribution, defined in eq. (\[cof\]). Here we should note that the gap functions $\Delta_{\pm}$ are still non zero even at $m=0$. The diffused part, however, give no contribution to the spin polarization from the viewpoint of the helicity eigenstates. The gap function in the massless limit becomes $$\Delta_\pm (p,\theta)=
\frac{p_t}{\sqrt{ (|p_z| \pm U_A)^2 + p_t^2 }} F,$$ see Fig. \[Delta\]a. The diffuseness from the above gap function has an equal contribution to the two complete Fermi spheres of chirality. Thus the total spin, which is obtained by summing up these momentum distributions, should be zero in the massless limit even if the CSC is taken into account.
To summarize we show a phase diagram for the quark mass and baryon number density in Fig. \[PD\] where we add the result of $\tilde{g}=0.26$MeV$^{-1}$ to see the dependence on the coupling constant.
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The lines indicate the critical mass at a fixed density (at regions above the lines spin polarization arises). We can confirm that the critical mass becomes smaller with the increase of the density, and spin polarization occurs at moderate densities ($\rho_B = 3 \sim 4 \rho_0$) if the coupling is strong enough even though quark mass is taken to be smaller as a simulation for change of dynamical mass (restoration of chiral symmetry).
Here we would like to understand how the gap function affects spin polarization and brings about a slight reduction of it. In the spin-polarized phase, the momentum distribution is deformed from the simple spherical shape. As mentioned in Ref. [@MaruTatsu], the deformation is induced by finite $U_A$ and feeds back to $U_A$ in a self-consistent manner. In the color superconducting phase, diffuseness caused by the Cooper pairing in the momentum distribution depends on the polar angle and then has an influence on the deformation. As can be expected from the polar-angle dependence of the gap function, diffuseness tends to obscure the deformation.
From the consideration of the spin expectation values by spinors (\[spinor\]) near the Fermi surfaces; $$\phi_\pm^\dagger (-\sigma_z) \phi_\pm=
\frac{U_A \pm \sqrt{p_z^2+m^2}}{\epsilon_\pm }\approx
\frac{U_A \pm \sqrt{p_z^2+m^2}}{\mu} \label{NearFermi},$$ where $\phi_\mp \equiv \phi_{1, 2}$ for two “spins”. The difference of the spin expectation value between two “spin” states $\phi_\pm$ is largely affected by high-$p_z$ regions or regions near both poles ($\theta=0,\pi$). Thus the large deformation along the $z$-axis seems to enhance spin polarization. In order to specify to what extent the Fermi sea is deformed, we calculate the quadrupole deformation of the momentum distribution defined by $$Q_2 \equiv 3\langle p_z^2 \rangle/\langle {{\mbox{\boldmath $p$}}}^2 \rangle-1 \label{quadru}.$$ In Fig. \[Q2\], we show $Q_2$ as a function of $U_A$ at $\mu=450$ MeV in the normal phase and in the color superconducting phase in which the gap functions are given by their equations for fixed $U_A$.
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From this result of $Q_2$ deformation, we can see that the diffused part near the Fermi surface obscures the deformation then gives an opposite effect against $Q_2$, and thus reduces spin polarization.
Nevertheless the gap function has another effect on spin polarization. It is to be noted that the qualitative relation, $\Delta_- \ge \Delta_+$, is always retained as seen from Fig. \[Delta\] and then has a effect to enlarge the difference of the state density between the two “spin” states. This effect is expected to enhance spin polarization since the difference of the spin expectation value by each spinor (\[NearFermi\]) near the equator ($\theta=\pi/2$), so that $p_z\approx0$, seems to depend only on the difference of the state density. To see it in both the normal and color superconducting phases, we define that $N_{up}$ ($N_{down}$) is the state density of the “spin”-up (-down) state and show their difference by $dN \equiv N_{down}-N_{up}$, only in the first two colors, as a function of $U_A$ in Fig. \[Frac\] at $\mu=450$ MeV. The result indicates that the gap functions slightly enhance $dN$ than normal phase.
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From the above discussions spin polarization is significantly influenced by both the deformation and the state density in each “spin” state. As a result of the self-consistent calculation in the color superconducting phase, the reduction effect on the deformation is slightly superior to the enhancement effect from the difference of the state densities, and the pairing effect finally reduces spin polarization than in the normal phase. It, however, should be noted that this qualitative conclusion about whether CSC enhances spin polarization than normal phase or not is very delicate and may be changed depending on the regularization scheme, as already mentioned. Moreover other types of pairing which are not considered here, e.g. pairing of the “spin” -up and -down states, may gives rise to qualitatively different results, while it is very difficult to see which type of pairing is energetically favored.
Finally we would like to comment on the coupling of the spin polarized quark matter with the external magnetic field; quark fields couple with the magnetic field through its anomalous magnetic moment. The magnetic interaction is described by the Gordon identity for the gauge coupling term: $g_L e^*/2m (\bar{\psi} \sigma_{\mu\nu} \psi) F^{\mu\nu}$ where $g_L$ is a form factor and $e^*$ an effective charge. [^4] In quark matter a magnetic moment is given as the expectation value $\langle \sigma_{i j} \rangle$ with respect to the ground state. In our model only $\langle \sigma_{12} \rangle$ is nonzero, and the magnetic moment per quark is given as
$$M_z \equiv \langle \sigma_{1 2}/ N_q \rangle =
\frac{1}{\rho_q}
\sum_{n=1, 2} \int \frac{d^3p}{(2\pi)^3} [ 2 v_n^2({{\mbox{\boldmath $p$}}})+\theta(\mu-\epsilon_n)]
\bar{\phi}_n({{\mbox{\boldmath $p$}}}) \sigma_{12} \phi_n({{\mbox{\boldmath $p$}}}).
\label{MZ}$$
Note that the expectation value of $\sigma_{12}$ by the spinor does not depend on $U_A$; $\bar{\phi}_\pm ({{\mbox{\boldmath $p$}}}) \sigma_{12} \phi_\pm ({{\mbox{\boldmath $p$}}})=\mp m/\beta_p$, so that $M_z$ reflects only the asymmetry in the momentum distribution due to the axial-vector mean-field.
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In Fig. \[Mag\], $M_z$ is given as a function of baryon number density. This indicates that resulting ground state also holds ferromagnetism (spontaneous magnetization).
Summary and Concluding remarks
==============================
In this paper we have examined spin polarization in quark matter in the color superconducting phase. We have introduced the axial-vector self-energy and the quark pair field (the gap function), whose forms are derived from the one-gluon-exchange interaction by way of the Fierz transformation under the zero-range approximation. Within the relativistic Hartree-Fock framework we have evaluated their magnitudes in a self-consistent manner by way of the coupled Schwinger-Dyson equations.
As a result of numerical calculations spontaneous spin polarization occurs at a high density for a finite quark mass in the absence of CSC, while it never appears for massless quarks as an analytical result. In the spin-polarized phase the single-particle energies corresponding to spin degrees of freedom, which are degenerate in the non-interacting system, are split by the exchange energy in the axial-vector channel. Each Fermi sea of the single-particle energy deforms in a different way, which causes an asymmetry in the two Fermi seas and then induces the axial-vector mean-field in a self-consistent manner. In the superconducting phase, however, spin polarization is slightly reduced by the pairing effect; it is caused by competition between reduction of the deformation and enhancement of the difference in the phase spaces of opposite “spin” states due to the anisotropic diffuseness in the momentum distribution.
In connection of the deformation with superconductivity it has recently been reported [@Muther] that in the superconducting asymmetric nuclear matter the Fermi sea may undergo a deformation even in the spin-saturated system due to the difference of the Fermi surface between neutrons and protons; the momentum distributions of neutrons and protons may deform respectively to enlarge the overlapped region in the phase space, which effectively contributes to the $np$- pairing. They have shown the possibility of the deformation in a variational way; the Fermi sea of the majority of nucleons deforms in a prolate shape, while the minority in an oblate shape. Thus the deformation property of the Fermi seas looks very similar to our case. Nevertheless, note that our deformation is produced by the relativistic effect. Anyway it would be interesting to look further into the common feature.
It is to be noted that if the effective coupling constant is strong enough to lower the critical quark mass, spin polarization (magnetization) has potential to appear at rather moderate densities such as in the core of neutron stars, even though CSC weakly works against it.
From the above observations it is suggested that spin polarization does not compete with CSC but can coexist with it, unlike in ordinary superconductors of the electron system with the $s$-wave and spin-singlet pairing. This reflects the fact that internal degrees of freedom of the quark field, e.g.the color, flavor and Dirac indices, have rich structures to satisfy the antisymmetric constraint on the quark-pair field.
The possibility of the coexistent phase might also give a clue as for the origin of the superstrong magnetic field observed in magnetars. We roughly estimate the expected magnetic field when magnetars are assumed to be quark stars. The maximum dipole magnetic field at the star surface reads $$B_{\rm max}=\frac{8\pi}{3}\mu_qn_q(\langle M_z\rangle/N_q),$$ with $\mu_q$ and $n_q$ being the quark magnetic moment and the quark number density, respectively;e.g., for $\langle M_z\rangle/N_q\sim (10^{-3})$ and $n_q\sim O(1{\rm fm}^{-3})$, we find $B_{\rm max}\sim O(10^{15}{\rm
G})$, which is comparable to that observed in magnetars [@MAG1; @MAG2].
In the present paper we have not taken into account chiral symmetry, which is one of the basic concepts in QCD. If chiral symmetry is restored at finite baryon number density, the quark mass becomes drastically smaller as density increases. In order to simulate it we have examined the quark mass dependence on spin polarization. In the future work we would like to consider an effect of the dynamical mass on the axial-vector self-energy.
[**[Acknowledgements.]{}**]{}
The present research of T.T. and T.M. is partially supported by the REIMEI Research Resources of Japan Atomic Energy Research Institute, and by the Japanese Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Culture, Sports, Science and Technology (11640272, 13640282).
Structure of spinor under the axial-vector mean-field $U_A$
===========================================================
In this Appendix we rewrite the spinor (\[spinor\]) in terms of the free quark one and the remainder characterized by $U_A$. We employ the free spinor $u_s({{\mbox{\boldmath $p$}}})$ in which the two-component Pauli spinors are given as eigenvectors of the spin matrix $\sigma_z$: $$\begin{aligned}
&& u_\pm({{\mbox{\boldmath $p$}}})=
\left(
\begin{array}{c}
\sqrt{\epsilon_0 +m} \xi_\pm\\
\frac{\sqrt{\epsilon_0 -m}}{|p|} {{\mbox{\boldmath $p$}}} \cdot \mbox{\boldmath $\sigma$} \xi_\pm
\end{array} \right)
~~~\mbox{with}~~~
\xi_+ =
\left(
\begin{array}{c}
1\\
0
\end{array} \right) ~~\mbox{and}~~
\xi_- =
\left(
\begin{array}{c}
0\\
1
\end{array} \right),\end{aligned}$$ where $\epsilon_0=\sqrt{{{\mbox{\boldmath $p$}}}^2+m^2}$ and $\{ \mbox{\boldmath $\sigma$} \}$ are the Pauli spin matrices.
The spinor $\phi_+({{\mbox{\boldmath $p$}}})\equiv \phi_2({{\mbox{\boldmath $p$}}})$ for the “spin”-up state is decomposed as follows : $$\begin{aligned}
&&\frac{ \beta_p({\rm \Delta}\epsilon -2U_A)
+m(\delta\epsilon +2\beta_p) }{p_t^2 p_z {\cal N}_+} \phi_+({{\mbox{\boldmath $p$}}})= \nonumber \\
&& ~~~2\beta_p\sqrt{\epsilon_0+m}
\left( \frac{\epsilon_+ -\beta_p-U_A}{p_x+ip_y}u_+({{\mbox{\boldmath $p$}}})
-\frac{\beta_p+m}{p_z}u_-({{\mbox{\boldmath $p$}}}) \right)
+\frac{\mbox{Rem$_1$}(U_A)}{p_z(p_x+ip_y)} ,\end{aligned}$$ where $\delta\epsilon \equiv \epsilon_- -\epsilon_+$, ${\rm \Delta}\epsilon \equiv \epsilon_- +\epsilon_+$ and $$\mbox{Rem$_1$}(U_A) =
\left(
\begin{array}{c}
p_z(\epsilon_+ -\beta_p-U_A)
\left[ \beta_p({\rm \Delta}\epsilon -2U_A -2\epsilon_0)
+m\delta\epsilon \right] \\
-(p_x+ip_y)(\beta_p+m)
\left[ \beta_p({\rm \Delta}\epsilon -2U_A -2\epsilon_0)
+m\delta\epsilon \right] \\
\delta\epsilon
[(\epsilon_+ -\beta_p-U_A)p_z^2 +2\beta_p^2(\beta_p+m)]
-2\beta_p(\beta_p+m)
\left[ (\epsilon_- -U_A)(\epsilon_+-U_A)
-\epsilon_0^2 \right] \\
p_z(p_x+ip_y)(\beta_p+m) \delta\epsilon
\end{array} \right). \nonumber$$ Note that the term $\mbox{Rem$_1$}(U_A)$ vanishes in the limit, $U_A \rightarrow 0$. Thus one can fined that $\phi_+({{\mbox{\boldmath $p$}}})$ is a mixture of the free spinors even when $U_A=0$.
A decomposition of the spinor $\phi_-({{\mbox{\boldmath $p$}}})\equiv \phi_1({{\mbox{\boldmath $p$}}})$ for the “spin”-down state can also be done in the similar way: $$\begin{aligned}
&&\frac{ \beta_p({\rm \Delta}\epsilon -2U_A)
+m(\delta\epsilon +2\beta_p) }{p_t^2 p_z {\cal N}_-} \phi_-({{\mbox{\boldmath $p$}}})= \nonumber \\
&& ~~~2\beta_p\sqrt{\epsilon_0+m}
\left( \frac{\epsilon_- +\beta_p-U_A}{p_x+ip_y}u_+({{\mbox{\boldmath $p$}}})
+\frac{\beta_p-m}{p_z}u_-({{\mbox{\boldmath $p$}}}) \right)
+\frac{\mbox{Rem$_2$}(U_A)}{p_z(p_x+ip_y)},\end{aligned}$$ where $$\mbox{Rem$_2$}(U_A) =
\left(
\begin{array}{c}
p_z(\epsilon_- +\beta_p-U_A)
\left[ \beta_p({\rm \Delta}\epsilon -2U_A -2\epsilon_0)
+m\delta\epsilon \right] \\
(p_x+ip_y)(\beta_p-m)
\left[ \beta_p({\rm \Delta}\epsilon -2U_A -2\epsilon_0)
+m\delta\epsilon \right] \\
\delta\epsilon
[(\epsilon_- +\beta_p-U_A)p_z^2 -2\beta_p^2(\beta_p-m)]
+2\beta_p(\beta_p-m)
\left[ (\epsilon_- -U_A)(\epsilon_+-U_A)
-\epsilon_0^2 \right] \\
-p_z(p_x+ip_y)(\beta_p-m) \delta\epsilon
\end{array} \right). \nonumber$$
Decomposition of $B_{\lowercase{n}}({\bf p})$ in terms of the Dirac gamma matrices
==================================================================================
The operator $B_n({\mbox{\boldmath $p$}})$ in eq. (\[Bn\]) consists of some gamma matrices; it is a linear combination of $\mbox{\boldmath $1$}$, $\mbox{\boldmath $\gamma$}$, $\gamma_5\mbox{\boldmath $\gamma$}$, $\sigma_{01}$ and $\sigma_{02}$, in the diquark field $\bar{\psi_c}B_n\psi=\psi^T C B_n \psi$. The last two matrices give the tensor diquark fields, while these terms have no influence on the gap equation (\[gap1i\]) due to axial symmetry of the Fermi seas around the $p_z$ axis: the integration of $B_n({{\mbox{\boldmath $p$}}})$ with respect to the azimuthal angle $\phi_p$ in eq. (\[gap1i\]) gives $$\tilde{B}_n({{\mbox{\boldmath $p$}}})\equiv
\int^{2 \pi}_0 \frac{{\rm d}\phi_p}{2 \pi}
B_n({{\mbox{\boldmath $p$}}})=
\frac{p_t}{4 |\epsilon_n({{\mbox{\boldmath $p$}}})| \beta_p}
\left[ (-1)^n p_z \gamma_3+ (-1)^n m \mbox{\boldmath $1$}
+\beta_p \gamma_5 \gamma_3 \right]. \label{deco1}$$ Thus tensor terms disappear because they are proportional to $\exp{(i\phi_p)}$ in $B_n({\mbox{\boldmath $p$}})$.
The first term in the right hand side has also no contribution after symmetric integration with respect to $p_z$. The remainders, $\{{\bf 1}, \gamma_5 \gamma_3\}$, imply the pseudo-scalar ($J^P=0^-$) and vector ($J^P=1^-$) diquark pairings in terms of the notation in ref. [@BL]. Please note that the CSC gap (\[deco1\]) results in a linear combination of different angular momentum pairs $0^-$ and $1^-$ because of the lack of rotation symmetry.
Since the diquark fields $\psi^TC({\bf 1}, \gamma_5 \gamma_3) \psi$ contain the off-diagonal matrices which connect the lower component with the upper one of the Dirac spinors, these pairings vanish in the non-relativistic limit or in the limit $m\rightarrow \infty$. Hence $B_n({\bf p})$ resembles $P$- wave pairing as is seen in eq. (50), although it has no correspondence in the non-relativistic limit: the gap function for (\[deco1\]) has the nodes (vanishing at $\theta=0,\pi$) due to the factor $p_t$, which is similar to $^3P$-pairing in the liquid $^3$He - A phase, but these nodes are entirely attributed to the genuine relativistic effect. This property survives even in the limit, $U_A \rightarrow 0$.
From eq. (\[deco1\]) we can also obtain the relation appearing in eq. (\[MN\]); $$\gamma_\mu\gamma_0\tilde{B}_n({{\mbox{\boldmath $p$}}})\gamma_0\gamma^\mu=
2 \tilde{B}_n({{\mbox{\boldmath $p$}}}) +2 m\left\{ m+(-1)^n \beta_p \right\}.$$
Parameterization of the gap function
====================================
In this Appendix we derive the parameterization (\[GAP1\]). The gap equation (\[para\]) is expanded as $$\begin{aligned}
\Delta_\pm(k) &=& \frac{2}{3}\tilde{g}^2
\int \frac{{\rm d}^3p}{(2\pi)^3}
\frac{k_t}{2\epsilon_\pm (k)} \left[
\frac{p_t}{\epsilon_+(p)}
\left(\pm \frac{2m^2}{\beta_k\beta_p}+1\right) \frac{\Delta_+(p)}{2 E_+(p)}
+\frac{p_t}{\epsilon_-(p)}
\left(\mp \frac{2m^2}{\beta_k\beta_p}+1\right) \frac{\Delta_-(p)}{2 E_-(p)} \right]. \end{aligned}$$ Introducing $\hat{\Delta}_\pm (k)$ through the equation, $$\Delta_\pm (k)=\frac{k_t}{\epsilon_\pm (k)}\hat{\Delta}_\pm (k),$$ we obtain the “gap” equation for $\hat{\Delta}_\pm (k)$, $$\hat{\Delta}_\pm (k) = \frac{2}{3}\tilde{g}^2
\int \frac{{\rm d}^3p}{(2\pi)^3}
\frac{p_t^2}{4} \left[
\mp \frac{2m^2}{\beta_k \beta_p}
\left(\frac{\hat{\Delta}_-(p)}{\epsilon_-(p)^2 E_-(p)}
-\frac{\hat{\Delta}_+(p)}{\epsilon_+(p)^2 E_+(p)} \right)
+\frac{\hat{\Delta}_-(p)}{\epsilon_-(p)^2 E_-(p)}
+\frac{\hat{\Delta}_+(p)}{\epsilon_+(p)^2 E_+(p)} \right]
\label{C1}.$$ Then we find the following properties, $$\hat{\Delta}_-(k)+\hat{\Delta}_+(k)=F ~~~\mbox{and}~~~
\hat{\Delta}_-(k)-\hat{\Delta}_+(k)=R \times \frac{m}{\beta_k},$$ where $F$ ($R$) is a constant which characterize the symmetric (asymmetric) combinations of the gap functions $\hat{\Delta}_{\pm}$. Thus we can further parameterize $\hat{\Delta}_s(k)$ as $$\hat{\Delta}_\pm (k)=\mp \frac{m}{\beta_k} R + F \label{C2}.$$ Substituting the above formula into eq. (\[C1\]), one can obtain the coupled equations for $F$ and $R$, eqs. (\[eqF\]) and (\[eqR\]).
Fierz transformation
====================
We present the Fock exchange energy term by the OGE interaction by the use of the Fierz transformation. The Green function with vertices in the right-hand side of eq. (\[self1\]) can be expanded as $$\begin{aligned}
\sum_a \left(\Gamma_a iG_{11} (p) \Gamma_a \right)_{ij}
&=&
\sum_a (\Gamma_a)_{ii'}
\langle \psi(p)_{i'} \bar{\psi}(p)_{j'} \rangle
(\Gamma_a)_{j'j}
=
\sum_{ab} C_{ab} (\Gamma_b)_{ij} {\rm Tr}(G_{11}\Gamma_b) \\
\mbox{with}~~
\Gamma_a &\equiv& {\gamma_\mu \otimes {\bf 1}_{flavor} \otimes \lambda_{color}}
~~ \mbox{and}~~
(\Gamma_a)_{ii'} (\Gamma_a)_{j'j} =
\sum_b C_{ab} (\Gamma_b)_{ij} (\Gamma_b)_{j'i'} \label{Fierz1},\end{aligned}$$ where $\{C_{ab}\}$ are coefficients of a Fierz transformation (\[Fierz1\]) for the Dirac matrices, the identity matrix in the flavor space and the Gell-Mann matrices in the color space; $$\begin{aligned}
(\gamma_\mu)_{ii'} (\gamma^\mu)_{j'j} &=&
\delta_{ij} \delta_{j'i'}
-\frac{1}{2}(\gamma_\mu)_{ij} (\gamma^\mu)_{j'i'}
-\frac{1}{2}(\gamma_5\gamma_\mu)_{ij} (\gamma_5\gamma^\mu)_{j'i'}
+(i\gamma_5)_{ij} (i\gamma_5)_{j'i'}\label{notensor}\\
\delta_{ii'} \delta_{j'j} &=&
\frac{1}{2}\left[\frac{2}{N_f}\delta_{ij} \delta_{j'i'}
+(\tau_a)_{ij} (\tau_a)_{j'i'}\right]\\
(\lambda_c)_{ii'} (\lambda_c)_{j'j} &=&
\frac{2}{N_c^2} (N_c^2-1)\delta_{ij} \delta_{j'i'}
-\frac{1}{N_c} (\lambda_c)_{ij} (\lambda_c)_{j'i'}.\end{aligned}$$ It is to be noted that there appears no tensor term in eq. (\[notensor\]) due to chiral symmetry in QCD. Thus, e.g., the coefficient for the color-singlet axial-vector self-energy reads $-4/9$ for $N_f=2$ and $N_c=3$.
We also present a Fierz transformation for diquark fields. The right hand side of eq. (\[gap1\]) can be expanded, in the similar way for $G_{11}$; $$\begin{aligned}
\sum_a (\bar{\Gamma}_a G_{21}(p) \Gamma_a)_{ij}
&=&
\sum_a (C \Gamma_a^T C^{-1})_{ii'}
\langle \psi_c(p)_{i'} \bar{\psi}(p)_{j'} \rangle
(\Gamma_a)_{j'j}
=
\sum_a (C)_{ik} (\Gamma_a)_{i'k}
\langle \bar{\psi}(-p)_{i'} \bar{\psi}(p)_{j'} \rangle
(\Gamma_a)_{j'j} \nonumber \\
&=& \sum_{ab} f_{ab}
{\rm Tr}\left( G_{21}(p) C^{-1}\Gamma_b^T C^{-1} \right) (C\Gamma_b^TC^T)_{ij} \\
\mbox{with} && (\Gamma_a)_{i'k} (\Gamma_a)_{j'j} =
\sum_b f_{ab} (\Gamma_bC^*)_{i'j'} (C\Gamma_b)_{jk}, \label{Fierz2}\end{aligned}$$ where $\{f_{ab}\}$ are coefficients of a Fierz transformation (\[Fierz2\]) and are explicitly given as $$\begin{aligned}
(\gamma_\mu)_{i'k} (\gamma^\mu)_{j'j} &=&
(C^*)_{i'j'} (C)_{jk}
-\frac{1}{2} (\gamma_\mu C^*)_{i'j'} (C\gamma^\mu)_{jk} -\frac{1}{2} (\gamma_\mu\gamma_5C^*)_{i'j'} (C\gamma^\mu\gamma_5)_{jk}
+(iC^*\gamma_5)_{i'j'} (iC\gamma_5)_{jk} \\
\delta_{i'k} \delta_{j'j} &=& \frac{1}{2}
\left[ \frac{2}{N_f} \delta_{i'j'} \delta_{jk}
+(\tau_a)_{i'j'} (\tau_a)_{jk} \right]\\
(\lambda_c)_{i'k} (\lambda_c)_{j'j}&=&
\left( 1-\frac{1}{N_c} \right)
\left[ \frac{2}{N_c} (\delta)_{i'j'} (\delta)_{jk}
+ (\lambda_c^S)_{i'j'} (\lambda_c^S)_{jk} \right]
-\left( 1+\frac{1}{N_c} \right)
(\lambda_c^A)_{i'j'} (\lambda_c^A)_{jk},\end{aligned}$$ where $\{ \lambda_c^{S(A)} \}$ are symmetric (antisymmetric) matrices of $\{\lambda_c\}$. The present gap function, $\Delta({\bf p})=\sum_n B_n({\bf p}) \Delta_n({\bf p}) $ which is a linear combination of the gamma matrices, can be obtained by taking projection on $B_n({\bf p})$.
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———————-
[^1]: We shall see that only the space component of the axial-vector mean-field is responsible for spin polarization. We,hereafter, take its direction along the $z$-axis without loss of generality.
[^2]: This is equivalent to the restriction of the sum over the index $n (n=1 - 4)$ to $1,2$, which correspond to the positive-energy states with different “spins” specified by the subscript $\mp$.
[^3]: This feature is very similar to $^3P$- pairing in liquid $^3{\rm He}$ [@He3] or nuclear matter [@NM3P].
[^4]: Here we needn’t consider the orbital angular momentum for uniform matter. But if a superconductor is of the ‘second’ type in which London’s penetration length is larger than the coherence length, a vortex lattice may be formed in response to the external field and then total magnetization is to undergo a qualitative change due to circulation of supercurrent [@Abri].
|
---
abstract: 'One of the main theories for explaining the formation of spiral arms in galaxies is the stationary density wave theory. This theory predicts the existence of an age gradient across the arms. We use the stellar cluster catalogues of the galaxies NGC 1566, M51a, and NGC 628 from the Legacy Extragalactic UV Survey (LEGUS) program. In order to test for the possible existence of an age sequence across the spiral arms, we quantified the azimuthal offset between star clusters of different ages in our target galaxies. We found that NGC 1566, a grand–design spiral galaxy with bisymmetric arms and a strong bar, shows a significant age gradient across the spiral arms that appears to be consistent with the prediction of the stationary density wave theory. In contrast, M51a with its two well–defined spiral arms and a weaker bar does not show an age gradient across the arms. In addition, a comparison with non–LEGUS star cluster catalogues for M51a yields similar results. We believe that the spiral structure of M51a is not the result of a stationary density wave with a fixed pattern speed. Instead, tidal interactions could be the dominant mechanism for the formation of spiral arms. We also found no offset in the azimuthal distribution of star clusters with different ages across the weak spiral arms of NGC 628.'
author:
- |
F. Shabani,$^{1}$[^1] E.K. Grebel,$^1$ A. Pasquali,$^1$ E. D’Onghia,$^{2,3}$ J.S. Gallagher III,$^2$ A. Adamo,$^4$ M. Messa,$^4$ B.G. Elmegreen,$^5$ C. Dobbs,$^6$ D.A. Gouliermis,$^{7,8}$ D. Calzetti,$^9$ K. Grasha,$^9$ D.M. Elmegreen,$^{10}$ M. Cignoni,$^{11,12,13}$ D.A. Dale,$^{14}$ A. Aloisi,$^{15}$ L.J. Smith,$^{16}$ M. Tosi,$^{13}$ D.A. Thilker,$^{17}$ J.C. Lee,$^{15,18}$ E. Sabbi,$^{15}$ H. Kim,$^{19}$ and A. Pellerin$^{20}$\
$^1$Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12–14, 69120 Heidelberg, Germany\
$^2$Dept. of Astronomy, University of Wisconsin- Madison, 475 N. Charter Street, Madison, WI 53076–1582, USA\
$^3$Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA\
$^{4}$Dept. of Astronomy, The Oskar Klein Centre, Stockholm University, Stockholm, Sweden\
$^{5}$IBM Research Division, T.J. Watson Research Center, Yorktown Hts., NY, USA\
$^{6}$School of Physics and Astronomy, University of Exeter, Exeter, United Kingdom\
$^{7}$Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik, Albert-Ueberle-Str.2, 69120 Heidelberg, Germany\
$^{8}$Max Planck Institute for Astronomy, Königstuhl17, 69117 Heidelberg, Germany\
$^{9}$Dept. of Astronomy, University of Massachusetts – Amherst, Amherst, MA 01003, USA\
$^{10}$Dept. of Physics and Astronomy, Vassar College, Poughkeepsie, NY, USA\
$^{11}$Dept. of Physics, University of Pisa, Largo B. Pontecorvo 3, 56127, Pisa, Italy\
$^{12}$INFN, Largo B. Pontecorvo 3, 56127, Pisa, Italy\
$^{13}$INAF - Osservatorio Astrofisico e di Scienza dello Spazio, Bologna, Italy\
$^{14}$Dept. of Physics and Astronomy, University of Wyoming, Laramie, WY, USA\
$^{15}$Space Telescope Science Institute, Baltimore, MD, USA\
$^{16}$European Space Agency/Space Telescope Science Institute, Baltimore, MD, USA\
$^{17}$Dept. of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD, USA\
$^{18}$Visiting Astronomer, Spitzer Science Center, Caltech. Pasadena, CA, USA\
$^{19}$Gemini Observatory, Casilla 603, La Serena, Chile\
$^{20}$Dept. of Physics and Astronomy, State University of New York at Geneseo, Geneseo, NY, USA\
\
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: Search For Star Cluster Age Gradients Across Spiral Arms of Three LEGUS Disk Galaxies
---
\[firstpage\]
galaxies: spiral galaxies: structure galaxies: indiviual: NGC 1566, M51, NGC 628
Introduction {#Introduction}
============
Understanding how spiral patterns form in disk galaxies is a long–standing issue in astrophysics. Two of the most influential theories to explain the formation of spiral structure in disk galaxies are named stationary density wave theory and swing amplification. The stationary density wave theory poses that spiral arms are static density waves [@Lindblad; @LS64]. In this scenario spiral arms are stationary and long–lived. The swing amplification proposes instead that spiral structure is the local amplification in a differentially rotating disk [@G; @JT; @SC; @Sellwood; @E11; @Elena13]. According to this theory indiviual spiral arms would fade away in one galactic year and should be considered transient features. Numerical experiments suggest that non–linear gravitational effects would make spiral arms fluctuate in density locally but be statistically long–lived and self–perpetuating [@Elena13].
To complicate the picture there is the finding that many galaxies in the nearby universe are grand–design, bisymmetric spirals. These galaxies may show evidence of a galaxy companion, suggesting that the perturbations induced by tidal interactions could induce spiral features in disks by creating localized disturbances that grow by swing amplification [@k; @B3; @Ga; @Elena16; @P16]. Some studies have been devoted to explore galaxy models with bar–induced spiral structure [@conto] and spiral features explained by a manifold [@conto; @A]. It is also possible that a combination of these models is needed to describe the observed spiral structure. We refer the interested reader to comprehensive reviews of different theories of spiral structure in [@DB14] and to [@Shu] for detailed explanations of the origin of spiral structure in stationary density wave theory.
The longevity of spiral structure can be tested observationally. In fact, in the stationary density wave theory, spiral arms are density waves moving with a single constant angular pattern speed. The angular speed of stars and gas equals the pattern speed at the corotation radius. Inside the corotation radius, material rotates faster than the spiral pattern. When the gas enters the higher–density region of spiral arms, it may experience a shock which may lead to star formation [@R69]. Consequently, the stars born in the molecular clouds in spiral arms eventually overtake the arms and move away from the spiral patterns as they age. This drift causes an age gradient across the spiral arms. If spiral arms have a constant angular speed, then we expect to find the youngest star clusters near the arm on the trailing side, and the oldest star clusters further away from the spiral arms inside the corotation radius [e.g., @M09]. Outside the corotation radius, the spiral pattern moves faster than the gas and leads to the opposite age sequence.
@DP10 carried out numerical simulations of the age distribution of star clusters in four different spiral galaxy models, including a galaxy with a fixed pattern speed, a barred galaxy, a flocculent galaxy, and an interacting galaxy. The results of their simulations show that in a spiral galaxy with a constant pattern speed or in a barred galaxy, a clear age sequence across spiral arms from younger to older stars is expected. In the case of a flocculent spiral galaxy, no age gradient can be observed in their simulation. Also in the case of an interacting galaxy, a lack of an age gradient as a function of azimuthal distance from the spiral arms is predicted. A simulation of an isolated multiple–arm barred spiral galaxy was performed by [@grand], who explored the location of star particles as a function of age around the spiral arms. Their simulation takes into account radiative cooling and star formation. They found no significant spatial offset between star particles of different ages, suggesting that spiral arms in such a spiral galaxy are not consistent with the long–lived spiral arms predicted by the static or stationary density wave theory. In a recent numerical study, [@D17] looked in detail at the spatial distribution of stars with different ages in an isolated grand–design spiral galaxy. They found that star clusters of different ages are all concentrated along the spiral arms without a clear age pattern.
A simple test of the stationary density wave theory consists of looking for a colour gradient from blue to red across spiral arms due to the progression of star formation. It is important to note that this method can be affected by the presence of dust. Several observational studies have tried to test the stationary density wave theory by looking for colour gradients across the spiral arms. In an early study of the ($B-V$) colours and total star formation rates in a sample of spiral galaxies with and without grand design patterns, [@Bruce86] found no evidence for an excess of star formation due to the presence of a spiral density wave, and explained the blue spiral arm colours as a result of a greater compression of the gas compared to the old stars, with star formation following the gas. [@M09] studied the colour gradients across the spiral arms of 13 SA and SAB galaxies. Ten galaxies in their sample present the expected colour gradient across their spiral arms.
A number of observational studies have used the age of stellar clusters in nearby galaxies as a tool to test the stationary density wave theory. @S09 studied the spatial distribution of 1580 stellar clusters in the interacting, grand–design spiral M51a from Hubble Space Telescope (HST) $UBVI$ photometry. They found no spatial offset between the azimuthal distribution of cluster samples of different age. Their results indicate that most of the young (age < 10 Myr) and old stellar clusters (age > 30 Myr) are located at the centers of the spiral arms. [@k10] also mapped the age of star clusters as a function of their location in M51a using HST data and found no clear pattern in the location of star clusters with respect to their age. Both above studies suggest that spiral arms are not stationary, at least for galaxies in tidal interaction with a companion. In order to study the spatial distribution of star–forming regions, [@Sanchez] produced an age map of six nearby grand–design and flocculent spiral galaxies. Only two grand–design spiral galaxies in their sample presented a stellar age sequence across the spiral arms as expected from stationary density wave theory.
In galaxies where spiral arms are long–lived and stationary as predicted by the static density wave theory, one would expect to find an angular offset among star formation and gas tracers of different age within spiral arms [@R69]. The majority of observational studies of the spiral density wave scenario have tried to examine such an angular offset [@vogel; @Rand]. [@Tam8] detected an angular offset between HI (a tracer of the cold dense gas) and 24 $\rm \mu$m emission (a tracer of obscured star formation) in a sample of 14 nearby disk galaxies. An angular offset between CO (a tracer of molecular gas) and H$\rm \alpha$ (a tracer of young stars) was detected for 5 out of 13 spiral galaxies observed by [@Egusa]. In another observational work, [@Foyle] tested the angular offset between different star formation and gas tracers including HI, $\rm H_{2}$, 24 $\rm \mu $m, UV (a tracer for unobscured young stars) and 3.6 $\rm \mu$m emission (a tracer of the underlying old stellar population) for 12 nearby disk galaxies. They detected no systematic trend between the different tracers. Similarly, [@F13] found no significant angular offset between H$\rm \alpha$ and UV emission in NGC 4321. [@L13] found a large angular offset between CO and H$\rm \alpha$ in M51a while no significant offsets have been found between HI, 21 cm, and 24 $\rm \mu$m emissions. These searches for offsets are based on the assumption that the different tracers represent a time sequence of the way a moving density wave interacts with gas and triggers star formation. [@Elmegreen2014] used the S$^4$G survey [@Sheth] and discovered embedded clusters inside the dust lanes of several galaxies with spiral waves, suggesting that star formation can sometimes start quickly.
In a recent observational study, [@S17] carried out a detailed investigation of a spiral arm segment in M51a. They measured the radial offset of the star clusters of different ages (< 3 Myr, and 3–10 Myr) and star formation tracers (HII regions and 24 $\rm \mu$m) from their nearest spiral arm. No obvious spatial offset between star clusters younger and older than 3 Myr was found in M51a. They also found no clear trend in the radial offset of HII regions and 24 $\mu$m. Similarly, [@chandar17] compared the location of star clusters with different ages (< 6 Myr, 6–30 Myr, 30–100 Myr, 100–400 Myr, and > 400 Myr) with the spiral patterns traced by molecular gas, dust, young and old stars in M51a. They found cold molecular gas and dark dust lanes to be located along the inner edge of the arms while the outer edge is defined by the old stars (traced with 3.6 $\rm \mu$m) and young star clusters. The observed sequence in the spiral arm of M51a is in agreement with the prediction from stationary density wave theory. [@chandar17] also measured the spatial offset between molecular gas, young (< 10 Myr) and old star clusters (100–400 Myr) in the inner (2.0–2.5 kpc) and outer (5.0–5.5 kpc) spiral arms in M51a. They found an azimuthal offset between the gas and star clusters in the inner spiral arm zone, which is consistent with the spiral density wave theory. In the outer spiral arms, the lack of such a spatial offset suggests that the outer spiral arms do not have a constant pattern speed and are not static. [@chandar17] found no star cluster age gradient along four gas spurs (perpendicular to the spiral arms) in M51a.
In conclusion, there have been numerous observational studies aiming to test the longevity of the spiral structure. In many cases, the conclusions show conflicting results and the nature of spiral arms is still an open question.
The main goal of this study is to test whether spiral arms in disk galaxies are static and long–lived or locally changing in density and locally transient. This work is based on the Legacy ExtraGalactic UV Survey (LEGUS)[^2] observations obtained with HST [@C15]. The paper is organized as follows: The survey and the sample galaxies are described in § \[The LEGUS Galaxy Samples\]. The selection of the star cluster samples is presented in § \[s3\]. We investigate the spatial distribution together with clustering of the selected clusters in § \[location\]. In § \[Azimutahl distribution\], we describe the results and analysis and how we measure the spatial offset of our star clusters across spiral arms. In § \[2arms\] we discuss whether the two spiral arms of our target galaxies have the same nature. In § \[chandra\], we use a non–LEGUS star cluster catalogue to measure the spatial offset of star clusters in M51a and we present our conclusions in § \[Summary\].
The sample galaxies {#The LEGUS Galaxy Samples}
===================
LEGUS is an HST Cycle 21 Treasury programme that has observed 50 nearby star–forming dwarf and spiral galaxies within 12 Mpc. High– resolution images of these galaxies were obtained with the UVIS channel of the Wide Field Camera Three (WFC3), supplemented with archival Advanced Camera for Surveys (ACS) imaging when available, in five broad band filters, $NUV\,(F275W)$, $U \,(F336W)$, $B \,(F438W)$, $V \,(F555W)$, and $I \,(F814W)$. The pixel scale of these observations is $ \rm 0.04^{\arcsec} \, pix^{-1}$. A description of the survey, the observations, the image processing, and the data reduction can be found in [@C15].
Face–on spiral galaxies with prominent spiral structures are interesting candidates to study stationary density wave theory. Therefore, three face–on spiral galaxies, namely NGC 1566, M51a, and NGC 628 were selected from the LEGUS survey for our study. The morphology, distance, corotation radius, and the pattern speed of each galaxy are listed in Table \[tab:properties of galaxies\]. The UVIS and ACS footprints of the pointings (red and yellow boxes, respectively) overlaid on Digitized Sky Survey (DSS) images of the galaxies are shown in Fig. \[fig:galaxies\] together with their HST red, green, and blue colour composite mosaics.
Galaxy Morphology D \[Mpc\] $\rm M_{\star} \, (M_{\sun})$ SFR (UV) $\rm(M_{\sun} \, yr^{-1}) $ $\rm R_{cr}$ \[$\mathrm{kpc}$\] $ \rm \Omega_{p}$ \[$\rm km\, s^{-1}\, \rm kpc^{-1}$\] Ref
---------- ------------ ----------- ------------------------------- -------------------------------------- --------------------------------- -------------------------------------------------------- ----- -- -- -- -- -- -- --
NGC 1566 SABbc 18 $\rm 2.7\times 10^{10}$ 2.026 10.6 23$\pm$2 1
M51a SAc 7.6 $\rm 2.4\times 10^{10}$ 6.88 5.5 38$\pm$7 2
NGC 628 SAc 9.9 $\rm 1.1\times 10^{10}$ 3.6 7 32$\pm$2 3
Column 1, 2: Galaxy name and morphological type as listed in the NASA Extragalactic Database (NED)\
Column 3: Distance\
Column 4: Stellar mass obtained from the extinction–corrected B–band luminosity\
Column 5: Star formation rate calculated from the GALEX far–UV, corrected for dust attenuation\
Column 6: Co–rotation radius\
Column 7: Pattern speed\
Column 8: References for the co–rotation radii and pattern speeds: 1- [@A04], 2- [@z4], 3- [@Sakhibov]\
NGC 1566
--------
NGC 1566, the brightest member of the Dorado group, is a nearly face–on (inclination = $\rm 37.3^{\circ}$) barred grand–design spiral galaxy with strong spiral structures [@Debra2]. The distance of NGC 1566 in the literature is uncertain and varies between 5.5 and 21.3 Mpc. In this study, we revised the distance of 13.2 Mpc listed in [@C15] and adopted a distance of 18 Mpc [@sabbi]. NGC 1566 has been morphologically classified as an SABbc galaxy because of its intermediate–strength bar. It hosts a low–luminosity active galactic nucleus (AGN) [@Combes]. The star formation rate and stellar mass of NGC 1566 are $ \rm 2.0 \, M_{\sun }yr^{-1}$and $\rm 2.7 \times 10^{10} \, M_{\sun }$, respectively within the LEGUS field of view [@sabbi].Two sets of spiral arms can be observed in NGC 1566. The inner arms connect with the star–forming ring at 1.7 kpc [@S15], which is covered by the LEGUS field of view (see Fig. \[fig:galaxies\], top panel). The outer arms beyond 100 arcseconds (corresponding to 8 kpc ) are weaker and smoother than the inner arms.
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M51a
----
M51a (NGC 5194) is a nearby, almost face–on (inclination = $\rm 22 ^{\circ} $) spiral galaxy located at a distance of 7.6 Mpc [@Tonry]. It is a grand design spiral galaxy morphologically classified as SAc with strong spiral patterns [@Debra2]. M51a is interacting with a companion galaxy, M51b (NGC 5195). M51a has a star formation rate and a stellar mass of $\rm 6.9 \, M_{\sun } yr^{-1}$and $\rm 2.4 \times 10^{10}M_{\sun }$, respectively [@Lee; @Both]. Five UVIS pointings in total were taken through LEGUS observations: 4 pointings cover the center, the north–east, and the south–west regions of M51a, and one covers the companion galaxy M51b.
NGC 628
-------
NGC 628 (M74) is the largest galaxy in its group. This nearby galaxy is seen almost face–on ($\rm i = 25.2 ^{\circ}$) and is located at a distance of 9.9 Mpc [@Oliver]. It has no bulge [@cor] and is classified as a SAc spiral galaxy. Its star formation rate and stellar mass obtained from the extinction–corrected B–band luminosity are $ \rm 3.6 \, M_{\sun } yr^{-1}$and $\rm 1.1 \times 10^{10}M_{\sun }$, respectively [@Lee; @Both]. NGC 628 is a multiple–arm spiral galaxy [@Debra] with two well–defined spiral arms. It has weaker spiral patterns than NGC 1566 and M51a [@Debra2]. The LEGUS UVIS observations of NGC 628 consist of one central and one east pointing that were combined into a single mosaic for the analysis.
Stellar cluster samples {#s3}
=======================
Selection from star cluster catalogues
--------------------------------------
In this section, we provide a detailed explanation of the process adopted to select star cluster candidates in our target galaxies. A general description of the standard data reduction of the LEGUS sample can be found in [@C15]. A careful and detailed description of the cluster extraction, identification, classification, and photometry is given in [@Angela17] and [@messa]. Stellar cluster candidates were extracted with SExtractor [@Bertin] in the five standard LEGUS filters. The resulting cluster candidate catalogues include sources with a $V$–band concentration index (CI)[^3] larger than the CI of star–like sources, which are detected in at least two filters with a photometric error $\leq$ 0.3 mag. The photometry of sources in each filter was corrected for the Galactic foreground extinction [@Schlafly]. In order to derive the cluster physical properties such as age, mass, and extinction, the spectral energy distribution (SED) of the clusters was fitted with Yggdrasil stellar population models [@Z11]. The uncertainties derived in the physical parameters of the star clusters are on average $\rm 0.1\, \rm dex$ [@Angela17]. For some of the LEGUS galaxies, star cluster properties were also estimated based on a Bayesian approach, using the Stochastically Lighting Up Galaxies (SLUG) code [@sila]. A detailed and complete explanation of the Bayesian approach can be found in [@krumholz].
Each source in the stellar cluster catalogue that is brighter than -6 mag in the $V$–band, and detected in at least four bands, has been morphologically classified via visual inspection by three independent members of the LEGUS team [@katie15; @Angela17]. The inspected clusters were divided into four morphological classes: Class 1 contains compact, symmetric, and centrally concentrated clusters. Class 2 includes compact clusters with a less symmetric light distribution, Class 3 represents less compact and multi–peak cluster candidates with asymmetric profiles, and Class 4 consists of unwanted objects like single stars, multiple stars, or background sources. Unclassified objects were labeled as Class 0.
In addition, a machine–learning (ML) approach was tested to morphologically classify the stellar clusters in an automated fashion. A forthcoming paper (Grasha et al., in prep.) will present the ML code that was used for cluster classification in the LEGUS survey and the degree of agreement with human classification. An initial comparison between human and ML classification in M51a was already discussed by @messa.
For our analysis, we use stellar cluster properties estimated with Yggdrasil deterministic models based on the Padova stellar libraries (see @Z11 for details) with solar metallicity, the Milky Way extinction curve [@Cardeli], and the [@Kroupa] stellar initial mass function (IMF). We also selected clusters based on human visual classification for NGC 628, a combination of human and machine learning classification in NGC 1566, and only machine learning for M51a. Star clusters classified as Class 4 and Class 0 are excluded from our analysis. Among our target galaxies, there is a total number of 1573, 3374, and 1262 star cluster candidates classified as Class 1, 2, and 3 in NGC 1566, M51a, and NGC 628, respectively.
A detailed description of the properties of the final cluster catalogues of M51a and NGC 628 and their completeness can be found in @messa and @Angela17.
[0.50]{} ![Distribution of ages and masses of the star clusters (class 1, 2, and 3) in NGC 1566, M51a, and NGC 628. The colours represent different age bins: blue (the young sample), green (the intermediate–age sample), red (the old sample), and black (excluded star clusters). The number of clusters in each sample is shown in parentheses. The horizontal dotted lines in NGC 1566 show the applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 100 Myr and $\rm 10^{4} \, \rm M_{\sun}$ up to the age of 200 Myr. The applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 200 Myr in M51a and NGC 628 are also by horizontal dotted lines. The solid black line shows the 90% completeness limit of 23.5 mag in the $V$–band in NGC 1566 and the magnitude cut of $\rm M_{V}$ = -6 mag in M51a, and NGC 628, respectively.[]{data-label="fig:age_mass"}](age_mass_NGC1566.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![Distribution of ages and masses of the star clusters (class 1, 2, and 3) in NGC 1566, M51a, and NGC 628. The colours represent different age bins: blue (the young sample), green (the intermediate–age sample), red (the old sample), and black (excluded star clusters). The number of clusters in each sample is shown in parentheses. The horizontal dotted lines in NGC 1566 show the applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 100 Myr and $\rm 10^{4} \, \rm M_{\sun}$ up to the age of 200 Myr. The applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 200 Myr in M51a and NGC 628 are also by horizontal dotted lines. The solid black line shows the 90% completeness limit of 23.5 mag in the $V$–band in NGC 1566 and the magnitude cut of $\rm M_{V}$ = -6 mag in M51a, and NGC 628, respectively.[]{data-label="fig:age_mass"}](age_mass_M51.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![Distribution of ages and masses of the star clusters (class 1, 2, and 3) in NGC 1566, M51a, and NGC 628. The colours represent different age bins: blue (the young sample), green (the intermediate–age sample), red (the old sample), and black (excluded star clusters). The number of clusters in each sample is shown in parentheses. The horizontal dotted lines in NGC 1566 show the applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 100 Myr and $\rm 10^{4} \, \rm M_{\sun}$ up to the age of 200 Myr. The applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 200 Myr in M51a and NGC 628 are also by horizontal dotted lines. The solid black line shows the 90% completeness limit of 23.5 mag in the $V$–band in NGC 1566 and the magnitude cut of $\rm M_{V}$ = -6 mag in M51a, and NGC 628, respectively.[]{data-label="fig:age_mass"}](age_mass_NGC628.pdf "fig:"){width="1\linewidth"}
Galaxy age (Myr) < 10 10 $ \leq $ age (Myr) < 50 50 $\leq $ age (Myr) $ \leq $ 200
---------- ------------------- ------------------------------- ----------------------------------- --
NGC 1566 392 679 124
M51a 361 441 979
NGC 628 77 111 302
Selection of star clusters of different ages {#selection of star clusters of different ages}
--------------------------------------------
In this study, we use the age of star clusters in our galaxy sample as a tool to find a possible age gradient across the spiral arms predicted by the stationary density wave theory. Therefore, we group star clusters into three different cluster samples according to their ages.
The estimated physical properties of star clusters based on the Yggdrasil deterministic models are inaccurate for low–mass clusters [@krumholz]. A comparison between the deterministic approach based on Yggdrasil models and the Bayesian approach with SLUG models presented by [@krumholz] suggests that the derived cluster properties are uncertain at cluster masses below 5000 $\rm M_{\sun}$. We adopted the same mass cut–off and for NGC 628 and M51a in our analysis. Using the luminosity corresponding to this mass, namely $\rm M_{V}$ = $-6$ mag ($\rm m_{V}$ = 23.4 and 23.98 mag for NGC 628 and M51a, respectively) results in an age completeness limit of $\rm \leq 200\, \rm Myr$. In @Angela17 and @messa the magnitude cut at $\rm M_{V}$ < $-6$ mag is a more conservative limit than the magnitude limit corresponding to 90% of completeness in the recovery of sources. We have tested our results using different mass cuts as well as by removing any constraint on the limiting mass, and we have not observed any significant change in the age distributions of the clusters as a function of azimuthal distances. Thus, the results presented in § \[Azimutahl distribution\] and § \[2arms\] are robust against uncertainties in the determination of cluster physical properties.
NGC 1566 is the most distant galaxy within our LEGUS sample. Due to the large distance of this galaxy, the 90% completeness limit ($\rm m_{V}$ = 23.5 mag) is significantly brighter than $\rm M_{V}$ = $-6$ mag. Therefore, in order to select star clusters in NGC 1566, we used the 90% completeness limit and a= mass cut of 5000 $\rm M_{\sun}$ for the cluster ages up to 100 Myr and $\rm 10^{4} \rm M_{\sun}$ for the 100–200 Myr old star clusters (see Fig. \[fig:age\_mass\]). Applying these two criteria reduced our cluster samples from 1573 to 1195 clusters for NGC 1566, from 3374 to 1781 clusters for M51a, and from 1262 to 490 for NGC 628.
Then, we selected three cluster samples of different ages for each galaxy as follows:
$\bullet$
: “Young” star clusters: age (Myr) < 10
$\bullet$
: “Intermediate–age” star clusters: 10 $\rm \leq$ age (Myr) < 50
$\bullet$
: “Old” star clusters: 50 $\rm \leq$ age (Myr) $\leq$ 200
The number of star clusters in the “young”, “intermediate–age”, and “old” samples is shown in Tab. \[tab:cluster sample\].
Fig. \[fig:age\_mass\] displays the age–mass diagram of star clusters in NGC 1566, M51a, and NGC 628. The young, the intermediate–age, and the old star cluster samples are shown in blue, green, and red colors, respectively. The excluded star clusters (due to the mass cut) are shown in black. The horizontal and vertical dotted lines show the applied mass cut of $ \rm 5000\, \rm M_{\sun}$ and its corresponding completeness limit at a stellar age of $ 200\, \rm Myr$, respectively.
Spatial distribution and clustering of star clusters {#location}
====================================================
In Fig. \[fig:clusters\], we plot the spatial distribution of star clusters of different ages in the galaxies NGC 1566, M51a, and NGC 628. The young, intermediate–age, and old stellar cluster samples are shown in blue, green, and red, respectively. In general, we observe a similar trend in our target galaxies: First, the young and the intermediate–age star clusters mostly populate the spiral arms rather than the interarm regions. This is particularly evident for NGC 1566 and M51a, which show strong and clear spiral structures in young and intermediate–age star clusters. Second, the old star clusters are less clustered and more widely spread compared to the young and intermediate–age star cluster samples.
Our findings are similar to other literature results on the spatial distribution of star clusters of different ages: [@D17], using LEGUS HST data found that in NGC 1566 the 100 Myr old star clusters clearly trace the spiral arms while in NGC 628 star clusters older than 10 Myr show only weak spiral structures. [@chandar17], using other HST data observed that M51a shows weak spiral structure in older star clusters (>100 Myr).
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Clustering of star clusters has been observationally investigated for a number of local star–forming galaxies [e.g., @Efremov; @EE]. In a detailed study of clustering of the young stellar population in NGC 6503 based on the LEGUS observations, [@d15] found that younger stars were more clustered compared to the older ones. [@katie15] investigated the spatial distribution of the star clusters in NGC 628 from the LEGUS sample. Their findings confirmed that the degree of the clustering increases with decreasing age. More recently, @grasha17a studied the hierarchical clustering of young star clusters in a sample of six LEGUS galaxies. Their results suggested that the youngest star clusters are strongly clustered and the degree of clustering quickly drops for clusters older than 20 Myr and the galactic shear appears to drive the largest sizes of the hierarchy in each galaxy @grasha17b.
Adopting a similar approach as [@katie15], we use the two–point correlation function to test whether or not the clustering distribution of the clusters in our selected age bins shows the expected age dependence. The two–point correlation function $\rm \omega (\theta)$ is a powerful statistical tool for quantifying the probability of finding two clusters with an angular separation $\rm \theta$ against a random, non–clustered distribution [@peebles]. Here we use the Landy–Szalay [@LS] estimator, which has little sensitivity to the presence of edges and masks in the data: $$\omega(\theta) = \frac{r (r-1)}{n (n-1)}\frac{DD}{RR} - \frac{(r-1)}{n}\frac{DR}{RR}+1,$$ where $ n$ and $r$ are the total number of data and random points, respectively. $ DD$, $ RR$, and $ DR$ are the total numbers of data–data, random–random, and data–random pair counts with a separation $\rm \theta \pm d\theta$, respectively. We construct a random distribution of star clusters that has the same sky coverage and masked regions (e.g., the ACS chip gap) as the images of each galaxy.
Fig. \[fig:two\_point\] displays the two–point correlation function for the star clusters in different age bins as defined for our galaxy samples. The blue, green, and red colours represent the young, intermediate–age, and old star cluster samples in each galaxy, respectively. The error bars on the two–point correlation function were estimated using a bootstrapping method with 1000 bootstrap resamples.
The general distribution of the star cluster samples in the target galaxies shows a similar trend: Independent of the presence of spiral arms, young clusters show hierarchical structure, whilst the old star clusters show a non–clustered, smooth distribution.
[0.50]{} ![ two–point correlation function for the star cluster samples of different ages as a function of angular distance (arcseconds) in NGC 1566, M51a, and NGC 628. The young, intermediate–age, and old star cluster samples are shown in blue, green, and red, respectively. The error bars were computed based on a bootstrapping method. The number of star clusters in each age bin are listed in parentheses.[]{data-label="fig:two_point"}](2p_NGC1566.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![ two–point correlation function for the star cluster samples of different ages as a function of angular distance (arcseconds) in NGC 1566, M51a, and NGC 628. The young, intermediate–age, and old star cluster samples are shown in blue, green, and red, respectively. The error bars were computed based on a bootstrapping method. The number of star clusters in each age bin are listed in parentheses.[]{data-label="fig:two_point"}](2p_M51.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![ two–point correlation function for the star cluster samples of different ages as a function of angular distance (arcseconds) in NGC 1566, M51a, and NGC 628. The young, intermediate–age, and old star cluster samples are shown in blue, green, and red, respectively. The error bars were computed based on a bootstrapping method. The number of star clusters in each age bin are listed in parentheses.[]{data-label="fig:two_point"}](2p_NGC628.pdf "fig:"){width="1\linewidth"}
Are the spiral arms static density waves? {#Azimutahl distribution}
=========================================
As discussed in § \[Introduction\], the stationary density wave theory foresees that the age of stellar clusters inside the corotation radius increases with increasing distance from the spiral arms. In other words, we expect to find a shift in the location of stellar clusters with different ages.
In order to test whether the distribution of star clusters of different ages in our target galaxies agrees with the expectations from the stationary density wave theory, we need to quantify the azimuthal offset between star clusters of different ages.
Spiral arm ridge lines definition
---------------------------------
First of all, we need to locate the spiral arms of our galaxy sample. We wish to define a specific location in each spiral arm so we can measure the relative positions of the star clusters in a uniform way. We use the dust lanes for this purpose because they are narrow and well–defined on optical images.
As gas flows into the potential minima of a density wave, it gets compressed and forms dark dust lanes in the inner part of the spiral arms, where star formation is then likely to occur [@R69]. We have used the $B$–band images for this purpose since most of the emission is due to young OB stars and dark obscuring dust lanes can be better identified in this band.
To better define the average positions of the dust lanes, we used a Gaussian kernel (with a 10 pixels sigma) to smooth the images, reduce the noise, and enhance the spiral structure. In the smoothed images the dust lanes are clearly visible as dark ridges inside the bright spiral arms. We defined these dark spiral arm ridge lines manually. For the remainder of this paper, we refer to the southern arm and northern arm as “Arm 1” and “Arm 2”, respectively. Fig. \[fig:arms\] presents the defined spiral arm ridge lines (red lines) overplotted on the smoothed $B$–band images of NGC 1566, M51a, and NGC 628.
[0.50]{} ![The location of spiral arm ridge lines is shown by red lines overplotted on the smoothed $B$—band images of NGC 1566, M51a, and NGC 628. We refer to the southern arm and northern arm as “Arm 1” and “Arm 2”, respectively. The two black dashed circles in each panel mark the onset of the bulge and the location of the co–rotation radius of the galaxies. The horizontal bar in the lower left corner denotes a length scale of 2 kpc. North is up and East to the left.[]{data-label="fig:arms"}](arms_NGC1566.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![The location of spiral arm ridge lines is shown by red lines overplotted on the smoothed $B$—band images of NGC 1566, M51a, and NGC 628. We refer to the southern arm and northern arm as “Arm 1” and “Arm 2”, respectively. The two black dashed circles in each panel mark the onset of the bulge and the location of the co–rotation radius of the galaxies. The horizontal bar in the lower left corner denotes a length scale of 2 kpc. North is up and East to the left.[]{data-label="fig:arms"}](arms_M51.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![The location of spiral arm ridge lines is shown by red lines overplotted on the smoothed $B$—band images of NGC 1566, M51a, and NGC 628. We refer to the southern arm and northern arm as “Arm 1” and “Arm 2”, respectively. The two black dashed circles in each panel mark the onset of the bulge and the location of the co–rotation radius of the galaxies. The horizontal bar in the lower left corner denotes a length scale of 2 kpc. North is up and East to the left.[]{data-label="fig:arms"}](arms_NGC628.pdf "fig:"){width="1\linewidth"}
Measuring azimuthal offset
--------------------------
Knowing the position of star clusters and spiral arm ridge lines in our target galaxies allowed us to measure the azimuthal distance of a star cluster from its closest spiral arm, assuming that it rotates on a circular orbit.
We limited our analysis to the star clusters located in the disk where spiral arms exist. The disk of a galaxy can be defined by its rotation curve. The rotational velocity increases when moving outwards from the central bulge–dominated part and becomes flat in the disk–dominated part of the galaxy. We derived a radius of 2 kpc for the bulge–dominated part of our galaxies using the rotation curves of [@k2000] for NGC 1566, [@sofi2; @sofi1] for M51a, and [@combes] for NGC 628. Furthermore, we limited our analysis to star clusters located inside the corotation radius. If stationary density waves are the dominant mechanism driving star formation in spiral galaxies we expect to find an age gradient from younger to older clusters inside the corotation radius. The bulge–dominated region and co–rotation radius of each galaxy are shown in Fig. \[fig:arms\]. The adopted corotation radii of the galaxies are listed in Tab. \[tab:properties of galaxies\].
Fig. \[fig:hist\] (left panels) shows the normalized distribution of the azimuthal distance of star clusters in the three age bins from their closest spiral arm ridge line in NGC 1566, M51a, and NGC 628. The error bars in each sample were calculated by dividing the square root of the number of clusters in each bin by the total number of clusters. We note that an azimuthal distance of zero degrees shows the location of the spiral arm ridge lines and not the center of the arms. Positive (negative) azimuthal distributions indicate that a cluster is located in front of (behind) the spiral arm ridge lines. Blue, green, and red colours represent the young, intermediate–age, and old star cluster samples, respectively.
Fig. \[fig:hist\] (right panels) shows the cumulative distribution function of star clusters as a function of the azimuthal distance. In order to test whether the samples come from the same distribution, we used a two–sample Kolmogorov–Smirnov test (hereafter K–S test). Since we aim at finding the age gradient in front of the spiral arms, the K–S test was only calculated for star clusters with positive azimuthal distances. The probability that two samples are drawn from the same distribution (p–values) and the maximum difference between pairs of cumulative distributions (D) are listed in Tab \[tab3\].
[0.49]{} {width="1\linewidth"}
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In the case of NGC 1566 (Fig. \[fig:hist\], top), we see that the young and intermediate–age star cluster samples are peaking close to the location of the spiral arm ridge lines (azimuthal distance of 0–5 degrees) while the old sample peaks further away from the ridge lines (azimuthal distances of 5–10 degrees). The derived p–values are lower than the test’s significance level (0.05) of the null hypothesis, i.e., that the two samples are drawn from the same distribution. As a consequence, our three star cluster samples are unlikely to be drawn from the same population. A clear age gradient across the spiral arms can be observed in NGC 1566, which is in agreement with the expectation from stationary density wave theory. The existence of such a pattern supports the picture of an age sequence in the model of a grand–design spiral galaxy and a barred galaxy suggested by [@DP10; @dimit17].
No obvious age gradient from younger to older is seen in the azimuthal distributions of the star cluster samples in M51a (Fig. \[fig:hist\], middle). What is remarkable here is that the older star clusters are located closer to the spiral arm ridge lines than the young and intermediate–age star clusters. The K–S test indicates that the probability that the young star cluster sample is drawn from the same distribution as the intemediate-age and old star cluster samples is more than 10%. The derived p–value for the intermediate–age and old cluster samples is lower than the significance level of the K–S test and rejects the null hypothesis that the two samples are drawn from the same distribution. The lack of an age pattern is consistent with the observed age trend for an interacting galaxy, modeled based on M51a, suggested by [@DP10]. Our result is compatible with a number of observational studies have found no indication for the expected spatial offset from the stationary density wave theory in M51a [@S09; @k10; @Foyle; @S17].
There is no evident trend in the azimuthal distribution of star clusters in NGC 628 (Fig. \[fig:hist\], bottom). The majority of the young star clusters tends to be located further away from the ridge lines (azimuthal distance of 20–25 degrees). The calculated p–values from the K–S test are larger than 0.05, which suggests weak evidence against the null hypothesis. As a result, the three young, intermediate–age, and old star cluster samples are drawn from the same distribution. The absence of an age gradient across the spiral arms in NGC 628 is consistent with a simulated multiple arm spiral galaxy by [@grand].
The origin of two spiral arms {#2arms}
=============================
An observational study by [@Egusa17], based on measuring azimuthal offsets between the stellar mass (from optical and near–infrared data) and gas mass distributions (from CO and HI data) in two spiral arms of M51a, suggest that the origin of these spiral arms differs. One spiral arm obeys the stationary density wave theory while the other does not.
In another recent study of M51a, [@chandar17] quantified the spatial distribution of star clusters with different ages relative to different segments of the two spiral arms of M51a traced in the 3.6 $\mu$m image. They observed a similar trend for the western and eastern arms: the youngest star clusters (< 6 Myr) are found near the spiral arm segments, and the older clusters (100–400 Myr) show an extended distribution.
In this section, we test whether measuring the azimuthal offset of star cluster samples from each spiral arm individually leads to different results. We assume that a star cluster whose distance from Arm 1 is smaller than its distance from Arm 2 belongs to Arm 1 and vice versa.
Fig. \[fig:age\_hist\] shows the normalized distribution of ages of star clusters associated with Arm 1 (shown in red) and Arm 2 (shown in blue) in each of the galaxies. No significant differences between the age distribution of star clusters belonging to the two spiral arms in our target galaxies can be observed. Also, the K–S test indicates that the age distributions of star clusters relative to Arm 1 and Arm 2 in each galaxy are drawn from the same population.
[0.50]{} ![The distribution of the age of star clusters associated with Arm 1 (red) and Arm 2 (blue) in NGC 1566, M51a, and NGC 628. The number of star clusters relative to the Arm 1 and Arm 2 is listed in parantheses.[]{data-label="fig:age_hist"}](NGC1566_arm_clusters.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![The distribution of the age of star clusters associated with Arm 1 (red) and Arm 2 (blue) in NGC 1566, M51a, and NGC 628. The number of star clusters relative to the Arm 1 and Arm 2 is listed in parantheses.[]{data-label="fig:age_hist"}](M51_arm_clusters.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![The distribution of the age of star clusters associated with Arm 1 (red) and Arm 2 (blue) in NGC 1566, M51a, and NGC 628. The number of star clusters relative to the Arm 1 and Arm 2 is listed in parantheses.[]{data-label="fig:age_hist"}](NGC628_arm_clusters.pdf "fig:"){width="1\linewidth"}
In Fig. \[fig:hist-arms\] we compare the normalized azimuthal distribution of the three young, intermediate–age, and old star cluster samples relative to Arm 1 (left panels) and Arm 2 (right panels) in our target galaxies. As before, our analysis was limited to the star clusters positioned in the disk and inside the corotation radius of our target galaxies.
[0.49]{} {width="1\linewidth"}
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The upper panels of Fig. \[fig:hist-arms\] exhibit a noticeable age gradient across both spiral arms of NGC 1566. The young star clusters are highly concentrated towards the location of Arm 1 and Arm 2 while the older ones are peaking further away from the two spiral arms.
The second row panels of Fig. \[fig:hist-arms\] show the azimuthal distance of star cluster samples across the two arms of M51a. This galaxy displays an offset in the location of young and old star clusters across Arm 1. The young star clusters culminate close to Arm 1 (at azimuthal distances of 2–6 degrees) while the old ones are positioned further away (at azimuthal distances of 6–10 degrees). Even though M51a shows an age gradient across the Arm 1 at first glance, the K–S test does not imply significant differences between the young and old star cluster samples (all derived p–values are larger than the test’s significance level). We do not observe any shift in the azimuthal distribution of the star cluster samples across Arm 2 in M51a.
In the case of NGC 628, no obvious age gradient across Arm 1 and Arm 2 is observed (the lower panels of Fig. \[fig:hist-arms\]). It is important to note that our results are inconclusive for the young star clusters associated with Arm 2 due to the small number statistics. Hence, we also explored the change in the azimuthal distribution of the star clusters by including clusters with masses < 5000 $\rm M_{\sun}$ and ages > 200 Myr. The observed differences are not significant and the general trend is the same as before.
Thus, measuring the azimuthal distance of the star clusters from the two individual spiral arms in each galaxy suggests that the two spiral arms of our target galaxies may have the same physical origin.
Comparison with the non–LEGUS cluster catalogue of M51 {#chandra}
======================================================
---------- ------ --------------------------- ------ --------------------------- ------ ---------------------------
p–value D p–value D p–value
NGC 1566 0.15 $ \rm 3.78\times 10^{-3}$ 0.31 $\rm 2.88 \times 10^{-5}$ 0.26 $\rm 6.19 \times 10^{-5}$
M51a 0.15 0.10 0.13 0.10 0.17 $\rm 2.4 \times 10^{-3}$
NGC 628 0.21 0.49 0.47 0.10
---------- ------ --------------------------- ------ --------------------------- ------ ---------------------------
In this section, we use the [@chandar16] catalogue (hereafter CH16 catalogue) to measure the azimuthal offsets of star clusters with different ages in M51a and to compare the results with our analysis based on the LEGUS catalogue. We caution that the south–eastern region of M51a is not covered by the LEGUS observations. We also investigated whether our results are biased due to the absence of star clusters from that region.
[@chandar16] provided a catalogue of 3816 star clusters in M51a based on HST ACS/WFC2 images obtained the equivalents of $UBVI$ and H$\rm \alpha$ filters. [@messa] compared the age distributions of star clusters in common between the LEGUS and CH16 catalogue. They observed that a large number of young star clusters (age < 10 Myr) in [@chandar16] have a broad age range (age: 1–100 Myr) in the LEGUS catalogue. They argued that the discrepancies in the estimated ages are due to the use of different filter combinations.
In Fig. \[fig:age\_mass\_chandar\], we show the distribution of ages and masses of star clusters in M51a from the CH16 catalogue. In order to be able to compare our results, we considered a mass–limited sample with masses > 5000 $\rm M_{\sun}$ and ages < 200 Myr and selected the same age bins as before: The young (< 10 Myr), intermediate–age (10–50 Myr), and old star cluster samples (50–200 Myr).
![The distribution of ages and masses of the 3816 star clusters in M51a, based on the CH16 catalogue. The young (<10 Myr ), intermediate–age (10–50 Myr), and old (50–200 Myr) star clusters are shown in blue, green, and red, respectively. The black points indicate excluded star clusters due to the applied mass cut and the imposed completeness limit. The number of clusters in each sample is listed in parentheses. The horizontal and vertical dotted lines show the applied mass cut of 5000 $\rm M_{\sun}$ and the corresponding detection completeness limit of 200 Myr, respectively. []{data-label="fig:age_mass_chandar"}](age_mass_chandar.pdf){width="49.00000%"}
In Fig. \[fig:clusters-ch16\], we plot the spatial distribution of the young, intermediate–age, and old star clusters based on the CH16 catalogue in M51a. As we can see, M51a displays a very clear and strong spiral pattern in the young star clusters. The intermediate–age star clusters tend to be located along the spiral arms while the old ones are more scattered and populate the inter–arm regions. Recently, [@chandar17] using the CH16 catalogue found that the youngest star clusters (< 6 Myr) are concentrated in the spiral arms (defined based on 3.6 $\mu$m observations). The older star clusters (6–100 Myr) are also found close to the spiral arms but they are more dispersed, and the spiral structure is not clearly recognisable in older star clusters (> 400 Myr).
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In order to quantify the possible spatial offset in the location of the three young, intermediate–age, and old star cluster samples from the CH16 catalogue across the spiral arms, we computed the normalized azimuthal distribution and corresponding cumulative distribution function of the star cluster samples in Fig. \[fig:azi\_ch\]. We applied our analysis only to the star clusters positioned in the disk and inside the co–rotation radius of M51a (2.0–5.5 kpc). Our result demonstrates that the three young, intermediate–age, and old star cluster samples peak at an azimuthal distance of 6 degrees from the location of the spiral arms. We observe no obvious offsets between the azimuthal distances of the three star cluster age samples in M51a. [@chandar17], using the same cluster catalogue, quantified the azimuthal offset of molecular gas (from PAWS and HERACLES) and young (<10 Myr) and intermediate–age (100–400 Myr) star clusters in the inner (2–2.5 kpc) and outer (5–5.5 kpc) annuli of the spiral arms. They found that in the inner annuli the young star clusters show an offset of 1 kpc from the molecular gas while there is no offset between the molecular gas and young and old star clusters in the outer portion of the spiral arms.
Adopting the CH16 catalogue, we found that there is no noticeable age gradient across the spiral arms of M51a, which is in agreement with our finding based on the LEGUS star cluster catalogue.
DISCUSSION AND CONCLUSIONS {#Summary}
==========================
The stationary density wave theory predicts that the age of star clusters increases with increasing distance away from the spiral arms. Therefore, a simple picture of the stationary density wave theory leads to a clear age gradient across the spiral arms. In this study, we are testing the theory that spiral arms are static features with constant pattern speed. For this purpose, we use the age and position of star clusters relative to the spiral arms.
We use high–resolution imaging observations obtained by the LEGUS survey [@C15] for three face–on LEGUS spiral galaxies, NGC 1566, M51a, and NGC 628. We have measured the azimuthal distance of the LEGUS star clusters from their closest spiral arm to quantify the possible spatial offset in the location of star clusters of different ages (< 10 Myr, 10–50 Myr, and 50–200 Myr) across the spiral arms. We found that the nature of spiral arms in our target galaxies is not unique. The main results are summarized as follows:
- Our detailed analysis of the azimuthal distribution of star clusters indicates that there is an age sequence across spiral arms in NGC 1566. NGC 1566 shows a strong bar and bisymmetric arms typical of a massive self–gravitating disk [@Elena15]. We speculate that when disks are very self–gravitating the bar and the two–armed features dominate a large part of the galaxy, producing an almost constant pattern speed. The observed trend is also in agreement with what was found by [@DP10] in simulations of a grand design and a barred spiral galaxy.
- We find no age gradient across the spiral arms of M51a. This galaxy shows less strong arms and a weaker bar and hence a less self–gravitating disk. The absence of an age sequence in M51a indicates that the grand–design structures of this galaxy are not the result of a steady–state density wave, with a fixed pattern speed and shape, as in the early analytical models. More likely, the spiral is a density wave that is still changing its shape and amplitude with time in reaction to the recent tidal perturbations. A possible mechanism to explain the formation and presence of grand–design structures in spiral galaxies is an interaction with a nearby companion [@Toomre; @72; @k; @B3]. Since such an interaction is obviously occurring in M51a, tidal interactions could be the dominant mechanism for driving its spiral patterns. [@DP10] simulated M51a with an interacting companion (M51b), and observed no age gradient across the tidally induced grand–design spirals arms. Our findings are consistent with the results of several other observational studies, which did not find age gradients as expected from the spiral density wave theory in M51a [@S09; @k10; @Foyle; @S17].
- NGC 628 is a multiple–arm spiral galaxy with weak spiral arms consistent with a pattern speed decreasing with radius and multiple corotation radii. In this case we find no significant offset among the azimuthal distributions of star clusters with different ages, which is consistent with the swing amplification theory. The lack of such an age offset is in agreement with an earlier analysis of NGC 628 [@Foyle], and consistent with the spatial distribution of star clusters with different ages in the simulated multiple–arm spiral galaxy by [@grand].
Acknowledgements {#acknowledgements .unnumbered}
================
This work is based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. These observations are associated with program 13364. Support for Program 13364 was provided by NASA through a grant from the Space Telescope Science Institute.
This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
A.A. acknowledges the support of the Swedish Research Council (Vetenskapsr[å]{}det) and the Swedish National Space Board (SNSB). D.A.G kindly acknowledges financial support by the German Research Foundation (DFG) through programme GO 1659/3–2.
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\[lastpage\]
[^1]: E-mail: f.shabani@stud.uni-heidelberg.de
[^2]: https://legus.stsci.edu
[^3]: the magnitude difference between apertures of radius 1 pixel and 3 pixels
|
---
abstract: 'In this paper, we investigate the classes of matroid intersection admitting a solution for the problem of partitioning the ground set $E$ into $k$ common independent sets, where $E$ can be partitioned into $k$ independent sets in each of the two matroids. For this problem, we present a new approach building upon the generalized-polymatroid intersection theorem. We exhibit that this approach offers alternative proofs and unified of previous results showing that the problem has a solution for the intersection of two laminar matroids and that of two matroids without $(k+1)$-spanned elements. Moreover, we newly show that the intersection of a laminar matroid and a matroid without $(k+1)$-spanned elements admits a solution. We also construct an example of a transversal matroid which is incompatible with the generalized-polymatroid approach.'
author:
- 'Kenjiro Takazawa[^1] and Yu Yokoi[^2]'
bibliography:
- 'myrefs.bib'
date: January 2019
title: |
A Generalized-Polymatroid Approach\
to Disjoint Common Independent Sets in Two Matroids
---
Introduction
============
For two matroids with a common ground set, the problem of partitioning the ground set into common independent sets is a classical topic in discrete mathematics. That is, extending bipartite edge-coloring theorem [@Konig16], described below, into general matroid intersection has been .
\[thm:Konig\] For a bipartite graph $G$ and a positive integer $k$, the edge set of $G$ can be partitioned into $k$ matchings if and only if the maximum degree of the vertices of $G$ is at most $k$.
Let $G=(U,V;E)$ be a bipartite graph. A subset $X$ of $E$ is a matching if and only if it is a common independent set of two partition matroids $M_{1}=(E,{\mathcal{I}}_{1})$ and $M_{2}=(E,{\mathcal{I}}_{2})$, where ${\mathcal{I}}_{1}$ (resp., ${\mathcal{I}}_{2}$) is the family of edge sets in which no two edges are adjacent at $U$ (resp., at $V$). The maximum degree of $G$ coincides with the minimum number $k$ such that $E$ can be partitioned into $k$ independent sets of $M_{1}$ and also into $k$ independent sets of $M_{2}$. We then naturally conceive the following problem for a general matroid pair on the common ground set.
\[prob:main\] Given two matroids $M_{1}=(E,{\mathcal{I}}_{1})$ and $M_{2}=(E,{\mathcal{I}}_{2})$ and a positive integer $k$ such that $E$ can be partitioned into $k$ independent sets of $M_{1}$ and also into $k$ independent sets of $M_{2}$, find a partition of $E$ into $k$ common independent sets of $M_{1}$ and $M_{2}$.
Solving Problem \[prob:main\] amounts to extending Theorem \[thm:Konig\] into matroid intersection. Such an extension is proved for arborescences in digraphs [@Edm73] and the intersection of two strongly base orderable matroids [@DM76], while such an extension is impossible for a simple example of the intersection of a graphic matroid and a partition matroid on the edge set of $K_4$ [@Schr03 Section 42.6c]. Indeed, Problem \[prob:main\] is known to be a challenging problem: we only have partial answers in the literature [@AB06; @DM76; @Edm73; @HKL11; @KZ05],
An interesting class of matroid intersection for which Problem \[prob:main\] admits a solution is introduced by Kotlar and Ziv [@KZ05]. For a matroid $M$ on ground set $E$ and a positive integer $k$, an element $e$ of $E$ is called *$k$-spanned* if there exist $k$ disjoint sets spanning $e$ (see Section \[sec:matroid\] for definition). Kotlar and Ziv [@KZ05] presented two sufficient conditions (Theorems \[thm:KZ1\] and \[thm:KZ2\]) for the common ground set $E$ of two matroids $M_1$ and $M_2$ to be into $k$ common independent sets, under the assumption that no element of $E$ is $(k+1)$-spanned in $M_1$ or $M_2$. Since the ground set of a matroid in which no element is $(k+1)$-spanned can be partitioned into $k$ independent sets (Lemma \[lem:single\_matroid\]), these two cases offer classes of matroid intersection for which Problem \[prob:main\] is solvable.
In this paper, we present a new approach to Problem \[prob:main\] building upon the integrality of *generalized polymatroids* [@Frank84; @Has82], a comprehensive class of polyhedra associated with a number of tractable combinatorial structures. This generalized-polymatroid approach is regarded as an extension of the polyhedral approach to bipartite edge-coloring [@Schr03 Section 20.3], and is indeed successful in *supermodular coloring* [@Tardos85], which is another matroidal generalization of bipartite edge-coloring (see [@Schr85; @Schr03]). Utilizing generalized polymatroids, we offer alternative proofs and unified for some special cases for which Problem \[prob:main\] admits solutions. To be more precise, we first prove the extension of Theorem \[thm:Konig\] for the intersection of two *laminar matroids*. Laminar matroids recently attract particular attention based on relation to the matroid secretary problem (see [@FO17] and references therein), and form a special case of strongly base orderable matroids. Thus, the generalized-polymatroid approach yields another proof for a special case of [@DM76]. We then show alternative proofs for the two cases of Kotlar and Ziv [@KZ05], which offer a new understanding of the tractability of the two cases. Moreover, we newly prove that Problem \[prob:main\] admits a solution for the intersection of a laminar matroid and a matroid in the two classes of Kotlar and Ziv [@KZ05]. Finally, we show a limit of the generalized-polymatroid approach by constructing an instance of a transversal matroid, another special class of strongly base orderable , which is incompatible with the generalized-polymatroid approach.
Preliminaries {#sec:pre}
=============
In this section, we review the definition and fundamental properties of matroids and generalized polymatroids. For more details, the readers are referred to [@FT88; @Fuj05; @Mbook; @Oxl11; @Schr03; @Wel76].
Matroid {#sec:matroid}
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Let $E$ be a finite set. For a subset $X\subseteq E$ and elements $e\in E\setminus X$, $e'\in X$, we denote $X+e=X\cup\{e\}$ and $X-e'=X\setminus\{e'\}$. For a set family ${\mathcal{I}}\subseteq 2^{E}$, a pair $(E, {\mathcal{I}})$ is called a [*matroid*]{} if ${\mathcal{I}}$ satisfies
- $\emptyset\in {\mathcal{I}}$,
- $X\subseteq Y\in {\mathcal{I}}$ implies $X\in {\mathcal{I}}$, and
- If $X,Y\in {\mathcal{I}}$ and $|X|<|Y|$, then $\exists e\in Y\setminus X: X+e\in {\mathcal{I}}$.
Each member $X$ of ${\mathcal{I}}$ is called an [*independent set*]{}. In particular, an independent set $B\in {\mathcal{I}}$ is called a [*base*]{} if it is maximal in ${\mathcal{I}}$ with respect to inclusion. It is known that all bases have the same size.
The [*rank function*]{} $r:2^{E}\to {\mathbf{Z}_{\geq 0}}$ of a matroid $M=(E, {\mathcal{I}})$ is defined by $r(A)=\max\{|X|\mid X\subseteq A,~X\in {\mathcal{I}}\}$ for any $A\subseteq E$. Then, it is known that ${\mathcal{I}}=\set{X| \forall A\subseteq E: |X\cap A|\leq r(A)}$ holds. A subset $X \subseteq E$ *spans* an element $e \in E$ if $r(X + e) = r(X)$.
For a matroid $M=(E,{\mathcal{I}})$ and a subset $S\subseteq E$, the restriction $M|S$ of $M$ to $S$ is a pair $(S, {\mathcal{I}}|S)$, where ${\mathcal{I}}|S=\set{X|X\in {\mathcal{I}},~X\subseteq S}$. For any $S\subseteq E$, the restriction $M|S$ is again a matroid.
For a matroid $M=(E,{\mathcal{I}})$ and a positive integer $k\in {\mathbf{Z}}$, we define a set family ${\mathcal{I}}^{k}\subseteq 2^{E}$ by $${\mathcal{I}}^{k}=\set{X\subseteq E|\text{$X$ can be partitioned into $k$ sets in ${\mathcal{I}}$}}.$$
Generalized Polymatroid
-----------------------
Let $E$ be a finite set. A function $b:2^{E}\to {\mathbf{R}}\cup\{\infty\}$ is called [*submodular*]{} if it satisfies the [*submodular inequality*]{} $$b(A)+b(B)\geq b(A\cup B)+b(B\cap A)
$$ for any $A, B\subseteq E$, where the inequality is to hold if the left-hand side is infinite. A function $p:2^{E}\to {\mathbf{R}}\cup\{-\infty\}$ is called [*supermodular*]{} if $-p$ is submodular. A pair $(p,b)$ is called [*paramodular*]{} if we have
- $p(\emptyset)=b(\emptyset)=0$,
- $p$ is supermodular, $b$ is submodular, and
- $p$ and $b$ satisfy the [*cross inequality*]{} $$b(A)-p(B)\geq b(A\setminus B)-p(B\setminus A) \label{eq:cross-ineq}$$ for any $A,B\subseteq E$, where the inequality is to hold if the left-hand side is infinite.
For a pair of set functions $p:2^{E}\to {\mathbf{R}}\cup\{-\infty\}$ and $b:2^{E}\to {\mathbf{R}}\cup \{\infty\}$, we associate a polyhedron $Q(p,b)$ defined by $$Q(p,b)=\set{x\in {\mathbf{R}}^{E}|\forall A\subseteq E: p(A)\leq x(A)\leq b(A)},$$ where $x(A)=\sum\set{{\textcolor{black}{x_e}}| e\in A}$. Here, $p$ serves as a lower bound while $b$ serves as an upper bound of the polyhedron $Q(p,b)$. A polyhedron $P\subseteq {\mathbf{R}}^{E}$ is called a [*generalized polymatroid*]{} (for short, a [*g-polymatroid*]{}) if $P=Q(p,b)$ holds for some paramodular pair $(p,b)$. It is known [@Frank84; @FT88] that such a paramodular pair is uniquely defined for any g-polymatroid.
We next introduce the concept of [*intersecting paramodularity*]{}, which is weaker than paramodularity but still yields g-polymatroids. We say that subsets $A,B\subseteq E$ are [*intersecting*]{} if none of $A\cap B$, $A\setminus B$ and $B\setminus A$ is empty. A function $b:2^{E}\to {\mathbf{R}}\cup\{\infty\}$ is called [*intersecting submodular*]{} if it satisfies the submodular inequality for any intersecting subsets $A, B\subseteq E$. A function $p:2^{E}\to {\mathbf{R}}\cup\{-\infty\}$ is called [*intersecting supermodular*]{} if $-p$ is intersecting submodular. A pair $(p,b)$ is called [*intersecting paramodular*]{} if $p$ and $b$ are intersecting super- and submodular functions, respectively, and the cross inequality holds for any intersecting subsets $A, B\subseteq E$. The following theorem, , states that an intersecting-paramodular pair $(p,b)$ defines a g-polymatroid. We say that a pair $(p,b)$ of set functions is [*integral*]{} if each of $b(A)$ and $p(A)$ is integer or infinite for any $A\subseteq E$. We say that a polyhedron is [*integral*]{} if each of its faces contains an integral point, so for pointed polyhedra, the vertices should be integral.
\[thm:intersecting\] For an intersecting-paramodular pair $(p,b)$ such that $Q(p,b)\neq \emptyset$, the polyhedron $Q(p,b)$ is a g-polymatroid, which is, in addition, integral whenever $(p,b)$ is integral.
In general, the intersection of two integral polyhedra $P_1$ and $P_2$ is not necessarily integral. For two integral g-polymatroids, however, the intersection preserves integrality as stated below. This fact plays a key role in our g-polymatroid approach to Problem 1.2.
\[thm:intersection\] For two integral g-polymatroids $P_{1}$ and $P_{2}$, the intersection $P_{1}\cap P_{2}$ is an integral polyhedron if it is nonempty.
As this paper studies partitions of finite sets, we are especially interested in vectors in the intersection of a g-polymatroid and the unit hypercube $[0,1]^{E}=\set{x\in {\mathbf{R}}^{E}|\forall e\in E:0\leq x(e)\leq 1}$. It is known that the intersection is again a g-polymatroid.
\[thm:unit\_cube\] For a g-polymatroid $P$, if $P\cap [0,1]^{E}$ is nonempty, then the intersection $P\cap [0,1]^{E}$ is again a g-polymatroid, which is, in addition, integral whenever $P$ is integral.
Similarly to the definition of $Q(p,b)$, for a pair of set functions $p:2^{E}\to {\mathbf{R}}\cup\{-\infty\}$ and $b:2^{E}\to {\mathbf{R}}\cup \{\infty\}$, we associate the following set family: $${\mathcal{F}}(p,b)=\set{X\subseteq E|\forall A\subseteq E: p(A)\leq |X\cap A|\leq b(A)}.$$
For a subset $Y\subseteq E$, its [*characteristic vector*]{} $\chi_{Y}\in \{0,1\}^{E}$ is defined by $\chi_{Y}(e)=1$ for $e\in Y$ and $\chi_{Y}(e)=0$ for $e\in E\setminus Y$. The following observation is derived from Theorem \[thm:unit\_cube\].
\[lem:convex\] For an integral intersecting-paramodular pair $(p,b)$, the polyhedron $Q(p,b)\cap [0,1]^{E}$ is a convex hull of the characteristic vectors of the members of ${\mathcal{F}}(p,b)$.
By Theorem \[thm:unit\_cube\], $Q(p,b)\cap [0,1]^{E}$ is integral, and hence all its vertices are $(0, 1)$-vectors. Also, by the definition of $Q(p,b)$ and ${\mathcal{F}}(p,b)$, we have $y\in Q(p,b)\cap \{0,1\}^{E}$ if and only if $y=\chi_{Y}$ for some $Y\in {\mathcal{F}}(p,b)$. Thus, the vertices of $Q(p,b)\cap [0,1]^{E}$ coincides with the characteristic vectors of the members of ${\mathcal{F}}(p,b)$.
Generalized-Polymatroid Approach {#sec:main}
================================
In this section, we exhibit some cases of matroid intersection for which a solution of Problem \[prob:main\] can be constructed by utilizing the g-polymatroid intersection theorem (Theorem \[thm:intersection\]). In Section \[sec:general\], we describe a general method to apply Theorem \[thm:intersection\] for solving Problem \[prob:main\]. In Section \[sec:laminar\], we use this method to prove an extension of Theorem \[thm:Konig\] to the intersection of two laminar matroids, a special case of intersection of two strongly base orderable matroids [@DM76]. In Section \[sec:k+1-spanned\], we utilize this method for alternative proofs for two classes of matroid intersection due to Kotlar and Ziv [@KZ05]. Section \[sec:new\] presents a new class of matroid intersection for which Problem \[prob:main\] admits a solution: intersection of a laminar matroid and a matroid in Kotlar and Ziv’s classes. Finally, in Section \[sec:complexity\], we explain an algorithmic implementation of our general method and analyze its time complexity.
General Method {#sec:general}
--------------
With the notations introduced in Section \[sec:matroid\], now Problem \[prob:main\] is reformulated as follows.
Given matroids $M_{1}=(E,{\mathcal{I}}_{1})$ and $M_{2}=(E,{\mathcal{I}}_{2})$ and a positive integer $k$ such that $E\in {\mathcal{I}}_{1}^{k}\cap {\mathcal{I}}_{2}^{k}$, find a partition $\{X_{1},X_{2},\dots,X_{k}\}$ of $E$ such that $X_{j}\in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$ for $j=1,2,\dots,k$.
[\[prob:main\]]{}
Our general method to solve Problem \[prob:main\] is to find $X \in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$ such that $E\setminus X\in {\mathcal{I}}^{k-1}_{1}\cap {\mathcal{I}}^{k-1}_{2}$ with the aid of g-polymatroid intersection, replace $E$ and $k$ with $E\setminus X$ $k-1$, respectively, and iterate. The following proposition, which can be proved by combining Theorems \[thm:intersecting\]–\[thm:unit\_cube\] and Lemma \[lem:convex\], shows a necessary condition that this method can be applied.
\[prop:g-matroid\_approach\] Let $M_{1}=(E,{\mathcal{I}}_{1})$, $M_{2}=(E,{\mathcal{I}}_{2})$ be matroids and $k\in {\mathbf{Z}}$ be a positive integer with $E\in {\mathcal{I}}_{1}^{k}\cap {\mathcal{I}}_{2}^{k}$. If there exists an integral intersecting-paramodular $(p_{i},b_{i})$ such that $$\begin{aligned}
\label{EQpb}
{\mathcal{F}}(p_i,b_i)=\set{X\subseteq E|X\in {\mathcal{I}}_i,~E\setminus X\in {\mathcal{I}}_i^{k-1}} \end{aligned}$$ for each $i=1,2$, then there exists a subset $X\subseteq E$ such that $X\in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$ and $E\setminus X\in {\mathcal{I}}_{1}^{k-1}\cap {\mathcal{I}}_{2}^{k-1}$.
Because $E\in {\mathcal{I}}_i^{k}$, there is a partition $\{X_{1}, X_{2},\dots,X_{k}\}$ of $E$ such that $X_{j}\in {\mathcal{I}}_i$ for each $j=1,2,\dots,k$. For each $j$, we have $X_{j}\in {\mathcal{I}}_i$ and $E\setminus X_{j}=\bigcup_{\ell:\ell\neq j}X_{\ell}\in {\mathcal{I}}^{k-1}$, and hence $X_{j}\in {\mathcal{F}}(p_i,b_i)$. As $\{X_{1}, X_{2},\dots,X_{k}\}$ is a partition of $E$, the vector $x:=\left(\frac{1}{k}, \frac{1}{k},\dots,\frac{1}{k}\right)^{\top}\in {\mathbf{R}}^E$ coincides with $\sum_{j=1}^{k}\frac{1}{k}\cdot \chi_{X_{j}}$, which is a convex combination of the characteristic vectors of $X_{j}\in {\mathcal{F}}(p_i,b_i)$ ($j=1,2,\ldots,k$). Then, Lemma \[lem:convex\] implies $x\in Q(p_i,b_i)\cap [0,1]^{E}$.
Now $Q(p_{1},b_{1})\cap Q(p_{2},b_{2})\cap [0,1]^{E}$ includes the vector $\left(\frac{1}{k}, \frac{1}{k},\dots, \frac{1}{k}\right)^{\top}$, and hence is nonempty. Then, by combining Theorems \[thm:intersecting\]–\[thm:unit\_cube\], we obtain that $Q(p_{1},b_{1})\cap Q(p_{2},b_{2})\cap [0,1]^{E}$ is an integral nonempty polyhedron, and hence it contains a $(0,1)$-vector $y$. Let $Y\subseteq E$ be the set satisfying $\chi_{Y}=y$. Then $y\in Q(p_{1},b_{1})\cap Q(p_{2},b_{2})\cap [0,1]^{E}$ implies $Y\in {\mathcal{F}}(p_{1},b_{1})\cap{\mathcal{F}}(p_{2},b_{2})$, which means $Y\in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$ and $E\setminus Y\in {\mathcal{I}}_{1}^{k-1}\cap {\mathcal{I}}_{2}^{k-1}$.
In order to use our method, $M_1$ and $M_2$ should belong to a class of matroids in which each member $M=(E,{\mathcal{I}})$ with $E\in {\mathcal{I}}^{k}$ admits an integral intersecting-paramodular pair $(p,b)$ satisfying and the restriction $M|(E\setminus X)$ with any $X\in {\mathcal{F}}(p,b)$ belongs to this class again with $k$ replaced by $k-1$. In the subsequent subsections, we show that the class of laminar matroids and the two matroid classes in [@KZ05] have this property.
\[REMindependent\] An advantage of our approach is that there is no constraint the two matroids $M_1$ and $M_2$. In other words, our approach can deal with any pair of matroids such that each of them admits an intersecting-paramodular pair required in Proposition \[prop:g-matroid\_approach\]. This some previous works [@DM76; @KZ05], which assume that the two matroids are in the same matroid class. Indeed, utilizing this fact, we provide (Theorems \[thm:lam+KZ1\] and \[thm:lam+KZ2\]) that is not included in previous works.
Intersection of Two Laminar Matroids {#sec:laminar}
------------------------------------
In this section, we prove that Problem \[prob:main\] is solvable for laminar matroids by our generalized-polymatroid approach. Since a laminar matroid is a generalization of a partition matroid, this extends the bipartite edge-coloring theorem of Kőnig [@Konig16]. On the other hand, since a laminar matroid is strongly base orderable, this proof amounts to another proof for a special case of strongly base orderable matroids by Davies and McDiarmid [@DM76].
We first define the concept of laminar matroids. A subset family ${\mathcal{A}}$ of a finite set $E$ is called *laminar* if $A_1,A_2 \in {\mathcal{A}}$ implies $A_1 \subseteq A_2$, $A_2 \subseteq A_1$, or $A_1 \cap A_2 = \emptyset$. Let ${\mathcal{A}}\subseteq 2^E$ be a laminar family and $q:{\mathcal{A}}\to {\mathbf{Z}_{\geq 0}}$ be a capacity function. Let ${\mathcal{I}}$ be a family of subsets $X$ satisfying all capacity constraints, i.e., $${\mathcal{I}}=\set{X\subseteq E|\forall A\in {\mathcal{A}}: |X\cap A|\leq q(A)}.$$ Then it is known that $(E,{\mathcal{I}})$ is a matroid, which we call the [*laminar matroid*]{} induced from .
It is known that a laminar matroid is a special case of a *strongly base orderable matroid* [@Bru70].
A matroid is [*strongly base orderable*]{} if for each pair of bases $B_{1}$, $B_{2}$ there exists a bijection $\pi:B_{1}\to B_{2}$ such that for each subset $X$ of $B_{1}$ the set $\pi(X)\cup (B_{1}\setminus X)$ is a base again.
Thus, it follows from the result of Davies and McDiarmid [@DM76] that Theorem \[thm:Konig\] can be extended to the intersection of laminar matroids.
\[thm:laminar-partition\] For laminar matroids $M_{1}=(E,{\mathcal{I}}_{1})$ and $M_{2}=(E,{\mathcal{I}}_{2})$ and a positive integer $k$ such that $E\in {\mathcal{I}}_{1}^{k}\cap {\mathcal{I}}_{2}^{k}$, there exists a partition $\{X_{1},X_{2},\dots,X_{k}\}$ of $E$ such that $X_{j}\in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$ for each $j=1,2,\dots,k$.
In the rest of this subsection, we present an alternative proof for this theorem via the generalized-polymatroid approach. We first observe some properties of laminar matroids. It is known and can be easily observed that the class of laminar matroids is closed under taking restrictions.
\[lem:restriction-laminar-matroid\] Let $M=(E,{\mathcal{I}})$ be a laminar matroid induced from a laminar family ${\mathcal{A}}$ and a capacity function $q:{\mathcal{A}}\to {\mathbf{Z}_{\geq 0}}$. Then for any subset $S\subseteq E$, the restriction $M|S$ of $M$ to $S$ is a laminar matroid induced from a laminar family and a capacity function $q_S \colon {\mathcal{A}}_S \to {\mathbf{Z}_{\geq 0}}$ defined by .
The next lemma[^3] states that, if $M=(E,{\mathcal{I}})$ is a laminar matroid induced from a laminar family ${\mathcal{A}}$, then $M^k = (E, {\mathcal{I}}^k)$ is also a laminar matroid induced from ${\mathcal{A}}$.
\[lem:sum-of-laminar-matroid\] Let $M=(E,{\mathcal{I}})$ be a laminar matroid induced from a laminar family ${\mathcal{A}}$ and a capacity function $q:{\mathcal{A}}\to {\mathbf{Z}_{\geq 0}}$. Then for a positive integer $k$, the matroid $M^{k}=(E,{\mathcal{I}}^{k})$ is a laminar matroid defined by $${\mathcal{I}}^{k}=\set{X\subseteq E|\forall A\in {\mathcal{A}}: |X\cap A|\leq k\cdot q(A)}.$$
We show that, for any $X\subseteq E$, there exists a partition $\{Y_{1},Y_{2},\dots,Y_{k}\}$ of $X$ with $Y_{j}\in {\mathcal{I}}~(j=1,\dots,k)$ if and only if $|X\cap A|\leq k\cdot q(A)$ for any $A\in {\mathcal{A}}$. The necessity is clear, because each $Y_{j}$ satisfies $|Y_{j}\cap A|\leq q(A)$ for any $A\in {\mathcal{A}}$. For the sufficiency, suppose $|X\cap A|\leq k\cdot q(A)$ for any $A\in {\mathcal{A}}$. Let $X=\{e_{1},e_{2},\dots,e_{|X|}\}$ (i.e., give indices for the elements in $X$), so that for all $A\in {\mathcal{A}}$ the elements in $X\cap A$ have consecutive indices. This can be done easily because ${\mathcal{A}}$ is a laminar family[^4]. For each $j\in\{1,2,\dots,k\}$, let . Then, $\{Y_{1},Y_{2},\dots,Y_{k}\}$ is a partition of $X$, and, for each $Y_{j}$ and $A\in {\mathcal{A}}$, we have $|Y_{j}\cap A|\leq \lceil |X\cap A|/k \rceil$ by the definition of the indices. Because $|X\cap A|\leq k\cdot q(A)$, this implies $|Y_{j}\cap A|\leq q(A)$ for all $A\in {\mathcal{A}}$. Thus, we have $Y_{j}\in {\mathcal{I}}$ for each $j\in \{1,2,\dots,k\}$.
The next lemma provides an integral intersecting-paramodular pair $(p,b)$ satisfying the condition in Proposition \[prop:g-matroid\_approach\] for a laminar matroid.
\[lem:g-polymatroid-approach-laminar\] Let $M=(E,{\mathcal{I}})$ be a matroid induced from a laminar family ${\mathcal{A}}$ and a function $q:{\mathcal{A}}\to {\mathbf{Z}_{\geq 0}}$ and suppose $E\in {\mathcal{I}}^{k}$ for a positive integer $k$. Define $p:2^{E}\to {\mathbf{Z}}\cup\{-\infty\}$ and $b:2^{E}\to {\mathbf{Z}}\cup \{\infty\}$ by $$\begin{aligned}
{2}
&p(A)=|A|-(k-1)\cdot q(A)\qquad &(A\in {\mathcal{A}}),\\
&b(A)=q(A) &(A\in {\mathcal{A}}),\end{aligned}$$ where $p(B)=-\infty$, $b(B)=\infty$ for all $B\in 2^{E}\setminus{\mathcal{A}}$. Then $(p,b)$ is an integral intersecting-paramodular pair satisfying ${\mathcal{F}}(p,b)=\set{X\subseteq E|X\in {\mathcal{I}},~E\setminus X\in {\mathcal{I}}^{k-1}}$.
Since ${\mathcal{A}}$ is laminar and the values of $p$ and $b$ are finite only on ${\mathcal{A}}$, there is no intersecting pair of subsets of $E$ both of which have finite function values. Thus, $(p,b)$ is trivially intersecting paramodular.
For any $X\subseteq E$, the condition $\forall A\in A: |X\cap A|\geq p(A)=|A|-(k-1)\cdot q(A)$ is equivalent to $\forall A\in {\mathcal{A}}: |(E\setminus X)\cap A|\leq (k-1)\cdot q(A)$, and hence equivalent to $E\setminus X\in {\mathcal{I}}^{k-1}$ by Lemma \[lem:sum-of-laminar-matroid\]. Also, $\forall A\in {\mathcal{A}}: |X\cap A|\leq b(A)=q(A)$ is equivalent to $X\in {\mathcal{I}}$. Thus we have $X\in {\mathcal{F}}(p,b)$ if and only if $X\in {\mathcal{I}}$ and $E\setminus X\in {\mathcal{I}}^{k-1}$ hold.
Now we show that Problem \[prob:main\] can be solved for any pair of laminar matroids using the generalized-polymatroid approach.
We show the theorem by induction on $k$. The case $k=1$ is trivial. Let $k\geq 2$ and suppose that the statement holds for $k-1$. By Lemma \[lem:g-polymatroid-approach-laminar\], for each $i=1,2$, there exists an integral intersecting-paramodular pair $(p_{i},b_{i})$ such that ${\mathcal{F}}(p_{i},b_{i})=\set{X\subseteq E|X\in {\mathcal{I}}_{i},~E\setminus X\in {\mathcal{I}}_{i}^{k-1}}$. Then, by Proposition \[prop:g-matroid\_approach\], there exists $X\in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$ satisfying $E\setminus X\in {\mathcal{I}}_{1}^{k-1}\cap {\mathcal{I}}_{2}^{k-1}$. By Lemma \[lem:restriction-laminar-matroid\], the restrictions $M'_{1}:=M_{1}|(E\setminus X)$ and $M'_{2}:=M_{2}|(E\setminus X)$ are laminar. Therefore, by the induction hypothesis, $E\setminus X$ can be partitioned into $k-1$ common independent sets of $M'_{1}$ and $M'_{2}$, and hence of $M_{1}$ and $M_{2}$. Thus, $E$ can be partitioned into $k$ common independent sets.
\[rem:general-case\] Here we mention an extension of Lemma \[lem:sum-of-laminar-matroid\]. For an intersecting-submodular function $b:2^{E}\to {\mathbf{Z}_{\geq 0}}\cup\{\infty\}$, define a family ${\mathcal{I}}_b=\set{X\subseteq E|\forall A\subseteq E: |X\cap A|\leq b(A)}$. Then, it is known [@Edmonds70] that $(E,{\mathcal{I}}_b)$ is a matroid. Actually, a laminar matroid is a special case of such matroids: When $(E,{\mathcal{I}})$ is a laminar matroid induced by a laminar family ${\mathcal{A}}$ and a capacity function $q:{\mathcal{A}}\to {\mathbf{Z}_{\geq 0}}$, then ${\mathcal{I}}= {\mathcal{I}}_b$ holds for an intersecting-submodular function $b$ defined by $b(A)=q(A)$ for $A\in {\mathcal{A}}$ and $b(A)=\infty$ for $A\in 2^E\setminus {\mathcal{A}}$. Lemma \[lem:sum-of-laminar-matroid\] can extends to this matroid class. That is, for any intersecting-submodular function $b$, the family ${\mathcal{I}}_b^k$ can be represented as ${\mathcal{I}}_b^k=\set{X\subseteq E|\forall A\subseteq E: |X\cap A|\leq k\cdot b(A)}$.
Now we show this claim. As shown by Edmonds [@Edmonds70], the rank function of $(E,{\mathcal{I}}_b)$ is given as $$\textstyle r_b(X)=\min\{|X\setminus(Y_1\cup Y_2\cup \cdots \cup Y_l)|+\sum_{i=1}^{l}b(Y_i)\mid Y_1, Y_2,\dots,Y_l \text{ are pairwise disjoint} \}.
\label{eq:truncation1}$$ Note that $k\cdot b$ is also an intersecting-submodular function on $E$. Hence, it defines a matroid $(E, {\mathcal{I}}_{k\cdot b})$ where ${\mathcal{I}}_{k\cdot b}:=\set{X\subseteq E|\forall A\subseteq E: |X\cap A|\leq k\cdot b(A)}$, and its rank function is given as $$\textstyle r_{k\cdot b}(X)=\min\{|X\setminus(Y_1\cup Y_2\cup\cdots \cup Y_l)|+\sum_{i=1}^{l}k\cdot b(Y_i)\mid Y_1, Y_2,\dots,Y_l \text{ are pairwise disjoint} \}.
\label{eq:truncation2}$$ On the other hand, by Theorem \[thm:matroid\_union\], the rank function $r_b^k$ of the matroid $(E, {\mathcal{I}}_b^{k})$, is given by using $r_b$. Substituting to , we can check that $r_b^k=r_{k\cdot b}$. Thus, ${\mathcal{I}}^k_b={\mathcal{I}}_{k\cdot b}$ is proved.
We remark that, even Lemma \[lem:sum-of-laminar-matroid\] extends to this matroid class, our approach for Problem \[prob:main\] does not extend because Lemma \[lem:g-polymatroid-approach-laminar\] fails to extend to this class.
Intersection of Two Matroids without $(k+1)$-Spanned Elements {#sec:k+1-spanned}
-------------------------------------------------------------
Let $M=(E,{\mathcal{I}})$ be a matroid and $k$ be a positive integer. Recall that an element $e\in E$ is said to be [*$k$-spanned*]{} in $M$ if there exist $k$ disjoint sets spanning $e$ (including the trivial spanning set $\{e\}$).
Consider a class of matroids such that no element is $(k+1)$-spanned. Kotlar and Ziv [@KZ05] provided two cases for which Problem \[prob:main\] admits solutions.
\[thm:KZ1\] Let $M_{1}=(E,{\mathcal{I}}_{1})$ and $M_{2}=(E,{\mathcal{I}}_{2})$ be two matroids with rank functions $r_{1}$ and $r_{2}$ and suppose $r_{1}(E)=r_{2}(E)=d$ and $|E|=k\cdot d$. If no element of $E$ is $(k+1)$-spanned in $M_{1}$ or $M_{2}$, then $E$ can be partitioned into $k$ common bases.
\[thm:KZ2\] Let $M_{1}=(E,{\mathcal{I}}_{1})$ and $M_{2}=(E,{\mathcal{I}}_{2})$ be two matroids. If no element of $E$ is $3$-spanned in $M_{1}$ or $M_{2}$, then $E$ can be partitioned into two common independent sets.
Note that, in Theorems \[thm:KZ1\] and \[thm:KZ2\], the condition $E\in {\mathcal{I}}_{1}^{k}\cap{\mathcal{I}}_{2}^{k}$ is not explicitly assumed. However, it can be easily proved by induction on $|E|$.
\[lem:single\_matroid\] If no element of a matroid $M=(E,{\mathcal{I}})$ is $(k+1)$-spanned, then $E\in {\mathcal{I}}^{k}$.
We provide unified proofs for Theorems \[thm:KZ1\] and \[thm:KZ2\] via the generalized-polymatroid approach, by constructing integral paramodular pairs satisfying in Proposition \[prop:g-matroid\_approach\]. We first show that the cross-inequality condition, which is required for paramodularity, is equivalent to a seemingly weaker condition.
\[lem:local-cross-ineq\] A pair $(p,b)$ of set functions satisfies the cross inequality for any $A, B\subseteq E$ if and only if it satisfies the following inequality for every pair of disjoint subsets $\tilde{A},\tilde{B}\subseteq E$ and element $e\in E\setminus(\tilde{A}\cup \tilde{B})$: $$b(\tilde{A}+e)-b(\tilde{A})\geq p(\tilde{B}+e)-p(\tilde{B}). \label{eq:cross-ineq2}$$
The necessity is obvious, since is obtained by substituting $A=\tilde{A}+e$ and $B=\tilde{B}+e$ into . For sufficiency, we show for arbitrary $A, B\subseteq E$ under the assumption of . Let $A\cap B=\{e_{1},e_{2},\dots,e_{m}\}$ where $m=|A\cap B|$ and define $\tilde{A}_{\ell}=(A\setminus B)\cup \{e_{1},e_{2},\dots,e_{\ell-1}\}$ and $\tilde{B}_{\ell}=(B\setminus A)\cup \{e_{\ell+1},e_{\ell+2},\dots,e_{m}\}$ for each $\ell\in\{1,2,\dots,m\}$. Then $\tilde{A}_{\ell}$ and $\tilde{B}_{\ell}$ are disjoint and $e_{\ell}\in E\setminus(\tilde{A}_{\ell}\cup \tilde{B}_{\ell})$, and hence we have $b(\tilde{A}_{\ell}+e_{\ell})-b(\tilde{A}_{\ell})\geq p(\tilde{B}_{\ell}+e_{\ell})-p(\tilde{B}_{\ell})$ for $\ell=1,2,\dots, m$. As we have $\tilde{A}_{1}=A\setminus B$, $\tilde{A}_{m}+e_{m}=A$, $\tilde{B}_{1}+e_{1}=B$, and $\tilde{B}_{m}=B\setminus A$, it follows that $$b(A)-b(A\setminus B)=\sum_{\ell=1}^{m}b(\tilde{A}_{\ell}+e_{\ell})-b(\tilde{A}_{\ell})
\geq \sum_{\ell=1}^{m}p(\tilde{B}_{\ell}+e_{\ell})-p(\tilde{B}_{\ell})=p(B)-p(B\setminus A).$$ Thus, $A$ and $B$ satisfy the cross inequality .
Lemma \[lem:local-cross-ineq\] states that, the range of the subsets $A,B \subseteq E$ in the cross inequality can be narrowed so that $|A \cap B|=1$. We also remark that the submodularity of $b$ and the supermodularity of $p$ are not assumed in Lemma \[lem:local-cross-ineq\].
Now an integral paramodular pair satisfying the condition in Proposition \[prop:g-matroid\_approach\] is constructed as follows.
\[lem:g-polymatroid-approach-k+1\] Let $M=(E,{\mathcal{I}})$ be a matroid with rank function $r:2^{E}\to {\mathbf{Z}_{\geq 0}}$. For a positive integer $k$, suppose that no element is $(k+1)$-spanned in $M$. Define $p:2^{E}\to {\mathbf{Z}}$ and $b:2^{E}\to {\mathbf{Z}}$ by $$\begin{aligned}
{2}
p(A)&=|A|-r^{k-1}(A)\quad&(A\subseteq E),\\
b(A)&=r(A) &(A\subseteq E).\end{aligned}$$ Then $(p,b)$ is an integral paramodular pair such that ${\mathcal{F}}(p,b)=\set{X\subseteq E|X\in {\mathcal{I}},~E\setminus X\in {\mathcal{I}}^{k-1}}$.
It directly follows from the definitions of $p$ and $b$ that (i) $p(\emptyset)=b(\emptyset)=0$, (ii) $p$ is supermodular, $b$ is submodular. Then, to prove that $(p,b)$ is paramodular, it remains to show the cross inequality for any $A,B\subseteq E$. By Lemma \[lem:local-cross-ineq\], it suffices to show for any disjoint $\tilde{A},\tilde{B}\subseteq E$ and element $e\in E\setminus(\tilde{A}\cup \tilde{B})$, where is rephrased as follows by the definitions of $p$ and $b$: $$\left(r(\tilde{A}+e)-r(\tilde{A})\right)+\left(r^{k-1}(\tilde{B}+e)-r^{k-1}(\tilde{B})\right)\geq 1.$$ Take a maximal independent set $X$ of $M$ subject to $X\subseteq \tilde{A}$ and a maximal independent set $Y$ of $M^{k-1}$ subject to $Y\subseteq \tilde{B}$. It is sufficient to show $X+e\in {\mathcal{I}}$ or $Y+e\in {\mathcal{I}}^{k-1}$, because they respectively imply $r(\tilde{A}+e)\geq r(\tilde{A})+1$ or $r^{k-1}(\tilde{B}+e)\geq r^{k-1}(\tilde{B})+1$. Note that $Y\in {\mathcal{I}}^{k-1}$ can be partitioned into $k-1$ independent sets $Y_{1},Y_{2},\dots,Y_{k-1}\in {\mathcal{I}}$. Also, $k+1$ subsets $\{e\}, X, Y_{1}, Y_{2},\dots,Y_{k-1}$ are all disjoint. Because no element is $(k+1)$-spanned in $M$, it follows that $e$ is not spanned by at least one of $X, Y_{1}, Y_{2},\dots,Y_{k-1}$. Note that $Y_{j}+e\in {\mathcal{I}}$ for some $j$ implies $Y+e\in {\mathcal{I}}^{k-1}$ by the definition of ${\mathcal{I}}^{k-1}$. We then have $X+e\in {\mathcal{I}}$ or $Y+e\in {\mathcal{I}}^{k-1}$. Thus, the paramodularity of $(p,b)$ is proved.
We next show ${\mathcal{F}}(p,b)=\set{X\subseteq E|X\in {\mathcal{I}},~E\setminus X\in {\mathcal{I}}^{k-1}}$. For any $X\subseteq E$, the condition $\forall A\subseteq E: |X\cap A|\geq p(A)=|A|-r^{k-1}(A)$ is equivalent to $\forall A\subseteq E: |(E\setminus X)\cap A|\leq r^{k-1}(A)$, and hence equivalent to $E\setminus X\in {\mathcal{I}}^{k-1}$. Also, $\forall A\subseteq E: |X\cap A|\leq b(A)=r(A)$ is equivalent to $X\in {\mathcal{I}}$. Thus we have $X\in {\mathcal{F}}(p,b)$ if and only if $X\in {\mathcal{I}}$ and $E\setminus X\in {\mathcal{I}}^{k-1}$ hold.
Combining Lemmas \[lem:single\_matroid\], \[lem:g-polymatroid-approach-k+1\] and Proposition \[prop:g-matroid\_approach\] yields the following proposition.
\[prop:induction\_step\] Let $M_{1}=(E,{\mathcal{I}}_{1})$ and $M_{2}=(E,{\mathcal{I}}_{2})$ be two matroids. If no element of $E$ is $(k+1)$-spanned in $M_{1}$ or $M_{2}$, then there exists a subset $X\subseteq E$ such that $X\in {\mathcal{I}}_{1}\cap{\mathcal{I}}_{2}$ and $E\setminus X\in {\mathcal{I}}_{1}^{k-1}\cap {\mathcal{I}}_{2}^{k-1}$.
Using this proposition, we can provide unified proofs for Theorems \[thm:KZ1\] and \[thm:KZ2\].
By Proposition \[prop:induction\_step\], there is $X\in {\mathcal{I}}_{1}\cap{\mathcal{I}}_{2}$ with $E\setminus X\in {\mathcal{I}}_{1}^{k-1}\cap {\mathcal{I}}_{2}^{k-1}$. Because $r_{1}(E)=r_{2}(E)=d$ and $|E|=k\cdot d$, the subsets $X$ and $E\setminus X$ should be common bases of $(M_{1}, M_{2})$ and $(M_{1}^{k-1}, M_{2}^{k-1})$, respectively. For each matroid $M_{i}$ ($i=1,2$), since every element in $E\setminus X$ is spanned by $X$ but not $(k+1)$-spanned, we see that no element in $E\setminus X$ is $k$-spanned in $M_{i}|(E\setminus X)$. Thus, $X\in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$ and restrictions $M_{1}|(E\setminus X)$ and $M_{2}|(E\setminus X)$ satisfy the assumption of Theorem \[thm:KZ1\] with $k$ replaced by $k-1$. By induction, $E\setminus X$ can be partitioned into $k-1$ common independent sets. Thus, the proof is completed
By just applying Proposition \[prop:induction\_step\] with $k=2$, we obtain a common independent set $X\in {\mathcal{I}}_{1}\cap{\mathcal{I}}_{2}$ satisfying $E\setminus X\in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$. Thus, the proof is completed
The original proofs for Theorems \[thm:KZ1\] and \[thm:KZ2\] [@KZ05] have no apparent relation. For these two theorems, we have shown unified proofs by our generalized-polymatroid approach. This offers a new understanding of the conditions in Theorems \[thm:KZ1\] and \[thm:KZ2\]: they are nothing other than conditions under which our induction method works.
Intersection of a Laminar Matroid and a Matroid without $(k+1)$-Spanned Elements {#sec:new}
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As mentioned in Remark \[REMindependent\], our g-polymatroid approach does not require the two matroids to be in the same matroid class, and thus can deal with an arbitrary pair of matroids which have appeared in this section. That is, we can obtain a solution of Problem \[prob:main\] for a new class of matroid intersection, i.e., the intersection of a laminar matroid and a matroid without $(k+1)$-spanned elements. The following theorems can be immediately derived from combining the proofs of Theorems \[thm:laminar-partition\], \[thm:KZ1\], and \[thm:KZ2\].
\[thm:lam+KZ1\] Let $k$ be a positive integer, $M_{1}=(E,{\mathcal{I}}_{1})$ be a laminar matroid such that , and $M_{2}=(E,{\mathcal{I}}_{2})$ be a matroid with rank function $r_{2}$ such that $|E|=k\cdot r_{2}(E)$ and no element is $(k+1)$-spanned in $M_{2}$. Then, there exists a partition $\{X_{1},X_{2},\dots,X_{k}\}$ of $E$ such that $X_{j}\in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$ for each $j=1,2,\dots,k$.
\[thm:lam+KZ2\] Let $M_{1}=(E,{\mathcal{I}}_{1})$ be a laminar matroid such that $E \in {\mathcal{I}}_1^2$ and $M_{2}=(E,{\mathcal{I}}_{2})$ be a matroid Then, there exists a partition $\{X_{1},X_{2}\}$ of $E$ such that $X_{1},X_2\in {\mathcal{I}}_{1}\cap {\mathcal{I}}_{2}$.
Time Complexity {#sec:complexity}
---------------
The proof for Proposition \[prop:g-matroid\_approach\] implies a polynomial-time algorithm to solve Problem \[prob:main\] for matroids mentioned above. Here we discuss the time complexity of the algorithm.
Suppose that $M_{1}=(E,{\mathcal{I}}_{1})$ and $M_{2}=(E,{\mathcal{I}}_{2})$ satisfy the assumptions in Proposition \[prop:g-matroid\_approach\], and membership oracles of ${\mathcal{I}}_{1}$ and ${\mathcal{I}}_{2}$ are provided. Let ${\mathcal{J}}_i:=\set{X| X\in {\mathcal{I}}_i, E\setminus X\in {\mathcal{I}}_i^{k-1}}$ for each $i=1,2$. We solve Problem \[prob:main\] by $k-1$ iterations of finding $X \in {\mathcal{J}}_1 \cap {\mathcal{J}}_2$, which can be done efficiently in the following manner.
By the assumption, ${\mathcal{J}}_i={\mathcal{F}}(p_{i},b_{i})$ holds for some intersecting-paramodular pair $(p_{i},b_{i})$, and hence $(E,{\mathcal{J}}_i)$ is a generalized matroid [@FT88; @Tardos85] (see Lemma \[lem:g-matroid\] below). , which is defined as follows. Let $m_{\rm max}$ and $m_{\rm min}$ be the maximum and minimum size of a subset in ${\mathcal{J}}_1\cup {\mathcal{J}}_2$ respectively, and let $U$ be a set disjoint from $E$ with $|U|=m_{\rm max}-m_{\rm min}$. As shown in [@Tardos85 Theorem 2.9], the family $$\tilde{{\mathcal{B}}}_i=\set{X\cup V|X\in {\mathcal{J}}_i, ~V\subseteq U, ~|X\cup V|=m_{\rm max}}$$ forms the base family of a matroid on $E\cup U$, which we denote by $\tilde{M}_i=(E\cup U, \tilde{{\mathcal{I}}}_i)$. Then we have ${\mathcal{J}}_i=\set{E\cap B|B\in \tilde{{\mathcal{B}}}_i}$ for each $i=1,2$, and hence ${\mathcal{J}}_1\cap {\mathcal{J}}_2=\set{E\cap B|B\in \tilde{{\mathcal{B}}}_1\cap \tilde{{\mathcal{B}}}_2}$.
Therefore, finding $X\in {\mathcal{J}}_1\cap {\mathcal{J}}_2$ is reduced to finding a common base of $\tilde{M}_1$ and $\tilde{M}_2$. This can be done by a standard matroid intersection algorithm [@Edmonds70], if membership oracles of $\tilde{{\mathcal{I}}}_1$ and $\tilde{{\mathcal{I}}}_2$ are available. Such oracles can be efficiently implemented as follows.
Since $\tilde{{\mathcal{I}}}_i$ consists of all subsets of the bases in $\tilde{{\mathcal{B}}}_i$, it follows from the definitions of $\tilde{{\mathcal{B}}}_i$ and ${\mathcal{J}}_i$ that $$\tilde{{\mathcal{I}}}_i=\set{X'\cup V|\exists X\in {\mathcal{I}}_i: X'\subseteq X,~|X\cup V|\leq m_{\rm max},~E\setminus X\in {\mathcal{I}}_i^{k-1} }.$$ Then, we have $X'\cup V\in \tilde{{\mathcal{I}}}_i$ if and only if $E\setminus X'$ is partitionable into $Z, Y_1,Y_2,\dots,Y_{k-1}$ such that $$Z\in {\mathcal{I}}^{\ast}_i:=\set{Z' \subseteq E \setminus X'|Z'\cup X'\in {\mathcal{I}}_i, ~|Z'\cup X'\cup V|\leq m_{\rm max}}$$ and $Y_j\in {\mathcal{I}}_i~(j=1,\dots, k-1)$. Note that ${\mathcal{I}}_i^{\ast}$ is also the independent set family of a matroid (as it is obtained from ${\mathcal{I}}_i$ by contraction and truncation). Therefore, by the matroid partition algorithm [@Edmonds65], we can decide whether $X'\cup V\in \tilde{{\mathcal{I}}}_i$ using the oracle of ${\mathcal{I}}_i$.
Here we show a complexity analysis of the above algorithm with the implementations of matroid intersection and matroid partition algorithms by Cunningham [@Cun86]. Let $n=|E|$, $r = \max\{r_1(E),r_2(E)\}$, where $r_i$ denotes the rank function of $M_i$ for $i=1,2$, and let $\tau$ denote the time for the membership oracles of ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$. Overall, we solve $k-1$ instances of the matroid intersection problem defined by $\tilde{M}_1$ and $\tilde{M}_2$. Each instance can be solved in $O(r^{1.5}n Q)$ time [@Cun86], where $Q$ is the time for an independence test in $\tilde{M}_1$ and $\tilde{M}_2$. Testing independence in $\tilde{M}_1$ and $\tilde{M}_2$ can be done in $O(n^{2.5} \tau)$ time [@Cun86]. Therefore, the total time complexity is $O(k r^{1.5} n^{3.5} \tau)$.
Example Incompatible with the G-polymatroid Approach {#sec:incompatible}
====================================================
As mentioned before, the class of strongly base orderable matroids admits a solution for Problem \[prob:main\] [@DM76], and our g-polymatroid approach can deal with laminar matroids, a special class of strongly base oderable matroids. Hence we expect that the g-polymatroid approach can be applied to strongly base orderable matroids. In this subsection, however, we by constructing an example of a transversal matroid, another simple special case of a strongly base orderable matroid, which admits no intersecting-paramodular pair required in Proposition \[prop:g-matroid\_approach\].
For a bipartite graph $G=(E,F;A)$ with color classes $E$, $F$ and edge set $A$, let ${\mathcal{I}}$ be a family of subsets $X$ of $E$ such that $G$ has a matching that is incident to all elements in $X$. Then it is known that $(E,{\mathcal{I}})$ is a matroid [@EF65], which we call the [*transversal matroid*]{} induced from $G$.
\[ex:1\]
![A bipartite graph that induces a transversal matroid $M=(E,{\mathcal{I}})$ in Example \[ex:1\][]{data-label="fig1"}](Figure1_revised-eps-converted-to.pdf){width="35mm"}
We show that transversal matroids are incompatible with our g-polymatroid approach by proving that the transversal matroid $M$ in Example \[ex:1\] admits no intersecting-paramodular pair $(p,b)$ satisfying ${\mathcal{F}}(p,b)=\set{X\subseteq E|X\in {\mathcal{I}},~E\setminus X\in {\mathcal{I}}^{k-1}}$, i.e., we cannot use Proposition \[prop:g-matroid\_approach\] for $M$. For this purpose, we prepare the following fact (see e.g., [@Frankbook; @Mbook; @MS99]).
\[lem:g-matroid\] For any integral intersecting-paramodular pair $(p,b)$, if ${\mathcal{J}}:={\mathcal{F}}(p,b)\neq \emptyset$, then $(E,{\mathcal{J}})$ is a [*generalized matroid*]{}, i.e., ${\mathcal{J}}$ satisfies the following axioms:
- If $X, Y \in{\mathcal{J}}$ and $e \in Y\setminus X$, then $X +e \in {\mathcal{J}}$ or $\exists e'\in X\setminus Y: X+e-e'\in {\mathcal{J}}$.
- If $X, Y \in{\mathcal{J}}$ and $e \in Y\setminus X$, then $Y-e \in {\mathcal{J}}$ or $\exists e'\in X\setminus Y: Y-e+e'\in {\mathcal{J}}$.
Now the incompatibility of the transversal matroid $M$ in Example \[ex:1\] is established by the following proposition.
For the transversal matroid $M=(E,{\mathcal{I}})$ given in Example \[ex:1\], there is no integral intersecting-paramodular pair $(p,b)$ satisfying ${\mathcal{F}}(p,b)=\set{X\subseteq E|X\in {\mathcal{I}},~E\setminus X\in {\mathcal{I}}^{k-1}}$ with $k=2$, while $E$ can be partitioned into two independent sets in $M$.
The latter statement is obvious: $E$ can be partitioned into two bases $\{e_{1}, e_{2},e_{3}\}$ and $\{e'_{1},e'_{2},e'_{3}\}$.
Suppose to the contrary that ${\mathcal{F}}(p,b)=\set{X\subseteq E|X\in {\mathcal{I}},~E\setminus X\in {\mathcal{I}}}$ holds for some integral intersecting-paramodular pair $(p,b)$. By Lemma \[lem:g-matroid\], then ${\mathcal{J}}:={\mathcal{F}}(p,b)=\set{X\subseteq E|X\in {\mathcal{I}},~\overline{X}\in {\mathcal{I}}}$ satisfies (J1) and (J2), where $\overline{X}$ denotes $E\setminus X$.
Let $X:=\{e_{1}, e'_{2}, e'_{3}\}$ and $Y:=\{e'_{1}, e'_{2}, e_{3}\}$. We can observe $X, \overline{X}, Y, \overline{Y}\in {\mathcal{I}}$, and hence $X,Y\in {\mathcal{J}}$. Apply (J1) to $X\in {\mathcal{J}}$ and $e_{3}\in Y\setminus X$. As $X+e_{3}\not \in {\mathcal{J}}$ and $X\setminus Y=\{e_{1},e'_{3}\}$, we must have $X+e_{3}-e_{1}\in {\mathcal{J}}$ or $X+e_{3}-e'_{3}\in {\mathcal{J}}$. However, it holds that $X+e_{3}-e_{1}=\{e'_{2}, e_{3}, e'_{3}\}\not\in {\mathcal{I}}$, implying $X+e_{3}-e_{1}\not\in {\mathcal{J}}$, and it also holds that $\overline{X+e_{3}-e'_{3}}=\{e'_{1}, e_{2}, e'_{3}\}\not\in {\mathcal{I}}$, implying $X+e_{3}-e'_{3}\not\in {\mathcal{J}}$, a contradiction.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Satoru Fujishige and Kazuo Murota for valuable comments. The authors are also obliged to anonymous referees for helpful comments, in particular on a smaller example of a transversal matroid. The first author is partially supported by JST CREST Grant Number JPMJCR1402, JSPS KAKENHI Grant Numbers JP16K16012, JP26280004, Japan. The second author is supported by JST CREST Grant Number JPMJCR14D2, JSPS KAKENHI Grant Number JP18K18004, Japan.
[^1]: Hosei University, Tokyo 184-8584, Japan. E-mail: [takazawa@hosei.ac.jp]{}.
[^2]: National Institute of Informatics, Tokyo 101-8430, Japan. E-mail: [yokoi@nii.ac.jp]{}.
[^3]:
[^4]: Let ${\mathcal{A}}_{X}=\{X\}\cup \set{X\cap A|A\in {\mathcal{A}}}\cup \set{\{e\}|e\in X}$. Since ${\mathcal{A}}$ is laminar, ${\mathcal{A}}_{X}$ is also laminar. Let $T$ be a tree representation of ${\mathcal{A}}_{X}$, i.e., the node sets of $T$ is ${\mathcal{A}}_{X}$ and a node $A$ is a child of $A'$ if $A\subsetneq A'$ and there is no $A''$ with $A\subsetneq A''\subsetneq A'$. Then each leaf is the singleton of an element in $X$. Let $X=\{e_{1},e_{2},\dots,e_{|X|}\}$ so that the indices represent the order in which the corresponding leaves are found in depth-first search from the root node $X$. These indices satisfy the required condition.
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abstract: 'Bohr-van Leeuwen theorem has been studied in non-commutative space where the space coordinates do not commute. It has been found that in non-commutative space Bohr-van Leeuwen theorem, in general, is not satisfied and a classical treatment of the partition function of charged particles in a magnetic field gives rise to non zero magnetization. **Keywords:** Non-commutative space · Statistical Mechanics · Magnetism'
author:
- 'Shovon Biswas$^{a}$[^1]'
title: 'Bohr-van Leeuwen theorem in non-commutative space'
---
Introduction
============
Due to the appearance of non-commutativity in string theories [@ref1] and the observation that ordinary and non-commutative gauge theories are equivalent [@ref2] there have been much studies of physics on a non-commutative space recently. For example, quantum hall effect and Landau diamagnetism in non-commutative space have been discussed in references [@dyl2; @dulat] and [@dyl1; @gam2]. Hydrogen atom spectrum and Lamb shift is discussed in references [@kang; @masud; @masud2]. A discussion on non-commutative quantum mechanics and non-commutative classical mechanics can be found in references [@gamboa; @gam3; @djema; @romero; @class; @therm].\
\
The non-commutative space can be characterized by the following commutation relations: $$\begin{aligned}
\begin{split}
[\hat{X}_j,\hat{X}_k] &=i\hbar\theta_{jk}\\
[X_j,P_k] &= i\hbar\delta_{jk} \\
[P_j,P_k] &= 0, \quad j,k=1,2,3
\end{split}\end{aligned}$$ where $\{\theta_{ij}\}$ is the totally antisymmetric matrix which represents the non-commutative property of the coordinates on a non-commutative space. Non-commutative operators $(\hat{X}_j,\hat{P}_j)$ can be expressed in terms of ordinary coordinate and momentum operators $(X_j,P_j)$ as [@li; @djema]: $$\begin{aligned}
\begin{split}
\hat{X_j} &= X_j -\frac{1}{2}
\theta_{jk}P_k, \quad k=1,2,3.\\
\hat{P_j} &=P_j
\end{split}\end{aligned}$$\
\
Here, $(X_j,P_j)$ satisfy the usual commutation relations: $$\begin{aligned}
\begin{split}
[X_j,X_k] &=0 \\
[X_j,P_k] &= i\hbar\delta_{jk} \\
[P_j,P_k] &= 0, \quad j,k=1,2,3
\end{split}\end{aligned}$$ It should be noted that the representations of non-commutative coordinates and momenta in terms of ordinary ones given in (2) have non trivial impacts. For example, the Schrodinger equation in the non-commutative space takes the form of the ordinary one but with a shifted potential having highly non-trivial consequences[@gam3].\
\
Bohr-van Leeuwen theorem[@van], on the other hand, is one of the very well known theorems of classical physics stating the nonexistence of magnetism in classical statistics. When statistical mechanics and classical mechanics are applied consistently, the thermal average of the magnetization is always zero[@aha]. This makes magnetism solely a quantum mechanical effect. In this work, based on the assumption that the phase space has a symplectic structure consistent with the rules of commutation of the non-commutative quantum mechanics given in (1), it will be proved that Bohr-van Leuween theorem is violated when classical partition function is written with respect to non-commutative phase space variables.\
\
The organization of this paper is as follows. In section 2 the classical Bohr-van Leeuwen theorem is reviewed. In section 3 the effect of non-commutivity of space coordinates on Bohr-van Leeuwen theorem is discussed.
Review of Bohr-van Leeuwen Theorem
===================================
As noted above the content of the Bohr–van Leeuwen theorem is the nonexistence of magnetism in classical statistics[@schwabl]. To prove the theorem let us evaluate the classical partition function for $N$ charged particles in the electromagnetic field [@schwabl] $$Z=\frac{\int d^{3N}p d^{3N}x}{(2\pi\hbar)^{3N}N!}e^{-\beta H\left( \{ \mathbf{p}^{(i)}-\frac{e}{c}\mathbf{A(\mathbf{x}^{(i)})}\},\{\mathbf{x}^{(i)}\}\right)}$$ Here $ H\left( \{ \mathbf{p}^{(i)}-\frac{e}{c}\mathbf{A(\mathbf{x}^{(i)})}\},\{\mathbf{x}^{(i)}\}\right)$ is the Hamiltonian which can be written as $$H=\sum_{i=1}^{N}\frac{1}{2m}\left( \mathbf{p}^{(i)}-\frac{e}{c}\mathbf{A(\mathbf{x}^{(i)}} \right)^2+W_{coul}$$ where $\mathbf{A}$ is the vector potential and $^{(i)}$ stands for $i$th particle. The last term in equation (5) stands for the Coulomb interaction of the particles with each other. If we substitute $$\mathbf{p'}^{(i)}=\mathbf{p}^{(i)}-\frac{e}{c}\mathbf{A(\mathbf{x}^{(i)})}$$ $Z$ becomes independent of $\mathbf{A}$ and thus of $\mathbf{B}$. With the free energy $$F=-\frac{1}{\beta}\ln Z$$ this leads to the magnetization $$\mathbf{M}=-\frac{\partial F}{\partial \mathbf{B}}=0.$$ Thus there can be no magnetism in classical physics.
Bohr-van Leeuwen Theorem in Non-commutative Space
==================================================
To study Bohr-van Leeuwen theorem in non-commutative space (NCS), we assume that the passage between NC classical mechanics and NC quantum mechanics can be realized via the following generalized Dirac quantization condition[@class; @therm] $$\{f,g\}=\frac{1}{i\hbar}[F,G]$$ Where $(F,G)$ stands for the operator associated with classical observables $(f,g)$ and $\{,\}$ stands for Poisson bracket. Using this rule we obtain from (1) $$\begin{aligned}
\begin{split}
\{\hat{x}_j,\hat{x}_k\}&=\theta_{jk}\\
\{\hat{x}_j,\hat{p}_k\}&=\delta_{jk}\\
\{\hat{p}_j,\hat{p}_k\}&=0\\
\end{split}\end{aligned}$$ It is easy to see that the following representation of NC coordinates $(\hat{x}_j,\hat{p}_j)$ in terms of commutative coordinates $(x_j,p_j)$ satisfies relations given in (10) $$\begin{aligned}
\begin{split}
\hat{x_j} &= x_j -\frac{1}{2}
\theta_{jk}p_k \\
\hat{p_j} &=p_j, \quad j,k=1,2,3.
\end{split}\end{aligned}$$ Now following the proposal that non-commutative observable $F^{NC}$ corresponding to the observable $F(x_,p)$ in commutative space can be defined by [@kang; @masud] $$F^{NC}=F(\hat{x}_j,\hat{p}_j)$$ we write the partition function (4) in terms of NC coordinates in non-commutative space as $$Z^{NC}=\frac{\int d^{3N}\hat{p} d^{3N}\hat{x}}{(2\pi\hbar)^{3N}N!}e^{-\beta H_{NC}\left( \{ \mathbf{\hat{p}}^{(i)}-\frac{e}{c}\mathbf{A(\mathbf{\hat{x}}^{(i)})}\},\{\mathbf{\hat{x}}^{(i)}\}\right)}$$ Using the representation (11), partition function (13) in NCS can be expressed in terms of ordinary coordinates as follows $$\begin{aligned}
Z^{NC}=\frac{\int d^{3N}p d^{3N}x }{(2\pi\hbar)^{3N}N!}e^{-\beta H\left( \{ \mathbf{p}^{(i)}-\frac{e}{c}\mathbf{A}(\mathbf{x}^{(i)}-\frac{1}{2}\mathbf{\tilde{p}}^{(i)})\},\{\mathbf{x}^{(i)}-\frac{1}{2}\mathbf{\tilde{p}}^{(i)})\}\right)}\end{aligned}$$ where $\mathbf{\tilde{p}}_j=\theta_{jk}p_k$. Now, since $\mathbf{A}$ depends on momenta in (13) one can not, in general, make a simple substitution like equation (6) to eliminate $\mathbf{A}$ from the partition function. Thus the partition function, in general, will be dependent on the magnetic field $\mathbf{B}$.\
\
As an example let us calculate the partition function of $N$ non-interacting electrons in a uniform magnetic field on non-commutative space. Since the electrons are non-interacting the partition function on NCS for $N$ electrons can be written as $$Z_N^{NC}=\frac{[Z_1^{NC}]^N}{N!}$$ where $Z_1^{NC}$ is the single particle partition function in NCS given by $$Z_1^{NC}=\frac{1}{(2\pi\hbar)^3}\int e^{-\beta H_{NC}}d^3\hat{p} d^3 \hat{x}$$ We let the magnetic field $\mathbf{B}$ be in the z-direction by choosing the symmetric gauge $\mathbf{A}=B(-\frac{y}{2},\frac{x}{2},0)$. For a single electron the Hamiltonian (5) in ordinary phase space takes the form $$H=\frac{1}{2m}\left[\left(p_x-\frac{eB}{2c}y\right)^2+\left( p_y+\frac{eB}{2c}x \right) ^2\right]+\frac{p_z^2}{2m}$$ Following the definition given in (12) and substituting the representations (11) in the Hamiltonian (17) we express the single particle partition function (16) in terms of ordinary coordinates and momenta as: $$\begin{aligned}
Z_1^{NC} &&=\frac{1}{(2\pi\hbar)^3}\int d^3p d^3 x e^{-\beta \left\{ \frac{1}{2m}\left[\left(\gamma p_x - \frac{eB}{2c}y \right)^2 +\left(\gamma p_y+\frac{eB}{2c}x \right)^2 \right] +\frac{p_z^2}{2m} \right\}}\end{aligned}$$ where $\gamma=1-\frac{eB\theta}{4c}$ and $c$ is the velocity of light. We have assumed non-commutivity in $XY$ plane only. It should be noted that similar expressions have been found for the non-commutative Hamiltonian appearing inside the curly braces in equation (18) in references [@dyl1; @dyl2] using star product definition. Performing the integration we obtain $$Z_1^{NC}=\frac{V}{\gamma^2\lambda}$$ where $V$ is the volume and $\lambda=\frac{h}{\sqrt{2m\pi k_B T}}$ is the thermal de Broglie wavelength. Thus the $N$ particle partition function (14) becomes $$Z_N^{NC}=\frac{1}{N!}\left( \frac{V}{\gamma^2\lambda} \right)^N$$ Now through $\gamma=1-\frac{eB\theta}{4c}$ the partition function depends on the magnetic field $B$. Also it should be noted that in the critical value of $B=\frac{4c}{e\theta}$ the partition function diverges. Thus the thermodynamic limit is applicable below this value. In fact, at the critical value of $B$ the first two quadratic momenta terms vanish in equation (18). This system is different from the system under consideration and should be studied separately [@dyl2]. Using formulae (7) and (8) we find the magnetization in non-commutative space to be $$M_{NC}=\frac{2Nk_B Te \theta}{4c-e\theta B}$$ Thus a non zero magnetization appears from classical Hamiltonian in non-commutative space indicating the violation of Bohr-van Leeuwen theorem. In case we switch off $\theta$, we have $M=0$ reproducing the classical result. Finally, using the formula for susceptibility[@schwabl] $\chi=-\frac{1}{V}\frac{\partial ^2 F}{\partial B^2}$, we find the expression for magnetic susceptibility of electrons in non-commutative space $$\chi_{NC}=-\frac{1}{V}\left[\frac{2Nk_BTe^2\theta^2}{(4c-e\theta B)^2}\right]$$ Thus electrons in a uniform magnetic field in NCS exhibit diamagnetism.
Conclusion
==========
We have found that Bohr-van Leeuwen theorem is not satisfied in non-commutative space defined by symplectic structure (10). This needs special attention because classical treatment of simplest systems can give rise to magnetism in non-commutative space even without the introduction of a spin degree of freedom. Such a scenario is completely absent in ordinary classical mechanics. As an example, we have analyzed the simple system of non-interacting electrons in a uniform magnetic field in non-commutative space and found that a classical treatment of the problem in NCS gives rise to diamagnetism of electrons. Finally, we note that the discussion presented here can be easily be extended to non-commutative phase space where the momenta just like as coordinates do not commute.
Acknowledgement
===============
The author would like to thank Professor Dr M. Arshad Momen and Professor Dr M. Chaichian for their valuable comments on the manuscript. The author expresses his gratitude to Mir Mehedi Faruk for his help to present the manuscript.
[5]{} A. Connes, M.R. Douglas, A. Schwarz, JHEP 9802 (1998) 003. N. Seiberg, E. Witten, JHEP 9909 (1999) 032. O.F. Dayi, A. Jellal, J. Math. Phys. 43, (2002) 4592 (Erratum-ibid. 45, (2004) 827 ) S. Dulat, K. Li, Eur. Phys. J. C60 , (2009) 163 O. F. Dayi, A. Jellal, Phys. Lett. A287 (2001), 349 K. Li and N. Chamoun, Chinese Physics Letters 23(5) (2006) 1122. M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, Physical Review Letters, 86(13) (2001), 2716. M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, arXiv preprint hep-th/0212259 (2002). J. Gamboa, M. Loewe, F. Méndez, J.C. Rojas, Mod. Phys. Lett. A 16, (2001) 2075 J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64 (2001) 067901. J. Gamboa, M. Loewe, F. Méndez, and J. C. Rojas, hep-th/0106125 A. E. F. Djemai and H. Smail, Commun. Theor. Phys. 41 (2004) 837. J. M. Romero, J. A. Santiago, and J. David Vergara, Physics Letters A 310.1 (2003): 9-12. Wei G.F. and al.,Chinese Physics C, 32(5), (2008) 338-341 M. Najafizadeh, M. Saadat, Chinese Journal of Physics 51 , (2013) 94 . K. Li, J. Wang, Chiyi Chen, Mod. Phys. Lett. A Vol. 20, No. 28(2005) 2165-2174 H.J., van Leeuwen, J.Phys.Radium 2,(1921) 362 A. Aharoni, Introduction to the Theory of Ferromagnetism, (Oxford University Press, 2001). F. Schwabl, Statistical Mechanics, 2nd Edition, (Springer, Berlin, Heidelberg, 2006). K. Huang, Statistical Mechanics, (Wiley, New York, 1967) R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, Oxford, 1996).
[^1]: shovon432@gmail.com
|
---
abstract: 'The inner surface of superconducting cavities plays a crucial role to achieve highest accelerating fields and low losses. The industrial fabrication of cavities for the European X-Ray Free Electron Laser (XFEL) and the International Linear Collider (ILC) HiGrade Research Project allowed for an investigation of this interplay. For the serial inspection of the inner surface, the optical inspection robot OBACHT was constructed and to analyze the large amount of data, represented in the images of the inner surface, an image processing and analysis code was developed and new variables to describe the cavity surface were obtained. This quantitative analysis identified vendor specific surface properties which allow to perform a quality control and assurance during the production. In addition, a strong negative correlation of $\rho= -0.93$ with a significance of $6\,\upsigma$ of the integrated grain boundary area $\sum{\mathrm{A}}$ versus the maximal achievable accelerating field $\mathrm{E_{acc,max}}$ has been found.'
address: 'Deutsches Elektronen Synchrotron, 22607 Hamburg, Notkestrasse 85'
author:
- Marc Wenskat
title: Optical Surface Properties and their RF Limitations of European XFEL Cavities
---
March 2017
[*Keywords*]{}: Niobium, Superconducting Cavities, Image Processing, Optical Inspection, Cavity Fabrication, Electron Beam Welding
Introduction
============
Superconducting niobium radio-frequency cavities are fundamental for the European XFEL and the ILC [@XFEL_TDR; @ILC_TDR]. To utilize the operational advantages of superconducting cavities, the inner surface has to fulfill quite demanding requirements. The surface roughness, welding techniques, and cleanliness improved over the last decades and with them, the achieved maximal accelerating gradient $\mathrm{E_{acc,max}}$. Still, limitations of the accelerating gradient are observed, which are not explained by localized geometrical defects or impurities. The method and results shown in this paper aim for a better understanding of these limitations in defect free cavities based on global, rather than local, surface properties.
Optical Inspection and Image Processing
=======================================
For this research, more than 100 cavities underwent subsequent surface treatments, cold RF tests, and optical inspections within the ILC-HiGrade research program and the European XFEL cavity production [@CORDIS; @Navitski2013; @Singer2016]. The optical inspection of the inner surface of superconducting RF cavities is a well-established tool at many laboratories [@Watanabe]. Its purpose is to characterize and understand field limitations and to allow optical quality assurance during cavity production. Theoretical calculations in [@Sara1995; @Xie2009] have shown that accelerating fields of $50\, \mathrm{MV/m}$ are achievable if surface structures and localized defects are below 10$\upmu \mathrm{m}$, hence the resolution of the system should be on that order.
An algorithm was developed which enables an automated surface characterization based on images taken by an optical inspection robot. This algorithm delivers a set of optical surface properties, which describe the inner cavity surface. These optical surface properties deliver a framework for a quality assurance of the fabrication procedures. Furthermore, they show promising results for a better understanding of the observed limitations in defect free cavities.
OBACHT
------
A fully automated robot for optical inspection, the “**O**ptical **B**ench for **A**utomated **C**avity inspection with **H**igh resolution on short **T**imescales” (OBACHT), has been continuously used at DESY. It is equipped with a high-resolution camera (Kyoto Camera System), which resolves structures down to $12\,\upmu \mathrm{m}$ for properly illuminated surfaces [@Iwashita2008; @Tajima2008; @Iwashita2009]. The details of OBACHT and the optical system are described in [@Lemke; @Sebastian; @Wenskat2015]. It consists of a camera tube with a diameter of 50mm to fit into the cavity without colliding with the irides or HOM antennas protruding into the cavity volume (see Fig. \[fig:sketch\]). In this tube the camera, together with a low-distortion lens, are installed. The camera system images the surface via a $ 45^\mathrm{o}$-tilted one way mirror which can be continuously adjusted to other angles in order to inspect other cavity regions. The focal distance of the camera to the cavity surface is controlled by a motor driven lead-screw. For illumination, acrylic strips (two LEDs per strip) attached to the camera tube around the camera opening are installed, together with three additional LEDs behind the one way mirror inside the camera tube.
![Schematic of the Kyoto Camera System used at DESY. The camera is viewing the inner surface via a $ 45^\mathrm{o}$-tilted one way mirror. Behind this one way mirror, three LEDs are mounted for the central illumination. $2 \times 10$ acrylic strips with LEDs are mounted left and right from the opening in the tube for a more detailed illumination. []{data-label="fig:sketch"}](sketch2.jpg){width="75.00000%"}
The highest magnetic field in a cavity, and hence the highest losses, are at the equatorial welding seam region including the heat affected zone. Therefore, the equatorial images are of main interest for this analysis. Figure \[OBACHT\] shows an image of the inner cavity surface taken with OBACHT.
![Image of the inner cavity surface with the equatorial welding seam in the image center taken with OBACHT. The image size is $9 \times 12\,\mathrm{mm^2}$. The red contours are examples of grain boundaries identified with the image processing algorithm in the welding seam region (WS) and the heat affected zone (HAZ).[]{data-label="OBACHT"}](InCavSurf.jpg){width="50.00000%"}
With given cavity geometry and optical set up, an individual image covers $5^\mathrm{o}$ of an equator. To have a small overlap at the edges of an image, an image is taken each $4.8^\mathrm{o}$. This results in 75 images per equator and 675 equator images per cavity. The objects of interest within an image of the inner cavity surface are grain boundaries. In order to identify and quantify those boundaries, an image processing and analysis algorithm has been developed.
Image Processing and Analysis
-----------------------------
The main goal of the image processing algorithm is to identify grain boundaries, regardless of their position within the image which shows a non-uniform illumination, as can be seen in Figure \[OBACHT\]. The approach of this algorithm is, to apply a sequence of high-pass filter and local contrast enhancements, to project pixels which belong to grain boundaries onto a gray scale interval, which is distinct to the background. After this projection, a histogram based segmentation of the processed image is performed. This segmentation assumes, that the image contains two classes of pixels (grain boundary and background), where the intensity values follow a bi-modal distribution, and calculates the optimum threshold separating the two classes. The output is a binary image with the same size as the input image. It will contain grain boundary pixels in white (logical one) and background pixels in black (logical zero). As a last step of the image processing, groups of connected white pixels which form a grain boundary have to be classified as a single object and a labeled binary image is obtained. An example of such a binary image is given in Figure \[GB\_area\].
![Left: a detail of an OBACHT image is shown. Right: the same detail after the image processing algorithm. Grain boundaries (white) are visible.[]{data-label="GB_area"}](Grainarea.png){width="60.00000%"}
The aim of the image analysis is to identify features in the binary image. Those are grain boundaries with varying width which are not symmetric. Hence, it is nontrivial to define important properties like diameter, centroid, eccentricity or orientation of an object. The method to overcome this problem, is to find an ellipse which has the same second central moment as the pixel distribution of the pixel [@Hu1962]. Within this framework, the grain boundary area is the total amount of pixels, consisting of a boundary. This number is retrieved from the binary image and then multiplied by the pixel size, which is a property of the optical system. At OBACHT, this value is $12.25\,\upmu \mathrm{m}^2$. With the given resolution at OBACHT, the experimentally obtained relative error for the grain boundary area is 3%, similar to [@Patil2011].
The orientation of an object is defined with respect to an axis perpendicular to the welding seam and an upper limit of the uncertainty of $ 5^\mathrm{o}$ is derived with [@Klette1999; @Liao1993].
In order to define a figure of merit for the roughness of an object with OBACHT, two assumptions were made. The first assumption is that the intensity of the reflected light is dependent on the roughness and structures of the cavity surface. This means that a change in the intensity is either caused by a geometric gradient or a change in reflectivity. A geometric gradient exists either at a grain boundary or a defect, while a change in reflectivity can be caused by an impurity. The second assumption is that the curvature of the elliptical cavity is negligible within the studied area and the surface seen by the image can be considered to be a flat surface. Based on the intensity of the original image, a quantity called $\mathrm{R_{dq}}$ is introduced. It is based on ISO 25178 for surface texture [@iso25178] and is the average of the intensity gradient of the boundary. A steeper slope of an edge or surface would imply a larger intensity gradient and hence a larger $\mathrm{R_{dq}}$.
A statistical noise arising from the Signal-to-Noise-Ratio (SNR) of the image sensor in the camera, which yields a $\delta \mathrm{R_{dq}}$ of $\frac{0.011}{\sqrt{N}} \frac{\mathrm{Bit}}{\mathrm{\upmu m}}$. A systematic uncertainty due to image focus was found to be $\frac{\delta \mathrm{R_{dq}}}{\mathrm{R_{dq}}}= 3\,\%$. For more details on the image processing algorithm and explicit definitions and discussion of the obtained variables see [@Wenskat2015].
Vendor dependent surface properties
===================================
As described in [@Singer2016; @Aderhold2010], the two cavity vendors, RI Research Instruments GmbH (RI) and Ettore Zanon S.p.A. (EZ), were qualified to produce cavities after two distinct procedures. Most notably is the electron beam welding (EBW) procedure as well as the final surface chemistry step since both have a significant influence on the final surface and therefore RF performance. Since only the standard fabrication procedure should be compared, cavities that underwent any repair were not used in this comparison, which reduced the usable data set since most of the inspected cavities were issued an inspection due to flaws in the fabrication. This data set than yields to a total number of nine RI and eleven EZ cavities which were inspected with OBACHT.
Electron Beam Welding Procedure
-------------------------------
The solidification dynamics of the weld puddle depends on parameters such as temperature gradient, crystalline growth rate, and chemical composition. Therefore, the granular microstructure which develops in the molten material varies and depends on the weld movement pattern, beam travel speed, and beam power. The histograms in Figure \[Orienthist\_vendors\] show the grain orientation in the welding seam region as obtained with the image processing algorithm for each vendor.
![The x-axis shows the boundary centroid position with respect to the image mirror axis, which is the welding seam ridge. The y-axis shows the boundary orientation $\phi$ in degrees w.r.t. an axis perpendicular to the welding seam. Only the welding seam region is shown. The left plot (a) represents boundaries in the welding seam region of a RI cavity, the right plot (b) of an EZ cavity. The white ellipses encircles the welding seam boundaries. The color depicts the counts per bin.[]{data-label="Orienthist_vendors"}](Orienthist_2d_pattern.jpg){width="100.00000%"}
As it can be seen, the boundaries in the welding seam region of EZ have angles of $ \pm\,30^\mathrm{o}$. The angles of the boundaries in the welding seam region of a RI cavity show a completely different distribution. At the edge of the welding seam, the boundaries have an angle of $ \pm\,30^\mathrm{o}$, similar to EZ, while the boundaries change their orientation towards the center of the welding seam.
Another observed difference between the vendors is shown in Figure \[Orienthist\_vendors\_patterns\].
![Within the histograms, the color depicts the counts per bin. The x-axis shows the boundary centroid position with respect to the image axis. The y-axis shows the boundary orientation $\phi$ in degrees w.r.t. an axis perpendicular to the welding seam. Only the welding seam region is shown. a) is for RI and b) for EZ. Each histogram represents a different equator along the cavity - equator 1 is on the left.[]{data-label="Orienthist_vendors_patterns"}](Orienthist_2d_pattern_v1.png){width="68.00000%"}
All equator orientation patterns for EZ point into the same direction while they flip between equator 4-5, 5-6, and 6-7 for RI.
Surface Roughness
-----------------
Cavities from both vendors underwent a bulk electro-polishing procedure (EP). In addition, the cavities produced by RI underwent a final electro-polishing procedure (EP) of 40$\upmu \mathrm{m}$ while the EZ cavities underwent a flash buffered chemical polishing (BCP) of 10$\upmu \mathrm{m}$. The difference between these surfaces, as parametrized by $\mathrm{R_{dq}}$, is shown in Figure \[Vendor\_Rdq\_Distribution\].
![The x-axis shows the grain boundary gradient $\mathrm{R_{dq}}$ for boundaries within the welding seam region and the y-axis shows the counts per bin. The plots show the average $\mathrm{R_{dq}}$ distribution for all cells with a one $\upsigma$ interval. The red distribution is for EZ (N=99) and the blue distribution is for RI (N=81).[]{data-label="Vendor_Rdq_Distribution"}](Vendor_Rdq_Distribution_WS.png){width="90.00000%"}
For a quantification, an exponential modified Gaussian distribution (EMG) is fitted to the observed distribution. The fit parameters are given in Table \[tab:Rdq\_emg\_fit\_vendor\] and the average value of the distribution in Table \[tab:Rdq\_emg\_fit\_vendor\_mlv\].
[l> p[1.8cm]{}> p[1.8cm]{}]{} &\
$\left[ 10^{-3} \frac{\mathrm{Bit}}{\upmu \mathrm{m}}\right]$ & **WS** & **HAZ**\
**RI** & $2.8 \pm 0.1 $ & $3.1 \pm 0.1 $\
**EZ** & $3.4 \pm 0.1 $ & $3.6 \pm 0.1 $\
As seen in the histograms and quantified by the values of the EMG parameters, the cavities produced by RI have, on average, a smaller $\overline{\mathrm{R_{dq}}}$ of 17% in the welding seam (WS) region in comparison to cavities produced by EZ. The biggest difference of the average roughness, quantified by $\upmu$, is found in the welding seam region. The heat affected zone (HAZ) of BCP cavities show a slightly larger amount of steep grain boundaries, quantified by $\uplambda^{-1}$.
Surface properties and RF performance
=====================================
The main motivation for the construction of an optical inspection for cavities was to gain a better understanding of their RF performance, respectively the quality factor $Q_0$ and the accelerating gradient $\mathrm{E_{acc,max}}$. Correlations between quenches or field emission during cold RF tests and localized defects seen in optical inspections are well known [@Watanabe; @Sebastian; @Moller2009; @Geng2009a; @Aderhold2010b; @Singer2010]. A systematic study on the correlation of global optical surface properties and RF performances has not been performed yet.
For a quantitative statement on the goodness of the relation between the obtained optical surface properties and a figure of merit of the RF performance, the Pearson correlation coefficient $\rho$ is used. Since both variables are subject to measurement uncertainties, the calculation of the correlation coefficient should include these uncertainties. With appropriate estimators, the corrected Pearson correlation coefficient can be deduced which includes the influence of the uncertainty on the correlation coefficient[@Pearson1966; @Darmstadt2011]. The 95% confidence interval of the correlation coefficient is calculated with a bootstrapping method [@efron1979; @efron1987].
Optical Surface Properties
--------------------------
The purpose of this investigation is to identify the RF limiting cell. Two assumptions were made for this analysis. Firstly, that the maximal accelerating field $\mathrm{E_{acc,max}}$ shows a negative correlation to an optical determined variable. Secondly, the optically worst cell (aka the cell with the maximum value of this variable) should also be the RF limiting cell, since one bad performing cell is sufficient for a cavity to show a bad performance. Hence, the maximum value of the yet to identify optical surface variable of the nine cells identifies the *optically conspicuous cell*. This maximum value is used to represent the whole cavity. The different loss models discussed in the literature predict different correlations between the RF performance and the surface properties, but mainly all of them see the grain boundaries as a potential source for limitations or losses [@Visentin2003; @Bauer2004; @Ciovati2007; @Ciovati2008a; @hylton1988; @Safa1999; @Knobloch1999a]. Hence a property called *integrated grain boundary area* $\sum{\mathrm{A}}$ in a cell as optical surface variable is used as a correlator within this work. This property is the sum of all grain boundary areas found in the 75 images of an equator.
Correlation with 2nd Sound Results
----------------------------------
To test the assumptions, that the *integrated grain boundary area* $\sum{\mathrm{A}}$ can be used as a correlator, a correlation of the quench location of a cavity and the image analysis is done, where a quench is the localized origin of the phase transition from the super- into the normal-conducting phase and limits the maximal accelerating field $\mathrm{E_{acc,max}}$. Only two of the cavities in the data set, CAV00518 and CAV00087, have been tested with the 2nd sound set up ([@2nd1; @2nd2]) and had a subsequent optical inspection . The quench location is obtained with a ray tracing method, described in [@Yegor2017]. The spot which minimizes the root-mean-square error (RMSE) between the theoretical and the experimentally 2nd sound velocity is the most likely origin of the 2nd sound wave and therefore the quench spot. The corresponding RMSE-maps for the two cavities are shown in figure \[fig:RMSEMap\_Yegor\].
![RMSE-Map of CAV00518 (left) and CAV00087 (right). The x-axis is the cell angle, the y-axis the longitudinal position along the cavity. The z-axis shows the color coded RMSE value. The most likely quench spot for CAV000518 is about 44mm below equator nine at ($330^\mathrm{o}\,\pm\,12^\mathrm{o}$) and for CAV00087 at equator nine at ($141^\mathrm{o}\,\pm\,16^\mathrm{o}$) [@Yegor2017].[]{data-label="fig:RMSEMap_Yegor"}](RMSEMap_Yegor.PNG){width="90.00000%"}
For both cavities, CAV0087 and CAV00518, the quench spot localized by 2nd sound was in cell nine. A visual inspection of the cavity surfaces using the 2nd sound system results as guidance did not reveal any local defect which could be identified as the origin of the quench, hence global surface properties are assumed to be the cause. The same equators were identified as optically conspicuous cells by the image analysis algorithm, as they had the largest *integrated grain boundary area* $\sum{\mathrm{A}}$ of their cavities. The probability for this observation to be a coincidence is 1.2%.
Optical Assessment of Cells
---------------------------
At the time of this analysis, 14 cavities from RI and 31 cavities from EZ were inspected. The observed grain boundary area distribution per equator is shown in figure \[GB\_area\_distribution\].
![Average grain boundary area distribution of an equator. The blue distribution shows the average of 126 RI equators, the red distribution shows the average of 279 EZ equators. The dotted lines are the 1$\,\upsigma$ confidence intervals of the average.[]{data-label="GB_area_distribution"}](Vendor_Area_Distribution.png){width="90.00000%"}
It can be seen that RI cavities exhibit a smaller number of boundaries with an area above $5 \cdot 10^3~\upmu \mathrm{m^2}$ than EZ cavities. Figure \[Hist\_EZ\] shows the histogram of the observed values of the integrated grain boundary area of the optically conspicuous cells for 31 cavities from EZ.
![Histogram of the values of the integrated grain boundary area for the 31 optically conspicuous cells from EZ.[]{data-label="Hist_EZ"}](Hist_occ_EZ_values.png){width="70.00000%"}
The integrated grain boundary area for EZ averages at $1600\,\pm 400\,\mathrm{mm^2}$, hence covering about $23\,\pm\,6$% of the weld area. For RI, the average is $1200\,\pm\,260\,\mathrm{mm^2}$ and the coverage rate is $17 \pm\,4$%. A priori, the longitudinal distribution of the optically conspicuous cell within a cavity should be a uniform distribution. The observed distribution of the optically conspicuous cells is shown in Figure \[Vendor\_worst\].
![Observed longitudinal distribution of the *optically conspicuous cells*. The blue distribution is derived from 14 RI cavities, the red distribution from 31 EZ cavities.[]{data-label="Vendor_worst"}](LongDist_9.png){width="80.00000%"}
For EZ, equator 9 is the conspicuous cell in 24 out of 31 cases. For RI, equators 6 and 7 are the conspicuous cells in 9 out of 14 cases. The $\chi^2$-test for both vendors showed a statistically significant deviation of the observed distribution from a uniform distribution with a $\chi^2$ of 17 for RI and a $\chi^2$ of 112 for EZ and the degrees of freedom (df) of 8 for both. The cell with the smallest integrated grain boundary area of a individual cavity is called the *optically best cell*. The distribution of the optically best cells within the cavity is shown in Figure \[Vendor\_best\].
![Observed longitudinal distribution of the *optically best cells*. The blue distribution is derived from 14 RI cavities, the red distribution from 31 EZ cavities.[]{data-label="Vendor_best"}](LongDist_1.png){width="80.00000%"}
For EZ, the $\chi^2$-test showed that the observed distribution is in agreement with a uniform distribution with a $\chi^2$ of 8.2 and dof equal to 8. For RI, equators 1 and 4 are favored. The $\chi^2$-test for a uniform distribution yields $\chi^2$ of 32 and dof equal to 8.
The integrated boundary area of three cells of EZ cavities exceed the average value by more than 25% and one cell is below the average value by 25%, see Figure \[Hist\_EZ\]. For a better understanding of the origin of this deviation, the integrated grain boundary area per image against the angular position of the image taken for these cells is shown in Figure \[Angular\_worst\_mean\].
![Integrated grain boundary area per image against the angular position of the image taken for the three cells with the largest values, the average of the whole set and the cell with the smallest value.[]{data-label="Angular_worst_mean"}](Angular_worst_mean.png){width="80.00000%"}
Images of the specific cells significantly exceeding the average are shown in Figures \[CAV518\] and \[CAV563\]. In comparison to this excess, Figure \[CAV532\] shows an image of the inner cavity surface with a smaller than average integrated grain boundary area per image.
![OBACHT image of the inner cavity surface of cavity CAV518, equator 9, 225.6 degree.[]{data-label="CAV518"}](CAV518_2_E9.jpg){width="50.00000%"}
![OBACHT image of the inner cavity surface of cavity CAV563, equator 9, 249.6 degree.[]{data-label="CAV563"}](CAV563_1_E9.jpg){width="50.00000%"}
![OBACHT image of the inner cavity surface of cavity CAV532, equator 6, 182.4 degree.[]{data-label="CAV532"}](CAV532_1_E6.jpg){width="50.00000%"}
CAV518 (Figure \[CAV518\]) shows some remnants of the machining procedure prior to welding. Although all equators of CAV00518 showed a larger integrated grain boundary area than average, equator nine was outstanding. The same equator was identified as quench origin.
CAV563 (Figure \[CAV563\]) shows prominent grain boundaries in the welding seam and heat affected zone. This topography was observed in several cavities, as well as in CAV579. It was usually observed in equator nine and in varying intensities and angular range.
CAV532 (Figure \[CAV532\]) is a cavity with an overall homogenous and smooth appearance.
In order to estimate the influence of the surface chemistry on the observed values of the integrated grain boundary area, images of the bulk niobium apart of the weld were analyzed. Those so called *cell images* are taken with an offset of $\pm\,11.5\,\mathrm{mm}$ relative to the equator images during the inspections. Hence, only an influence of the surface chemistry and not from the electron beam welding in these images is expected. In contrast to the equator images, where a huge spread of the integrated grain boundary area was observed, the values for the inner surface in the cell are comparable within uncertainties for all inspected cavities of both vendors.
Correlation with RF Performance
-------------------------------
Table \[Values\] shows a comparison of the achieved accelerating field of a cavity against the average integrated grain boundary area per image of the optically conspicuous cell of this cavity.
**Cavity** **$E_{\mathrm{acc,max}}~ [\mathrm{MV/m}]$** **$<\upsigma \mathrm{A}>_{\mathrm{im}}~[\mathrm{mm^2}]$**
----------------- --------------------------------------------- -----------------------------------------------------------
CAV518 22 28
CAV563 20 26
CAV579 25 24
$<\mathrm{EZ}>$ 28 $\pm$ 7 20 $\pm$ 2
CAV532 35 15
: Comparison of the RF results and the integrated grain boundary area per image.[]{data-label="Values"}
For a more quantitatively approach, a larger set of cavities was chosen. In order not to be affected by local defects and to deduce an unbiased correlation between optical surface properties and the RF performance of a cavity, a set of cavities are selected by the following criteria:
1. No surface chemistry between optical inspection and the cold RF test,
2. Optical inspection shows no local defect,
3. No field emission during the RF test.
The first criterion is needed to assure that the results of the two methods, optical inspection and cold RF test, can be correlated. A local defect, which is more likely to be the cause of a possible limitation of the cavity RF performance, has to be avoided in order to study the correlation between global surface properties and the RF performance. This is the reason for the second criterion. The last criterion prevents a falsification of the RF performance, because field emission is caused by a local defect in the highest electric field region, which is the iris region for TESLA type cavities, and introduces a different loss mechanism. A total number of 17 cavities from the XFEL production fulfill the before mentioned criteria, nine RI and eight EZ cavities. To increase the data set, but also to improve the universality of this study, three so-called large grain cavities were included, namely AC151, AC153 and AC154. They fulfill the above mentioned criteria and increase the data set to a total number of 20 cavities. Figure \[fig:Eacc\_sumarea\_wLG\] shows the measured maximum accelerating field as a function of the integrated grain boundary area.
![On the x-axis, the integrated grain boundary area of the optically conspicuous cell, which stands for a complete cavity, is shown, while the y-axis depicts the maximum accelerating field achieved by the respective cavity. The red squares display large grain cavities, the blue circles fine grain cavities.[]{data-label="fig:Eacc_sumarea_wLG"}](Corr_Area.png){width="100.00000%"}
The correlation coefficients are given in Table \[tab:Eacc\_sum\_area\].
A very strong negative correlation of $\rho\,=\,-0.93$ between the two variables $\sum{\mathrm{A}}$ and $\mathrm{E_{acc,max}}$ was found. The result is consistent for different subgroups, which were tested to reveal a systematic origin of the observed correlation. Those three subgroups are the (1) nine RI cavities, (2) the eight EZ cavities, and (3) eleven cavities from both vendors but from the same niobium supplier (Sheets). The large grain cavities are only included in the complete sample. The statistical significance of the correlation was found to be $6\,\upsigma$. The reduced significance in the subgroups is caused by the reduced dataset.
Discussion
==========
The observed vendor specific grain boundary orientation as seen in Figure \[Orienthist\_vendors\] can be explained by the vendor specific EBW parameters. The travel speed influences the overall bead shape as the shape changes from elliptical to tear drop shaped as the welding speed increases (see Figure \[WeldingFeed\] taken from [@EBW1997]). The grains will grow in the direction of the thermal gradient, starting at the base metal and into the liquified niobium.
![Comparison of welding puddle shapes. Travel speeds: (a)slow, (b) fast. Note that the orientation of the grains change during growth in (a) while the orientation is constant in (b). This is a consequence of the spatial orientation of the thermal gradient. While in (a) and (b) the thermal gradient at point A is $90^\mathrm{o}$ to the weld axis, in (a) the thermal gradient at B is parallel to the weld axis while it changes the orientation only slightly (b).[]{data-label="WeldingFeed"}](Crystallization.png){width="60.00000%"}
Comparing this with the observed pattern shows that RI has a beam travel speed slower than EZ, while the exact values are not public. But given the data and experience in [@Udomphol2007; @EBWBuch], it can be estimated that the welding speed for EZ has to be on the order of $16\,\pm\,1\,\frac{\mathrm{mm}}{\mathrm{s}}$ while for RI it has to be on the order of $5\,\pm\,1\,\frac{\mathrm{mm}}{\mathrm{s}}$. Those values are in good agreement with available data on welding speeds for different cavities and their grain boundary orientations [@Singer2017; @Geng1999; @Schmidt10].
The specific series of welding patterns which are shown in Figure \[Orienthist\_vendors\_patterns\] are identical for all cavities from a vendor and can be explained with the vendor specific assembly procedure (see Figure \[VendorAssembly\]).
![The upper schematic sketches the assembly procedure of a cavity at the two vendors and allows for the assignment of the patterns to the corresponding cells. The lower sketch depicts the orientation pattern as found in the respective cell.[]{data-label="VendorAssembly"}](Vendor_Welding_Procedure.PNG){width="60.00000%"}
While EZ assembles all dumbbells in one group and weld them in one EBW run - where the equators are welded subsequently in an alternating sequence - the assembling process for RI is split up in four steps.
The first two steps are the assembly of two different structures consisting of three dumbbells and one end group, which are welded together (\[E1-3\] and \[E7-9\]),but performed separately.
The third step is the weld of one double-dumbbell (E5). Between each step, the fabricated parts are brought out of the EBW machine. For the final step, the weld of equator 4 and 6, one end cell group is rotated by $180^\mathrm{o}$ with respect to the other. The different welding sequences result in a distinct welding pattern along the cavity for each vendor. A single image of the first equator can identify the cavity vendor.
The observed vendor-specific surface roughness (see Figure \[Vendor\_Rdq\_Distribution\] and Table \[tab:Rdq\_emg\_fit\_vendor\] and \[tab:Rdq\_emg\_fit\_vendor\_mlv\]) are in good agreement with the surface topology of EP- and BCP-treated cavities. It is known that EP leads to a smaller average roughness than BCP [@Padamsee2008; @Ciovati2008], as well as smaller boundary step heights and slopes [@Geng1999; @Xu2011], which is reflected in the average $\mathrm{R_{dq}}$. In addition, a spatial inhomogeneous removal of material by EP has been observed [@Geng1999], where the welding seam regions are more affected than the heat affected zone due to the inhomogeneous electric field along the cavity axis. This effect is visible in the mean of the Gaussian component, $\upmu$. The BCP treated cavities show a uniform roughness value while the values for EP treated cavities differ for the regions.
The agreement between the quench location obtained with the 2nd sound system (see Figure \[fig:RMSEMap\_Yegor\]) and the image analysis algorithm in two of two tests supports the assumption that the *integrated grain boundary area* $\sum{\mathrm{A}}$ can be used as a correlator with the RF performance. Although the underlying mechanism for this correlation is not known, it is plausible that grain boundaries influence the RF performance. Given current developments for high quality factor and high accelerating gradient treatments, the influence of grain boundaries are yet to be investigated, although first results in the serial production of nitrogen doped cavities show such a dependency [@Grassellino2013; @Grassellino2017; @TTC1].
The observed non-uniform longitudinal position of the optically conspicuous cell (see figure \[Vendor\_worst\]) is unexpected. The most likely optically conspicuous cells are cells with a special position in the vendor specific welding sequence. For EZ, equator 9, which is the optically conspicuous cell in 24 of 31 cavities, is the first weld to be welded while the other welds follow in an alternating sequence. At RI, the first step is the fabrication of assembly groups in which the assembly and equatorial welds of two 3.5 cell cavities \[E1-3\] and \[E7-9\] and one double dumbbell \[E5\] are done. In a final assembly and welding step of those assembly groups, equators 4 and 6 are welded. Equators 6 and 7 are the optically conspicuous cells in 9 of 14 cavities and equator 4 the optically best cell in 6 of 14 cavities.
To better understand this behavior, exemplary cavities were investigated further, namely CAV518, CAV563, CAV579, and CAV532. These specific welding seam surfaces show distinct grain boundary topographies. The inspection of the cell images, containing the bulk niobium with no influence of the welding procedure, showed no comparable topography nor a vendor specific surface structure, which should be the case if the surface chemistry alone is the cause for the deviations found. This leads to the conclusion that the assembly and electron beam welding procedure have the most significant influence on the integrated grain boundary area.
The observed correlation, quantified in Table \[tab:Eacc\_sum\_area\] and shown in Figure \[fig:Eacc\_sumarea\_wLG\], suggest that higher $E_{\mathrm{acc,max}}$ can be reached when the integrated grain boundary area $\sum{\mathrm{A}}$ is small. As mentioned, the underlying mechanism of this correlation is not clear, especially since the mentioned models describe loss mechanisms and do not discuss achievable accelerating gradients. This is a weak point of this analysis, since it can only state a correlation. To improve this analysis another approach was to model the achieved maximal gradient as the result of the interplay of a given quality factor at low fields and loss mechanisms depending on the integrated grain boundary area $\sum{\mathrm{A}}$. This model showed no correlation and was rejected. The next steps will be to include the observed surface roughness $\mathrm{R_{dq}}$, the orientation of the grain boundaries, material parameters of the cavities, and other parametrization of the RF performance in the analysis.
Conclusion
==========
In the scope of this work, a framework has been developed which allows an automated analysis of images of the inner surface of SRF Tesla type cavities by means of the optical inspection robot OBACHT. A first application of the newly developed framework was an investigation of optical surface properties of the two cavity vendors for the European XFEL and significant differences in the quantitative characterization have been identified. Each difference turns out to be related to the vendor specific cavity assembly, electron beam welding and surface treatment procedures. In order to study the interplay between the optical surface properties and RF limitations, the concept of an optically conspicuous cell, which is assumed to be the limiting cell of the RF performance, was introduced. A crosscheck with 2nd sound tests of two cavities support the assumptions of the identity of the optically conspicuous cell and the limiting cell of the RF performance. A noteworthy observation has been made. A non-uniform distribution of the optically conspicuous cell was observed and in fact, this non-uniform distribution is vendor specific. This observation can be explained by the vendor specific assembly procedure and equatorial welding sequence. In addition, correlations of optical surface properties versus the maximal achievable accelerating field $\mathrm{E_{acc,max}}$ of 20 cavities has been investigated. A strong negative correlation of the integrated grain boundary area $\sum{\mathrm{A}}$ versus the maximal achievable accelerating field $\mathrm{E_{acc,max}}$ has been found. In conclusion, a quantitative analysis and characterization of a cavity surface by means of optical methods has been achieved, which can be adapted and used for the quality assurance of a future large scale cavity production.
Acknowledgments
===============
I would like to thank B. van der Horst, J. Iversen, A. Matheisen, W.-D. Möller, D. Reschke and W. Singer (DESY) for their support, insights and valuable discussions. Furthermore, I would like to thank S. Aderhold (now FNAL), S. Karstensen (DESY), A. Navitski (now RI), J. Schaffran (DESY), L. Steder (DESY) and Y. Tamashevich (now HZB) for their work. Otherwise, mine would have been impossible. This work is funded from the EU 7th Framework Program (FP7/2007-2013) under grant agreement number 283745 (CRISP), “Construction of New Infrastructures - Preparatory Phase”, ILC-HiGrade, contract number 206711, BMBF project 05H12GU9, and from the Alexander von Humboldt Foundation.
References {#references .unnumbered}
==========
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|
---
abstract: 'SagittariusA$^\star$ ([SgrA$^\star$]{}) is the supermassive black hole residing at the center of the Milky Way. It has been the main target of an extensive multiwavelength campaign we carried out in April 2007. Herein, we report the detection of a bright flare from the vicinity of the horizon, observed simultaneously in X-rays ([*XMM-Newton*]{}/EPIC) and near infrared ([*VLT*]{}/NACO) on April 4$^{\rm th}$ for 1–2 h. For the first time, such an event also benefitted from a soft $\gamma$-rays ([*INTEGRAL*]{}/ISGRI) and mid infrared ([*VLT*]{}/VISIR) coverage, which enabled us to derive upper limits at both ends of the flare spectral energy distribution (SED). We discuss the physical implications of the contemporaneous light curves as well as the SED, in terms of synchrotron, synchrotron self-Compton and external Compton emission processes.'
author:
- 'G. Trap$^{1,2}$, A. Goldwurm$^{1,2}$, R. Terrier$^{2}$, K. Dodds-Eden$^{3}$,'
- 'S. Gillessen$^{3}$, R. Genzel$^{3}$, E. Pantin$^{2,4}$, P.O. Lagage$^{2,4}$,'
- 'P. Ferrando$^{1,2}$, G. Bélanger$^{5}$, D. Porquet$^{6}$, N. Grosso$^{6}$,'
- 'F. Yusef-Zadeh$^{7}$, F. Melia$^{8}$'
title: |
Soft gamma-ray constraints\
on a bright flare from\
the Galactic Center\
supermassive black hole
---
=1
Black hole physics ,Radiation mechanisms: non-thermal ,Galaxy: center ,Gamma rays: observations ,Infrared: general ,X-rays: general 04.70.Bw ,95.85.Gn ,95.85.Hp ,95.85.Nv ,95.85.Pw ,97.60.Lf ,98.35.Jk ,98.35.Mp
=0.5 cm
Introduction
============
From the discovery of a compact radio source, [SgrA$^\star$]{}, at the Galactic Center (GC) in 1974 [@balick74] to the near infrared (NIR) tracking of stars in Keplerian motion around [SgrA$^\star$]{} three decades later [@schodel02; @ghez03], the evidence for a $\sim$$4\times10^{6}$ $M_\odot$ black hole with very slow proper motion at the dynamical center of our galaxy [@reid08] gradually piled up [see @melia07 for a general review and references therein].
Yet, the long quest for the high energy emission pertaining to the black hole has only been achieved recently. [SgrA$^\star$]{} was resolved as a notably dim ($2.4\times10^{33}$ [ergs$^{-1}$]{}, 2–10 keV) and slightly extended (1.4$''$) point source with the [*Chandra*]{} satellite in 1999 [@baganoff03a]. One year later, the same instrument witnessed the source exhibiting an X-ray flare for $\sim$3 h [@baganoff01]. A $\sim$10 min long substructure within the light curve of the eruption and light time travel arguments imply that this event took place close to the event horizon ($<15~R_{\rm S}$). Many other detections of X-ray flares followed, either with [*XMM-Newton*]{} or [*Chandra*]{} [see e.g. @goldwurm03a; @baganoff03b; @porquet03; @belanger05], and established that the duty cycle of the black hole is nearly one X-ray flare per day. The origin of these events is still unclear, in spite of all the efforts aimed at their monitoring in different energy ranges. In 2003, NIR flares from [SgrA$^\star$]{} were indeed discovered with the [*VLT*]{} [@genzel03], and later confirmed by the [*Keck*]{} [@ghez04] and the [*HST*]{} [@yusef-zadeh06a]. They occur more frequently than the X-ray ones (around four per day) and have been observed in many NIR atmospheric pass bands (H, K, L, M). Each new infrared flare has generally induced either spatial [@clenet05], spectral [@eisenhauer05; @ghez05; @gillessen06; @krabbe06; @hornstein07], polarimetric [@eckart06b; @meyer06; @meyer07; @trippe07], or timing studies [@meyer08; @do09]. Numerous multiwavelength campaigns showed that an X-ray flare always comes along with a simultaneous NIR one[^1] [@eckart04; @eckart06a; @eckart08a; @yusef-zadeh06a; @hornstein07], and maybe a delayed submm one [@marrone08; @eckart08b; @yusef-zadeh08] caused by plasmon expansion [@liu04; @yusef-zadeh06b].
Above 6 $\mu$m, in the mid infrared (MIR), no detection of [SgrA$^\star$]{} has been reported so far. Recent upper limits on the black hole flux at 8.6 $\mu$m were set by the [*VLT*]{}/VISIR instrument during low level NIR variability by @schodel07, who argued that a detection would be reachable in case of a strong NIR flare.
Above 20 keV, repeated surveys of the heart of the Milky Way in soft $\gamma$-rays with the [*INTEGRAL*]{} satellite unveiled a persistent pointlike source compatible with [SgrA$^\star$]{} location (within the 1$'$ error radius), [IGRJ17456–2901]{} [@belanger04; @belanger06]. The nature of the source is still uncertain, and a possible association with the supermassive black hole remains conceivable. Given the limited angular resolution of the soft $\gamma$-ray telescope [*INTEGRAL*]{}/IBIS/ISGRI ($\sim$12$'$ FWHM), the best way to unequivocally identify the mysterious [IGRJ17456–2901]{} with [SgrA$^\star$]{} is the detection of correlated variability between soft $\gamma$-rays and other wavelengths.
To tackle the above puzzles and investigate the correlated X-ray/NIR variability of [SgrA$^\star$]{} in more details, a coordinated multiwavelength campaign on the GC was conducted in spring 2007. It involved in particular the [*XMM-Newton*]{} and [*INTEGRAL*]{} satellites for the high energies, as well as the [*VLT*]{}/ NACO and [*VLT*]{}/VISIR ground instruments to cover the NIR and MIR part of the spectrum, respectively. Their results are presented in Sect. \[obs\] and interpreted in Sect. \[SED\]. Note that the X-ray and infrared findings have already been published by @porquet08 and @dodds09, respectively.
We will not discuss here the short term variability of [SgrA$^\star$]{} in April 2007 at cm, mm, and submm wavelengths, which will be reported in another article, along with NIR results obtained by the Hubble Space Telescope [@yusef-zadeh09][^2].
Throughout this paper we adopt a GC distance of 8 kpc [@reid93] and a black hole mass $M_{\bullet}=4\times10^{6}$ $M_\odot$ [@ghez08], for which the Schwarzschild radius is $R_{\rm S}=1.2\times10^{12}$ cm.
Observations & results {#obs}
======================
X-rays {#X-rays}
------
The [*XMM-Newton*]{} satellite [@jansen01] was pointed towards the GC during $\sim$2.5 consecutive revolutions, from March 30$^{\rm th}$ to April 4$^{\rm th}$ 2007. The data of the EPIC/PN [@struder01] and EPIC/MOS1–2 cameras [@turner01] were processed and analyzed through the procedure described in @porquet08.
We produced an image of the last revolution of the campaign (rev-1340, 97.6 ks exposure), cleaned for out of time events in the 2–10 keV band (see Fig. \[ima\], middle). Two GC transient X-ray binaries and bursters, active at the time, stand out prominently: [GRS1741.9–2853]{} [@trap09] and [AXJ1745.6–2901]{} [@grosso08]. [SgrA$^\star$]{} is clearly apparent in the middle since this observation contains several flares from the vicinity of the black hole, enhancing its average luminosity.
Indeed, on April 4$^{\rm th}$, a high level of flaring activity from [SgrA$^\star$]{} was caught. A bright flare (Fig. \[imax\])—the second brightest ever recorded ($\sim$100 times the quiescent level) in the X-ray band (2–10 keV)—was rapidly followed by three moderate ones. The bright event lasted $\sim$1 h; its PN light curve has a rather symmetrical morphology and no apparent substructures (Fig. \[lc\]). Note that the 10$''$ radius area used to extract this light curve not only contains [SgrA$^\star$]{} but other X-ray sources as well: a pulsar wind nebula candidate, [G359.95–0.04]{} \
[@wang06], the star cluster IRS13, and diffuse emission [@baganoff03a], which all, however, provide a constant contribution.
From a spectral point of view, this outburst was rather soft. The best fit to the data with an absorbed power-law model, including dust scattering, yields the following parameters: a spectral photon index $\Gamma = 2.3 \pm 0.3$ ($N(E)\propto E^{-\Gamma}$) and a column density $N_{\rm H} = 12\pm2 \times 10^{22}$ [cm$^{-2}$]{}. In Sect. \[SED\], we will use an equivalent definition of the spectral index, $\beta_{\rm X}$, easier to compare to other multiwavelength spectra: $\beta_{\rm X}=-\Gamma+2=-0.3\pm0.3$ with $\nu F_{\nu}^{\rm X} \propto \nu^{\beta_{\rm X}}$. The unabsorbed mean flux of the flare was $16\pm3 \times 10^{-12}$ [ergs$^{-1}$cm$^{-2}$]{} (2–10 keV), which translates to a luminosity of $2.4\pm4 \times 10^{35}$ [ergs$^{-1}$]{} at the GC distance. Albeit luminous relative to previously observed X-ray flares, it was still $\sim$9 orders of magnitude below the Eddington luminosity for a supermassive black hole of this kind.
As pointed out by @porquet08, this rapid train of flares in just a few hours challenges disruption mechanisms of the accretion flow as the origin of the outbursts, since they rely on temporary storage of mass/energy. This energy should indeed be released at once during the outburst, with a radiation efficiency of a few percent. But, the weak accretion rate of the black hole seems insufficient ($\sim$$10^{16-17}$ gs$^{-1}$, @melia07) to accumulate the required energy on such short timescales. In contrast, scenarios based on the stochastic infall and tidal disruption of gas clumps [@tagger06; @falanga07; @falanga08] or small bodies [@cadez08] do not encounter this issue.
Near infrared {#NIR}
-------------
The [*VLT*]{}/NACO (NAOS+CONICA) set of instruments [@lenzen03; @rousset03] installed on the ESO/[*VLT*]{} unit telescope Yepun (UT4) at Paranal, Chile, observed the Galactic nucleus every nights from April 1$^{\rm st}$ to April 6$^{\rm th}$ in multiple NIR bands: L’ (3.8 $\mu$m), K$_{\rm S}$ (2.1 $\mu$m), and H (1.6 $\mu$m). The details of the data reduction and analysis are presented in @dodds09.
In particular, from 5:00 to 7:00 (UT) on April 4$^{\rm th}$, NACO followed the strong X-ray flare mentionned in Sect. \[X-rays\] in the L’ band. We constructed an image of the GC during this flare period (Fig. \[ima\], bottom, left, and Fig. \[imanir\]), in which [SgrA$^\star$]{} is confused with the star S17 and a small cloud of dust [@clenet05]. On Fig. \[lc\], we display the light curve of the flare. Substructures on a timescale of $\sim$20 min within the light curve are evident. The shortest variation in the light curve ($\Delta t\sim1$ min) constrains the emitting zone to a size no bigger than $c/\Delta t \sim 1.5~R_{\rm S}$. This is the first time that such features are visible in the L’ band. Regarding the total duration of the eruption, it lasted distinctively longer in the NIR than in X-rays ($\sim$2 h vs. $\sim$1 h), even though the X-ray background could hide the rising and decaying flanks of the flare. Note that the X-ray/NIR flare detected by [*Chandra*]{}/[*Keck*]{} on July 17$^{\rm th}$ 2006 [@hornstein07] also indicated a longer NIR duration, even if [*Chandra*]{} has a smaller X-ray background than [*XMM-Newton*]{} thanks to its better point spread function (PSF). As for the peak of the flare, there is no time lag bigger than $\sim$3 min between NIR wavelengths and X-rays.
Assuming an extinction $A_{\rm L}= 1.8$ mag, the NIR flare peaked at $\sim$30 mJy (dereddened), which makes it one of the most powerful NIR flare ever captured, and definitely the brightest one detected simultaneously in X-rays. To allow comparison with other wavelengths, we also computed the background subtracted, extinction corrected, mean flux of the flare over the period of MIR observations (see Sect. \[MIR\]): $19.1\pm3.6$ mJy at 3.8 $\mu$m. No direct NIR spectral information are available for this flare.
Mid infrared {#MIR}
------------
VISIR, the [*VLT*]{} Imager and Spectrometer for the mid Infrared mounted on the ESO/[*VLT*]{} unit telescope Melipal (UT3) [@lagage04; @pantin05], pointed the GC from 2007-04-04 5:29:00 to 2007-04-04 10:34:00 (UT) as part of a guaranteed time program. Data were acquired with the imager and [PAH2\_2]{} filter on, at $11.88\pm0.37~\mu$m in the atmospheric window N. The Small Field mode ([SF]{}) was employed, which resulted in a field of view of $256 \times 256$ pixels ($19.2''\times19.2''$), each pixel corresponding to $0.075''\times0.075''$. Data reduction and analysis techniques are given in @dodds09.
No point source at the position of [SgrA$^\star$]{} is detected in either the individual images or the collapsed image of the entire night (Fig. \[ima\], bottom, right). We also performed a Lucy-Richardson deconvolution with HD102461 as point spread function without success. The flux from a box of $0.375''\times0.375''$ centered on the position of [SgrA$^\star$]{} is fairly constant (see the light curve on Fig. \[lc\]) with an average value of $123\pm6$ mJy (not dereddened). This flux may be attributed to the faint and diffuse dust ridge on which [SgrA$^\star$]{} lies. Our measured value is consistent with previous VISIR observations [@eckart04; @schodel07] and other instruments before [@stolovy96; @cotera99; @morris01].
However, for the first time, the measurements presented here were concurrent with a bright X-ray/NIR flare from [SgrA$^\star$]{} as shown in Sect. \[X-rays\] and \[NIR\]. We estimate that [SgrA$^\star$]{} could not have been brighter than $\sim$12 mJy at 11.88 $\mu$m (3$\sigma$, not dereddened). This value is compatible with VISIR empirical sensitivity at this wavelength: 7 mJy/10$\sigma$/1 h (median value for different atmospheric conditions). We note also that similar constraints were obtained with VISIR during NIR variability by @gillessen06, @schodel07, and @haubois08.
The value of the extinction correction, $A_{\rm \lambda}$, in the MIR depends critically on the strength and shape of the silicate absorption feature at $\sim$10 $\mu$m. In the literature, $A_{\rm \lambda}$ is usually given as ratios relative to $A_{\rm V}$ or $A_{\rm K}$, so we use $A_{\rm K} = 2.8$ mag ($A_{\rm V} = 25$ mag) to ensure consistency across our multiwavelength observations. The closest extinction measurement to $\lambda = 11.88$ $\mu$m was made by @lutz99 for a wavelength of $\sim$12.4 $\mu$m. We consider the recent theoretical model of @chiar06 for the extinction profile in the silicate region to allow us to extrapolate the value measured at 12.4 $\mu$m to 11.88 $\mu$m. When normalized to the @lutz99 values, the model predicts $A_{\rm 11.88\,\mu m}=1.7 \pm 0.2$ mag and hence the dereddened 3$\sigma$ upper limit on the MIR emission of [SgrA$^\star$]{} during the flare is $\sim$57 mJy. @schodel07 found an upper limit of $\sim$22 mJy at 8.6 $\mu$m during quiescence of [SgrA$^\star$]{} and predicted a [*bright*]{} flare was likely to come out of the noise in the MIR. Yet, using the dereddening used here, their limit would be a bit higher, $\sim$32 mJy, and, since we did not detect this strong X-ray/NIR flare, we speculate that no flaring counterpart of [SgrA$^\star$]{} will be detected with the current settings of VISIR.
![From top to bottom, light curves of [IGRJ17456–2901]{} + [AXJ1745.6–2901]{}, [SgrA$^\star$]{} + [G359.95–0.04]{} + IRS13 + diffuse emission, [SgrA$^\star$]{}+ S17, and diffuse emission.[]{data-label="lc"}](Fig4.pdf){width="14cm"}
Gamma-rays
----------
The [*INTEGRAL*]{} satellite [@winkler03] monitored the GC in parallel to the other instruments in April 2007, for a total effective exposure of $\sim$212 ks for IBIS/ISGRI (20–100 keV) [@ubertini03; @lebrun03] and $\sim$46 ks for JEM-X 1 (3–20 keV) [@lund03][^3] at [SgrA$^\star$]{} position. Measurements were spread over two consecutive revolutions, 545 and 546, from 2007-04-01 12:58:00 to 2007-04-02 21:32:34 and 2007-04-03 11:48:14 to 2007-04-04 20:26:59 (UT), respectively. In total, data of 74 individual pointings (science windows, ScWs) were acquired, lasting $\sim$2930 s each. The whole dataset was reduced with OSA 7.0, the Offline Science Analysis Software, distributed by the [*INTEGRAL*]{} Science Data Center (ISDC) [@courvoisier03], with algorithms described in @goldwurm03b for IBIS/ISGRI and @westergaard03 for JEM-X.
To search for a counterpart above 20 keV of the aforementioned flare, we selected the two consecutive ScWs of ISGRI that covered the flare time interval, 054600220010 and 054600230010, and created a combined mosaic of the individual images, in two energy bands: 20–40 and 40–100 keV. None of these mosaics contained a distinctive source at the position of the black hole. Hence, no high energy counterpart of the flare was found. By considering the variance in [SgrA$^\star$]{} pixel, we derive 3$\sigma$ upper limits on the flare of 1.17 and 1.11 [ctss$^{-1}$]{} in the 20–40 and 40–100 keV bands, respectively. Assuming a power-law spectral shape of index $\Gamma=2.3$ (Sect. \[X-rays\]), these rates convert to flux limits of 5.76 and $11.1\times10^{-11}$ [ergs$^{-1}$cm$^{-2}$]{}, respectively.
Regarding JEM-X 1 data in the 3–20 keV band, we report similar results. There was no detection in the combined mosaic, and given a flare duration of $\sim$3000 s, the sensitivity curves of JEM-X[^4] provide 5$\sigma$ upperlimits of 10 and $7\times10^{-11}$ [ergs$^{-1}$cm$^{-2}$]{} in the 3–10 and 10–25 keV energy ranges, respectively.
On Fig. \[lc\] (top panel), we plotted the ISGRI light curves of the pixel at the position of [SgrA$^\star$]{}, built with individual ScW. It is noteworthy that no source was significantly detected in any individual exposure.
In contrast to individual ScWs, the total ISGRI and JEM-X 1 mosaics of the observation dataset both reveal a significant excess at the position of [SgrA$^\star$]{} (see Fig. \[ima\], top, left, and right). The significances of these signals are both 13.7$\sigma$. In view of the [*XMM-Newton*]{} image (Fig. \[ima\], middle), the transient neutron star low-mass X-ray binary [AXJ1745.6–2901]{}, located just 1.5$'$ from [SgrA$^\star$]{} in projection, was markedly the dominant source of the region. Given JEM-X angular resolution of $\sim$3$'$ (FWHM), we can safely associate its 3–20 keV excess with the binary. In the ISGRI mosaic (Fig. \[ima\], top, left), the PSF is $\sim$13$'$ (FWHM), and so does not allow us to disentangle [AXJ1745.6–2901]{} from [IGRJ17456–2901]{}, the persistent hard X-ray source discovered by [*INTEGRAL*]{}/ISGRI[^5] [@belanger04; @belanger06]. To assess the contribution of the transient binary to the ISGRI signal though, we compared the April 2007 20–40 keV mosaic with another equivalent GC map, constructed with data spanning four months, from August to November 2006. During this latter period, we know for sure that the transient binary was in quiescence and undetected at high energies, thanks to a regular [*Swift*]{}/XRT monitoring of the GC [@degenaar09]. So, the total count rate of $0.86\pm0.03$ [ctss$^{-1}$]{} we measured in the central pixel of the excess in the 2006 mosaic, can be entirely attributed to [IGRJ17456–2901]{}. In April 2007, we found that the total count rate increased to $0.97\pm0.07$ [ctss$^{-1}$]{}, so that, presuming [IGRJ17456–2901]{} remained constant, the photons from the 20–40 keV excess visible in Fig. \[ima\] (top, left) came at $\sim$90% from [IGRJ17456–2901]{} and $\sim$10% from [AXJ1745.6–2901]{}.
This is the first time [*INTEGRAL*]{} was gazing the GC during a period of known flaring activity from [SgrA$^\star$]{}. Previous X-ray/$\gamma$-ray coordinated campaigns in 2004 were, indeed, inconclusive, since the X-ray flares detected then by [*XMM-Newton*]{} occured at times when [*INTEGRAL*]{} was crossing the radiation belts with all its intruments in standby mode [@belanger06].
As indicated above, we did not identify any $\gamma$-ray counterpart of the intense X-ray flare from April 4$^{\rm th}$. This proves once again that [SgrA$^\star$]{} does not release the bulk of its emission in soft $\gamma$-rays [@goldwurm94]. This result is also somewhat reminescent of the 2005 [*Chandra*]{}/[*HESS*]{} joint campaign, which demonstrated that the TeV source of the GC, [HESSJ1745–290]{}, stayed still during an X-ray outburst seen by [*Chandra*]{} [@hinton07a; @aharonian08].
On Fig. \[sed\], we display the broad band quiescent spectral energy distribution (SED) of [SgrA$^\star$]{} in dark gray. We also overplot in blue the spectral information on the 2007 April 4$^{\rm th}$ flare. By extrapolating the X-ray power-law, one expects fluxes of 3.9 and $4.1\times10^{-12}$ [ergs$^{-1}$cm$^{-2}$]{} in the 20–40 and 40–100 keV, respectively. These expected values are roughly one order of magnitude below the 3$\sigma$ constraints worked out above, which suggest that the next generation of hard X-ray focusing instruments, like [*Simbol–X*]{} [@ferrando08], will be able to extend spectral measurements on a flare of the GC supermassive black hole to above 20 keV [@goldwurm08].
Concerning [IGRJ17456–2901]{}, it was relatively improbable to find it flare up in April 2007, based on the long ISGRI exposures targeted at the source in 2003 and 2004. These did not reveal any sign of variability on any timescale [@belanger06], despite a temporal artefact in the early light curves of [IGRJ17456–2901]{} [@belanger04], that was later attibuted to a poor correction of the background [@belanger06]. Note, however, that variability on a single ScW duration basis cannot really be excluded, since this time interval is too short to convincingly detect the source [IGRJ17456–2901]{}.
The provenance of [IGRJ17456–2901]{} thus remains enigmatic. We showed that the activity of the luminous transient binary [AXJ1745.6–2901]{} did not amount to more than $\sim$10% of the total 20–40 keV flux of [IGRJ17456–2901]{}, contrary to what was alluded to in @revnivtsev04. The absence of variability and the fact that the flux of [IGRJ17456–2901]{} is two orders of magnitude above the quiescent emission of [SgrA$^\star$]{} as measured by [*Chandra*]{}, supports the idea that the hard X-ray photons visible in [*INTEGRAL*]{}’s mosaics are unlikely to be produced in the inner region of the accretion/ejection flow around the black hole. Instead, these photons should arise from a diffuse, and yet compact (a few arcminutes), zone, or maybe result from the sum of unresolved hard X-ray point sources [@revnivtsev06]. A possible connection between [IGRJ17456–2901]{} and [HESSJ1745–290]{} is another option. @hinton07b put forward that $\sim$10–100 TeV electrons permeating the inner 20 pc may be responsible for the combined [*XMM-Newton*]{}/[*INTEGRAL*]{} spectrum of the central 8$'$ radius region [@belanger06] via synchrotron emission, as well as [HESSJ1745–290]{} through inverse Compton (IC) processes. These authors favor the pulsar wind nebula candidate [G359.95–0.04]{} [@wang06] as the X-ray counterpart of [HESSJ1745–290]{}, though. In their scenario, the TeV photons come about in the compact nebula, just 0.3 pc from [SgrA$^\star$]{}, by the IC boosting of ambient photons by relativistic electrons originating from the pulsar. Nevertheless, [IGRJ17456–2901]{} does not fit within this frame, as its flux is too high to be the simple hard X-ray extension of [G359.95–0.04]{} soft X-ray flux as determined by [*Chandra*]{} [@wang06]. The increased angular resolution and sensitivity in the hard X-ray range of the next generation of instruments will also help address the question of [IGRJ17456–2901]{} true nature.
Radiative processes {#SED}
===================
The extremely low quiescent luminosity of [SgrA$^\star$]{} (10 orders of magnitude below the Eddington luminosity) is a long standing puzzle, that has stimulated numerous theoretical studies based on a combination of a low accretion rate, a radiatively inefficient accretion flow and outflows ejecting out the matter that just flowed in [See @melia01 for a review and references therein]. We will not discuss here the quiescent state and rather concentrate on the flaring emission, for which many models have also been proposed: jet models [@markoff01], ADAF like models [@yuan03; @yuan04] and accretion/stochastic acceleration models [@liu02; @liu04; @liu06a; @liu06b]. Each of them usually invokes either synchrotron self-Compton (SSC), external Compton (EC) or synchrotron broken power-laws (SB) processes as radiation mechanisms. In the subsequent discussion, we will explore these different possibilities for the April 4$^{\rm th}$ event, with no [*a priori*]{} assumption about the true nature of the engine behind the flare. We will examine the case of power-law distributions of electrons and highlight the natural synchrotron self-Compton component of each model.
Synchrotron self-Compton {#ssc}
------------------------
NIR flares are traditionally thought to arise from synchrotron emission since they are highly polarized [@eisenhauer05; @gillessen06; @krabbe06; @hornstein07] and have power-law spectral shapes [@eckart06b; @meyer06; @meyer07; @trippe07]. Here, we could not obtain a direct NIR spectrum of the flare, but we do have a stringent MIR upper limit. Hence, supposing a power-law spectral shape from MIR to NIR, $\nu F_{\nu}^{\rm IR} \propto \nu^{\beta_{\rm IR}}$, we have $\beta_{\rm IR}>0.04$. This is consistent with the index $\beta_{\rm NIR}=0.4$ published in previous NIR studies [@genzel03; @gillessen06; @hornstein07]. The submm bump of [SgrA$^\star$]{} quiescent SED has a slope $\beta_{\rm submm}^{\rm thick}>0$ below $\sim$$10^{12}$ Hz, which is thought to arise from an optically thick regime, and a slope $\beta_{\rm submm}^{\rm thin}<0$ above $\sim$$10^{12}$ Hz, presumably coming from an optically thin regime, judging from polarization measurements [@aitken00]. The fact that $\beta_{\rm IR}>0$ around $10^{13}$ Hz, shows that the NIR flare did not come from a global shift upward of the submm bump, but from a distinct population of particles creating a new rising hump in the SED in the IR band (see Fig. \[sed\]).
In an SSC model for the flares, the NIR photons are produced by a momentarily accelerated population of electrons radiating in the NIR band via a synchrotron process. In the following we will use the simple parametrization of @kraw04 in which a spherical homogeneous source of synchrotron radiation with a radius $R$ and a volumic electron density $n_{\rm e}$, pervaded by a magnetic field $B$, has a power-law energy distribution:
$$n(\gamma) \propto \gamma^{-p}~~~{\rm for}~~~\gamma_{\rm min} < \gamma < \gamma_{\rm max}\;.$$
We set $p=2$ in what follows[^6]. The electron density is thus determined by the normalization factor, $n_0$, of the power-law distribution by:
$$n_{\rm e}=\int^{\gamma_{\rm max}}_{\gamma_{\rm min}} n(\gamma)\, d\gamma = \int^{\gamma_{\rm max}}_{\gamma_{\rm min}} n_0 \gamma^{-2}\, d\gamma = -n_0 (\gamma_{\rm max}^{-1}-\gamma_{\rm min}^{-1}) \;.$$
The energy density of the electrons, $w_{\rm e}$, used by @kraw04 as normalization is also linked to $n_0$ through:
$$w_{\rm e}=\int^{\gamma_{\rm max}}_{\gamma_{\rm min}} \gamma m_{\rm e} c^2 n(\gamma)\, d\gamma=n_0 m_{\rm e} c^2 \ln{\gamma_{\rm max} \over \gamma_{\rm min}} \;.$$
The resulting synchrotron photon spectrum is optically thin and has a power-law shape [@rybicki79]:
$$\nu F_{\nu}^{\rm sync} \propto n_{\rm e} R^3 B^{(1+p)/2} \nu^{(3-p)/2}~~~{\rm for}~~~\nu_{\rm min}^{\rm sync} < \nu < \nu_{\rm max}^{\rm sync}\;,$$
with $\nu_{\rm min(max)}^{\rm sync}\propto \gamma_{\rm min(max)}^2 \nu_{g}$, where $\nu_{\rm g}=\frac{eB}{2\pi m_{\rm e}c}$ is the gyration frequency and $m_{\rm e}$ the mass of the electron. Below $\nu_{\rm min}^{\rm sync} $, the photon spectrum has a power-law shape, $\nu F_{\nu}^{\rm sync} \propto \nu^{4\over3}$, due to the lowest energetic electrons which have a Lorentz factor $\gamma_{\rm min}$, and above $\nu_{\rm max}^{\rm sync} $ it has an exponential cut-off.
In this SSC scheme, the X-ray flare is provoked by the inverse Compton boosting of the NIR flare photons by the same electrons that have just given rise to the NIR photons. The inverse Compton spectrum has the same morphology as the synchrotron one and scales like the Thomson optical depth of the sphere, $n_{\rm e} R \sigma_{\rm T}$, times $\nu F_{\nu}^{\rm sync}$:
$$\nu F_{\nu}^{\rm ic} \propto n_{\rm e} R \sigma_{\rm T} \times \nu F_{\nu}^{\rm sync} \propto n_{\rm e}^2 R^4 B^{(1+p)/2} \nu^{(3-p)/2}~~~{\rm for}~~~\nu_{\rm min}^{\rm ic}<\nu<\nu_{\rm max}^{\rm ic}\;,$$
where $\sigma_{\rm T}$ stands for the Thomson cross-section[^7] and:
$$\nu_{\rm min(max)}^{\rm ic}\propto \gamma_{\rm min(max)}^2 \nu_{\rm min(max)}^{\rm sync}\;.$$
Hence, this model has six free parameters: $p$, $\gamma_{\rm min}$, $\gamma_{\rm max}$, $B$, $R$, and $n_{\rm e}$. We arbitrarily fix $p=2$, $\gamma_{\rm min}=1$, and assess the four other parameters by considering four observables: the two frequencies of the synchrotron, $\nu_{\rm max}^{\rm sync}$, and inverse Compton, $\nu_{\rm max}^{\rm ic}$, peaks and their two respective amplitudes, $\nu F_{\nu}^{\rm sync,max}$ and $\nu F_{\nu}^{\rm ic,max}$. Indeed, on the one hand, $\gamma_{\rm max}$ is given by:
$$\label{gamma}
\nu_{\rm max}^{\rm ic}/\nu_{\rm max}^{\rm sync}\simeq \gamma_{\rm max}^2\;,$$
and $B$ by [@rybicki79][^8]:
$$\label{B}
\nu_{\rm max}^{\rm sync}\simeq 2.8 \left( \frac{B}{1~{\rm G}}\right) \gamma_{\rm max}^2 \times 10^{6}~{\rm Hz}\;.$$
On the other hand, $R$ and $n_{\rm e}$ can be deduced from the relations:
$$\left\{ \begin{array}{ll}
\displaystyle\frac{\nu F_{\nu}^{\rm ic,max}}{\nu F_{\nu}^{\rm sync,max}} & \propto n_{\rm e}R \\
\nu F_{\nu}^{\rm sync,max} & \propto n_{\rm e}R^3
\end{array} \right.\;.$$
Yet, our measurements do not provide us these four observables [*per se*]{}. We know that the X-ray spectral slope is softer than the IR one, which suggests that our [*XMM-Newton*]{} measurement at $\nu_{\rm X}\approx10^{18}$ Hz lies between the inverse Compton peak and the cut-off. Regarding the synchrotron peak, we only know it has a frequency $\nu_{\rm max}^{\rm sync}>10^{14}$ Hz. We will presume that this peak occurs at $\sim$$10^{14}$ Hz, in order to keep the magnetic field, $B$, as low as possible. As a result, we do not have to introduce the Klein-Nishina cross sections since the upscattering of the seed photons statisfies the Thomson regime condition, $\gamma_{\rm max} h \nu_{\rm seed} \ll m_{\rm e}c^2$ (the transition to the Klein-Nishina regime is at $\sim$$10^{18}$ Hz).
With the set of parameters listed in Tab. \[table\] we obtain a good fit of the flare SED, as displayed in orange on Fig. \[sed\]. However the values of $B$ and $n_{\rm e}$ necessary to accommodate the data are extremely high. First, synchrotron cooling of the radiating particles has a characteristic timescale:
$$\tau_{\rm cool} = \left(\frac{1}{6 \pi} \frac{\sigma_{\rm T} B^2}{m_{\rm e} c} \gamma (1-\gamma^{-2}) \right)^{-1}
=1.3 \left(\frac{\nu}{1~{\rm Hz}}\right)^{-\frac{1}{2}} \left(\frac{B}{1~{\rm G}}\right)^{-\frac{3}{2}}\times 10^{12}~{\rm s}\;.$$
This leads here, for such $B$, to $\tau_{\rm cool}\approx 5$ s at $\nu_{\rm NIR}\approx 10^{14}$ Hz. Sustained injection of particles is therefore required to power the flare during 1–2 h. On top of that, the electron density is so high that synchrotron self-absorption (SSA) comes into play right below $8\times10^{13}$ Hz, with an optically thick power-law $\nu F_{\nu}^{\rm SSA}\propto\nu^{{7}\over{2}}$ [@rybicki79]. As a consequence, we cannot observe the power-law with index $(3-p)/2$ for the synchrotron hump in contrast to the inverse Compton one, which also suffers from SSA but only at low frequencies. In any case the expected values for $B$ and $n_{\rm e}$ in the inner accretion flow around the black hole are orders of magnitude smaller ($B\approx10$ G and $n_{\rm e} \approx 10^7$ cm$^{-3}$ for @yuan03), even though interestingly the magnetic field at equipartition for particles with a typical energy $\gamma_{\rm max}=100$ and density $n_{\rm e}=2.2\times 10^{12}$ cm$^{-3}$ is $B_{\rm eq}\simeq \sqrt{8\pi \gamma_{\rm max} n_{\rm e} m_{\rm e}c^2}\approx7\times10^4$ G, which is almost of the order of magnitude of the SSC magnetic field.
Another weakness of the model is that it predicts in sync variation for the NIR and X-ray light curves, i.e. there should also be visible substructure in the X-ray light curve.
But an SSC scenario succeeds in explaining the simultaneity of the X-ray and NIR flares as well as the difference in their widths. Indeed, the SSC X-ray flux goes quadradically in $n_{\rm e}$ whereas the NIR flux goes linearly in $n_{\rm e}$. So, if one suppose the evolution of $n_{\rm e}(t)$ in time has a gaussian profile, then $n_{\rm e}(t)^2$ will have a width $1/\sqrt{2}$ times narrower than $n_{\rm e}(t)$, which could explain the discrepancies of the X-ray and NIR light curves.
We expose here the results for a power-law energy distribution of electrons, but other distributions such as a relativistic Maxwellian (thermal distribution of typical Lorentz factor $\theta_{\rm e} = \frac{kT_{\rm e}}{m_{\rm e}c^2}$) have been explored in past works [e.g. @liu06b]. @dodds09 applied this distribution in an SSC pattern to this flare and found similar results for the physical parameters $B$, $R$, and $n_{\rm e}$. This is because both the power-law and the relativistic maxwellian distributions have a characteristic peak energy, $\gamma_{\rm max}$ or $\theta_{\rm e}$, which determines the relative positions of the synchrotron and inverse Compton bumps.
For the first time, we find it difficult to explain a simultaneous X-ray/NIR flare with SSC emission. As a matter of fact, past observations dealt with weaker flares and poorer spectral information. @eckart06a modeled their contemporaneous X-ray/NIR flares with SSC but had no individual X-ray and NIR indices, so in particular the position of the inverse Compton peak was free, which relaxed the constraints on $B$. In contrast, @marrone08 obtained individual NIR and X-ray spectra, though they were not exactly simultaneous. The flare was fainter than the one presented here and the X-ray spectral index was consequently poorly constrained ($\beta_{\rm X}=0.0^{+1.6}_{-1.0}$). The SED could thus be accommodated with a hard X-ray power-law, which again relaxed the constraints on $B$ and yielded acceptable SSC physical parameters.
Parameters SSC EC SB
------------------------- -------------------- --------------------- --------------------
$p$ 2 2 2–3
$\gamma_{\rm min}$ 1 1 1
$\gamma_{\rm max}$ $10^2$ $10^3$ $9\times10^4$
$\gamma_{\rm br}$ — — $9\times10^2$
$B$ (G) $10^3$ 40 50
$n_{\rm e}$ (cm$^{-3}$) $2.2\times10^{12}$ $>1.8\times10^{10}$ $7.6\times10^{6}$
$R$ (cm) $1.3\times10^{10}$ $<1.6\times10^{11}$ $1.4\times10^{12}$
: Parameters of the radiative processes that match the SED of the April 2007 flare from [SgrA$^\star$]{}.
\[table\]
External Compton
----------------
Another alternative is that transiently accelerated relativistic electrons initiate the NIR flare through synchrotron and upscatter ambient low energy photons to the keV range, thus causing the X-ray flare. The most abundant source of photons around [SgrA$^\star$]{} are the ones from the submm bump at $\nu_{\rm submm}\approx10^{12}$ Hz (see Fig. \[sed\]); we will designate by external Compton (EC) the comptonization of these photons by NIR electrons. We keep the same parametrization as for the SSC case in Sect. \[ssc\]. We can estimate the maximal Lorentz factor of the particle and magnetic field as we did in the previous section, by switching respectively $\nu_{\rm max}^{\rm sync}$ and $\nu_{\rm max}^{\rm ic}$, by $\nu_{\rm submm}$ and $\nu_{\rm X}$ in Eq. \[gamma\] and \[B\]. By this means we find $\gamma_{\rm max}\approx 10^3$ and $B\approx 40$ G. Such a magnetic field is more reasonable than in the SSC picture, the cooling time is $\sim$10 min, and we no longer have to worry about SSA. Besides we know that whenever synchrotron and inverse Compton occur at the same place, the respective luminosities are linked via:
$$\frac{L^{\rm ic}}{L^{\rm sync}}=\frac{U_{\rm seed}}{U_{\rm B}}$$
where $U_{\rm B}=\frac{B^2}{8\pi}$ is the magnetic energy density and $U_{\rm seed}$ is the seed photons energy density. If $A=4\pi R_{\rm Q}^2$ denotes the surface area of the region of particles driving the quiescent submm lumninosity $L^{\rm submm}$, then $U_{\rm seed}=\frac{L^{\rm submm}}{cA}$. Thereby, we can assess the radius of the quiescent emission:
$$R_{\rm Q} \simeq 0.016 \times 10^{12} \left( \frac{ L_{\rm NIR} } { L_{\odot} } \right)^{1\over2} \left( \frac{ L_{\rm submm} } { L_{\odot} } \right)^{1\over2} \left( \frac{ L_{\rm X} } { L_{\odot} } \right)^{-{1\over2}} \left( \frac{ B } { 40~{\rm G} } \right)^{-1}~{\rm cm}\;,$$
where the solar luminosity is $L_{\odot}=3.8\times10^{33}~{\rm erg\,s}^{-1}$. The resulting radius is $R_{\rm Q} \approx 0.1~R_{\rm S}$, and this is probably the main weakness of the EC scenario, because VLBI measurements of [SgrA$^\star$]{} at 1.3 mm give an intrinsic size for the quiescent region of the order of the Schwarzschild radius [@doeleman08]. To compute the electron density $n_{\rm e}$ and the size of the flaring region $R$, a detailed treatment of the quiescence spectrum has to be taken into account, as done by @dodds09 with the quiescent RIAF model of @yuan03. Just to get a feeling of these parameters, we will consider that the flare is embedded in the submm region so that $R<R_{\rm Q}$ and consequently we find that $n_{\rm e}$ must be at least $1.8\times 10^{10}$ cm$^{-3}$ to fit the NIR synchrotron data.
To complement the study of @dodds09, we have computed the synchrotron self-Compton emission that will naturally come along in the EC scheme. Interestingly, this contribution peaks in the soft gamma-ray band, less than one order of magnitude below the *INTEGRAL* upper limits (see Fig. \[sed\], dashed line). If this scenario is real, it would tend to flatten the X-ray spectral slope towards high energies, thus making the prospects for future soft gamma-ray missions promising.
As in the SSC scheme, EC ensures simultaneity of the flare in the X-ray and NIR light curves. It could also provide an explanation for the absence of substructures in the X-ray light curve. Indeed, if one naively neglect the synchrotron losses, then the synchrotron luminosity is proportional to $B^{3\over2}$ whereas the inverse Compton luminosity is, [*a priori*]{}, independent of $B$ because it depends on $L_{\rm submm}$ and not on $L_{\rm NIR}$ as for SSC. So the NIR flare should be subject to $B$ variations contrary to the X-ray flare. A clumpy magnetic field in which the flaring region moves is maybe the key to these observed or unobserved features.
@yusef-zadeh06a proposed that their synchronous observation of a flare with the [*HST*]{} and [*XMM-Newton*]{} in 2004 resulted from EC with an acceptable size of the submm quiescence region of $\sim$$10~R_{\rm S}$. This was possible only because the spectrum of the X-ray flare was hard, $\beta_{\rm X}\approx0.5$ [@belanger05], and therefore allowed for a larger $\gamma_{\rm max}$, a lower $B$ and larger $R_{\rm Q}$. These authors also pointed out that the quiescent electrons reponsible for the submm bump were also likely to upscatter the NIR flare photons to keV energies as well, even though this may be a second order effect [@dodds09].
Synchrotron with break
----------------------
From our flare observations, it is clear that $\beta_{\rm NIR} \neq \beta_{\rm X}$ so that we cannot fit the entire SED of the flare with a single synchrotron power-law. But broad synchrotron power-laws are known to exhibit breaks of several kinds. In particular, a natural break comes from the synchrotron cooling of the electrons, which generates a difference of slopes of $|\Delta p |=1$ in the electrons distribution (and $|\Delta \beta |=0.5$ in the photons distribution) between the power-law below and above the break. This simple “synchrotron with a break” model (SB) would actually suit our measurements. The frequency at which the break is supposed to occur depends upon the modeling adopted. Here, we assume a “leaking box” model in which a constant injection of fresh particles is balanced by the synchrotron cooling of this electrons on a timescale $\tau_{\rm cool}$ and their escape on another typical timescale. For the latter variable, we will take the the dynamical timescale $\tau_{\rm dyn}=\sqrt{\frac{r^3}{2GM_{\bullet}}}\approx 5$ min, where $r=3~R_{\rm S}$ is the radius of the last stable orbit for a non spinning black hole. The condition $\tau_{\rm cool} = \tau_{\rm dyn}$ provides the frequency of the spectral break:
$$\label{break}
\nu_{\rm br}=6.37 \left(\frac{B}{1~\rm{G}}\right)^{-3}\times10^{18}~{\rm Hz}\;.$$
In Tab. \[table\] we list a sketch of physical parameters for SB that match the SED well. Again we chose $\gamma_{\rm min}=1$, and $\gamma_{\rm max}=9\times10^4$ to engender photons up to the X-ray range. $B$ was chosen to satisfy Eq. \[break\] for a break at $\sim$$10^{14}$ Hz. Finally, $n_{\rm e}$ and $R$ were adjusted to normalize the spectrum with reasonable values and in order not to violate our $\gamma$-ray constraints with the natural SSC component (see the rising SSC hump in red on bottom right of Fig. \[sed\]) coming along with synchrotron radiation. Note once more on Fig. \[sed\] the SSA below $\sim$$10^{12}$ Hz.
SB is appealing because, compared to SSC and EC, it yields less extreme values in terms of $B$, $n_{\rm e}$, and $R$. However, in the SB case, there is no obvious justification for the differents durations of the flare in X-rays and NIR, and the presence/absence of substructures in the light curves. Here, we only discuss the average spectrum of the flare, where SB certainly requires a more detailed examination of the time evolution of the phenomenon, beyond the scope of this paper.
Conclusions
===========
This paper complements a series of articles about the April 2007 synchronous observations of the Galactic Center from radio to $\gamma$-rays [@porquet08; @dodds09; @yusef-zadeh09]. Here, we have recapped the results on the brightest flare ever detected simultaneously at NIR and X-ray frequencies. We have also reported for the first time $\gamma$-ray constraints on such an event, which, added to our MIR/NIR/X-ray spectral measurements, constitute the broadest simulaneous spectrum of a flare ever achieved. The essential observational conclusions may be summarized as follows:
- the peaks of the X-ray and NIR emissions are coincident within 3 min;
- the width of the NIR flare light curve is broader than the X-ray one by a factor $\sim$2;
- the NIR light curve is substructured on a timescale of $\sim$20 min while the X-ray light curve is rather smooth;
- there is no detectable MIR counterpart;
- the soft $\gamma$-ray source [IGRJ17456–2901]{} is non variable.
The high quality of the spectral information we gathered allowed for a discussion of the several classical radiative processes models employed to explain the flares: SSC, EC, and SB. Yet, none of these mechanisms is entirely satisfactory to meet our observations. The theoretical inquiries to come will have to take into account the time evolution of the phenomenon and the aging of the radiating particles to better connect the light curves and spectra. From an observational stand point, it will be useful to repeat such NIR/X-ray measurements in a near future to get two respective individual and fully contemporaneous spectra, which has never been accomplished thus far. As we have seen, one key probe of what powers the flares, is a better determination of the X-ray spectral slope. In a more distant future, thanks to a broad X-ray sensitivity over the 1–80 keV band and a high angular resolution above 10 keV, [*Simbol–X*]{} should address this issue and resolve the GC region in soft $\gamma$-rays.
Acknowledgements {#acknowledgements .unnumbered}
================
GT acknowledges M. Falanga, D. Götz, and J. Chenevez for help with the [*INTEGRAL*]{} data analysis, Y. Clénet, B. Draine, and M. Morris for useful discussions about NIR/MIR measurements, and CEA Saclay for financial support to attend the 37$^{\rm th}$ COSPAR meeting in Montreal.
Part of this work has been funded by the french Agence Nationale pour la Recherche through grant ANR-06-JCJC-0047. At Arizona, this work was also supported by NASA grants NNX08AX33G and NNX08AX34G.
[*INTEGRAL*]{} is an ESA project with instruments and science data center funded by ESA member states (especially the PI countries: Denmark, France, Germany, Italy, Switzerland, and Spain), the Czech Republic, and Poland, and with the participation of Russia and the US.
The [*XMM-Newton*]{} project is an ESA Science Mission with instruments and contributions directly funded by ESA Member States and the USA (NASA).
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[^1]: The converse is not true, some NIR flares have no X-ray counterpart [@hornstein07].
[^2]: For further discussion of the past variability of [SgrA$^\star$]{} in cm and mm bands, see for example @zhao01 [@herrnstein04] and @tsuboi99 [@zhao03], respectively.
[^3]: Compared to IBIS, JEM-X field of view is narrower, so given the rectangular dithering pattern used ($5 \times 5$ pointings = 24 off-source + 1 on-source), the GC was invisible to JEM-X most of the time, which explains the discrepancy in the exposures.
[^4]: See JEM-X user manual:\
http://isdc.unige.ch/Soft/download/osa/osa\_doc/prod/osa\_um\_ibis-7.0.pdf
[^5]: Notice that [IGRJ17456–2901]{} has not been significantly detected with JEM-X yet.
[^6]: Such an index could be the natural consequence of a Fermi II acceleration process for instance.
[^7]: $\sigma_{\rm T} = \frac{8}{3}\pi r_{\rm e}^2$, where $r_{\rm e}=e^2/(m_{\rm e} c^2)= 2.82 \times 10^{-13}$ cm is the classical radius of the electron.
[^8]: This is an upper limit obtained for a pitch angle of $\pi/2$.
|
---
abstract: 'A recent trend in compressed sensing is to consider non-convex optimization techniques for sparse recovery. A general class of such optimizations, called $F$-minimization, has become of particular interest, since its exact reconstruction condition (ERC) in the noiseless setting can be precisely characterized by null space property (NSP). However, little work has been done concerning its robust reconstruction condition (RRC) in the noisy setting. In this paper we look at the null space of the measurement matrix as a point on the Grassmann manifold, and then study the relation of the ERC and RRC sets, denoted as $\Omega_J$ and $\Omega_J^r$. It is shown that $\Omega_J^r$ is the interior of $\Omega_J$. From this characterization, a previous result of the equivalence of ERC and RRC for $\ell_p$-minimization follows easily as a special case. Moreover, when $F$ is non-decreasing, it is shown that $\overline{\Omega}_J\setminus\operatorname*{int}(\Omega_J)$ is a set of measure zero and of the first category. As a consequence, the probabilities of ERC and RRC are the same if the measurement matrix $\mathbf{A}$ is randomly generated according to a continuous distribution. Quantitatively, if the null space $\mathcal{N}(\bf A)$ lies in the “$d$-interior” of $\Omega_J$, then RRC will be satisfied with the robustness constant $C=\frac{2+2d}{d\sigma_{\min}(\mathbf{A}^{\top})}$; and conversely if RRC holds with $C=\frac{2-4d}{d\sigma_{\max}(\mathbf{A}^{\top})}$, then $\mathcal{N}(\bf A)$ must lie in $d\mbox{-}\operatorname*{int}(\Omega_J)$. Based on this result, Gordon’s escape through the mesh theorem is applied to study the tradeoff between measurement rate and robustness in the asymptotic region. Finally, we present several rules for comparing the performances of different cost functions, which potentially provide guiding principles for the design of the $F$ function.'
author:
- 'Jingbo Liu, Jian Jin, and Yuantao Gu, [^1]'
bibliography:
- 'refs1.bib'
title: 'Robustness of Sparse Recovery via $F$-minimization: A Topological Viewpoint'
---
[Shell : Robustness of Sparse Recovery via $F$-minimization: A Topological Viewpoint]{}
Reconstruction algorithms, compressed sensing, minimization methods, robustness, null space
Introduction
============
Sensing is a method of recovering a sparse signal from a set of under-determined linear measurements. Ideally, the sparsest solution is given by the $\ell_0$-norm minimization method: $$\min_{\mathbf{x}\in \mathbb{R}^n}~\|\mathbf{x}\|_0~\textrm{s.t.}~\bf y=Ax,$$ where $\mathbf{A}$ is an $m\times n$ measurement matrix, $\mathbf{y}\in \mathbb{R}^m$ is the linear measurements, and we assume that $m<n$. It is well known that exactly solving the $\ell_0$-minimization is computational intractable since it is a hard combinatorial problem [@DC]. Therefore, many algorithms have been proposed to reduce the computational complexity. Roughly speaking, these algorithms fall into two categories: 1) minimization techniques, where the sparse solution is retrieved by minimizing an appropriate cost function [@Donoho; @zap], and 2) greedy pursuits, a representative of which is the orthogonal matching pursuit (OMP) [@OMP].
In general, the greedy algorithms often incur less computational complexity, but the minimization techniques are more advantageous in terms of accuracy. The most basic minimization technique is the $\ell_1$-minimization, or Basis Pursuit (BP) [@DC; @Donoho; @donoho2]: $$\label{min1}
\min_{\mathbf{x}\in \mathbb{R}^n}\|\mathbf{x}\|_1\quad \textrm{s.t.}~\mathbf{y}=\mathbf{A}\mathbf{x},$$ which is a simple convex optimization and can be recast as a linear program. Recently there is trend to consider minimizing non-convex cost functions. Examples include:
$\bullet$ $\ell_p$ cost function. The $\ell_p$-minimization ($0<p<1$) [@chartrand; @Foucart1; @Gribonval; @anotherlq] considers an optimization problem similar to (\[min1\]) but the cost function is replaced with $\|\mathbf{x}\|^p_p$.
$\bullet$ Approximate $\ell_0$ cost functions, such as those in the zero point attracting projection (ZAP), [@zap], and smooth $\ell_0$ algorithm [@sl0]. Also for statisticians, smoothly clipped absolute deviation (SCAD) penalty [@Fan] and the minimax concave penalty (MCP) [@MCP] are familiar concave penalties used for variable selection.
Although the non-convex nature of these cost functions makes it difficult to exactly solve the corresponding optimization problems, various practical algorithms can be adapted to these non-convex problems, including the iteratively re-weighted least squares minimization (IRLS) [@chartrand1; @irls], iterative thresholding algorithm (IT) [@it], which are based on fixed point iteration; and the zero point attracting projection algorithm (ZAP) [@zap; @wang; @laming], which is based on Newton’s method for solving nonlinear optimization. In general the non-convex algorithms have empirically outperformed BP in the various respects, because nonlinear cost functions can better promote sparsity than the $\ell_1$ cost function. Thus, a detailed study of the reconstruction properties of these sparse recovery methods remain important.
Most of these non-convex optimizations can be subsumed in a general category called “$F$-minimization” [@lqnsp], in which the cost function satisfies some desirable properties, such as subadditivity. The precise definition of the class of cost functions of our interest will be given in the next section.
Two concepts arise naturally in the compressed sensing problem: The *exact recovery condition* (ERC) in the noiseless setting and the *robust recovery condition* (RRC) in the noisy setting. In the literature, ERC typically requires that all sparse signals can be exactly recovered. In addition to this, RRC requires that if the measurement is noisy, the reconstruction error is bounded by the norm of the noise vector multiplied by a constant factor.
While the rigorous definitions of ERC and RRC are deferred to Section \[sec2\], we remark here in passing that RRC trivially implies ERC, because ERC can be seen as a special case of RRC where the measurement is free of noise. Conversely, it is not obvious whether ERC also implies RRC, or RRC is *strictly* stronger than ERC. Early work in compressed sensing have provided sufficient conditions for ERC and RRC of the $\ell_1$-minimization, based on the so-called restricted isometry property (RIP) [@DC], and those sufficient conditions appear to be identical. However, analysis based on RIP generally fails to provide exact (necessary and sufficient) condition for ERC and RRC. Another line of research has considered the null space property (NSP), which gives a both necessary and sufficient condition for ERC of the $\ell_p$-minimization. In addition, [@Foucart] provided a sufficient condition, called NSP’, for RRC of $\ell_p$-minimization. Later Aldroubi et al proved in [@lqnsp] that NSP and NSP’ are in fact equivalent. Hence, we have that ERC and RRC are actually the same condition for $\ell_p$-minimization.
In contrast to the special case of $\ell_p$-minimization, the robust recovery condition for the more general case of $F$-minimization has been recognized as “not easy to establish” [@lqnsp], merely based on the idea of NSP. The fundamental issue of robustness in $F$-minimization has remained relatively unexplored.
The purpose of this paper is to give an exact characterization of the relationship between ERC and RRC in the general $F$-minimization problem. We first show that ERC and RRC depends only on the configuration of the null space of the measurement matrix (the entire entries of the matrix is of course sufficient, but not necessary, information). Moreover, since the null spaces are linear subspaces of the Euclidean space, they can be viewed as points on a Grassmann manifold, which has a natural topological structure, hence concepts such as open sets and interior are well defined for collections/sets of the null spaces. We denote by $\Omega_J$ and $\Omega_J^{r}$ the sets that consist of the null spaces satisfying ERC and RRC for the $F$-minimization, respectively. We show that $\Omega_J^{r}$ is exactly the interior of $\Omega_J$ (Theorem \[th2\]). Hence we can give an alternative proof of the equivalence of ERC and RRC in $\ell_p$-minimization, by simply showing that $\Omega_J$ is open in this special case. We would like to emphasize that this analytical framework also gives rise to new ideas and results, including:
$\bullet$ Equivalence of ERC and RRC in probability. Under some mild assumptions we show that $\Omega_J$ and $\Omega_J^{r}$ are “almost equal” in the sense that the difference set is of measure zero and of the first category. Building on this, we show that ERC and RRC hold true with the same probability if the measurement matrix is randomly generated according to a continuous distribution.
$\bullet$ Comparison between different sparseness measures. It is interesting and valuable to know how the performances between different sparseness measures compare. Gribonval et al [@Gribonval Lemma 7] provided a condition when one spareness measure is better than another in the sense of ERC. Combining this with our result, we show that this condition also provides a comparison in terms of RRC. Moreover, with the concept of measure zero set on the Grassmannian, we are able to provide addition comparison rules which guarantee one sparse measure is better than the other in terms of probability of ERC/RRC.
$\bullet$ Tradeoff between measurement rate and robustness. We show that a matrix whose null space falls in the “$d$-interior” of $\Omega_J$ satisfies RRC with the robustness constant $C=\frac{2(1+d)}{d\sigma_{\min}(\mathbf{A}^{\top})}$. This can be seen as a quantitative version of the aforemention interior characterization of the RRC set, and Gordon’s escape through the mesh theorem can be used to upper bound the measure of $d\mbox{-}\operatorname*{int}(\Omega_J)$. To illustrate this method, we computed the tradeoff between measurement rate and the robustness in the special case in the asymptotic linear growth region. The tradeoff is explicitly expressed in terms of the scaling of the matrix dimensions and the sparsity level, in contrast to the previous results expressed in terms of RIP constants in the literature.
The organization of the paper is as follows. In Section \[sec2\] we present the mathematical formulation of the problem and a brief introduction to null space property and the Grassmann manifold. Section \[secrelation\] studies the relationship between ERC and RRC: Part A gives an exact characterization of RRC set as the interior of ERC set on the Grassmannian; in Part B we show than the ERC and RRC sets differ by a set of measure zero and of the first category; Part C provides quantitative results of the robustness of the measurement matrix whose null space lies in $d\mbox{-}\operatorname*{int}(\Omega_J)$, and illustrates a method of estimating the probability of $d\mbox{-}\operatorname*{int}(\Omega_J)$. In Section \[secrules\] we provide some rules for comparing the performance of different sparse measures. Section V compares our approach and definitions with similar ones in the literature. Finally in Section \[conclusion\] we conclude by reviewing the results and pointing out possible directions for future work.
Problem Setup and Key Definitions {#sec2}
=================================
This section provides the mathematical formulation of the problem and the definitions of some key concepts. We shall use lower case bold letters for vectors, and upper case bold letters for matrices. Notation $\mathbb{M}(m,n)$ denotes the set of $m\times n$ real matrices. Throughout the paper we suppose the observation matrix is $m\times n$, and set $l:=n-m$, unless otherwise indicated. $\|\mathbf{x}\|_0$ refers to the $\ell_0$ norm[^2] of $\mathbf{x}$, i.e., the number of non-zero elements in the vector, and $\|\mathbf{x}\|_p:=(\sum_k|x(k)|^p)^{1/p}$ denotes the $\ell_p$ norm of $\mathbf{x}$.
Basic Model
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Let $\mathbf{\bar{x}}\in \mathbb{R}^n$, $\mathbf{A}\in \mathbb{M}(m,n)$, $\mathbf{v}\in \mathbb{R}^m$ be the sparse signal, the measurement matrix, and the additive noise, respectively. Let $T:=\operatorname*{supp}(\mathbf{\bar{x}})$ be the support of $\mathbf{\bar{x}}$. Vector $\mathbf{\bar{x}}$ is called $k$-sparse if $|T|\le k$. The linear measurement $\mathbf{y}$ is given by $$\mathbf{y}=\mathbf{A}\mathbf{\bar{x}}+\mathbf{v}.$$
We consider the problem of recovering $\mathbf{\bar{x}}$ through an optimization. Supposing $F:[0,+\infty)\to[0,+\infty)$ is a given function, we define the cost function $$\label{cost}
J({\mathbf{x}}):=\sum_{k=1}^n F(|x(k)|).$$ With a slight abuse of the notation, we shall also use the notations: $$\begin{aligned}
J(\mathbf{x}_T):&=\sum_{k\in T} F(|x(k)|),\nonumber\\
J({\mathbf{x}}_{T^c}):&=\sum_{k\in T^c} F(|x(k)|),\nonumber\end{aligned}$$ where $\mathbf{x}_T\in\mathbb{R}^{|T|},~
\mathbf{x}_{T^c}\in\mathbb{R}^{n-|T|}$ denote the restriction of $\mathbf{x}$ on the set $T,~T^c$, respectively. Clearly (\[cost\]) is a very general model: For example, if one chooses $F(x)=1_{x>0}$ then $J(\mathbf{x})=\|\mathbf{x}\|_0$; if $F(x)=x^p$ then $J(\mathbf{x})=\|\mathbf{x}\|^p_p$.
The conditions ERC and RRC are commonly formulated as follows, see for example [@lqnsp].
In the noiseless case, the sparse signal is retrieved via the following optimization: $$\label{minimization}
\min_{\mathbf{x}\in\mathbb{R}^n}J({\mathbf{x}})\quad \textrm{s.t.}~\mathbf{A}\mathbf{x}=\mathbf{y}.$$ We say $\mathbf{A},~J$ satisfy the *exact recovery condition* (ERC) if for any measurement $\mathbf{y}=\mathbf{A}\mathbf{\bar{x}}$, where $\mathbf{\bar{x}}$ is $k$-sparse, the vector $\mathbf{\bar{x}}$ is also the unique solution to (\[minimization\]).
In the noisy measurement ($\mathbf{v}\neq \mathbf{0}$) case, the sparse signal is retrieved via the following optimization: $$\label{minimization2}
\min_{\mathbf{x}\in\mathbb{R}^n}J({\mathbf{x}})\quad \textrm{s.t.}~\|\mathbf{A}\mathbf{x}-\mathbf{y}\|<\epsilon,$$ where $\epsilon\in \mathbb{R}^+$ is a constant chosen to tolerate the noise. We say that the *robust recovery condition* (RRC) is satisfied if the following holds. For any $k$-sparse signal $\mathbf{\bar{x}}$, noise $\mathbf{v}$ and $\epsilon$ satisfying $\|\mathbf{v}\|\le \epsilon$, and feasible solution $\mathbf{\hat{x}}$ satisfying $J(\mathbf{\hat{x}})\le J(\mathbf{\bar{x}})$, we have $$\label{defc}
\|\mathbf{\bar{x}}-\mathbf{\hat{x}}\|<C\epsilon,$$ where $C$ is a constant.
We end this part by remarking that ERC, RRC, and the constant $C$ in the definition of RRC all depend only on $\mathbf{A},k,J$.
Null Space Property
-------------------
The null space property [@gribonval1; @nsp; @Gribonval] is useful for the analysis of a special class of cost functions, which we introduce as follows:
\[def1\] Function $$F:[0,+\infty)\to[0,+\infty)$$ is called a *sparseness measure* if the following two conditions are satisfied:
$\bullet$ $F(|\cdot|)$ is subadditive on $\mathbb{R}$, i.e. $F(|x+y|)\le F(|x|)+F(|y|)$ for all $x,y,z\in \mathbb{R}$;
$\bullet$ $F(x) = 0$ if and only if $x=0$.\
We denote by $\mathcal{M}$ the set of all sparseness measures.[^3]
In this paper we assume that the function $F$ is a sparseness measure as in Definition \[def1\]. This is a rather loose assumption, so that the key optimization problems in many of the sparse recovery algorithms can be subsumed in our framework, including $\ell_p$-minimization and ZAP algorithm. The definition is also quite natural, since it can be checked that $F$ is a sparseness measure if and only if its corresponding cost function $J$ induces a metric on $\mathbb{R}^n$ via $d(\mathbf{x},\mathbf{y})
:=J(\mathbf{x}-\mathbf{y})$.
When $F\in\mathcal{M}$, the *null space property* (NSP) turns out to be equivalent with ERC:
\[nspcond\] If $F\in\mathcal{M}$, then a necessary and sufficient condition for ERC is $$J(\mathbf{z}_T)<J(\mathbf{z}_{T^c}),\quad\forall \mathbf{z}\in \mathcal{N}(\mathbf{A})\setminus\{\mathbf{0}\},~T:|T|\le k.$$ where $\mathcal{N}(\mathbf{A})$ denotes the null space of $\mathbf{A}$.
It’s useful to define the *null space constant [@Gribonval]*, especially when one wants to study $\ell_p$-minimization or to compare it with $F$-minimization:
\[de\_nsc\] Suppose $F\in \mathcal{M},~q\in(0,1]$. Define the null space constant is defined as: $$\label{nsc}
\theta_{J}:=\sup_{\mathbf{z}\in\mathcal{N}(\mathbf{A})\setminus\{\mathbf{0}\}}
\max_{|T|\le k}\frac{J(\mathbf{z}_{T})}{J(\mathbf{z}_{T^c})}.$$ In the same spirit, we denote by $\theta_{\ell_p}$ the null space constant associated with $\ell_p$ cost function.
The null space constant is closely associated with NSP, and hence characterizes the performance of $F$-minimization. We have the following result, which is a direct consequence of Definition \[de\_nsc\] and Lemma \[nspcond\].
\[nspcond2\] \
1) $\theta_J\le 1$ is a necessary condition for ERC;\
2) $\theta_J <1$ is a sufficient condition for ERC.
In the case of $\ell_p$-minimization, one can obtain the following characterization (c.f.[@Foucart]), which is more exact than the case of $F$-minimization as described in Lemma \[nspcond2\]:
\[le1\] For $\ell_p$ cost functions, $\theta_{\ell_p}< 1$ is a both necessary and sufficient condition for ERC.
Preliminaries of the Grassmann Manifold {#pc}
---------------------------------------
In this part we briefly review some relevant properties of the Grassmann manifold. More detailed treatment of the subject can be found in many standard texts, such as [@boothby; @milnor]. The main thrust for considering this object is that, the property of exact recovery of a particular measurement matrix is completely determined by its null space, from Lemma \[nspcond\]. Of course, $\mathcal{N}(\mathbf{A})$ is an $l:=n-m$ dimensional linear subspace of $\mathbb{R}^n$ when $\mathbf{A}$ is of full rank.
Geometrically, the Grassmann manifold $G_l(\mathbb{R}^n)$ can be conceived as the collection of all the $l$ dimensional subspaces ($l$-planes) of $\mathbb{R}^n$. One can introduce a topology on $G_l(\mathbb{R}^n)$ by defining a metric on it: for arbitrary $\nu,\nu'\in G_l(\mathbb{R}^n)$, the distance between $\nu,\nu'$ can be defined as [@mattila]: $$\label{def_dis}
\operatorname*{dist}(\nu,\nu'):=\|\mathbf{P}_{\nu}-\mathbf{P}_{\nu'}\|,$$ where $\mathbf{P}_{\nu}$ (resp. $\mathbf{P}_{\nu'}$) is the projection matrix onto $\nu$ (resp. $\nu'$), and $\|\cdot\|$ denotes the spectral norm. The Grassmann manifold is then a compact metric space.
We shall next define the coordinates on $G_l(\mathbb{R}^n)$ to introduce its differential manifold structure. Let $F(n,l)$ be the set of all non-degenerate (invertible) $n\times l$ matrices, and let $\sim$ be the following equivalence relation: If $\mathbf{X},\mathbf{Y}\in F(n,l)$, then $\mathbf{X}\sim \mathbf{Y}$ if there is an $l\times l$ invertible matrix $\mathbf{V}$ such that $\mathbf{Y = XV}$. Hence the Grassmann manifold can be defined as a quotient space $G_l(\mathbb{R}^n):=F(n,l)/\sim$, for which we denote by $\pi: F(n,l)\to G_l(\mathbb{R}^n)$ the associated natural projection. For any arbitrary collection of indices $1\le i_1<i_2<\dots<i_l\le n$, let $1\le \bar{i}_1<\bar{i}_2<\dots<\bar{i}_{n-l}\le n$ be the remaining indices. Given an index set $I=\{i_1,i_2,\dots,i_l\}$, we denote by $\mathbf{X}_I$ the $l\times l$ sub-matrix formed by the rows of $\mathbf{X}$ indexed by $I$. Define $$V_{I}:=\{\mathbf{X}\in F(n,l)~|~\det \mathbf{X}_{I}\neq 0\},\quad
U_{I}:=\pi(U_{I}).$$ Then $\{U_{I}\}$ constitutes an open covering of $G_l(\mathbb{R}^n)$. For $\mathbf{Y}\in\pi^{-1}(\nu)$, where $\nu\in U_{I}$, the matrix $\mathbf{X}=\mathbf{Y}\mathbf{Y}_I^{-1}$ is an invariant of $\nu$, meaning that for any other $\tilde{\mathbf{Y}}\in \pi^{-1}(\nu)$, $\tilde{\mathbf{Y}}(\tilde{\mathbf{Y}}_I)^{-1}=\mathbf{X}$. Since $\mathbf{X}_I$ is the $l\times l$ identity matrix, $\mathbf{X}$ is determined by $\mathbf{X}_{I^c}$. Define $\phi_{I}: U_{I}\to \mathbb{M}(n-l,l), v\mapsto\mathbf{X}_{I^c}$. We call each $(U_{I},\phi_{I})$ a chart. Then $\{(U_{I},\phi_{I})~|~1\le i_1<\dots<i_l\le n\}$ forms an atlas of $G_l(\mathbb{R}^n)$, meaning that $U_{I}$ covers $G_l(\mathbb{R}^n)$ and any two charts in this collection are $C^{\infty}$ compatible.
Concepts such as open sets and interior are well-defined once a topology on $G_{l}(\mathbb{R}^n)$ has been unambiguously chosen. One might notice that there are possibly two topologies defined on $G_{l}(\mathbb{R}^n)$: the metric topology arising from the metric defined in (\[def\_dis\]), and the manifold topology (which is connected to the standard topology on $\mathbb{R}^{ml}$ by all the homeomorphisms $\{\phi_I\}$). Unsurprisingly these two topologies agree, since standard calculations would show that the metric on $U_I$ induced from the Euclidean metric on $\phi_I(U_I)$ is topologically equivalent to the metric defined in (\[def\_dis\]). Further, since $G_{l}(\mathbb{R}^n)$ is a $C^\infty$ (therefore differentiable) manifold, the concept of measure zero set can be defined as follows:
\[def2\][@milnor Definition 1.16] A subset $A$ of a differentiable manifold has *measure zero* if $\phi(A\cap U)$ has Lebesgue measure zero for every chart $(U,\phi)$.
The differentiability of the manifold ensures that the definition of the measure zero set is “consistent” for the various choices of $\phi$. In particular, to check that a set $A$ is of measure zero, one only needs to pick a collection of homeomorphisms $\{\phi_I\}_{I\in S}$ whose domains cover $U$, and check that $\phi_I(A\cap U_I)$ has measure zero for each $I\in S$.
There is a unique probability measure that’s invariant with respect to the orthogonal group’s action on $G_l(\mathbb{R}^n)$, that is, the action on the quotient space $G_l(\mathbb{R}^n)$ induced from the action of left multiplication of $n\times n$ orthogonal matrix on $F(n,l)$. The requirement that a set $A$ has zero Haar measure agrees with Definition \[def2\] [@boothby]. The Haar measure is of practical importance, since it coincides with the distribution of the null space of $\mathbf{A}$ when it is a Gaussian random matrix. We use $\mu$ to denote the normalized Haar measure on $G_l(\mathbb{R}^n)$, which can also be understood as a probability in the case of Gaussian random matrices.
The Relationship between ERC and RRC {#secrelation}
====================================
Equivalence Lost: $\Omega_J^r=\operatorname*{int}(\Omega_J)$ {#pa}
------------------------------------------------------------
We have mentioned earlier that NSP is a necessary and sufficient condition for ERC. If $\mathbf{A}\in\mathbb{M}(m,n)$ is in a general position (i.e., the rows of $\mathbf{A}\in\mathbb{M}(m,n)$ are linearly independent), then $\mathbf{A}$ is of full rank, and $\mathcal{N}(\mathbf{A})$ is a $l$-dimensional subspace in $\mathbb{R}^n$ (recall that $l=n-m$). Therefore almost every measurement matrix (except for the set of $\mathbf{A}$’s not in a general position, which is of Lebesgue measure zero) corresponds to an element in $G_{l}(\mathbb{R}^n)$; and this element is sufficient to determine whether NSC, and therefore ERC, is satisfied. By Lemma \[nspcond\], the set of null spaces such that ERC is satisfied is as follows: $$\begin{aligned}
\label{exactdef}
\Omega_{J}(n,k,l):=&\{\nu\in G_{l}(\mathbb{R}^n)~|~ J(\mathbf{z}_T)<J(\mathbf{z}_{T^c}),\forall \mathbf{z}\in \nu\setminus \{\mathbf{0}\},T:|T|\le k\}.\end{aligned}$$ For simplicity we shall omit the arguments $n,k,l$ throughout this paper when there is no confusion. If two cost functions induced from the sparseness measures $F,G\in \mathcal{M}$ satisfy the following condition $$\label{e_comp}
\Omega_{J_G}\subseteq\Omega_{J_F},$$ then ERC for $G$-minimization implies ERC for $F$-minimization, i.e., $F$ is a better sparseness measure than $G$ in the sense of ERC. In the light of this we can describe and compare the performances of different sparseness measures in terms of ERC by a simple set inclusion relation like (\[e\_comp\]).
In Lemma \[nspcond\], the necessary and sufficient condition for exact recovery is fully characterized by the structure of the null space. Inspired by this fact we now provide a necessary and sufficient condition for robust recovery:
\[suf2\] Consider the minimization problem in (\[minimization2\]). Let $\sigma_{\min},\sigma_{\max}$ be the least and largest singular values of $\mathbf{A}^\top$, respectively. Then RRC holds with constant $C=2(1+d)/(d\sigma_{\min})$ if there exists a $d>0$, such that for each $\mathbf{z}\in \mathcal{N}(\mathbf{A})\setminus \{\mathbf{0}\}$, $\mathbf{n}\in\mathbb{R}^n$, $T\subseteq\{1,...,n\}$ satisfying $\|\mathbf{n}\|<d\|\mathbf{z}\|$, and $|T|\le k$, we have the following: $$\label{eq16}
J(\mathbf{z}_T+\mathbf{n}_T)<J(\mathbf{z}_{T^c}+\mathbf{n}_{T^c}).$$ Conversely, if RRC holds with $C=2(1-d)/(d\sigma_{\max})$ for some $0<d<1$, then for any $\mathbf{z}\in \mathcal{N}(\mathbf{A})\setminus \{\mathbf{0}\}$, $\mathbf{n}\in\mathbb{R}^n$, $T\subseteq\{1,...,n\}$ satisfying $\|\mathbf{n}\|<d\|\mathbf{z}\|$, and $|T|\le k$, the relation (\[eq16\]) is true.
See Appendix \[p\_suf2\].
An immediate corollary is the following, the proof of which is omitted:
\[corsuf2\] Consider the minimization problem in (\[minimization2\]). The RRC holds *if and only if* there exists a $d>0$, such that for each $\mathbf{z}\in \mathcal{N}(\mathbf{A})\setminus \{\mathbf{0}\}$, $\mathbf{n}\in\mathbb{R}^n$, $T\subseteq\{1,...,n\}$ satisfying $\|\mathbf{n}\|<d\|\mathbf{z}\|$, and $|T|\le k$, we have the following: $$\label{eq166}
J(\mathbf{z}_T+\mathbf{n}_T)<J(\mathbf{z}_{T^c}+\mathbf{n}_{T^c}).$$
RRC easily implies ERC, as can be seen in their definitions (Letting $\mathbf{v}=\mathbf{0}$ in the definition of RRC would result in the definition of the ERC), as well as in Corollary \[corsuf2\] (Letting $\mathbf{n}=\mathbf{0}$).
From Corollary \[corsuf2\] it is clear that the property of robust recovery of a particular matrix is also completely determined by its null space. Moreover, it implies that the subset of $G_{l}(\mathbb{R}^n)$ that guarantees RRC is the following: $$\begin{aligned}
\label{robdef}
\Omega^r_J:=&\{\nu\in G_{l}(\mathbb{R}^n)~|~ \exists d>0,\textrm{s.t.}~J(\mathbf{z}_T+\mathbf{n}_T)<J(\mathbf{z}_{T^c}+\mathbf{n}_{T^c}),\forall \mathbf{z}\in \nu\setminus \{\mathbf{0}\}, \mathbf{n}:\|\mathbf{n}\|<d\|\mathbf{z}\|,T:|T|\le k\}.\end{aligned}$$
It is not immediately clear from Lemma \[nspcond\] and Corollary \[corsuf2\] the connection between ERC and RRC. However there is a nice relation between these two conditions once taking a perspective from the point set topology:
\[th2\] With the standard topology on $G_{l}(\mathbb{R}^n)$, the following relation holds. $$\Omega^r_J=\operatorname*{int}(\Omega_J).$$
See Appendix \[p\_th2\].
Two questions then arise: are the conditions ERC and RRC equivalent for generic cost functions? If not, how much do they differ from each other? We shall first address the former question in the remainder of this part, while the second question will be discussed in Part B. In the special case of $\ell_p$-minimization, these two conditions are indeed equivalent [@lqnsp], as discussed in the introductory section. In view of Theorem \[th2\], we can show this result by simply proving that the $\Omega$ is an open set in the case of $\ell_p$-minimization.
\[th5\] If $0<p\le 1$, then $\Omega_{\ell_p}$ is open, hence $\Omega_{\ell_p}^r=\Omega_{\ell_p}$.
The equivalence result of $\Omega_{\ell_p}^r=\Omega_{\ell_p}$ in the above is essentially ‘non-topological’, since it does not involve the concept of open sets on the Grassmann manifold. A comparison of different proof methods can be found in Section \[comp\], Part A.
Next we shall show an example in which RRC is strictly stronger than ERC, i.e., $\Omega^{r}_J\varsubsetneqq\Omega_J$.
\[ex1\] The function $$\label{fex1}
F(t):=t+1-\textrm{e}^{-t}$$ defined on $[0,+\infty)$ is a spareness measure. Suppose that $x,y>0,~z=x+y$, $k=1$, and that the null space of the measurement matrix is the following one dimensional linear sub-space of $\mathbb{R}^3$ $$\mathcal{N}:=\mathbb{R}(x,y,z)^\top,$$ Conclusion: in this setting ERC is satisfied, but not RRC.
The cost function in (\[fex1\]) has two salient properties: strict subadditivity (i.e., $F(x)+F(y)>F(x+y)$ for $x,y>0$) and existence of a derivative at the origin. In appendix \[appex\] we shall prove the assertions in the Counter-example using these properties.
Equivalence Regained: $\Omega_J\setminus\Omega_J^r$ is zero measure and meagre
------------------------------------------------------------------------------
While strict equivalence of ERC and RRC is lost when passing from $\ell_p$ cost functions to generic sparseness measures, as demonstrated in Counter-example \[ex1\], we will show in this part that the difference is only a set of measure zero on the Grassmann manifold, at least for non-decreasing sparseness measures. First we take a closer look at Counter-example \[ex1\]. Using the subadditivity property and the Taylor expansion of $F$ at the origin, one can explicitly write out: $$\label{om1}
\Omega_J=\left\{[x_1,x_2,x_3]:2\max_{i=1,2,3}|x_i|\le\sum_{i=1,2,3}|x_i|\right\},$$ and $$\label{om2}
\Omega_J^{r}=\left\{[x_1,x_2,x_3]:2\max_{i=1,2,3}|x_i|<\sum_{i=1,2,3}|x_i|\right\}.$$ We recall that $\mu$ denotes the Haar measure on $G_{l}(\mathbb{R}^n)$. From (\[om1\]) and (\[om2\]) it is intuitively clear in this simple case that $\mu(\Omega_J)=\mu(\Omega_J^{r})$, i.e. the set of null spaces satisfying ERC and the set of null spaces satisfying RRC differ at most by a set of measure zero. Recall that the Haar measure agrees with the probability measure in the case of i.i.d. Gaussian random entries, as described in Section \[sec2\], Part C. This means that if $\mathbf{A}$ is a Gaussian random matrix, then the probability of ERC and RRC are the same, even though the former is implied by the latter. The general case tends to be much more complicated. Indeed, there exists Euclidean set $A$ such that $\mu(\operatorname*{int}(A))<\mu(A)$, so the relation $\Omega^r_J=\operatorname*{int}(\Omega_J)$ alone by no means implies that $\mu(\Omega^r_J)=\mu(\Omega_J)$. In fact merely $F\in\mathcal{M}$ does not guarantee $\mu(\Omega_J^r)=\mu(\operatorname*{int}(\Omega_J^r))$, as we shall see.
However, the set $\Omega_J\setminus \Omega_J^r$ is still guaranteed to be “small” if we assume in addition that $F$ is non-decreasing. The smallness may be described in two distinct senses, namely measure and Baire category. A measure zero set is of course negligible since its corresponding probability is zero. On the other hand, Baire category has nothing to do with the probability; but it is a purely topological concept, so it’s worth pointing out the smallness in this sense given the topic of this paper. A set is said to be of *first category* (or *meagre*) if it’s a countable union of nowhere dense sets, which are defined as sets whose closures have empty interiors.
\[probeq\] Suppose $F\in\mathcal{M}$ is a non-decreasing function, then $\overline{\Omega}_J\setminus\operatorname*{int}(\Omega_J)$ is zero measure and of the first category.[^4]
The technical proof of this general result will be given in Appendix \[p\_probeq\]. In the following we discuss the intuition behind the monotonicity assumption in the theorem through a specific low dimensional example, which does not require any background in measure theory.
We first make some comments on the notations. In the remaining of this section we always consider the set $\Omega_J$ associated with a particular $J$, so we shall just write the set as $\Omega$ for brevity. Define the set $\Omega_T$ for each $T:|T|\le k$ as follows: $$\begin{aligned}
\label{ot}
\Omega_T:=\{\nu\in G_{l}(\mathbb{R}^n)~|~J(\mathbf{z}_T)<J(\mathbf{z}_{T^c}),\forall \mathbf{z}\in \nu\setminus \{\mathbf{0}\}\},\end{aligned}$$ hence $\Omega=\bigcap_{T:|T|=k}\Omega_T$. Note that this definition is not to be confused with the definitions of $\Omega_J$ or $\Omega_{\ell_p}$.
Next we shall make some preparations for the proof. Notice that $$\begin{aligned}
\overline{\Omega}\setminus\operatorname*{int}(\Omega)
&=\overline{\bigcap_{T:|T|=k}\Omega_T} \setminus \operatorname*{int}(\bigcap_{T:|T|=k}\Omega_T)\nonumber\\
&\subseteq \bigcap_{T:|T|=k}\overline{\Omega}_T\setminus\bigcap_{T:|T|=k}\operatorname*{int}(\Omega_T)\label{explain1}\\
&\subseteq \bigcup_{T:|T|=k}(\overline{\Omega}_T \setminus\operatorname*{int}(\Omega_T)),\end{aligned}$$ where (\[explain1\]) is because $\overline{\bigcap_{T:|T|=k}\Omega_T}\subseteq \bigcap_{T:|T|=k}\overline{\Omega}_T$ and $\operatorname*{int}(\bigcap_{T:|T|=k}\Omega_T)=\bigcap_{T:|T|=k}\operatorname*{int}(\Omega_T)$. Also, define $$\begin{aligned}
\label{defs}
S&=\{\nu\in G_{l}(\mathbb{R}^n)~|~ \textrm{$\forall \mathbf{x}\in \nu$ has at most $l-1$ zero entries}\}\nonumber\\
&=
\bigcap_{
\begin{subarray}{c}
I\subseteq\{1,\dots,n\}\\
|I|=l
\end{subarray}
}U_I.\end{aligned}$$ Then $S^c$ is a closed set with empty interior and of measure zero. Therefore, we only need to show that $[\overline{\Omega}_T\setminus\operatorname*{int}(\Omega_T)]\cap S$ is of measure zero and of the first category (as a subset of the subspace $S$) for each $T:|T|\le k$. Since $\phi_{I}$ preserves measure zero sets and the topology, we in turn only need to show that the Euclidean set $$\begin{aligned}
\label{27}
\phi_{I}([\overline{\Omega}_T\setminus\operatorname*{int}(\Omega_T)]\cap S)\end{aligned}$$ is of measure zero and of the first category for some fixed $I$ and for every $T:|T|\le k$.
Now we are ready to prove that $\overline{\Omega}_T\setminus\operatorname*{int}(\Omega_T)$ is of measure zero in the special case of $n=3,k=1,m=1,T=\{3\}$, and this proof will demonstrate some basic ideas of the proof of general case in the appendix. It is enough to show that $\phi_I([\overline{\Omega}_T\setminus\operatorname*{int}(\Omega_T)]\cap U_I)$ is of measure zero (in $\mathbb{R}^2$) for $I=\{1,2\}$. For an arbitrary $\nu\in U_I\subseteq G_2(\mathbb{R}^3)$, define $(a,b):=\phi_I(\nu)$. Then $\nu$ is the subspace spanned by the columns of the matrix $\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
a & b \\
\end{array}
\right)
$, so $\nu\in U_I\setminus\Omega_T$ if and only if $$\begin{aligned}
\label{lowdim}
\exists~(x,y)\in \mathbb{R}^2\setminus\{{\bf 0}\},~\textrm{s.t}.~F(x)+F(y)\le F(ax+by).\end{aligned}$$ If $F$ is a non-decreasing subadditive sparseness measure, from the subadditivity it’s easily seen that (\[lowdim\]) holds if and only if $|a|\ge 1$ or $|b|\ge 1$, and the statement easily follows. However we will a prove for the case where $F$ is not necessarily subadditive (while possessing other properties of sparseness measures and being non-increasing), because the idea of this proof will hint on the idea of the general result in theorem \[probeq\]. By symmetry, we first note that it suffices to consider the region where $a,b\ge0$, in which case $\nu\in U_I\setminus\Omega_T$ if and only if $$\begin{aligned}
\label{lowdim1}
\exists~x,y\ge 0,(x,y)\neq(0,0),~\textrm{s.t.}~F(x)+F(y)\le F(ax+by).\end{aligned}$$ Let’s call the set of $(a,b)$ specified by (\[lowdim1\]) the region A, and its complement in $[0,\infty)^2$ the region B. Then the task is just to show that the boundary between A and B has measure zero (by boundary we mean a point belonging to the closures of both region A and B). This is not always true when $A$ is an arbitrary subset of $[0,\infty)^2$. But since $F$ is non-decreasing, from (\[lowdim1\]) we deduce the following property:
\(P) If $(a,b)$ is in region A, then for any $a_+\ge a$, $b_+\ge b$ the point $(a_+,b_+)$ is also in region A.
Also notice that the points $(1,0)$ and $(0,1)$ are on the boundary of $A$,$B$, and from (P) it’s easy to see that the boundary is a subset of $[0,1]^2$. Therefore the boundary of $A$,$B$ looks like the curve depicted in Figure \[fig1\] (but we don’t actually need a notion of “curve” for this proof.) To measure the area of the boundary, divide $[0,1]^2$ into $m$ rows and $n$ columns uniformly, so that $[0,1]^2$ is covered by small $1/m$ by $1/n$ (closed) rectangles. According to (P), there are only three possibilities concerning the vertices of a rectangle:
1\) Both its upper right and lower left vertices belong to $A$;\
2) It upper right vertex belongs to $A$ and lower left vertex belongs to $B$;\
3) Both its upper right and lower left vertices belong to $B$.
Clearly, the union of rectangles of type 1), 2) is a closed set containing $A$, so it also contains $\overline{A}$. Similarly, the union of rectangles of type 2), 3) contains $\overline{B}$. Therefore the boundary set, $\overline{A}\cap\overline{B}$ is contained in the intersection of these two unions, whose measure is total area of type 2) rectangles. However, property (P) implies that the number of type 2) rectangles is at most $m+n$ (one way of seeing this is to note that for any two adjacent columns, the rows for which the rectangles are colored have at most one overlap). Therefore total area of type 2) rectangles is at most $(m+n)/mn$ which converges to zero as $m,n\to\infty$. Thus the measure of $\overline{A}\cap\overline{B}$ must be zero.
Readers familiar with fractal geometry may also realize that (P) implies that $\overline{A}\cap\overline{B}$ is actually a porous set, meaning that there exists $0<\alpha<1$ and $r_0>0$ such that for any $0<r<r_0$ and $(a,b)\in \overline{A}\cap\overline{B}$, there is some $(a',b')\in [0,\infty)^2$ such that the ball centered at $(a',b')$ with radius $\alpha r$ is a subset of the ball centered at $(a,b)$ with radius $r$. In our example we can choose, say, $a'=a+r/100, b'=b+r/100$, and $\alpha=1/200$. A porous Euclidean set is necessarily of measure zero and of the first category. In higher dimensions, the idea of proof is again based on porosity, although the construction is more complicated than in this low dimensional example.
![The set $[0,1]^2$ is uniformly dived into $10$ columns and $10$ rows. Region A is the region above the curve, and region B the one below it. The type 2) rectangles in the discussion correspond to the colored squares in the figure, the number of which does not exceed $10+10=20$.[]{data-label="fig1"}](intuition "fig:"){width="3in"}\
Almost all commonly used cost functions that promote sparsity (e.g. the cost functions for $\ell_p$-minimization, ZAP, SCAD, or MCP) satisfy the requirement of $F$ being non-decreasing, hence the non-increasing assumption in the theorem is a very mild and reasonable one. Indeed, intuitively the cost function should penalize more as an entry moves farther away from zero. On the other hand, the non-decreasing requirement is also essential for the validity of Theorem \[probeq\]. To see this, we construct an example where the ERC set is almost the entire Grassmannian whereas the RRC set is empty.
Define $$\begin{aligned}
\label{eq_F}
F(x)=\left\{\begin{array}{cc}
0 & x=0; \\
0.1 & \textrm{$x>0$ and $x$ is rational};\\
1 & \textrm{$x>0$ and $x$ is irrational},
\end{array}
\right.\end{aligned}$$ and set the dimensions and sparsity to be $m=2,n=3,k=1$. it can then be verified that $F$ satisfies the definition of sparseness measure in Definition \[def1\]. Then $\mu(\Omega_J)=1$ but $\Omega_J^r=\emptyset$.
For arbitrary $x_1,x_2\in \mathbb{R}\setminus \{0\}$, denote by $x_1\simeq x_2$ if the equivalence relation $x_1/x_2\in\mathbb{Q}$ holds[^5]. Recall the set $S\subseteq G_1(\mathbb{R}^3)$ as defined in (\[defs\]) is of full measure, and for any $\nu\in S$ and $\mathbf{z}\in \nu\setminus \{\bf 0\}$ we have $z_i\neq 0$, $i=1,2,3$. Then the three coordinates of $\mathbf{z}$ can be partitioned into equivalent classes according to $\simeq$, and the type of partition is a property of $\nu$, i.e. it is independent of the choice of $\mathbf{z}\in\nu$ (remember that $\nu$ is a now one dimensional subspace). Moreover, whether $\nu$ is in $\Omega_J$ or not is completely determined by the type of partition, according to the construction of $F$. For example, we say $\nu$ is of type $(1,1,2)$ if the first two coordinates of $\mathbf{z}$ are from a same equivalence class and the third coordinate is from another equivalence class. From the null space property we can check that the type $(1,2,3)$ is in $\Omega_J$, since for each $\mathbf{z}\in \nu\setminus\{0\}$, there is at least one $i\in T^c$ such that $z_i$ is irrational. However, any type $(1,1,2)$ null space $\nu$ is not in $\Omega_J$, because there exists a $\mathbf{z}\in\nu\setminus\{\bf 0\}$ such that $z_1,z_2$ are rational and $z_3$ is irrational, in which case null space property fails when choosing $T=\{1\}$. Since the null spaces of the type $(1,2,3)$ is of measure $1$, we have that $\mu(\Omega_J)=1$. On the other hand, since the set of one dimensional subspaces corresponding to the type $(1,1,2)$ is dense in $G_1(\mathbb{R}^3)$ but does not intersect $\Omega_J$, the interior of $\Omega_J$ must be vacuous.
A trivial observation from Theorem \[probeq\] is that the probability of ERC and RRC are the same if the observation matrix $\mathbf{A}$ has i.i.d. Gaussian entries, since in this case the probability agrees with the measure $\mu$. More generally, suppose $P$ is the probability measure corresponding to the distribution of the null space of $\mathbf{A}$, and $P$ is absolutely continuous with respect to $\mu$,[^6] then $P(\Omega_J\setminus\Omega_J^r)=0$. Then it is not counter-intuitive that this should be true if the entries of $\mathbf{A}$ are i.i.d. generated from a certain continuous distribution, which is a common practice used in generating the observation matrix. Nevertheless, the above speculation requires a formal justification. We formulate this result as a corollary, the proof of which is deferred to Appendix \[app\_F\].
\[co2\] Suppose $F\in\mathcal{M}$ is a non-decreasing function, and the distribution of the matrix $\mathbf{A}$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{M}(m,n)$. Then the probability of ERC and RRC are the same. This holds true in particular when $\mathbf{A}$ has i.i.d. entries drawn from a continuous distribution.
Apart from the one described in Corollary \[co2\], another popular method for the generation of $\mathbf{A}$ is by randomly selecting $m$ rows in the $n\times n$ Fourier transform matrix [@tao1; @tao2]. However in this scheme the probability of ERC and RRC may not agree, since the probability distribution of the null space is not continuous on $G_l(\mathbb{R}^n)$.
Escape through the Mesh
-----------------------
Although the characterization in Theorem \[th2\] is simple and accurate, it fails to convey any quantitative information about robustness: given a subspace in $\Omega_J^r=\operatorname*{int}(\Omega_J)$, we still do not know how large the constant $C$ in the definition of RRC is. To overcome this drawback, consider the “$d$-interior” of $\Omega_J$, defined as $$\begin{aligned}
d\mbox{-}\operatorname*{int}(\Omega_J):=\{\nu\in G_l(\mathbb{R}^n)|\nu'\in \Omega_J,~\forall~\nu':\operatorname*{dist}(\nu,\nu')\le d \}.\end{aligned}$$ We remark that by definition, $\operatorname*{int}(\Omega_J)=\bigcup_{d>0} d\mbox{-}\operatorname*{int}(\Omega_J)$. Now a “quantitative” version of Theorem \[th2\] is as follows:
\[th22\] If $\mathcal{N}(\mathbf{A})\in d\mbox{-}\operatorname*{int}(\Omega_J(n,k,\operatorname*{rank}(\bf A)))$, then RRC is satisfied for $({\bf A},k,J)$ with the robustness constant $C=2(1+d)/(d\sigma_{\min})$. Conversely, if RRC holds for $({\bf A},k,J)$ with $C=2(1-2d)/(d\sigma_{\max})$ for some $0<d<1/2$, then $\mathcal{N}(\mathbf{A})\in d\mbox{-}\operatorname*{int}(\Omega_J(n,k,\operatorname*{rank}(\bf A)))$.
See Appendix \[p\_th2\].
In principal, the supremum of $d$ such that $\mathcal{N}({\bf A})\in d\mbox{-}\operatorname*{int}(\Omega_J)$ is completely determined by $({\bf A},k,J)$. However, exactly computing $d$ for a given $\bf A$ seems to be out of reach since $T$ may take $\left(
\begin{array}{c}
n \\
k \\
\end{array}
\right)
$ number of values. In practical applications we may be interested in the probability that a randomly generated measurement matrix falls into $d\mbox{-}\operatorname*{int}(\Omega_J)$. The key idea of our analysis is Gordon’s *escape through the mesh* theorem [@gordon], which was employed in the study of exact reconstruction of sparse signals via $\ell_1$-minimization by Rudelson and Vershynin [@rudelson]. With some additional observations, we can use this approach to bound the robustness constant $C$. Define the sets $$\begin{aligned}
\mathcal{D}_J(n,k):=\{\mathbf{z}\in \mathbb{R}^n\setminus \{\mathbf{0}\}|
J(\mathbf{z}_T)\ge J(\mathbf{z}_{T^c}),~\exists
T\subseteq\{1,...,n\}:|T|\le k\},\end{aligned}$$ $$\begin{aligned}
\mathcal{D}_{J,d}(n,k):&=\{\mathbf{z}\in \mathbb{R}^n\setminus \{\mathbf{0}\}|
J(\mathbf{z}_T+\mathbf{n}_T)\ge J(\mathbf{z}_{T^c}+\mathbf{n}_{T^c}),
~\exists
\mathbf{n}\in\mathbb{R}^n:\|\mathbf{n}\|<d\|\mathbf{z}\|,
T\subseteq\{1,...,n\}:|T|\le k\}\\
&=\{\mathbf{z}\in \mathbb{R}^n\setminus \{\mathbf{0}\}|
\mathbf{z}+\mathbf{n}\in\mathcal{D}_d(n,k),
~\exists
\mathbf{n}\in\mathbb{R}^n:\|\mathbf{n}\|<d\|\mathbf{z}\|,
T\subseteq\{1,...,n\}:|T|\le k\}.\label{33}\end{aligned}$$ Again, we shall omit the subscript $J$ when there is no confusion from the context. Define the cones $$\begin{aligned}
\mathcal{C}(n,k)&:=\{\mathbf{x}\in\mathbb{R}^n~|~\exists t\in \mathbb{R}~{\rm s.t.}~t\mathbf{x}\in \mathcal{D}(n,k)\}\end{aligned}$$ $$\begin{aligned}
\mathcal{C}_d(n,k):=\{\mathbf{x}\in\mathbb{R}^n~|~\exists t\in \mathbb{R}~{\rm s.t.}~t\mathbf{x}\in \mathcal{D}_d(n,k)\};\label{def4}\end{aligned}$$ Also define the following subsets of unit sphere in $\mathbb{R}^n$: $$\mathcal{K}(n,k):=\mathcal{C}(n,k)\cap S^{n-1},$$ $$\mathcal{K}_d(n,k):=\mathcal{C}_d(n,k)\cap S^{n-1}.$$ Then it’s easy to see that $$\begin{aligned}
\label{3940}
\Omega_J&=\{\nu\in G_l(\mathbb{R}^n)|\mathcal{K}(n,k)\cap\nu=\emptyset\}.\end{aligned}$$ Define the set $$\begin{aligned}
\label{ehat}
\widehat{\Omega_d} &:=\{\nu\in G_l(\mathbb{R}^n)|\mathcal{K}_d(n,k)\cap\nu=\emptyset\}.\end{aligned}$$ From the proof of Theorem \[th22\], it will be seen that $d\mbox{-}\operatorname*{int}(\Omega_J)\subseteq\widehat{\Omega_d}\subseteq\frac{d}{1+d}\mbox{-}\operatorname*{int}(\Omega_J)$. However what’s more essential is that by Theorem \[suf2\], subspaces in $\widehat{\Omega_d}$ guarantee RRC with $C=2(1+d)/(d\sigma_{\min})$.
For any vector $\mathbf{g}\in \mathbb{R}^n$ and $\epsilon>0$, there exists $\mathbf{x}'\in \mathcal{K}_d(n,k)$ so that $$\begin{aligned}
\label{ineq1}
\sup_{\mathbf{x} \in \mathcal{K}_d(n,k)}\mathbf{g}^\top\mathbf{x}
\le \mathbf{g}^\top\mathbf{x}'+\epsilon\end{aligned}$$ By (\[def4\]) and (\[33\]), there exists $t\neq 0$ and $\mathbf{n}':\|\mathbf{n}'\|<d\|t\mathbf{x}'\|$ such that $t\mathbf{x}'+\mathbf{n}'\in \mathcal{D}(n,k)$. Therefore $\mathbf{x}'+\mathbf{n}\in \mathcal{C}(n,k)$ where $\mathbf{n}:=t^{-1}\mathbf{n}'$. Let $\mathbf{y}$ be the projection of $\mathbf{x}'$ onto the one dimensional subspace spanned by $\mathbf{x}'+\mathbf{n}$. Then $\mathbf{y}\in \mathcal{C}(n,k)$ because $\mathcal{C}(n,k)$ is a cone. Also $\|\mathbf{y}-\mathbf{x}'\|\le \|{\bf (x'+n)-x'}\|\le d\|{\bf x'}\|=d$, $\|\mathbf{y}\|\le 1$ by properties of the projection. Thus $$\begin{aligned}
\mathbf{g}^\top\mathbf{x}'&=\mathbf{g}^\top(\mathbf{x}'-\mathbf{y})+\mathbf{g}^\top\mathbf{y}\\
&\le \|\mathbf{g}\|\|\mathbf{x}'-\mathbf{y}\|+|\mathbf{g}^\top\mathbf{y}/\|\mathbf{y}\||\cdot\|\mathbf{y}\|\\
&\le d\|\mathbf{g}\|+|\mathbf{g}^\top\mathbf{y}/\|\mathbf{y}\||\\
&\le d\|\mathbf{g}\|+\sup_{\mathbf{x} \in \mathcal{K}(n,k)}\mathbf{g}^\top\mathbf{x}\label{ineq2}\end{aligned}$$ The last inequality used the fact that $\pm\mathbf{y}/\|\mathbf{y}\|\in\mathcal{C}(n,k)$, since $\mathcal{C}(n,k)$ is centrally symmetric. Now (\[ineq1\]), (\[ineq2\]) and arbitrariness of $\epsilon$ give $$\label{ineq3}
\sup_{\mathbf{x} \in \mathcal{K}_d(n,k)}\mathbf{g}^\top\mathbf{x}
\le d\|\mathbf{g}\|+\sup_{\mathbf{x} \in \mathcal{K}(n,k)}\mathbf{g}^\top\mathbf{x}$$ This result will be useful soon in connecting the two sets $\mathcal{K}_d(n,k)$ and $\mathcal{K}(n,k)$.
The *Gaussian width* of a subset of $\mathcal{K}\subseteq S^{n-1}$ is defined as $$w(\mathcal{K})=\mathbb{E}\sup_{x \in \mathcal{K}}\mathbf{g}^\top\mathbf{x}$$ where $\mathbf{g}$ is a random vector in $\mathbb{R}^n$ whose components are independent $\mathcal{N}(0,1)$ random variables.
From (\[ineq3\]), the Gaussian width of the extended set $\mathcal{K}_d(n,k)$ is upper bounded by $$\label{48}
w(\mathcal{K}_d(n,k))
\le w(\mathcal{K}(n,k))+d\mathbb{E}\|\mathbf{n}\|
\le w(\mathcal{K}(n,k))+d\sqrt{\mathbb{E}\|\mathbf{n}\|^2}
\le w(\mathcal{K}(n,k))+d\sqrt{n}.$$
A small Gaussian width implies that a random linear subspace of $\mathbb{R}^n$ is not likely to intersect with it:
Let $\mathcal{K}$ be a subset of the unit Euclidean sphere $S^{n-1}$ in $R^n$. Let $\nu$ be a random $(n-m)$-dimensional subspace of $R^n$, distributed uniformly in the Grassmannian with respect to the Haar measure. Assume that $$\label{wcond}
w(\mathcal{K})<\sqrt{m}.$$ Then $\nu\cap \mathcal{K}=\emptyset$ with probability at least $$1-2.5\exp(-(m/\sqrt{m+1}-w(\mathcal{K}))^2/18).$$
As noted in [@rudelson], the original coefficient $3.5$ in [@gordon] can be replaced with $2.5$ shown above. Nevertheless, its exact value does not matter for the our purpose.
From (\[3940\]), (\[ehat\]), (\[48\]) and Gordon’s Theorem, one immediately obtains the following estimate of the probability of $\widehat{\Omega_d}$ in the case of Gaussian random measurement matrix:
\[k\] If $w(\mathcal{K}_J)<\sqrt{m}-d\sqrt{n}$, then $$\mu(\widehat{\Omega_d})\ge 1-2.5\exp(-(m/\sqrt{m+1}-w(\mathcal{K}_J)-d\sqrt{n})^2/18).$$
Note that in Theorem \[k\], the results rely on the Gaussian width $\mathcal{K}_J$, which is essentially determined by $J$. In the remainder of this section we shall focus on the case where $J$ is the $\ell_1$ norm, and demonstrate how to analyze asymptotic performance from Theorem \[k\]. As remarked earlier, the asymptotic analysis of $\mathcal{K}_{\ell_1}$ was carried out by [@rudelson] in the study of exact recovery property. Lemma 4.4 and 4.5 in [@rudelson] combined yield the following upper bound on the Gaussian width of $\mathcal{K}_{\ell_1}(n,k)$: $$w(\mathcal{K}_{\ell_1}(n,k))\le 2\sqrt{k(3+2\log(n/k))}\cdot\zeta(n,k)$$ where $$\zeta(n,k)=\exp\left(\frac{\log(1+2\log(en/k))}{4\log(en/k)}+\frac{1}{24k^2\log(en/k)}\right).$$ we shall consider the asymptotic case where $k,n,m$ scales linearly, i.e., $n=\lfloor\beta k\rfloor$, $m=\lceil\gamma k\rceil$ for some constants $\beta>\gamma\ge1$. Then $w(\mathcal{K}_{\ell_1}(n,k))$ satisfies the condition $w(\mathcal{K}_{\ell_1})<\sqrt{m}-d\sqrt{n}$ in Theorem \[k\] for large $k,n,m$ if the scaling parameters $\beta,\gamma$ and the number $d$ satisfy $$2\sqrt{(3+2\log(\beta))}\cdot\exp\left(\frac{\log(1+2\log(e\beta))}{4\log(e\beta)}\right)<\sqrt{\gamma}-d\sqrt{\beta}.$$ Define $$\label{delta}
\delta(\beta,\gamma)=\frac{1}{\sqrt{\beta}}\left(\sqrt{\gamma}-
2\sqrt{(3+2\log(\beta))}\cdot\exp\left(\frac{\log(1+2\log(e\beta))}{4\log(e\beta)}\right)\right).$$ Notice that $\delta(\beta,\gamma)>0$ when $\gamma>4(3+2\log(\beta))\cdot\exp\left(\frac{\log(1+2\log(e\beta))}{2\log(e\beta)}\right)$. If this is the case, the escape through the mesh theorem implies that $\mu(\widehat{\Omega_d})$ tends to one as $k,n,m\to \infty$, if $d<\delta(\beta,\gamma)$. On the other hand, theory of random matrix (see for example [@DC] and the references therein) shows that if the entries of $\mathbf{A}$ are i.i.d. Gaussian with zero mean and variance $1/n$, then $\sigma_{\min}(\mathbf{A}^\top)$ converges to $1-\sqrt{\gamma/\beta}$ almost surely as $m,n,k\to\infty$. Thus by Theorem \[th22\], we have:
\[as\] Suppose $n=\lfloor\beta k\rfloor$, $m=\lceil\gamma k\rceil$ for some constants $\beta>\gamma\ge1$, and the entries of the measurement matrix are i.i.d. Gaussian with zero mean and variance $1/n$. If $\delta(\beta,\gamma)$ defined in (\[delta\]) is positive, then with probability converging to one as $k\to\infty$, the $\ell_1$-minimization satisfies RRC with the robustness constant $C=\frac{2(1+\delta(\beta,\gamma))}{\delta(\beta,\gamma)(1-\sqrt{\gamma/\beta})}$, where $\delta(\beta,\gamma)$ is defined in (\[delta\]).
The proof of Corollary \[as\] follows directly from the preceding discussion. Notice that $\beta$ characterizes the sparsity, which is determined by the nature of the signal; and $\gamma/\beta$ is the measurement rate, which is may be controlled by the designer. If we view $\beta$ as a fixed parameter, then Corollary \[as\] can be interpreted as the tradeoff between measurement rate and robustness: there is phase transition point for $\gamma$ where $\ell_1$-minimization becomes robust; after $\gamma$ exceeds that point, the robustness constant $C$ may continue to drop, but no further qualitative changes may occur.
Notice that according to Corollary \[as\], the robustness constant $C$ is $O(1)$ as $k\to \infty$. On the other hand, for an oracle that knows the true support $T$, the ML estimation error should be $\|(\mathbf{A}_T^\top\mathbf{A}_T)^{-1}\mathbf{A}_T^{\top}\mathbf{v}\|\le \lambda_{\min}^{-1/2}(\mathbf{A}^{\top}_T\mathbf{A}_T)\|\mathbf{v}\|$, which is upper bounded by $(1-\sqrt{\gamma^{-1}})^{-1}\|\mathbf{v}\|$ almost surely as $m,n,k\to\infty$, which means that the robustness constant of the oracle estimator is approximately $(1-\sqrt{\gamma^{-1}})^{-1}$ as $k\to\infty$. Thus the robustness constant $C$ in Corollary \[as\] is optimal, in the sense that it has the same scaling with the robustness constant of an oracle estimator, with overwhelming probability as $k\to\infty$.
\[rem8\] In the case where $F$ is a generic sparseness measure, i.e. $J$ not the $\ell_1$ norm, it may still be possible to perform the asymptotic analysis of $\mathcal{K}_J$, and then substitute into Theorem \[k\] to obtain rate-robustness tradeoff results for $J$. For example, in [@stojnic Section 3], an upper bound of $\mathcal{K}_{\ell_p}$ was derived, although numerical optimizations needs to be solved in order to compute that bound. Another approach is consider the relationship between $\Omega_J$ and $\Omega_{\ell_p}$. It’s possible to provide some sufficient conditions such that $\Omega_J\subseteq\Omega_{\ell_p}$. If this is the case, then it immediately follows that $d\mbox{-}\operatorname*{int}(\Omega_J)\subseteq d\mbox{-}\operatorname*{int}(\Omega_{\ell_p})$, so that the robustness of $F$-minimization and $\ell_p$-minimization can be quantitatively compared. This will be the topic of the next section.
Comparison of Different Sparseness Measures {#secrules}
===========================================
In this section we provide some methods to compare the performance between two sparseness measures in terms of ERC or RRC. Since $\Omega_J$, $\operatorname*{int}(\Omega_J)$, and $d\mbox{-}\operatorname*{int}(\Omega_J)$ are shown to correspond to the measurement matrices satisfying ERC, RRC, or with a particular robustness constant, it’s easy to compare the performances of two sparseness measures if an inclusion relation such as $\Omega_{J_1}\subseteq\Omega_{J_2}$ is available. However, sometimes its not true that $\Omega_{J_1}\subseteq\Omega_{J_2}$, but it may still be possible to show $\Omega_{J_1}\subseteq\overline{\Omega}_{J_2}$. The second relation is not terribly different than the first one, since we have shown that $\overline{\Omega}_{J_2}\setminus\Omega_{J_2}$ is negligible when $F_2$ is non-decreasing. Therefore, both the topological characterization of RRC and the probabilistic (measure-theoretic) viewpoint become particularly useful when passing from the $\ell_p$ cost functions to general sparseness measures.
The following lemma comes from the corresponding result for ERC in [@Gribonval] and our interior point characterization of RRC:
\[le2\] Suppose $F,G\in\mathcal{M}$. If $F,G$ are non-decreasing and $F/G$ is non-increasing on $\mathbb{R}^+$, then we have $\Omega_{J_G}\subseteq\Omega_{J_F}$ and $\Omega^{r}_{J_G}\subseteq\Omega^{r}_{J_F}$.
The fact that $\Omega_{J_G}\subseteq\Omega_{J_F}$ comes from [@Gribonval Lemma 7]. It then follows that $\Omega^{r}_{J_G}\subseteq\Omega^{r}_{J_F}$ from Theorem \[th2\].
The set inclusions formulas in Lemma \[le2\] means that the sparseness measure $F$ is better than $G$, in the sense that whenever the cost function $J_G$ guarantees ERC/RRC, so does the $J_F$. By letting $G(x):=x^q$ in this lemma we can obtain the following result:
\[co1\] Suppose $F\in\mathcal{M},~p\in(0,1]$. If $F$ is non-decreasing and $F(x)/x^p$ is non-increasing on $\mathbb{R}^+$, then we have $\Omega_{\ell_p}\subseteq\Omega_{J_F}$ and $\Omega^{r}_{\ell_p}\subseteq\Omega^{r}_{J_F}$.
Corollary \[co1\] gives a condition such that $J_F$ is better than $\ell_p$ in the sense of ERC and RRC. Conversely, we shall show that the asymptotic of $F$ around $0^+$ and $+\infty$ gives a sufficient condition that $\ell_p$ is better than $J_F$ in terms of probability.
The following result implies that, in some sense, it’s not good to design an $F$ which is differentiable (or Holder continuous) at zero or infinity, as far as the worst case performance is concerned:
\[th1\] Suppose $F\in\mathcal{M},~p\in(0,1]$. If $\lim_{x\to 0^+}F(x)/x^p$ or $\lim_{x\to\infty}F(x)/x^p$ exist and is positive, then $\Omega_{J_F}\subseteq\overline{\Omega}_{\ell_p}$, and $\mu(\Omega_{J_F})\le \mu(\Omega_{\ell_p})$.
See Appendix \[ap2\].
We Remark that $\mu(\Omega_{J_F})\le \mu(\Omega_{\ell_p})$ in Theorem \[th1\] cannot be replaced by the stronger set inclusion relation $\Omega_{J_F}\subseteq\Omega_{\ell_p}$, which holds for $\ell_p$ cost functions but fails for general sparseness measures. Thus the measure-theoretic viewpoint allows us to restore a comparison criteria when extending $\ell_p$-minimization to the $F$-minimization.
As an illustration, we demonstrate how to derive the relation between ZAP [@zap] and $\ell_1$-minimization from above results. Consider the typical form of sparseness measure used in the ZAP algorithm (which is essentially the same as the minimax concave penalty (MCP) [@MCP] familiar to statisticians): $$\begin{aligned}
\label{MCPP}
F(x)=
\left\{
\begin{array}{cc}
\alpha x-\alpha^2x^2 & x<1/\alpha;\\
1 & \textrm{otherwise},\end{array}
\right.\end{aligned}$$ where the tuning parameter $\alpha$ is usually chosen as the inverse of the standard deviation of the non-zero entries in $\mathbf{\bar{x}}$. Our following result says that, while ZAP performs far better than $\ell_1$-minimization in the average case, as shown in the numerical experiments [@zap], the worst case performance (requiring all sparse vectors can be constructed) of the two cost functions are the same:
\[co\_zap\] $$\mu(\Omega_{ZAP})=\mu(\Omega_{\ell_1}).$$
Using Corollary \[co1\] and Theorem \[th1\] with $p=1$ one can obtain both the lower and upper bound on $\mu(\Omega_{ZAP})$ respectively.
The result of Corollary \[co\_zap\] is not in contradiction with the proverbial fact that concave penalties induce smaller risks, since we are using different benchmarks of performance. When the parameter $\alpha$ can be tuned according to the statistics of the variables, concave penalties usually have better average performance (risk); but this is irrelevant to our worst case analysis. We end this section by summarizing the relationship between the various requirements on $F$ appeared in this section:
\[prop1\] Assuming that $0\le p\le1$, $F:[0,+\infty)\to[0,+\infty)$, and $F(0)=0$, we have\
(1) $F$ is concave $\Longrightarrow$ $F(t)/t$ is non-increasing;\
(2) $F(t)/t^p$ is non-increasing $\Longrightarrow$ $F(t)/t$ is non-increasing;\
(3) $F(t)/t$ is non-increasing $\Longrightarrow$ $F$ is subadditive.
See Appendix \[prop1\_proof\].
Comparison with Other Works {#comp}
===========================
The ERC/RRC Equivalence for $\ell_p$-minimization
-------------------------------------------------
To the best of our knowledge, the *exact* characterization of robustness of $\ell_p$-minimization first appeared in [@Foucart], where the definition of robustness is the same as in our paper. In [@Foucart] a variant of the null space property, called NSP’, was proposed as a sufficient condition for the robustness of $\ell_p$ minimization. The NSP’ is obviously stronger than NSP, but the reverse situation is not immediately clear. Later Aldroubi et al adopted the same approach in [@lqnsp], and proved that NSP and NSP’ are in fact equivalent (see also [@lqharmonic]). The proof method in [@lqnsp] requires a lemma from matrix analysis [@lqnsp Lemma 2.1]. We remark that this lemma, from a slightly more general viewpoint, can be seen as a classical application of the open mapping theorem in functional analysis [@stein_func Chapter 4, Corollary 3.2]. Thus it is established that NSP, NSP’, ERC and RRC are all equivalent for $\ell_p$-minimization.
While the NSP’ approach is nice for the $\ell_p$ case, it is hard to be extended to the general $F$-minimization problem. This is because NSP’ consists of a homogeneous inequality, which appears to work well only for homogeneous cost functions such as the $\ell_p$ norm. In contrast, the heart of our approach is the interior point characterization of RRC (Theorem \[th2\]) for the general $F$-minimization problem. Then our proof of the ERC/RRC equivalence for $\ell_p$-minimization, although involves some basic facts about topological spaces, follows almost immediately as a corollary. Note this application is particularly interesting since the statement of ERC/RRC equivalence does not involve topology at all. Nevertheless, we emphasize that the significance of Theorem \[th2\] is to provide a simple, accurate, and general characterization of the robustness of $F$-minimization; and the proof of ERC/RRC equivalence for $\ell_p$ is one of its applications in a special setting.
The Notion of Sparseness Measure
--------------------------------
The sparseness measure defines the class of cost functions of our interest, and is therefore of great importance. In general we want to consider a class wide enough to cover most applications, but also small enough to possess important recovery properties. Intuitively, the cost function should penalize non-zero coefficients, and not penalize the zero coefficients. However there are additional reasonable requirements, the precise definitions of which differ in the literature. For clarifications we compare these different requirements on $F$ as follows (Recall that $\mathcal{M}$ denotes the set of sparseness measures defined in Definition \[def1\]):
$\bullet$ $F\in\mathcal{M}$. This is the class of functions mainly considered in our paper as well as [@lqnsp]. This seems to be most general class of functions that can be studied by the null space property.
$\bullet$ $F\in\mathcal{M}$ and $F$ is non-decreasing. This requirement appears in Theorem \[probeq\]. As shown in the counter example in the remark following the theorem, the assumption that $F$ being non-decreasing cannot be dropped.
$\bullet$ $F\in\mathcal{M}$, $F$ is non-decreasing, and $F(t)/t$ is non-increasing[^7]. This requirement is considered in [@Gribonval; @troppdis], and it guarantees that the cost function $J_F$ is better than $\ell_1$ norm in the sense of ERC. There is also another nice property relating to the composition of two functions in this class [@Gribonval Lemma 7]. Finally, $\ell_1$ norm is the only convex cost function whose corresponding $F$ satisfies this definition of sparseness measure [@troppdis Proposition 2.1].
About the Robustness Constant
-----------------------------
A remarkable feature of Corollary \[as\] (which is based on Theorem \[as\]) is that the robustness constant does not blow up in the linear scaling asymptotic region. Moreover, the same method may be applicable to the case of general sparseness measures, as long as an upper bound of $\mathcal{K}_J$ is available, as discussed in Remark \[rem8\]. On the other hand, the robustness constant obtained in [@lqnsp] is (in the notation of our paper) $n^{1/p-1/2}(\frac{4}{1-\theta_{\ell_p}})^{1/p}\sqrt{\frac{2}{\sigma^2_{\min}(\mathbf{A}^{\top})}}$, which blows up in the linear scaling setting in Corollary \[as\] as $n\to\infty$.
In the special case of $\ell_1$-minimization, our problem setting and the notion of robustness is also the same as the classical paper [@candes2006stable] by Candès. In Theorem 1 of that paper, it is shown that (in the notation of our paper) if the RIP constants satisfy $\delta_{3k}+3\delta_{4k}<2$ then RRC holds while robustness constant $C$ may depend on $\delta_{4k}$. Later, the same author provided a similar but improved result on robustness in [@candes2008restricted], where the assumption depends on $\delta_{2k}$ instead of $\delta_{4k}$. However it is will known that the RIP constant is hard to compute or provide a precise estimate. Moreover, according to a comparative study of [@blanchard2011compressed], performance estimates based on RIP is usually not as sharp as analysis based on Gordon’s theorem in the proportional growth setting.
Conclusion
==========
$F$-minimization refers to a broad family of non-convex optimizations for sparse recovery, which is known to outperform conventional $\ell_1$ minimization experimentally. However because of some technical difficulties, the robustness of $F$-minimization was not fully understood before, even though its exact recovery property has been studied by using the null space property. The novel approach of this paper is to view the collection of null spaces as a topological manifold, called the Grassmann manifold, and provide an exact characterization of the relationship between robust recovery condition (RRC) and exact recovery condition (ERC): the set of null spaces of measurement matrix $\mathbf{A}$ satisfying RRC is the interior of the one satisfying ERC. Building on this characterization, the previous result of the equivalence of exact recovery and robust recovery in the $\ell_p$-minimization follows as an easy consequence. Although the RRC set is in general a proper subset of the ERC set, there difference is only a set of measure zero and of the first category, provided that $F$ satisfies the mild condition of being non-decreasing. The practical significance of this result is that ERC and RRC will occur with equal probability when the measurement matrix is randomly generated according to a continuous distribution. On the quantitative side, a desired level of robustness can be guaranted if the null space of $\mathbf{A}$ is drawn from the “$d$-interior” of $\Omega_J$ for a certain $d>0$. Specifically, the null spaces in $d\mbox{-}\operatorname*{int}(\Omega_J)$ satisfies RRC with $C=\frac{2+2d}{d\sigma_{\min}(\mathbf{A}^{\top})}$; and null spaces outside of $d\mbox{-}\operatorname*{int}(\Omega_J)$ cannot satisfy RRC with $C=\frac{2-4d}{d\sigma_{\max}(\mathbf{A}^{\top})}$. From this, the relation of robustness, measurement rate and sparsity may be analyzed using Gordon’s escape through the mesh theorem. Some simple rules for comparing the performances of different choices of the sparseness measure $F$ are available, which may provide guidelines for the design of sparseness measure.
Further improvements may include finding more general conditions on $F$ than being non-decreasing in order that Theorem \[probeq\] still holds. Although the non-decreasing condition seems to be mild enough for applications, it is still mathematically interesting to know how much the condition can be further relaxed. Studies of the robustness under perturbation in the measurement matrix within this Grassmannian framework may also be of interest. Moreover, as demonstrated in Corollary \[as\] in the case of $\ell_1$-minimization, one can potentially derive the tradeoff between robustness and measurement rate for other sparseness measures, provided that an estimate of the Gaussian width of $\mathbf{K}_J$ is available. As discussed in Remark \[rem8\], upper bounds on the Gaussian width of $\mathbf{K}_{\ell_p}$ are already available in the literature.
Proof of Theorem \[suf2\] {#p_suf2}
=========================
For the direct part, assume that (\[eq16\]) is true. Suppose $\mathbf{\hat{x}}$ is a feasible vector with $J(\mathbf{\hat{x}})\le J(\mathbf{\bar{x}})$, and we want to show that $\mathbf{\hat{x}}$ is close to $\mathbf{\bar{x}}$. From the constraint of the optimization we have $$\|\mathbf{A}(\mathbf{\hat{x}}-\mathbf{\bar{x}})\|\le
\|\mathbf{A}\mathbf{\hat{x}}-\mathbf{y}\|
+\|\mathbf{A}\mathbf{\bar{x}}-\mathbf{y}\|
\le 2\epsilon.$$ Define $\mathbf{u}:=\mathbf{\bar{x}}-\mathbf{\hat{x}}$; we find that $$\begin{aligned}
J({\bf u}_T)
&\ge J(\bar{\bf x}_T)-J(\hat{\bf x}_T) \label{step1}\\
&= J(\bar{\bf x})-J(\hat{\bf x}_T) \label{step2}\\
&\ge J(\hat{\bf x})-J(\hat{\bf x}_T) \label{step3}\\
&= J(\hat{\bf x}_{T^c})\nonumber\\
&= J({\bf u}_{T^c})\nonumber\end{aligned}$$ Where (\[step1\]) is from subadditivity of $F$, (\[step2\]) is because $\bar{\bf x}$ is supported on $T$, and (\[step3\]) is from the assumption of $\hat{\bf x}$. Decompose $\mathbf{u}=\mathbf{z}+\mathbf{n}$, such that $\mathbf{z}\in\mathcal{N}(\mathbf{A})$, $\mathbf{n}\in\mathcal{N}(\mathbf{A})^{\bot}$. The above inequality is in contradiction with (\[eq16\]), hence from the assumption we must have: $$\|\mathbf{n}\|\ge d\|\mathbf{z}\|,$$ which by triangular inequality implies that $\|{\bf n}\|\ge d(\bf \|u\|-\|n\|)$, or $\|{\bf n}\|\ge\frac{d}{1+d}\|\bf u\|$. Therefore $$\begin{aligned}
2\epsilon
&\ge\|\mathbf{A}(\mathbf{\hat{x}}-\mathbf{\bar{x}})\|\nonumber\\
&=\|\mathbf{A}\mathbf{n}\|\nonumber\\
&\ge \sigma_{\min}\|\mathbf{n}\|\nonumber\\
&\ge \sigma_{\min}\frac{d}{1+d}\|\mathbf{u}\|\nonumber\\
&=\sigma_{\min}\frac{d}{1+d}\|\mathbf{\hat{x}}-\mathbf{\bar{x}}\|
,\nonumber\end{aligned}$$ where $\sigma_{\min}$ is the smallest singular value of $\mathbf{A}^{\top}$. Thus RRC holds with $C=2(1+d)/(d\sigma_{\min})$.
For the converse part, suppose that the statement is not true, i.e., for some fixed $d>0$, $$\begin{aligned}
\label{cond4}
&\exists \mathbf{z}\in \mathcal{N}(\mathbf{A})\setminus\{\mathbf{0}\}, \mathbf{n}:\|\mathbf{n}\|<d\|\mathbf{z}\|, T:|T|\le k \nonumber\\
&\textrm{s.t.~} J(\mathbf{z}_T+\mathbf{n}_T)\ge J(\mathbf{z}_{T^c}+\mathbf{n}_{T^c}),\end{aligned}$$ we will show that RRC with $C=\frac{2(1-d)}{d\sigma_{\max}}$ is impossible. To do this, we will construct $\hat{\mathbf{x}}$, $\bar{\mathbf{x}}$ with $J(\bar{\mathbf{x}})\ge J(\hat{\mathbf{x}})$, and $\mathbf{v},\epsilon$ with $\|\mathbf{v}\|=\epsilon$, $\|\mathbf{A}\hat{\mathbf{x}}-(\mathbf{A}\bar{\mathbf{x}}+\mathbf{v})\|=\epsilon$; but $$\label{eq3}
\|\hat{\mathbf{x}}-\bar{\mathbf{x}}\|> \frac{2(1-d)\epsilon}{d\|\mathbf{A}\|},$$ where $\|\mathbf{A}\|=\sigma_{\max}$ denotes the operator norm of matrix $\mathbf{A}$. Now suppose $\mathbf{n},\mathbf{z}$ are as in (\[cond4\]). Define[^8] $\bf u:=z+n$, $\bar{\mathbf{x}}:=(\mathbf{u}_T)^T$, $\hat{\mathbf{x}}:=-(\mathbf{u}_{T^c})^{T^c}$, $\mathbf{v}:=\mathbf{A}(\hat{\mathbf{x}}-\bar{\mathbf{x}})/2$, $\epsilon:=\|\mathbf{v}\|$. Then feasibility is satisfied since $\|\mathbf{A}\hat{\mathbf{x}}-(\mathbf{A}\bar{\mathbf{x}}+\mathbf{v})\|=\epsilon$. The relation $J(\bar{\mathbf{x}})\ge J(\hat{\mathbf{x}})$ is true because of (\[cond4\]). Also $$\begin{aligned}
2\epsilon&=\|\mathbf{A}(\hat{\mathbf{x}}-\bar{\mathbf{x}})\|\nonumber\\
&=\|\mathbf{A}\mathbf{n}\|\nonumber\\
&\le \|\mathbf{A}\|\|\mathbf{n}\|\nonumber\\
&< \|\mathbf{A}\|\frac{d}{1-d}\|\mathbf{u}\|,\nonumber\end{aligned}$$ where the last step is because $\|{\bf u}\|+\|{\bf n}\|\ge\|{\bf z}\|>\frac{1}{d}\|{\bf n}\|$, which implies that $\|{\bf u}\|>(\frac{1}{d}-1)\|{\bf n}\|$. Thus the relation (\[eq3\]) holds, as desired.
Proof of Theorem \[th2\] and Theorem \[th22\] {#p_th2}
=============================================
The proof of the theorems will be based on the following result:
\[lmth2\] Suppose $\nu\in G_{l}(\mathbb{R}^n)$. For all $\mathbf{z}\in \nu\setminus \{\mathbf{0}\},\|\mathbf{n}\|<d\|\mathbf{z}\|$, there exists $\nu'\in G_{l}(\mathbb{R}^n)$ such that $\mathbf{z}+\mathbf{n}\in \nu'$ and $\operatorname*{dist}(\nu,\nu')<d$.
If $d>1$, then any $\nu'\in G_l(\mathbb{R}^n)$ will satisfy $\operatorname*{dist}(\nu,\nu')\le1<d$ as desired. Now suppose $d\le 1$, so that $\bf z+n\neq 0$. Let $\nu_0\subseteq \nu$ be the subspace such that $\dim(\nu_0)=l-1$ and ${\bf z}\bot \nu_0$. Define $\nu'=\operatorname*{span}({\bf z+n})\oplus \nu_0\in G_l(\mathbb{R}^n)$. [^9] Let $\theta_i, i=1,\dots, l$ be the principal angles between $\nu$ and $\nu'$ in the ascending order. Then by the construction we have $$\begin{aligned}
\theta_i&=0,\quad \forall i=1,\dots,l-1;\\
\theta_l&=\angle({\bf z},{\bf P}_{\nu_0^{\bot}}(\bf z+n))=\angle({\bf z},{\bf z}+{\bf P}_{\nu_0^{\bot}}{\bf n}).\end{aligned}$$ where ${\bf P}_{\nu_0^{\bot}}$ denotes the projection matrix onto the orthogonal complement of $\nu_0$. Then, $$\begin{aligned}
\operatorname*{dist}(\nu,\nu')&=\|\bf P_{\nu}-P_{\nu'}\| \\
&=\sin(\theta_l) \label{l61}\\
&=\sin(\angle({\bf z},{\bf z}+{\bf P}_{\nu_0^{\bot}}{\bf n}))\\
&\le \|{\bf P}_{\nu_0^{\bot}}{\bf n}\|/\|\bf z\|\label{l62}\\
&\le \|{\bf n}\|/\|\bf z\|\\
&\le d,\end{aligned}$$ where (\[l61\]) is from a basic property of the principal angles, see for example [@optheory], and (\[l62\]) is from elementary geometry. The lemma is proved.
If $\nu\in d\mbox{-}\operatorname*{int}(\Omega_J)$, then by definition we have $$\label{leth21}
\nu'\in \Omega_J,\quad\forall d:\operatorname*{dist}(\nu,\nu')<d.$$ Now for any $\mathbf{z}\in \nu\setminus \{\mathbf{0}\}$, and $\mathbf{n}$ satisfying $\|\mathbf{n}\|<d\|\mathbf{z}\|$, there exist $\nu'$ such that $\mathbf{z}+\mathbf{n}\in \nu'$ and $\operatorname*{dist}(\nu,\nu')<d$ by Lemma \[lmth2\]. Define $\bf z' = z + n\in \nu'$. Since $\nu'\in\Omega_J$ by (\[leth21\]), we have $J({\bf z}'_T ) < J({\bf z}'_{T^c})$ for all $T$ such that $|T|\le k$, which is exactly (\[eq16\]).
Conversely, supposing RRC is satisfied for some $\bf A$ with $C=\frac{2(1-d)}{d\sigma_{\max}}$ for some $0<d<1$, we will show that $\mathcal{N}({\bf A})\in \frac{d}{1+d}\mbox{-}\operatorname*{int}(\Omega_J)$, which will recover the converse statement after the substitution $d\mapsto\frac{d}{1-d}$. Let $\nu:=\mathcal{N}(\bf A)$. It suffices to prove that the neighbourhood $U=\{\nu'~|~\operatorname*{dist}(\nu,\nu')<d/(1+d)\}$ is a subset of $\Omega_J$, i.e. any $\nu'$ in this neighbourhood satisfies the condition in (\[exactdef\]). This is because for any $\nu'\in U$ and $\mathbf{z'}\in \nu'\setminus \{\mathbf{0}\}$, one can find $\mathbf{z}:=\mathbf{P}_{\nu}\mathbf{z}'\in \nu$ such that $\|\mathbf{z}-\mathbf{z}'\|/\|\mathbf{z}'\|<d/(1+d)$, which implies that $\frac{1+d}{d}\|{\bf z-z'}\|<\|{\bf z'}\|\le\|{\bf z}\|+\|{\bf z-z'}\|$. Therefore $\|\mathbf{z}-\mathbf{z}'\|/\|\mathbf{z}\|<d$, and in particular $\bf z\neq 0$. Since $\mathbf{z}\in \mathcal{N}(\bf A)\setminus\{\bf 0\}$ and $({\bf A},k,J)$ satisfies RRC with $C=\frac{2(1-d)}{d\sigma_{\max}}$, setting $\bf n=z'-z$ in Theorem \[suf2\] shows that $J(\mathbf{z}'_T)<J(\mathbf{z}'_{T^c})$ for every $|T|\le k$. Hence $\nu'\subseteq \Omega_J$ follows from the arbitrariness of $\mathbf{z}'$, as desired.
Since $\operatorname*{int}(\Omega_J)=\bigcup_{d>0}d\mbox{-}\operatorname*{int}(\Omega_J)$, Theorem \[th2\] follows directly from Theorem \[th22\].
Proof of Corollary \[th5\] {#p_cont}
==========================
We first note the following basic fact about generic continuous functions. (It is stated in a slightly stronger and more complete manner than needed for proving Corollary \[th5\]).
\[cont\] Suppose $\mathcal{X},\mathcal{M}$ are metric spaces, and $\overline{\mathbb{R}}=\mathbb{R}\cup\{+\infty,-\infty\}$ be the extended real line. If $f:~\mathcal{X}\times \mathcal{M}\to\overline{\mathbb{R}}$ is continuous, then $g:~\mathcal{X}\to\overline{\mathbb{R}},~x\mapsto\sup_{y\in \mathcal{M}}f(x,y)$ is lower semicontinuous on $\mathcal{X}$. Further, if $\mathcal{M}$ is compact, then $g$ is also continuous.
The lower semi-continuity of $g$ follows from the fact that $g$ is defined as the supremum of a collection of continuous functions [@rudin P38 (c)]. To show that $g$ is also upper semicontinuous when $\mathcal{M}$ is compact, we will prove that $g$ is upper semicontinuous at an arbitrary $x_0\in \mathcal{X}$: let $y_0$ be a point in $\mathcal{M}$ such that $g(x_0)=f(x_0,y_0)$ (Here we used the compactness of $\mathcal{M}$). Suppose otherwise, that $g$ is not supper semicontinuous at $x_0$, then there exists $\epsilon> 0$ such that: $$\limsup_{x\to x_0}g(x) > g(x_0) + \epsilon.$$ This implies that we can find sequences $x_n,y_n(n\ge 1)$ such that $\lim_{n\to \infty}x_n=x_0$ and the following holds: $$f(x_n,y_n) > g(x_0) + \epsilon.$$ Since $\mathcal{M}$ is compact, we can find a subsequence $y_{n_k},(k\ge 1)$ converging to some point $y^*\in \mathcal{M}$. Hence $$\begin{aligned}
g(x_0)&=f(x_0,y_0)\nonumber\\
&\ge f(x_0,y^*)\nonumber\\
&=\lim_{k\to\infty}f(x_{n_k},y_{n_k})\nonumber\\
&\ge g(x_0)+\epsilon,\nonumber\end{aligned}$$ which is an apparent contradiction.
In the above proof, the assumption that $\mathcal{X},\mathcal{M}$ are metrical spaces rather than topological spaces is useful only when showing the existence of the sequences $x_n,y_n,(n\ge1)$. Therefore, the result actually holds when $\mathcal{X},\mathcal{M}$ are topological spaces satisfying the first countable theorem [@munk].
It then follows the following result about the null space constant $\theta_J$, now conceived as a map from $G_l(\mathbb{R}^n)$ to the real numbers:
\[le3\] If $F$ is continuous, then $\theta_{J}:G_{l}(\mathbb{R}^n)\to [0,+\infty)$ is a lower semicontinuous function. Further, $\theta_{\ell_p}:G_{l}(\mathbb{R}^n)\to [0,+\infty)$ is a continuous function.
It suffices to show that $\theta_{J}$ is lower semicontinuous or continuous on each $U_I$. Without loss of generality, we may assume that $I=\{1,\dots,l\}$. For generic $F$, let $\mathcal{X}=\mathbb{M}(n-l,l)$, $\mathcal{M}=\mathbb{R}^l\setminus\{\bf 0\}$, and $f: \mathcal{X}\times\mathcal{M}\to\overline{\mathbb{R}},(\mathbf{X},\mathbf{y})
\mapsto\frac{J(\mathbf{z}_T)}{J(\mathbf{z}_{T^c})}$, where $\bf z:=\left(
\begin{array}{c}
{\bf I} \\
{\bf X} \\
\end{array}
\right)y
$. Then $\theta_J(\phi_I^{-1}(\mathbf{X}))=\sup_{{\bf y}\in\mathcal{M}}f(\bf X,y)$, which by Lemma \[cont\] implies that the composition map $\theta_J\circ \phi_I^{-1}$ is lower semicontinuous. Since $\phi_I$ is a homeomorphism, we conclude that $\theta_J$ is also lower semicontinuous.
For the case of $\ell_p$-minimization, we can define $\mathcal{M}:=S^{l-1}$, while $\mathcal{X}$ and $f$ are as before. By homogeneity we still obtain $\theta_J(\phi_I^{-1}(\mathbf{X}))=\sup_{{\bf y}\in\mathcal{M}}f(\bf X,y)$. But since $\mathcal{M}$ is compact in this case, we conclude that $\theta_{\ell_p}$ is continuous.
The openness of $\Omega_{\ell_p}$ then follows easily, from the very definition of continuous functions: that the pre-images of open sets are open.
By Lemma \[le3\], function $\theta_{\ell_p}$ is continuous with respect to $\nu$. Since $\Omega_{\ell_p}$ is the pre-image of $(-\infty,1)$ under the continuous mapping of $\theta_{\ell_p}$ (Lemma \[le1\]), we conclude that $\Omega_{\ell_p}$ is open, hence $\Omega^r_{\ell_p}=\operatorname*{int}(\Omega_{\ell_p})=\Omega_{\ell_p}$.
Proof of assertions in Counter-example \[ex1\] {#appex}
==============================================
For any $\mathbf{w}\in\mathcal{N}$ we can write $\mathbf{w}=(xt,yt,zt)^\top$ for some $t\in\mathbb{R}$. Notice that $|zt|=|xt|+|yt|$, $|xt|<|zt|+|yt|$, $|yt|<|xt|+|zt|$, and $F$ is nondecreasing and strictly subadditive. Therefore for any $T$ such that $|T|=1$ we have: $$J(\mathbf{w}_T)<J(\mathbf{w}_{T^c}).$$ Hence NSP is satisfied, and ERC must hold. On the other hand, the above inequality fails under arbitrarily small perturbation: for any $0<d<1$, Taylor expansion yields $F((1-d)xt)+F(yt)=2(1-d)xt+2yt+o(t^2)=2zt-2dxt+o(t^2)$ and $F(zt)=2zt+o(t^2)$ (for small $t$), so there exist $t>0$ such that $$\label{eq5}
F((1-d)xt)+F(yt)<F(zt).$$ Now in Corollary \[corsuf2\], take $\mathbf{z}=(xt,yt,zt)^\top$, $T=\{3\}$, and $\mathbf{n}=(-dxt,0,0)$. On the one hand we have $\|\mathbf{n}\|/\|\mathbf{z}\|\le d$; on the other hand (\[eq16\]) doesn’t hold because of (\[eq5\]). Therefore RRC is not fulfilled as a result of Corollary \[corsuf2\].
Proof of Theorem \[probeq\] {#p_probeq}
===========================
Before the proof of Theorem \[probeq\], we shall make a digression on some concepts in real analysis.
[[@stein]]{} Suppose $E$ is a measurable set in $\mathbb{R}^L$, the *Lebesgue density* of $E$ at a point $\mathbf{x}\in \mathbb{R}^L$ is defined as $
\lim_{r\to 0}\frac
{\lambda(B(\mathbf{x},r)\cap E)}
{\lambda(B(\mathbf{x},r))}
$ where $\lambda$ denotes the Lebesgue measure. If the density exists and is equal to $1$, $\mathbf{x}$ is said to *have the Lebesgue density of $E$*.
The following result can be found in standard textbook on measure theory, such as [@stein Chapter 3, Corollary 1.5].
If $E$ is a measurable set in $\mathbb{R}^L$, then almost all $\mathbf{x}\in E$ (except for a set of Lebesgue measure zero) has the Lebesgue density of $E$.
\[defporous\] We say a set $E\subseteq\mathbb{R}^L$ is *porous* at a point $\mathbf{x}\in \mathbb{R}^L$, if there exists $r_0>0$, $0<\alpha< 1$ such that for each $0<r<r_0$, there exists some $\mathbf{y} \in \mathbb{R}^L$ such that the ball $B(\mathbf{y},\alpha r)\subseteq B(\mathbf{x},r)\setminus E$.
If $E\subseteq\mathbb{R}^L$ is porous at each point in $E$, then $E$ must be of measure zero (a direct consequence of Lebesgue density theorem) and of the first category. See for example [@brucknerthomson].
\[remporous\] As in many other problems from analysis, a trivial observation is that the “ball of radius $a$” (where $a=r$ or $\alpha r$) in Definition \[defporous\] can be replaced with, say, a “hypercube[^10] of edge length $a$” (with $\alpha$ taking a possibly different value), since any hypercube of edge length $a$ is contained in a ball of radius $\lambda_{\max}a$, and contains a ball of radius $\lambda_{\min}a$, where $0<\lambda_{\min}<\lambda_{\max}$ are some constants independent of $a$. Another observation based on the same mechanism is that porosity is preserved under invertible linear transforms, since the image of a unit ball under the linear transform must contain a ball whose radius is the least singular value of the linear transform. Further, since a smooth function can be locally approximated by a linear map by Taylor expansions, we see that any $C^{\infty}$ function which has a $C^{\infty}$ inverse also preserves porosity.
We shall adopt the notation in (\[ot\]). For any $\nu\in S\setminus\operatorname*{int}(\Omega_T)$, there exists a sequence $\nu^l\in \Omega^c_T, l=1,2,\dots$ with the properties:
1\) $\nu^l\to\nu$ as $l\to\infty$.\
2) For each $l$ there exist $\mathbf{z}^l\in \nu^l\setminus\{{\bf 0}\}$ such that $J(\mathbf{z}^l_T)\ge J(\mathbf{z}^l_{T^c})$.\
3) The sequence $\mathbf{\bar{z}}^l:=\mathbf{z}/\|\bf z\|$ converges to some $\mathbf{x}\in S^{n-1}$.
Property 1) is from the fact that $\nu$ is in the closure of $\Omega^c_T$. Property 2) is from the definition of $\Omega_T$. As for property 3), we note that the compactness of $S^{n-1}$ implies there exists a convergent subsequence of $\mathbf{\bar{z}}^l$. So if the $\mathbf{\bar{z}}^l$ sequence itself is not convergent, we can redefine $\mathbf{\bar{z}}^l$ to be its convergent subsequence, and then redefine the sequences $\nu^l$, $\mathbf{z}^l$ to be their corresponding subsequences. As a result, Properties 1), 2), 3) can always be satisfied by some sequence $\{\nu^l\}_{l=1}^{\infty}$. Notice that $$\begin{aligned}
\|\mathbf{P}_{\nu}\mathbf{x}-\mathbf{x}\|&\le \|\bar{\mathbf{z}}^l-\mathbf{x}\|+\|\mathbf{P}_{\nu}\mathbf{x}-\bar{\mathbf{z}}^l\|\\
&\le \|\bar{\mathbf{z}}^l-\mathbf{x}\|+\|\mathbf{P}_{\nu}\bar{\mathbf{z}}^l-\bar{\mathbf{z}}^l\|\\
&= \|\bar{\mathbf{z}}^l-\mathbf{x}\|+\|\mathbf{P}_{\nu}\bar{\mathbf{z}}^l-\mathbf{P}_{\nu^l}\bar{\mathbf{z}}^l\|\\
&\le \|\bar{\mathbf{z}}^l-\mathbf{x}\|+\|\mathbf{P}_{\nu}-\mathbf{P}_{\nu^l}\|\\
&\to 0,\quad\textrm{as~}l\to\infty,\end{aligned}$$ therefore $\|\mathbf{P}_{\nu}\mathbf{x}-\mathbf{x}\|=0$ and so $\mathbf{x}\in \nu$.
Suppose $\mathbf{x}=(x_1,\dots,x_n)^\top$. Since $\nu\in S$, at most $l-1$ of the entries of $\mathbf{x}$ can be zero. Hence there exists an $l$-element index set $I_0\subseteq\{1,\dots,n\}$ such that $\{x_i~|~i\in I_0\}$ has at least one non-zero element and $\{x_i~|~i\notin I_0\}$ has no zero element. Without loss of generality, let us assume that $I_0=\{1,\dots,l\}$ and $x_1\neq 0$. Consider the chart $({\bf B}\circ \phi_{I_0},U_{I_0})$, where ${\bf B}\circ\phi_{I_0}$ is the composite mapping of $\phi_{I_0}:U_{I_0}\to\mathbb{M}(n-l,l)$, and the right multiplication of matrix $$\mathbf{B}:=
\left(
\begin{array}{c|c}
x_1 & 0 \\
\hline
x_2 & ~ \\
\vdots & {\bf I}_{(l-1)\times(l-1)} \\
x_l & ~ \\
\end{array}
\right).$$ Then for each $\nu\in G_l(\mathbb{R}^n)$, it holds that $$\label{e83}
\pi(\left(
\begin{array}{c}
\mathbf{B} \\
\hline
\mathbf{B}\circ\phi_{I_0}(\nu) \\
\end{array}
\right)
)
=\pi(\left(
\begin{array}{c}
\mathbf{I} \\
\hline
\phi_{I_0}(\nu) \\
\end{array}
\right)
)
=\nu,$$ where we recall that $\pi$ is the projection map to the subspace spanned by the column vectors of a matrix. Notice that the first column of matrix $\left(
\begin{array}{c}
\mathbf{B} \\
\hline
\mathbf{B}\circ\phi_{I_0}(\nu) \\
\end{array}
\right)$ is $\mathbf{x}$, because its first $l$ elements of this column agree with $\mathbf{x}$ by definition of $\mathbf{B}$, and then the rest of the $n-l$ elements must also agree with $\mathbf{x}$ since $\mathbf{x}$ is a unique linear combination of columns of $\left(
\begin{array}{c}
\mathbf{B} \\
\hline
\mathbf{B}\circ\phi_{I_0}(\nu) \\
\end{array}
\right)$. Now define $$\begin{aligned}
\label{pv}
V:=\{\mathbf{M}\in\mathbb{M}(n-l,l)|~|M_{i1}|>|x_{i+l}|~\textrm{if}~i+l\in T {\rm~and~}
|M_{i1}|<|x_{i+l}|~\textrm{otherwise, where $1\le i\le n-l$}\}.\end{aligned}$$ Since by assumption $|x_{i+l}|>0$ for $i=1,\dots,n-l$, the set $V$ is not empty.
Next we shall show that for each $\mathbf{M}\in V$, we have $$\begin{aligned}
\label{star}
\pi(\left(
\begin{array}{c}
\mathbf{B} \\
\mathbf{M} \\
\end{array}
\right)
)
\notin \Omega_T,\end{aligned}$$ which, by setting $\nu:=(\mathbf{B}\circ\phi_{I_0})^{-1}(\mathbf{M})$ in (\[e83\]), will imply that $(\mathbf{B}\circ\phi_{I_0})^{-1}(\mathbf{M})\in U_{I_0}\setminus \Omega_T$, or $\mathbf{M}\in \mathbf{B}\circ\phi_{I_0}(U_{I_0}\setminus\Omega_T)$. This shows that $V\subseteq \mathbf{B}\circ\phi_{I_0}(U_{I_0}\setminus\Omega_T)$. Since $V$ is open, this in turn implies that $$\begin{aligned}
\label{89}
V\subseteq \operatorname*{int}(\mathbf{B}\circ\phi_{I_0}(U_{I_0}\setminus\Omega_T))
=\mathbf{B}\circ\phi_{I_0}(\operatorname*{int}(U_{I_0}\setminus\Omega_T))
=\mathbf{B}\circ\phi_{I_0}(U_{I_0}\setminus\overline{\Omega}_T).\end{aligned}$$ To see (\[star\]), consider a vector $\mathbf{c}_{\epsilon_1,\epsilon_2}:=(1+\epsilon_1s_1,0,\dots,0)^{\top}+\epsilon_2(0,s_2,\dots,s_l)^{\top}\in\mathbb{R}^l$, where $$\begin{aligned}
s_1:=\left\{
\begin{array}{cc}
1 & \textrm{if }1\in T; \\
-1 & \textrm{otherwise},
\end{array}
\right.\end{aligned}$$ and for $1<i\le l$, $$\begin{aligned}
s_i:=
\left\{
\begin{array}{cc}
\operatorname*{sign}(x_i) & \textrm{if }x_i\neq0, i\in T; \\
-\operatorname*{sign}(x_i) & \textrm{if }x_i\neq0, i\in T^c; \\
0 & \textrm{if }x_i=0, i\in T^c; \\
1 & \textrm{if }x_i=0, i\in T.
\end{array}
\right.\end{aligned}$$ Then define the vector $\mathbf{v}:=\left(
\begin{array}{c}
\mathbf{B} \\
\mathbf{M} \\
\end{array}
\right)
\mathbf{c}_{\epsilon_1,\epsilon_2}
\in \pi(\left(
\begin{array}{c}
\mathbf{B} \\
\mathbf{M} \\
\end{array}
\right)
)
$. For fixed $\mathbf{M}\in V$, we wish to show that there exist $0<\epsilon_1<<\epsilon_2<<1$ such that the components of $\mathbf{v}$ satisfy
a\) $|v_i|>|x_i|$, if $i\in T$;\
b) $|v_i|<|x_i|$, if $i\notin T$ and $|x_i|\neq 0$;\
c) $v_i=0$, if $i\notin T$ and $|x_i|= 0$.\
For $l<i\le n$, since $|x_i|\neq0$, we only have to verify a), b). By the construction of (\[pv\]), properties a) b) hold when $\epsilon_1=\epsilon_2=0$, hence they also hold for small enough $\epsilon_1,\epsilon_2$ by continuity. As for $1\le i\le l$, if $i\in T$, we compute $$\begin{aligned}
v_i&=(1+\epsilon_1)x_1, \textrm{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~for $i=1$;}\\
v_i&=\left\{\begin{array}{cc}
(1+\epsilon_1s_1)x_i+\operatorname*{sign}(x_i)\epsilon_2 & \textrm{ if } x_i\neq0, \\
\epsilon_2 & \textrm{if } x_i=0,
\end{array}
\right.
\textrm{for } 1<i\le l,\end{aligned}$$ and similarly if $i\notin T$, $$\begin{aligned}
v_i&=(1-\epsilon_1)x_1, \textrm{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~for $i=1$;}\\
v_i&=\left\{\begin{array}{cc}
(1+\epsilon_1s_1)x_i-\operatorname*{sign}(x_i)\epsilon_2 & \textrm{ if } x_i\neq0, \\
0 & \textrm{if } x_i=0,
\end{array}
\right.
\textrm{for } 1<i\le l,\end{aligned}$$ hence a), b), c) are guaranteed if $0<\epsilon_1\max_{1\le i\le l}|x_i|<\epsilon_2<\frac{1}{2}\min_{1\le i\le l:x_i\neq 0}|x_i|$. Therefore we can choose $0<\epsilon_1<<\epsilon_2<<1$ such that a), b), c) hold in either case. Since $\lim_{l\to\infty}\mathbf{\bar{z}}^l=\mathbf{x}$, by a), b), c) there exists $l$ such that $$\begin{aligned}
|v_i|\ge|\bar{z}^l_i|, \textrm{ if }i\in T;\\
|v_i|\le|\bar{z}^l_i|, \textrm{ if }i\notin T.\end{aligned}$$ Therefore from the monotonicity of $F$, we get $$\begin{aligned}
J(\|\mathbf{z}^l\|\mathbf{v}_T)&=\sum_{i\in T}F(\|\mathbf{z}^l\|v_i)\\
&\ge\sum_{i\in T}F(\|\mathbf{z}^l\|\bar{z}^l_i)\\
&=\sum_{i\in T}F(z^l_i)\\
&\ge\sum_{i\in T^c}F(z^l_i)\\
&=\sum_{i\in T^c}F(\|\mathbf{z}^l\|\bar{z}^l_i)\\
&\ge \sum_{i\in T^c}F(\|\mathbf{z}^l\|v_i)\\
&=J(\|\mathbf{z}^l\|\mathbf{v}_{T^c}).\end{aligned}$$ Since $\|\mathbf{z}^l\|\mathbf{v}\in\pi(\left(
\begin{array}{c}
\mathbf{B} \\
\mathbf{M} \\
\end{array}
\right)
)$, (\[star\]) is proved.
Finally, from (\[pv\]) we see that for each hypercube $Q\subseteq\mathbb{M}(m,l)$ centered at $\mathbf{B}\circ \phi_{I_0}(\nu)=\left(
\begin{array}{cc}
x_{l+1} & ~ \\
\vdots & \dots \\
x_n & ~ \\
\end{array}
\right)
$ with edge length less than $\min_{l<i\le n}\{|x_i|\}$, it must contain a hypercube $Q'\subseteq V$ whose edge length is half of $Q$, such that $\mathbf{B}\circ \phi_{I_0}(\nu)$ is a vertex of $Q'$. By (\[89\]), the set $V$ does not intersect with $\mathbf{B}\circ\phi_{I_0}(S\cap(\overline{\Omega}_T\setminus\operatorname*{int}(\Omega_T)))$, nor does $Q'$. Then we can take $\alpha=1/2$ in Remark \[remporous\], showing that $\mathbf{B}\circ\phi_{I_0}(S\cap(\overline{\Omega}_T\setminus\operatorname*{int}(\Omega_T)))$ is porous at $\mathbf{B}\circ \phi_{I_0}(\nu)$. But the map $\phi_I\circ\phi_{I_0}^{-1}\circ\mathbf{B}^{-1}:\mathbf{B}\circ\phi_{I_0}(S)\to\phi_{I}(S)$ and its inverse are $C^{\infty}$, therefore from the observation in Remark \[remporous\], the set $\phi_{I}(S\cap(\overline{\Omega}_T\setminus\operatorname*{int}(\Omega_T))$ is also porous at $\phi_{I}(\nu)$. Recalling that $\nu$ is arbitrarily chosen from $S\cap(\overline{\Omega}_T\setminus\operatorname*{int}(\Omega_T))$, we see $\phi_I(S\cap(\overline{\Omega}_T\setminus\operatorname*{int}(\Omega_T)))$ must be of measure zero and of the first category. Then by the observation made in (\[27\]), the proof is complete.
Proof of Corollary \[co2\] {#app_F}
==========================
Suppose $\mu(\Omega_J\setminus\Omega_J^r)=0$. Let $H\subseteq G_m(\mathbb{R}^n)$ be the set of orthogonal complements of the $l$-dimensional subspaces in $\Omega_J\setminus\Omega_J^r$. Since $G_m(\mathbb{R}^n)$ is isomorphic to $G_l(\mathbb{R}^n)$ (recall that $l:=n-m$), we have $\mu(H)=0$ as well[^11]. Then for each $U_I$ (here $I\subseteq\{1,\dots,n\}$ and $|I|=m$), $H\cap U_I$ is a measure zero subset of $U_I$. By the property of product measure we have that $\phi_I(H\cap U_I)\times \mathbb{M}(m,m)$ is a measure zero subset of $\mathbb{M}(l,m)\times\mathbb{M}(m,m)$. Without loss of generality, let’s assume that $I=\{1,\dots,m\}$. Define a $C^{\infty}$ map: $$\begin{aligned}
f:\mathbb{M}(l,m)\times\mathbb{M}(m,m)&\to \mathbb{M}(n,m)\\
(\mathbf{M},\mathbf{V})&\mapsto \left(
\begin{array}{c}
\mathbf{I} \\
\mathbf{M} \\
\end{array}
\right)
\mathbf{V}\end{aligned}$$ Then $\pi^{-1}(U_I\cap H)=f(\phi_I(U_I\cap H),\mathbb{M}(m,m))$ is a measure zero subset of $\pi^{-1}(U_I)$. Finally $\pi^{-1}(H)=\bigcup_{I}\pi^{-1}(H\cap U_I)$ is a measure zero subset of $\pi^{-1}(G_m(\mathbb{R}^n))\subseteq\mathbb{M}(n,m)$. This shows that $\mathcal{N}(\mathbf{A})\in(\Omega_J\setminus\Omega^r_J))$ only if $\mathbf{A}$ falls into a Lebesgue measure zero set on $\mathbb{M}(n,m)$, which is of probability zero if the probability distribution of $\mathbf{A}$ is absolutely continuous (with respect to the Lebesgue measure on $\mathbb{M}(n,m)$).
Proof of Theorem \[th1\] {#ap2}
========================
Since the value of $\theta_J$ depends on the null space of the measurement matrix, in the following it is considered as a function of $\nu\in G_{l}(\mathbb{R}^n)$.
Let $F\in\mathcal{M},q\in(0,1]$. If $\lim_{x\downarrow 0}F(x)/x^q$ or $\lim_{x\to\infty}F(x)/x^q$ exists and is positive, then $\theta_{\ell_q}\le\theta_J$ for any $\nu\in G_{l}(\mathbb{R}^n)$.
We only prove for the case where $\lim_{t\downarrow 0}F(t)/t^q$ exists and is positive, because the case where $\lim_{t\to \infty}F(t)/t^q$ exists and is positive is essentially similar. By definition we only have to prove the following for any $\mathbf{z}\in \nu\setminus \{\textbf{0}\}$ and $T$ satisfying $|T|\le k$: $$\label{tosee}
\frac{\|\mathbf{z}_{T}\|^q_q}{\|\mathbf{z}_{T^c}\|^q_q}\le \theta_{J}.$$ Notice that for any $t\in \mathbb{R}$, vector $t\mathbf{z}$ still belongs to $\mathcal{N}(\mathbf{A})$, hence $$\begin{aligned}
\textrm{left side of}(\ref{tosee})&=\lim_{t\downarrow 0}
\frac{J(t\mathbf{z}_{T})}{J(t\mathbf{z}_{T^c})}\label{lhospital}\\
&\le\theta_{J},\label{supdef}\end{aligned}$$ where (\[lhospital\]) is because $\lim_{t\downarrow 0}F(t)/t^q$ exists and is positive, and (\[supdef\]) is from the definition of supremum.
\[cl\] $\overline{\Omega_{\ell_q}}=\{\nu~|~\theta_{\ell_q}\le 1\}$.
First we prove that $\overline{\Omega_{\ell_q}}\subseteq\{\nu~|~\theta_{\ell_q}\le 1\}$. This is because Lemma \[le3\] shows that $\{\nu~|~\theta_{\ell_q}\le 1\}$ is closed, and we have the inclusion $\Omega_{\ell_q}\subseteq\{\nu~|~\theta_{\ell_q}\le 1\}$. On the other hand, from the definition of $\theta_{\ell_q}$ it is obvious that $\{\nu~|~\theta_{\ell_q}<1\}$ is dense at any point in $\{\nu~|~\theta_{\ell_q}\le 1\}$, which means that $\{\nu~|~\theta_{\ell_q}\le 1\}\subseteq\overline{\Omega_{\ell_q}}$. The proof is complete.
Given $\nu\in G_{l}(\mathbb{R}^n)$, if $\theta_{\ell_q}\le\theta_J$, then $\Omega_J\subseteq \overline{\Omega_{\ell_q}}$.
By Lemma \[nspcond2\], the assumptions imply that $$\Omega_J\subseteq\{\nu~|~\theta_J\le 1\}\subseteq\{\nu~|~\theta_{\ell_q}\le 1\}=\overline{\Omega_{\ell_q}}.$$
Theorem \[th1\] then follows immediately from the following lemma:
$$\mu(\overline{\Omega_{\ell_q}})
=\mu(\Omega_{\ell_q}).$$
This follows immediately from Theorem \[probeq\], with $J$ being the $\ell_p$ norm.
Proof of Proposition \[prop1\] {#prop1_proof}
==============================
Suppose $0<t_1<t_2$. From concavity we have $$F(t_1)\ge \frac{t_2-t_1}{t_2}F(0)+\frac{t_1}{t_2}F(t_2)=\frac{t_1}{t_2}F(t_2).$$ Therefore $F(t_1)/t_1\ge F(t_2)/t_2$, which implies that $F(t)/t$ is non-increasing on $(0,+\infty)$.
For arbitrary $0<t_1<t_2$ we have $$\begin{aligned}
\frac{F(t_1)}{t}&=\frac{F(t_1)}{t_1^p}\cdot t_1^{p-1}\\
&\ge \frac{F(t_2)}{t_2^p}\cdot t_1^{p-1}\\
&=\frac{F(t_2)}{t_2}\cdot (\frac{t_2}{t_1})^{1-p}\\
&\ge \frac{F(t_2)}{t_2}\end{aligned}$$ where the inequalities used the fact that $F(t)/t^p$ is non-increasing, and that $1-p\ge 0$. Thus $F(t)/t$ is also non-increasing.
For arbitrary $t_1,t_2>0$, the the assumption that $F(t)/t$ is non-increasing implies that $$\begin{aligned}
F(t_1+t_2)&=\frac{F(t_1+t_2)}{t_1+t_2}\cdot(t_1+t_2)\\
&\ge (\frac{t_1}{t_1+t_2}\frac{F(t_1)}{t_1}+\frac{t_2}{t_1+t_2}\frac{F(t_2)}{t_2})\cdot(t_1+t_2)\\
&=F(t_1)+F(t_2).\end{aligned}$$ Also $F(t_1+t_2)=F(t_1)+F(t_2)$ clearly holds in the case where $t_1=0$ or $t_2=0$. Thus $F$ is subadditive.
[^1]: J. Liu is now with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: jingbo@princeton.edu). Main part of this work was done when he was with the Department of Electronic Engineering, Tsinghua University, Beijing, 100084 China. J. Jin and Y. Gu are both with the Department of Electronic Engineering, Tsinghua University (e-mail: jinjian620@gmail.com, gyt@tsinghua.edu.cn). The corresponding author of this paper is Yuantao Gu.
This paper was presented in part at ISIT 2013.
[^2]: Strictly speaking, the $\ell_0$ norm and $\ell_p(0<p<1)$ norm defined here do not satisfy the definition of norm in mathematics.
[^3]: For our purpose, the definition of sparseness measure in this paper does not need to require that $F(x)/x$ is non-increasing. A comparison with other definitions of the sparseness measure is given in Section \[comp\], Part B.
[^4]: Here the notation ‘$\setminus$’ denotes the set minus.
[^5]: $\mathbb{Q}$ denotes the set of rational numbers
[^6]: The measure $\mu_1$ is said to be absolutely continuous with respect to the measure $\mu_2$ if $\mu_2(E)=0$ implies $\mu_1(E)=0$, for arbitrary measurable set $E$.
[^7]: The assumption of $F(t)/t$ being non-increasing guarantees that $F$ is subadditive, as shown in Proposition \[prop1\].
[^8]: For $\mathbf{x}\in \mathbb{R}^{|T|}$, we denote by $\mathbf{x}^T\in \mathbb{R}^n$ the $n$-vector supported on $T$ satisfying $(\mathbf{x}^T)_T=\mathbf{x}$.
[^9]: Here $\oplus$ denotes the direct sum of linear subspaces.
[^10]: For convenience, in this proof by a “hypercube” we are always referring to an open hypercube in which each edge is parallel to one of the basis vectors. For example, a hypercube in $\mathbb{M}(m,l)$ centered at the origin with edge length $a>0$ is the set $\{\mathbf{M}\in\mathbb{M}(m,l)\mid |M_{i,j}|< a/2,\textrm{ for $1\le i\le m$, $1\le j\le l$}\}$.
[^11]: Here we abused the notation by denoting $\mu$ the Haar measure both on $G_l(\mathbb{R}^n)$ and on $G_m(\mathbb{R}^n)$, since the two manifolds are isomorphic.
|
---
abstract: 'Motivated by the matrix form of the DDVV conjecture in submanifold geometry which is an optimal inequality involving norms of commutators of several real symmetric matrices and takes an important role in the proof of the well-known Simons inequality for closed minimal submanifolds in spheres, in this paper we first derive a similar optimal inequality of real skew-symmetric matrices, then we apply it to establish a Simons-type inequality for Riemannian submersions, which shows another “evidence" of the duality between submanifold geometry and Riemannian submersions.'
address: 'School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, P.R. CHINA'
author:
- Jianquan Ge
title: An optimal matrix inequality and its applications to geometry of Riemannian submersions
---
Introduction {#sec1}
============
Let $M^n$ be an immersed submanifold of a real space form $N^{n+m}(c)$ of constant sectional curvature $c$. Given an orthonormal basis $\{e_1,\cdots,e_n\}$ (resp. $\{\xi_1,\cdots,\xi_m\}$) of $T_pM$ (resp. $T^{\bot}_pM$), the normalized scalar curvature $\rho$ and the normal scalar curvature $\rho^{\bot}$ of $M^n$ at p are defined by $$\rho=\frac{2}{n(n-1)}\sum^n_{1=i<j}\langle R(e_i,e_j)e_j,e_i\rangle,$$ $$\rho^{\bot}=\frac{2}{n(n-1)}\Big(\sum^n_{1=i<j}\sum^m_{1=r<s}\langle R^{\bot}(e_i,e_j)\xi_r, \xi_s\rangle^2\Big)^{\frac{1}{2}}=\frac{2}{n(n-1)}|R^{\bot}|,$$ where $R$ and $R^{\bot}$ are curvature tensors of the tangent and normal bundles of $M$ respectively. Denote by $h$ the second fundamental form and $H=\frac{1}{n}Tr(h)=\frac{1}{n}\sum_{i=1}^nh(e_i,e_i)$ the mean curvature vector field. The DDVV conjecture raised by [@DDVV] says that there is a pointwise inequality among $\rho$, $\rho^{\bot}$ and $|H|^2$ as the following: $$\label{DDVV ineq}
\rho+\rho^{\bot}\leq |H|^2+c.$$ Due to the Gauss and Ricci equations, this conjecture can be translated into the following algebraic inequality (cf. [@DFV]): $$\label{DDVV sym}
\sum_{r,s=1}^m\|[B_r,B_s]\|^2\leq \Big(\sum_{r=1}^m\|B_r\|^2\Big)^2,$$ where $\{B_1, \cdots ,B_m\}$ are arbitrary real symmetric $(n\times
n)$-matrices, $[\cdot,\cdot]$ is the commutator operator and $\|\cdot\|$ is the standard norm of matrix.
The inequality (\[DDVV sym\]) (and thus the DDVV conjecture (\[DDVV ineq\])) has been proved independently and differently by [@GT; @Lu]. In particular, the equality condition given in [@GT] shows that the inequality (\[DDVV sym\]) is an optimal inequality. As for the classification problem of submanifolds attaining the equality of (\[DDVV ineq\]) everywhere, we refer to [@DT] for a big advance. In this paper, by a similar method as in [@GT], we obtain the following optimal inequality of real skew-symmetric matrices in the form of the inequality (\[DDVV sym\]), which has been previously reviewed in the survey paper [@GT2].
Throughout this paper, a $K:=O(n)\times O(m)$ action on $(B_1,\cdots,B_m)$ means that $$(P,R)\cdot
(B_1,\cdots,B_m):=(PB_1P^t,\cdots,PB_mP^t)\cdot R,\quad for~~
(P,R)\in K.$$
\[thm1\] Let $B_1,\cdots,B_m$ be ($n\times n$) real skew-symmetric matrices.\
$(i)$ If $n=3$, then we have $$\sum_{r,s=1}^m\|[B_r,B_s]\|^2\leq
\frac{1}{3}\Big(\sum_{r=1}^m\|B_r\|^2\Big)^2,$$ where the equality holds if and only if under some $K$ action all $B_r$’s are zero except $3$ matrices which can be written as $$C_1:=\left(\begin{array}{ccc}0&
\lambda& 0\\-\lambda& 0& 0\\0& 0& 0
\end{array}\right),\quad C_2:=\left(\begin{array}{ccc}0&
0& \lambda\\0& 0& 0\\-\lambda& 0& 0
\end{array}\right),\quad C_3:=\left(\begin{array}{ccc}0& 0& 0\\0&
0& \lambda\\0& -\lambda& 0
\end{array}\right).$$ $(ii)$ If $n\geq 4$, then we have $$\sum_{r,s=1}^m\|[B_r,B_s]\|^2\leq
\frac{2}{3}\Big(\sum_{r=1}^m\|B_r\|^2\Big)^2,$$ where the equality holds if and only if under some $K$ action all $B_r$’s are zero except $3$ matrices which can be written as $diag(D_1, 0)$, $diag(D_2, 0)$, $diag(D_3, 0)$, where $0\in M(n-4)$ is the zero matrix of order $n-4$ and $$D_1:=\left(\begin{array}{cccc}0& \lambda& 0&0\\-\lambda& 0& 0&0\\0&
0& 0&\lambda\\0&0&-\lambda&0
\end{array}\right),
D_2:=\left(\begin{array}{cccc}0& 0&\lambda& 0\\0& 0&
0&-\lambda\\-\lambda& 0& 0&0\\0&\lambda&0&0
\end{array}\right),
D_3:=\left(\begin{array}{cccc}0& 0&0&\lambda\\0& 0&\lambda&0\\0&
-\lambda& 0&0\\-\lambda&0&0&0
\end{array}\right).$$
In sight of the geometric origin of the inequality (\[DDVV sym\]), *i.e.*, the DDVV inequality (\[DDVV ineq\]) in submanifold geometry, we get interested in applications to geometry of this “dual" matrix inequality. Our attention will be focused on the geometry of Riemannian submersions which in some sense is also a “dual" theory of submanifold geometry. It turns out rather inspiring that, in analogy with the important role the symmetric matrix inequality takes in the proof of the well-known Simons inequality for closed minimal submanifolds in spheres (cf. [@chern; @Lu; @Simons]), the skew-symmetric matrix inequality also takes crucial effect in deducing a Simons-type inequality for Riemannian submersions. In order to state the result we first recall some notions about Riemannian submersions. The notions in Chapter 9 of the book [@Be] will be used throughout this paper.
Let $M^{n+m}$ and $B^n$ be (connected) Riemannian manifolds. A smooth map $\pi: M\rightarrow B$ is called a *Riemannian submersion* if $\pi$ is of maximal rank and $\pi_{*}$ preserves the lengths of horizontal vectors, *i.e.*, vectors orthogonal to the fibre $\pi^{-1}(b)$ for $b\in B$. Let $\mathscr{V}$ denote the *vertical distribution* consisting of vertical vectors (tangent to the fibres) and $\mathscr{H}$ denote the *horizontal distribution* consisting of horizontal vectors on $M$. The corresponding projections from $TM$ to $\mathscr{V}$ and $\mathscr{H}$ are denoted by the same characters. For Riemannian submersions there are two fundamental tensors $T$ and $A$ on $M$ defined by O’Neill [@O] as follows. For vector fields $E_1$ and $E_2$ on $M$, $$\label{A T}
\begin{array}{ll}
T_{E_1}E_2:=\mathscr{H}D_{\mathscr{V}E_1}\mathscr{V}E_2+\mathscr{V}D_{\mathscr{V}E_1}\mathscr{H}E_2,\\
A_{E_1}E_2:=\mathscr{H}D_{\mathscr{H}E_1}\mathscr{V}E_2+\mathscr{V}D_{\mathscr{H}E_1}\mathscr{H}E_2,&
\end{array}$$ where $D$ is the Levi-Civita connection on $M$. In fact, $T$ is the second fundamental form along each fibre if it is restricted to vertical vectors, while $A$ measures the obstruction to integrability of the horizontal distribution $\mathscr{H}$ and hence it is called the *integrability tensor* of $\pi$. Moreover, some analogues of the Gauss-Codazzi equations for a Riemannian submersion obtained by O’Neill [@O] are expressed in terms of $T$ and $A$ as well as their covariant derivatives. These equations will be recovered in Section \[simons-sec\] by moving frame method, which is an effective method firstly used to the study of Riemannian submersions though widely adopted in submanifold geometry. More details about $T$ and $A$ can be found in [@Be; @O]. Next we introduce the notion of *Yang-Mills* which has been intensely studied both in physics and in mathematics and also found important for Einstein Riemannian submersions (see for example [@At-Hi-Si; @Be; @Don] and references therein). Here we use the presentation given in [@Be]. Let $X_1,\cdots,X_n$ be a local orthonormal basis of the horizontal distribution $\mathscr{H}$. Define a co-differential operator $\check{\delta}$ over tensor fields on $M$ by $$\check{\delta}E:=-\sum_{i=1}^n(D_{X_i}E)_{X_i}.$$ Then we say that $\mathscr{H}$ satisfies the *Yang-Mills* condition if, for any vertical vector $U$ and any horizontal vector $X$, we have $$\langle\check{\delta}A(X),U\rangle-\langle A_X,T_U\rangle=0,$$ where the bracket $\langle\cdot,\cdot\rangle$ denotes the metric of $M$ and also its induced metric on tensors. As pointed out in [@Be], this condition depends only on $\mathscr{H}$ and the metric of $B$ and not on the family of metrics on the fibres. By properties of $T$ and $A$, it is not hard to verify that when the fibres are totally geodesic, *i.e.*, $T=0$, this condition is equivalent to $$\check{\delta}A=0,$$ which is one of the three sufficient and necessary conditions for $M$ to be Einstein in this case. To be coherent with that in [@Be], we define the square norm of $A$ by $$\label{A norm}
|A|^2:=\sum_{i,j=1}^n\langle A_{X_i}X_j,A_{X_i}X_j\rangle=\sum_{i=1}^n\sum_{r=1}^m\langle A_{X_i}U_r,A_{X_i}U_r\rangle,$$ where $\{U_1,\cdots,U_m\}$ is a local orthonormal basis of the vertical distribution $\mathscr{V}$. Besides several references cited in [@Be], it is noteworthy that the square norm of $A$ has been also studied by Chen ([@Chen], *etc.*) who denoted it by $\breve{A}_{\pi}$ and obtained its sharp upper bound for an arbitrary isometric immersion from $M$ (with totally geodesic fibres) into a unit sphere in terms of square norm of the mean curvature of the immersion.
Now we are ready to state the main result as follows. For $x\in M$, we denote by $\check{\kappa}(x)$ the largest eigenvalue of the curvature operator $\check{R}:\bigwedge^2TB\rightarrow\bigwedge^2TB$ of $B$ at $\pi(x)\in B$, $\check{\lambda}(x)$ the lowest eigenvalue of the Ricci curvature $\check{r}$ of $B$ at $\pi(x)\in B$ (thus $\check{\kappa}$, $\check{\lambda}$ are constant along any fibre), and $\hat{\mu}(x)$ the largest eigenvalue of the Ricci curvature $\hat{r}$ of the fibre at $x$.
\[Thm-simonstype ineq\] Let $\pi: M^{n+m}\rightarrow B^n$ be a Riemannian submersion with totally geodesic fibres and Yang-Mills horizontal distribution, i.e., $T=0$ and $\check{\delta}A=0$. Suppose that $M$ is closed. Then the following cases hold:
- If $n=2$, then we have $$\int_M~|A|^2\hat{\mu}~dV_M\geq0;$$
- If $m=1$, then we have $$\int_M~|A|^2(\check{\kappa}-\check{\lambda})~dV_M\geq0;$$
- If $m\geq2$ and $n=3$, then we have $$\int_M~|A|^2(\frac{1}{6}|A|^2+2\hat{\mu}+\check{\kappa}-\check{\lambda})~dV_M\geq0;$$
- If $m\geq2$ and $n\geq4$, then we have $$\int_M~|A|^2(\frac{1}{3}|A|^2+2\hat{\mu}+\check{\kappa}-\check{\lambda})~dV_M\geq0.$$
Moreover, if $A\neq0$, or equivalently, $M$ is not locally a Riemannian product $B\times F$, then we have the following conclusions about the equality conditions:
- In each case, if the equality holds, then each fibre has flat normal bundle in $M$ and $|A|^2\equiv Const=:C>0$, which implies further the following:
- In case (i), $\hat{\mu}\equiv0$;
- In case (ii), $\check{\kappa}-\check{\lambda}\equiv0$;
- In case (iii), $\hat{\mu}\equiv\frac{1}{12}C$, $\check{\kappa}-\check{\lambda}\equiv\frac{-1}{3}C$;
- In case (iv), $\hat{\mu}\equiv\frac{1}{6}C$, $\check{\kappa}-\check{\lambda}\equiv\frac{-2}{3}C$.
- If the equality in (iii) or (iv) holds, then $m\geq3$ and at each point of $M$ there exist an orthonormal vertical basis $\{U_1,\cdots,U_m\}$ and an orthonormal horizontal basis $\{X_1,\cdots,X_n\}$ such that the $(n\times n)$ skew-symmetric matrices $$A^r:=\Big(\langle A_{X_i}U_r,X_j\rangle\Big)_{n\times n},\quad r=1,\cdots,m,$$ are in the forms of the matrices in the equality conditions of (i) or (ii) of Theorem \[thm1\] respectively. Furthermore, under these basis, the following decompositions hold $$\begin{aligned}
&&\hat{r}=\hat{\mu}I_3\oplus \hat{r}', \nonumber\\
&&\check{R}\equiv\check{\kappa}I_3,\quad
\check{r}\equiv2\check{\kappa}I_3,\quad in~~case~~(iii),\nonumber\\
&&\check{R}=\check{\kappa}I_6\oplus \check{R}',\quad
\check{r}\equiv\check{\lambda}I_4\oplus\check{r}',\quad
in~~case~~(iv),\nonumber\end{aligned}$$ where $\hat{r}'=\hat{r}|_{span\{U_4,\cdots,U_m\}}$, $\check{R}'=\check{R}|_{span\{X_i\wedge X_j|1\leq i\leq n,~5\leq
j\leq n\}}$, and $\check{r}'=\check{r}|_{span\{X_5,\cdots,X_n\}}$. In particular, when $m=3$, the fibres have constant sectional curvature. Similarly, when $3\leq n\leq5$, the base manifold $B^n$ has constant sectional curvature. More precisely and specifically, we have the following (c-d).
- When $m=3$, if the equality in (iii) holds, then there exist some $a>0$ such that
- all fibres are isometric to a manifold $F^3$ of constant sectional curvature $a$;
- the base manifold $B^3$ has constant sectional curvature $8a$;
- the following identities hold: $$\begin{aligned}
&&|A|^2\equiv24a,\nonumber\\
&&K_{rs}\equiv a,\quad K_{ij}\equiv-4a,\quad
K_{ir}=\Big\{\begin{array}{ll} 0&
for~~(i,r)=(1,3),(2,2),(3,1)\\4a&otherwise,
\end{array}\nonumber\\
&&R_{rs}\equiv10a\delta_{rs},\quad R_{ij}\equiv0,\quad
R_{ir}\equiv0,\nonumber\end{aligned}$$ where $K_{rs}$, $K_{ij}$, $K_{ir}$ (resp. $R_{rs}$, $R_{ij}$, $R_{ir}$) are sectional curvatures (resp. Ricci curcatures) of $M$ on the $2$-planes spanned by $\{U_r,U_s\}$, $\{X_i,X_j\}$, $\{X_i,U_r\}$, respectively, under the basis $\{U_r\}$ and $\{X_i\}$ given in case (b).
- When $m=3$, if the equality in (iv) holds, then there exist some $a>0$ such that all fibres are isometric to a manifold $F^3$ of constant sectional curvature $a$. In addition,
- if $n=4$, then the submersion $\pi$ is covered by the Hopf fibration $\pi_0:S^7(\frac{1}{\sqrt{a}})\rightarrow
S^4(\frac{1}{2\sqrt{a}})$, i.e., there are two covering maps $\pi_1: S^7(\frac{1}{\sqrt{a}})\rightarrow M^7$ and $\pi_2: S^4(\frac{1}{2\sqrt{a}})\rightarrow B^4$ such that $\pi_2\circ\pi_0=\pi\circ\pi_1$;
- if $n=5$, then the base manifold $B^5$ has constant sectional curvature $\frac{8}{3}a$, and the following identities hold (with the same notations as in (c3)): $$\begin{aligned}
&&|A|^2\equiv12a,\nonumber\\
&&K_{rs}\equiv a,\quad
K_{ij}=\Big\{\begin{array}{ll}
\frac{-1}{3}a & for~~1\leq i< j\leq4\\
\frac{8}{3}a & for~~1\leq i< j=5,
\end{array}\quad
K_{ir}=\Big\{\begin{array}{ll}
a & for~~1\leq i\leq4\\
0 & for~~i=5,
\end{array}\nonumber\\
&&R_{rs}\equiv6a\delta_{rs},\quad
R_{ij}=\Big\{\begin{array}{ll}
\frac{14}{3}a\delta_{ij} & for~~1\leq i,j\leq4\\
\frac{32}{3}a\delta_{ij} & for~~1\leq i\leq j=5,
\end{array}\quad
R_{ir}\equiv0.\nonumber\end{aligned}$$
As we mentioned previously, the Yang-Mills condition is implied by the Einstein condition of $M$ when the fibres are totally geodesic. Therefore, examples satisfying our assumptions of the theorem are plentiful (cf. [@Be]). Note that the corresponding pointwise inequalities with the same equality conclusions also hold when $M$ is not closed provided that $|A|^2$ is constant on $M$, which is also implied by the Einstein condition of $M$.
Besides the classification problem, searching examples of Riemannian submersions in (c) and (d2) of the theorem might make sense to itself. For instance, the base manifold $B^3$ can not be simply connected in case (c) due to the facts that any principal $G$-bundle over $S^3$ is trivial if $G$ is a Lie group (in which case $A\equiv0$) and that any Riemannian submersion from a complete manifold $M$ with totally geodesic fibres is a fibre bundle associated to a principal $G$-bundle for some Lie group $G$ (cf. Remark 9.57 in [@Be], and [@He; @Na]). Therefore, searching examples in (c) should start with a non-simply-connected 3-dimensional base manifold of constant sectional curvature.
To conclude the introduction, we remark that as the Chern conjecture, the classification of the equality case and Peng-Terng pinching theorems based on the Simons inequality in submanifold geometry (cf. [@chern; @CCK; @DX; @Law; @Lu; @Peng-T1; @Peng-T2], *etc.*), one can now ask the “dual" version for Riemannian submersions with square norm of the integrability tensor $A$ instead of square norm of the second fundamental form.
DDVV-type skew-symmetric matrix inequality
==========================================
Notations and preparing lemmas
------------------------------
Throughout this section, we denote by $M(m,n)$ the space of $m\times n$ real matrices, $M(n)$ the space of $n\times n$ real matrices and $\mathfrak{o}(n)$ the $N:=\frac{n(n-1)}{2}$ dimensional subspace of skew-symmetric matrices in $M(n)$. 0.05cm For every $(i,j)$ with $1\leq
i<j\leq n$, let $\tilde{E}_{ij}:=\frac{1}{\sqrt{2}}(E_{ij}-E_{ji})$, where $E_{ij}\in M(n)$ is the matrix with $(i,j)$ entry $1$ and all others $0$. Clearly $\{\tilde{E}_{ij}\}_{i<j}$ is an orthonormal basis of $\mathfrak{o}(n)$. Let us take an order of the indices set $S:=\{(i,j)| 1\leq i< j\leq n\}$ by $$\label{order}
(i,j)<(k,l) \hskip 0.2cm if\hskip 0.1cm and\hskip 0.1cm only\hskip
0.1cm if \hskip 0.2cm i<k\hskip 0.1cm or\hskip 0.1cm i=k<j<l.$$ In this way we can identify $S$ with $\{1,\cdots,N\}$ and write elements of $S$ in Greek, i.e. for $\alpha=(i,j)\in S$, we can say $1\leq\alpha\leq N$.
For $\alpha=(i,j)<(k,l)=\beta$ in $S$, direct calculations show that $$\label{E tilde norm}
\|[\tilde{E}_{\alpha}, \tilde{E}_{\beta}]\|^2=
\begin{cases}
\frac{1}{2}, \quad i<j=k<l ~ or ~
i=k<j<l ~ or ~ i<k<j=l;\\
0, \quad otherwise,
\end{cases}$$ and for any $\alpha, \beta\in S,$ $$\label{E tilde inner}
\sum_{\gamma\in S}~ \langle~ [\tilde{E}_{\alpha},
\tilde{E}_{\gamma}],~ [\tilde{E}_{\beta}, \tilde{E}_{\gamma}]~
\rangle= (n-2)\delta_{\alpha\beta},$$ where $\delta_{\alpha\beta}=\delta_{ik}\delta_{jl}$, and $\langle
\cdot , \cdot\rangle$ is the standard inner product of $M(n)$.
Let $\{\tilde{Q}_{\alpha}\}_{\alpha\in S}$ be any orthonormal basis of $\mathfrak{o}(n)$. There exists a unique orthogonal matrix $Q\in
O(N)$ such that $(\tilde{Q}_1,\cdots,\tilde{Q}_N)=(\tilde{E}_1,\cdots,\tilde{E}_N)Q$, i.e. $\tilde{Q}_{\alpha}=\sum_{\beta}q_{\beta\alpha}\tilde{E}_{\beta}$ for $Q=(q_{\alpha\beta})_{N\times N}$. If we set $\tilde{Q}_{\alpha}=(\tilde{q}^{\alpha}_{ij})_{n\times n}$, then $\tilde{q}^{\alpha}_{ij}=-\tilde{q}^{\alpha}_{ji}=
\frac{1}{\sqrt{2}}q_{\beta\alpha}$ for $\beta=(i,j)\in S$. Henceforth, this correspondence between an orthonormal basis $\{\tilde{Q}_{\alpha}\}_{\alpha\in S}$ of $\mathfrak{o}(n)$ and an orthogonal matrix $Q\in O(N)$ is regarded known.
Let $\lambda_1,\cdots,\lambda_{[\frac{n}{2}]}$ be $[\frac{n}{2}]$ real numbers satisfying $\sum_i\lambda_i^2=\frac{1}{2}$ and $\lambda_1\geq \cdots\geq \lambda_{[\frac{n}{2}]}\geq 0$. Denote by $I:=\{(i,j)\in S|(\lambda_i+\lambda_j)^2>\frac{2}{3}\}$ and $n_0$ the number of elements of $I$. It is easily seen that $n_0=0$ when $n=3$. Moreover, we have
\[lem1\] If $I$ is not empty, i.e. $n_0\geq 1$, then $$I=\{1\}\times \{2,\cdots,n_0+1\}, \quad n_0+1\leq [\frac{n}{2}].$$
Obviously, by the assumptions of $\lambda_i$’s, $(1,2)\in I$ if $I$ is not empty. It suffices to prove that $(2,3)$ is not in $I$. Otherwise, we have $$(\lambda_1+\lambda_2)^2\geq(\lambda_1+\lambda_3)^2\geq (\lambda_2+\lambda_3)^2>\frac{2}{3},$$ and thus $$4(\lambda_1^2+\lambda_2^2+\lambda_3^2)\geq(\lambda_1+\lambda_2)^2+(\lambda_1+\lambda_3)^2+(\lambda_2+\lambda_3)^2>2,$$ which contradicts with $\lambda_1^2+\lambda_2^2+\lambda_3^2\leq\sum_i\lambda_i^2=\frac{1}{2}.$
\[lem2\] We have $$\sum_{(i,j)\in I}\Big((\lambda_i+\lambda_j)^2-\frac{2}{3}\Big)\leq \frac{1}{3},$$ where the equality holds if and only if $n_0=1$, $\lambda_1=\lambda_2=\frac{1}{2}$ and all other $\lambda_j$’s 0.
By Lemma \[lem1\], $$\begin{aligned}
\sum_{(i,j)\in I}[(\lambda_i+\lambda_j)^2-\frac{2}{3}]
&=&\sum_{j=2}^{n_0+1}(\lambda^2_1+\lambda^2_j+2\lambda_1\lambda_j)-\frac{2}{3}n_0\nonumber\\
&=&n_0\lambda^2_1+\sum_{j=2}^{n_0+1}\lambda^2_j+2\lambda_1\sum_{j=2}^{n_0+1}\lambda_j-\frac{2}{3}n_0\nonumber\\
&\leq&(n_0+1)\lambda^2_1+\sum_{j=2}^{n_0+1}\lambda^2_j+\Big(\sum_{j=2}^{n_0+1}\lambda_j\Big)^2-\frac{2}{3}n_0\nonumber\\
&\leq&(n_0+1)\Big(\lambda^2_1+\sum_{j=2}^{n_0+1}\lambda^2_j\Big)-\frac{2}{3}n_0\nonumber\\
&\leq&(n_0+1)\sum_{i}\lambda^2_i-\frac{2}{3}n_0=
\frac{n_0+1}{2}-\frac{2}{3}n_0\leq \frac{1}{3},\nonumber\end{aligned}$$ where the equality condition is easily seen from the proof.
\[lem3\] For any $Q\in O(N)$, $\alpha\in S$ and any subset $J_{\alpha}\subset S$, we have $$\sum_{\beta\in J_{\alpha}}\Big( \|[\tilde{Q}_{\alpha},
\tilde{Q}_{\beta}]\|^2-\frac{2}{3}\Big)\leq \frac{2}{3}.$$
Given $\alpha\in S$, under some $O(n)\subset K$ action, without loss of generality, we can assume $$\tilde{Q}_{\alpha}=diag\left(\left(\begin{array}{cc}0& \lambda_1
\\-\lambda_1& 0
\end{array}\right),\cdots,\left(\begin{array}{cc}0& \lambda_{[\frac{n}{2}]}
\\-\lambda_{[\frac{n}{2}]}& 0
\end{array}\right),0\right),$$ where $\lambda_1\geq\cdots\geq\lambda_{[\frac{n}{2}]}\geq 0$, $\sum_i\lambda^2_i=\frac{1}{2}$ and the last $0$ exists only if $n$ is odd.
Put $$\label{U matrix}
U:=diag\left(\left(\begin{array}{cc}
\frac{1}{\sqrt{2}}&\frac{\sqrt{-1}}{\sqrt{2}}
\\\frac{\sqrt{-1}}{\sqrt{2}}& \frac{1}{\sqrt{2}}
\end{array}\right),\cdots,\left(\begin{array}{cc}
\frac{1}{\sqrt{2}}&\frac{\sqrt{-1}}{\sqrt{2}}
\\\frac{\sqrt{-1}}{\sqrt{2}}& \frac{1}{\sqrt{2}}
\end{array}\right),1\right),$$ where the last $1$ exists only if $n$ is odd. Set $\check{Q}_{\gamma}:=U\sqrt{-1}\tilde{Q}_{\gamma}U^*:=(\check{q}^{\gamma}_{ij})$ for $\gamma\in S$, where $U^*$ denotes the conjugate transpose. Then the following identities can be easily verified for $k,l=1,\cdots,[\frac{n}{2}]$ and $k<l$: $$\begin{aligned}
&&\check{q}^{\gamma}_{2k-1,2k-1}=-\check{q}^{\gamma}_{2k,2k}=\tilde{q}^{\gamma}_{2k-1,2k},\quad \check{q}^{\gamma}_{n,n}=0\quad if~n~is~odd;\\
&&\check{q}^{\gamma}_{2k-1,2k}=\check{q}^{\gamma}_{2k,2k-1}=0;\\
&&\check{q}^{\gamma}_{2k-1,2l-1}=-\overline{\check{q}^{\gamma}_{2k,2l}}=\frac{1}{2}\{(\tilde{q}^{\gamma}_{2k-1,2l}-\tilde{q}^{\gamma}_{2k,2l-1})+\sqrt{-1}(\tilde{q}^{\gamma}_{2k-1,2l-1}+\tilde{q}^{\gamma}_{2k,2l})\};\\
&&\check{q}^{\gamma}_{2k-1,2l}=-\overline{\check{q}^{\gamma}_{2k,2l-1}}=\frac{1}{2}\{(\tilde{q}^{\gamma}_{2k-1,2l-1}-\tilde{q}^{\gamma}_{2k,2l})+\sqrt{-1}(\tilde{q}^{\gamma}_{2k,2l-1}+\tilde{q}^{\gamma}_{2k-1,2l})\};\\
&&\check{q}^{\gamma}_{2k-1,n}=\sqrt{-1}\overline{\check{q}^{\gamma}_{2k,n}}=\overline{\check{q}^{\gamma}_{n,2k-1}}=-\sqrt{-1}\check{q}^{\gamma}_{n,2k}=\frac{1}{\sqrt{2}}(-\tilde{q}^{\gamma}_{2k,n}+\sqrt{-1}\tilde{q}^{\gamma}_{2k-1,n})~~ if~n~is~odd.\end{aligned}$$ In particular, $$\check{Q}_{\alpha}=
diag(\lambda_1,-\lambda_1,\cdots,\lambda_{[\frac{n}{2}]},-\lambda_{[\frac{n}{2}]},0)=:diag(u_1,u_2,\cdots,u_n).$$ For any $(i,j)\in
\acute{S}:=\{(i,j)\in S|(i,j)\neq (2k-1,2k), 1\leq k\leq
[\frac{n}{2}]\}$, it follows from the identities above that $$\sum_{\gamma\in S}|\check{q}^{\gamma}_{ij}|^2=\frac{1}{2}.$$ As for the proof, we take $(2k-1,2l-1)\in\acute{S}$ for example as the following: $$\begin{aligned}
\sum_{\gamma\in S}|\check{q}^{\gamma}_{2k-1,2l-1}|^2
&=&\sum_{\gamma\in S}\frac{1}{4}\Big((\tilde{q}^{\gamma}_{2k-1,2l})^2+(\tilde{q}^{\gamma}_{2k,2l-1})^2+(\tilde{q}^{\gamma}_{2k-1,2l-1})^2+(\tilde{q}^{\gamma}_{2k,2l})^2\\
&&-2\tilde{q}^{\gamma}_{2k-1,2l}\tilde{q}^{\gamma}_{2k,2l-1}+2\tilde{q}^{\gamma}_{2k-1,2l-1}\tilde{q}^{\gamma}_{2k,2l}\Big)\\
&=&\sum_{\gamma\in S}\frac{1}{8}\Big((q_{(2k-1,2l)\gamma})^2+(q_{(2k,2l-1)\gamma})^2+(q_{(2k-1,2l-1)\gamma})^2+(q_{(2k,2l)\gamma})^2\\
&&-2q_{(2k-1,2l)\gamma}q_{(2k,2l-1)\gamma}+2q_{(2k-1,2l-1)\gamma}q_{(2k,2l)\gamma}\Big)\\
&=&\frac{1}{8}(1+1+1+1+0+0)=\frac{1}{2}.\end{aligned}$$ Denote by $\check{S}:=\{(i,j)\in
\acute{S}|(u_i-u_j)^2>\frac{2}{3}\}$. Since $\sum_i\lambda^2_i=\frac{1}{2}$, we find that $u_iu_j<0$ for $(i,j)\in\check{S}$ and hence $(u_i,u_j)=(\lambda_k,-\lambda_l)$ or $(-\lambda_k,\lambda_l)$ for some $(k,l)\in I$. Then by the preceding identities and Lemma \[lem2\], we complete the proof of the lemma as follows: $$\begin{aligned}
\sum_{\beta\in J_{\alpha}}\Big( \|[\tilde{Q}_{\alpha},
\tilde{Q}_{\beta}]\|^2-\frac{2}{3}\Big)&=&\sum_{\beta\in
J_{\alpha}}\Big(
\|[\check{Q}_{\alpha}, \check{Q}_{\beta}]\|^2-\frac{2}{3}\Big)\nonumber\\
&=&\sum_{\beta\in
J_{\alpha}}\sum^n_{i,j=1}\Big((u_i-u_j)^2-\frac{2}{3}\Big)|\check{q}^{\beta}_{ij}|^2\nonumber\\
&\leq&\sum_{\beta\in
J_{\alpha}}2\sum_{i<j}\Big((u_i-u_j)^2-\frac{2}{3}\Big)|\check{q}^{\beta}_{ij}|^2\nonumber\\
&=&2\sum_{\beta\in
J_{\alpha}}\sum_{(i,j)\in \acute{S}}\Big((u_i-u_j)^2-\frac{2}{3}\Big)|\check{q}^{\beta}_{ij}|^2\nonumber\\
&\leq&2\sum_{(i,j)\in
\check{S}}\Big((u_i-u_j)^2-\frac{2}{3}\Big)\sum_{\beta\in
J_{\alpha}}|\check{q}^{\beta}_{ij}|^2\nonumber\\
&\leq&2\sum_{(i,j)\in
\check{S}}\Big((u_i-u_j)^2-\frac{2}{3}\Big)\sum_{\beta\in S}|\check{q}^{\beta}_{ij}|^2\nonumber\\
&\leq& 4\sum_{(k,l)\in
I}\Big((\lambda_k+\lambda_l)^2-\frac{2}{3}\Big)\frac{1}{2}\nonumber\\
&\leq&\frac{2}{3}.\nonumber\end{aligned}$$
\[lem4\] For any $Q\in O(N)$ and $\alpha\in S$, we have $$\sum_{\beta\in
S}\|[\tilde{Q}_{\alpha}, \tilde{Q}_{\beta}]\|^2=n-2.$$
It follows from $(\ref{E tilde inner})$ that $$\begin{aligned}
\sum_{\beta\in S}\|[\tilde{Q}_{\alpha},
\tilde{Q}_{\beta}]\|^2&=&\sum_{\beta\gamma\tau\xi\eta}q_{\gamma\alpha}q_{\xi\alpha}q_{\tau\beta}q_{\eta\beta}\langle~[\tilde{E}_{\gamma},
\tilde{E}_{\tau}],~
[\tilde{E}_{\xi}, \tilde{E}_{\eta}]~ \rangle\nonumber\\
&=&\sum_{\gamma\tau\xi\eta}q_{\gamma\alpha}q_{\xi\alpha}\delta_{\tau\eta}\langle~[\tilde{E}_{\gamma},
\tilde{E}_{\tau}],~
[\tilde{E}_{\xi}, \tilde{E}_{\eta}]~ \rangle\nonumber\\
&=&\sum_{\gamma\xi}q_{\gamma\alpha}q_{\xi\alpha}\sum_{\tau}\langle~
[\tilde{E}_{\gamma}, \tilde{E}_{\tau}],~
[\tilde{E}_{\xi}, \tilde{E}_{\tau}]~ \rangle\nonumber\\
&=&\sum_{\gamma\xi}q_{\gamma\alpha}q_{\xi\alpha}(n-2)\delta_{\gamma\xi}
=(n-2)\sum_{\gamma}q^2_{\gamma\alpha}= n-2.\nonumber\end{aligned}$$
\[lem5\] Let $A,B$ be $(n\times n)$ real skew-symmetric matrices.\
$(i)$ If $n=3$, then we have $$\|[A,B]\|^2\leq\frac{1}{2}\|A\|^2\|B\|^2,$$ where the equality holds if and only if there is a $P\in O(3)$ such that $$PAP^t=C_1,\quad PBP^t=aC_2+bC_3,$$ where $C_1,C_2,C_3$ are the matrices in Theorem \[thm1\] and $a,b$ are real numbers.\
$(ii)$ If $n\geq4$, then we have $$\|[A,B]\|^2\leq\|A\|^2\|B\|^2,$$ where the equality holds if and only if there is a $P\in O(n)$ such that $$PAP^t=diag(D_1,0),\quad PBP^t=a\cdot diag(D_2,0)+b\cdot
diag(D_3,0),$$ where $D_1,D_2,D_3$ are the matrices in Theorem \[thm1\] and $a,b$ are real numbers.
$(i)$ As $A$ is now a $(3\times 3)$ real skew-symmetric matrix, there is a $P\in O(3)$ such that $$PAP^t=\left(\begin{array}{ccc}0&
\lambda& 0\\-\lambda& 0& 0\\0& 0& 0
\end{array}\right)=C_1.$$ Denote by $PBP^t:=(b_{ij})\in \mathfrak{o}(3)$. Then direct computation shows that $$[PAP^t,PBP^t]=\left(\begin{array}{ccc}0&
0& \lambda b_{23}\\0& 0& -\lambda b_{13}\\ -\lambda b_{23}&\lambda
b_{13} & 0
\end{array}\right).$$ Thus $$\|[A,B]\|^2=\|[PAP^t,PBP^t]\|^2=2\lambda^2(b_{23}^2+b_{13}^2)\leq\frac{1}{2}\|A\|^2\|B\|^2,$$ where the equality holds if and only if $b_{12}=0$, *i.e.*, $PBP^t$ lies in $Span\{C_2,C_3\}$.\
$(ii)$ As $A$ is now a $(n\times n)$ real skew-symmetric matrix, there is a $P\in O(n)$ such that $$PAP^t=diag\left(\left(\begin{array}{cc}0& \lambda_1
\\-\lambda_1& 0
\end{array}\right),\cdots,\left(\begin{array}{cc}0& \lambda_{[\frac{n}{2}]}
\\-\lambda_{[\frac{n}{2}]}& 0
\end{array}\right),0\right),$$ where $\lambda_1\geq\cdots\geq\lambda_{[\frac{n}{2}]}\geq 0$ and the last $0$ exists only if $n\geq4$ is odd.
Let $U$ be the unitary matrix defined in (\[U matrix\]). Then we have $$\check{A}:=U\sqrt{-1}PAP^tU^{*}=
diag(\lambda_1,-\lambda_1,...,\lambda_{[\frac{n}{2}]},-\lambda_{[\frac{n}{2}]},0):=diag(u_1,u_2,...,u_n).$$ Put $$\check{B}:=U\sqrt{-1}PBP^tU^{*}:=(b_{ij}),\quad sgn(n)=\Big\{
\begin{array}{ll}
1 & for~~n~odd,\\
0 & for~~n~even.
\end{array}$$ Then it follows from the proof of Lemma \[lem3\] that $b_{2k-1,2k}=0$ and $$\begin{aligned}
\|[A,B]\|^2&=&\|[\check{A},\check{B}]\|^2=\sum_{i,j=1}^n(u_i-u_j)^2|b_{ij}|^2\nonumber\\
&=&2\Big(\sum_{k<l}[(\lambda_k-\lambda_l)^2(|b_{2k-1,2l-1}|^2+|b_{2k,2l}|^2)+(\lambda_k+\lambda_l)^2
(|b_{2k-1,2l}|^2+|b_{2k,2l-1}|^2)]\Big)\nonumber\\
&~&+2(sgn(n))\sum_k\lambda_k^2(|b_{2k-1,n}|^2+|b_{2k,n}|^2)\nonumber\\
&\leq&2\Big(\sum_{k<l}(\lambda_1+\lambda_2)^2(|b_{2k-1,2l-1}|^2+|b_{2k,2l}|^2+|b_{2k-1,2l}|^2+|b_{2k,2l-1}|^2)\Big)
\nonumber\\
&~&+2(sgn(n))\sum_k\lambda_1^2(|b_{2k-1,n}|^2+|b_{2k,n}|^2)\nonumber\\
&\leq&2\|A\|^2\Big(\sum_{k<l}(|b_{2k-1,2l-1}|^2+|b_{2k,2l}|^2+|b_{2k-1,2l}|^2+|b_{2k,2l-1}|^2)\Big)\nonumber\\
&~&+\|A\|^2(sgn(n))\sum_k(|b_{2k-1,n}|^2+|b_{2k,n}|^2)\nonumber\\
&\leq&\|A\|^2\|B\|^2.\nonumber\end{aligned}$$ Analyzing these inequalities, we find that the equality in this case holds if and only if $\lambda_1=\lambda_2=\frac{1}{2}\|A\|$, $\lambda_j=0$ for $j>2$, and all $b_{ij}$’s are zero except $b_{14}=\bar{b}_{41}$ and $b_{23}=\bar{b}_{32}$, which is equivalent to that $PAP^t, PBP^t$ are in the forms specified in the lemma.
Now let $\varphi : M(m,n)\longrightarrow M(C_m^2,C_n^2)$ be the map defined by $\varphi(A)_{(i,j)(k,l)}:=A(_{i\hskip 0.1cm j}^{k\hskip
0.1cm l})$, where $C_m^2=\frac{m(m-1)}{2}$, $1\leq i<j\leq m$, $1\leq k<l\leq n$ and $A(_{i\hskip 0.1cm j}^{k\hskip 0.1cm l})=a_{ik}a_{jl}-a_{il}a_{jk}$ is the determinant of the sub-matrix of $A:=(a_{ij})$ with the rows $i, j$, the columns $k, l$, arranged with the same ordering as in (\[order\]). It is easily seen that $\varphi(I_n)=I_{C_n^2}$ (preserving identity matrices), $\varphi(A)^t=\varphi(A^t)$ and the following
\[lem6\] The map $\varphi$ preserves the matrix product, i.e. $\varphi(AB)=\varphi(A)\varphi(B)$ holds for $A\in M(m,k)$, $B\in
M(k,n)$.
We will also need the following exercise of linear algebra in the proof of the equality case of Theorem \[thm1\].
\[lem7\] Let $A$, $B$ be two matrices in $M(m,n)$. Then $AA^t=BB^t$ if and only if $A=BR$ for some $R\in O(n)$.
Proof of Theorem \[thm1\]
-------------------------
Let $B_1,\cdots,B_m$ be any $(n\times n)$ real skew-symmetric matrices. Their coefficients under the standard basis $\{\tilde{E}_{\alpha}\}_{\alpha\in S}$ of $\mathfrak{o}(n)$ are determined by a matrix $B\in M(N,m)$ as $(B_1,\cdots,B_m)=(\tilde{E}_1,\cdots,\tilde{E}_N)B$. Taking the same ordering as in (\[order\]) for $1\leq r<s\leq m$ and $1\leq
\alpha<\beta\leq N$, we arrange $\Big\{[B_r, B_s]\Big\}_{r<s}$, $\Big\{[\tilde{E}_{\alpha}, \tilde{E}_{\beta}]\Big\}_{\alpha<\beta}$ into $C_m^2$, $C_N^2$-dimensional vectors respectively. We first observe that $$([B_1, B_2],\cdots,[B_{m-1}, B_m])=([\tilde{E}_1, \tilde{E}_2],\cdots,[\tilde{E}_{N-1}, \tilde{E}_N])\cdot \varphi(B).$$
Let $C(\tilde{E})$ denote the matrix in $M(C_N^2)$ defined by $C(\tilde{E})_{(\alpha,\beta)(\gamma,\tau)}:=\langle~
[\tilde{E}_{\alpha}, \tilde{E}_{\beta}],~ [\tilde{E}_{\gamma},
\tilde{E}_{\tau}]~ \rangle$, for $1\leq\alpha<\beta\leq N$, $1\leq\gamma<\tau\leq N$. Moreover we will use the same notation for $\{B_r\}$ and $\{\tilde{Q}_{\alpha}\}$, *i.e.*, $C(B)$ and $C(Q)$ respectively. Then it is obvious that $$C(B)=\varphi(B^t)C(\tilde{E})\varphi(B), \hskip 0.3cm C(Q)=\varphi(Q^t)C(\tilde{E})\varphi(Q).$$ Since $BB^t$ is a $(N\times N)$ semi-positive definite matrix, there exists an orthogonal matrix $Q\in SO(N)$ such that $BB^t=Q~
diag(x_1,\cdots,x_N)~ Q^t$ with $x_{\alpha}\geq 0$, $1\leq\alpha\leq
N.$ Thus $$\sum_{r=1}^m\|B_r\|^2=\|B\|^2 =\sum_{\alpha=1}^Nx_{\alpha}$$ and hence by Lemma \[lem6\], $$\begin{aligned}
\sum_{r,s=1}^m\|[B_r, B_s]\|^2&=&2Tr~C(B)=2Tr\hskip 0.1cm
\varphi(B^t)C(\tilde{E})\varphi(B)=2Tr~
\varphi(BB^t)C(\tilde{E})\nonumber\\&=&2Tr~
\varphi(diag(x_1,\cdots,x_N))C(Q)=\sum_{\alpha,\beta=1}^Nx_{\alpha}x_{\beta}\|[\tilde{Q}_{\alpha},
\tilde{Q}_{\beta}]\|^2.\nonumber\end{aligned}$$
We are now ready to prove Theorem \[thm1\].\
Proof of Theorem \[thm1\].
Put $d(n):= \frac{1}{3}$ if $n=3$ and $\frac{2}{3}$ if $n\geq4$. It follows from the arguments above that the inequalities of the theorem are equivalent to the following $$\label{ineq-poly}
\sum_{\alpha,\beta=1}^Nx_{\alpha}x_{\beta}\|[\tilde{Q}_{\alpha},
\tilde{Q}_{\beta}]\|^2\leq
d(n)\Big(\sum_{\alpha=1}^Nx_{\alpha}\Big)^2, ~~~~for~any~ x\in
\mathbb{R}^N_{+}, ~Q\in SO(N),$$ where $\mathbb{R}^N_{+}:=\{0\neq
x=(x_1,...,x_N)\in\mathbb{R}^N~|~x_{\alpha}\geq0, 1\leq\alpha\leq
N\}$.
For $n=3$, $N=\frac{n(n-1)}{2}=3$ and by Lemma \[lem5\], we have $\|[\tilde{Q}_{\alpha},\tilde{Q}_{\beta}]\|^2\leq\frac{1}{2}$ and thus $$\sum_{\beta\in S}\|[\tilde{Q}_{\alpha},\tilde{Q}_{\beta}]\|^2\leq\frac{1}{2}\times2=1.$$ On the other hand, it follows from Lemma \[lem4\] that $\sum_{\beta\in
S}\|[\tilde{Q}_{\alpha},\tilde{Q}_{\beta}]\|^2=n-2=1$. Therefore, we get $$\|[\tilde{Q}_{\alpha},\tilde{Q}_{\beta}]\|^2=\frac{1}{2},~~~~ for~any~ \alpha\neq\beta\in S.$$ In fact, this equality just says that the cross product of two orthogonal unit vectors in $\mathbb{R}^3$ is still a unit vector if we identify $\mathfrak{o}(3)$ with $\mathbb{R}^3$ and correspond the commutator operator to the cross product. So in this case, the inequality (\[ineq-poly\]) is equivalent to $$x_1x_2+x_2x_3+x_3x_1\leq\frac{1}{3}(x_1+x_2+x_3)^2,~~~~for~any~ x\in
\mathbb{R}^3_{+},$$ which is easily verified by $$x_1x_2+x_2x_3+x_3x_1-\frac{1}{3}(x_1+x_2+x_3)^2=-\frac{1}{6}\Big((x_1-x_2)^2+(x_2-x_3)^2+(x_3-x_1)^2\Big)\leq0.$$ Note that the equality above holds if and only if $x_1=x_2=x_3:=\lambda^2$, *i.e.*, $BB^t=\lambda^2I_3$, which, by Lemma \[lem7\], is equivalent to that there is a $R\in O(m)$ such that $$(B_1,\cdots,B_m)=(\tilde{E}_{12},\tilde{E}_{13},\tilde{E}_{23})\cdot \Big(\lambda I_3,0_{3\times(m-3)}\Big)R=(C_1,C_2,C_3,0,\cdots,0)R.$$ This completes the proof of (i) of Theorem \[thm1\].
Now we consider the case (ii). Put $$f_Q(x)=F(x,Q):=\sum_{\alpha,\beta=1}^Nx_{\alpha}x_{\beta}\|[\tilde{Q}_{\alpha},
\tilde{Q}_{\beta}]\|^2-\frac{2}{3}\Big(\sum_{\alpha=1}^Nx_{\alpha}\Big)^2.$$ Then $F$ is a continuous function defined on $\mathbb{R}^N\times
SO(N)$ and thus uniformly continuous on any compact subset of $\mathbb{R}^N\times SO(N)$. Let $\bigtriangleup:=\{x\in\mathbb{R}^N_{+}~|~\sum_{\alpha}x_{\alpha}=1\}$ and for any sufficiently small $\varepsilon>0$, $\bigtriangleup_{\varepsilon}:=\{x\in
\bigtriangleup~|~x_{\alpha}\geq \varepsilon, 1\leq\alpha\leq N\}$. Also let $$G:=\{Q\in SO(N)~|~f_Q(x)\leq 0, ~for~all~ x\in
\bigtriangleup\},$$ $$G_{\varepsilon}:=\{Q\in SO(N)~|~f_Q(x)< 0,
~for~all~ x\in \bigtriangleup_{\varepsilon}\}.$$ We claim that $G=\lim_{\varepsilon\rightarrow 0}G_{\varepsilon}=SO(N).$ Note that this implies (\[ineq-poly\]) and thus proves the inequality. In fact we can show $$\label{G epsilon}
G_{\varepsilon}=SO(N) ~ for ~ any~ sufficiently ~ small ~
\varepsilon>0.$$ To prove (\[G epsilon\]), we use the continuity method, in which we must prove the following three properties:
- \[step1\] $I_N\in G_{\varepsilon}$ (and thus $G_{\varepsilon}\neq\emptyset$);
- \[step2\] $G_{\varepsilon}$ is open in $SO(N)$;
- \[step3\] $G_{\varepsilon}$ is closed in $SO(N)$.
Since $F$ is uniformly continuous on $\triangle_{\varepsilon}\times SO(N)$, (b) is obvious.\
**Proof of (a)**. For any $x\in \bigtriangleup_{\varepsilon}$, $f_{I_N}(x)=\sum_{\alpha,\beta=1}^Nx_{\alpha}x_{\beta}\|[\tilde{E}_{\alpha},
\tilde{E}_{\beta}]\|^2-\frac{2}{3}\Big(\sum_{\alpha=1}^Nx_{\alpha}\Big)^2$. 0.05cm It follows from (\[E tilde norm\]) that $$\begin{aligned}
f_{I_N}(x)&=&\sum_{i<j<k}(x_{ij}x_{jk}+x_{ij}x_{ik}+x_{ik}x_{jk})-\frac{2}{3}\Big(\sum_{i<j}x_{ij}\Big)^2\nonumber\\
&<&\sum_{i<j<k}(x_{ij}x_{jk}+x_{ij}x_{ik}+x_{ik}x_{jk})-\frac{2}{3}\sum_{i<j<k}2(x_{ij}x_{jk}+x_{ij}x_{ik}+x_{ik}x_{jk})\nonumber\\
&<&0,\nonumber\end{aligned}$$ which means $I_N\in G_{\varepsilon}$. $\Box$\
**Proof of (c)**. We only need to prove the following a priori estimate: Suppose $f_Q(x)\leq 0$ for every $
x\in\bigtriangleup_{\varepsilon}$. Then $f_Q(x)< 0$ for every $
x\in\bigtriangleup_{\varepsilon}$.
The proof of this estimate is as follows: If there is a point $y\in
\bigtriangleup_{\varepsilon}$ such that $f_Q(y)= 0$, we can assume without loss of generality that $$y\in
\bigtriangleup^{\gamma}_{\varepsilon}:=\{x\in\bigtriangleup_{\varepsilon}~|~x_{\alpha}>\varepsilon~
for~ \alpha\leq \gamma ~and~ x_{\beta}=\varepsilon~
for~\beta>\gamma\}$$ for some $1\leq \gamma\leq N$. Then $y$ is a maximum point of $f_Q(x)$ in the cone spanned by $\bigtriangleup_{\varepsilon}$ and an interior maximum point in $\bigtriangleup^{\gamma}_{\varepsilon}$. Hence there exist numbers $b_{\gamma+1},\cdots,b_N$ and a number $a$ such that $$\label{partial f}
\begin{array}{ll}
\Big(\frac{\partial f_Q}{\partial x_1}(y),\cdots,\frac{\partial
f_Q}{\partial x_{\gamma}}(y)\Big)=2a(1,\cdots,1),&\\
\Big(\frac{\partial f_Q}{\partial
x_{\gamma+1}}(y),\cdots,\frac{\partial f_Q}{\partial
x_{N}}(y)\Big)=2(b_{\gamma+1},\cdots,b_N)&
\end{array}$$ or equivalently $$\label{partial f 2}
\sum_{\beta=1}^Ny_{\beta}(\|[\tilde{Q}_{\alpha},
\tilde{Q}_{\beta}]\|^2)-\frac{2}{3}=\Big\{
\begin{array}{ll}
a & \alpha\leq\gamma,\\
b_{\alpha}& \alpha>\gamma.
\end{array}$$ Hence $$f_Q(y)=\Big(\sum_{\alpha=1}^{\gamma}y_{\alpha}\Big)a+\Big(\sum_{\alpha=\gamma+1}^Nb_{\alpha}\Big)\varepsilon
=0\quad and \quad
\sum_{\alpha=1}^{\gamma}y_{\alpha}+(N-\gamma)\varepsilon=1.$$ Meanwhile, we see $\frac{\partial f_Q}{\partial \nu}(y)=2(a\gamma
+\sum_{\alpha=\gamma+1}^Nb_{\alpha})\leq 0$, where $\nu=(1,\cdots,1)$ is the vector normal to $\bigtriangleup$ in $\mathbb{R}^N$. For any sufficiently small $\varepsilon$ (such as $\varepsilon<1/N$), it follows from the above three formulas that $a\geq 0$. Without loss of generality, we assume $y_1=max\{y_1,\cdots,y_{\gamma}\}>\varepsilon$. Let $J:=\{\beta\in
S~|~ \|[\tilde{Q}_{1}, \tilde{Q}_{\beta}]\|^2\geq \frac{2}{3}\}$, and let $n_1$ be the number of elements of $J$. Now combining Lemma \[lem3\] , Lemma \[lem4\] and Equation (\[partial f 2\]) will give a contradiction as follows: $$\begin{aligned}
\frac{2}{3}\leq
\frac{2}{3}+a&=&\sum_{\beta=2}^Ny_{\beta}\|[\tilde{Q}_1,
\tilde{Q}_{\beta}]\|^2\nonumber\\
&=&\sum_{\beta\in J}y_{\beta}(\|[\tilde{Q}_1,
\tilde{Q}_{\beta}]\|^2-\frac{2}{3})+\frac{2}{3}\sum_{\beta\in
J}y_{\beta}+\sum_{\beta\in
S/J}y_{\beta}\|[\tilde{Q}_1, \tilde{Q}_{\beta}]\|^2\nonumber\\
&\leq&y_1\sum_{\beta\in J}(\|[\tilde{Q}_1,
\tilde{Q}_{\beta}]\|^2-\frac{2}{3})+\frac{2}{3}\sum_{\beta\in
J}y_{\beta}+\sum_{\beta\in S/J}y_{\beta}\|[\tilde{Q}_1,
\tilde{Q}_{\beta}]\|^2\nonumber\\
&\leq&\frac{2}{3}y_1+\frac{2}{3}\sum_{\beta\in
J}y_{\beta}+\sum_{\beta\in S/J}y_{\beta}\|[\tilde{Q}_1,
\tilde{Q}_{\beta}]\|^2 \leq
\frac{2}{3}\sum_{\beta=1}^Ny_{\beta}=\frac{2}{3}.\label{line}\end{aligned}$$ Thus $$\label{n1N}
a=0\quad and \quad \sum_{\beta\in J}\|[\tilde{Q}_1,
\tilde{Q}_{\beta}]\|^2=\frac{2}{3}(n_1+1)\leq n-2<\frac{2}{3}N.$$ Hence $S/(J\cup\{1\})\neq\emptyset$, and the second “$\leq$" in line (\[line\]) should be “$<$" by the definition of $J$ and the positivity of $y_{\beta}$ for $\beta\in S/(J\cup\{1\})$.$\Box$
Now we consider the equality condition of (ii) of Theorem \[thm1\] in view of the proof of the a priori estimate.
If there is an orthogonal matrix $Q$ and a point $y\in
\bigtriangleup$ such that $f_Q(y)= 0$, we can assume without loss of generality that $$y\in
\bigtriangleup^{\gamma}:=\{x\in\bigtriangleup~|~x_{\alpha}>0~
for~all~\alpha\leq \gamma ~and~ x_{\beta}=0~
for~all~\beta>\gamma\}$$ for some $2\leq \gamma\leq N$. Then $y$ is a maximum point of $f_Q(x)$ in $\mathbb{R}^N_{+}$ and an interior maximum point in $\bigtriangleup^{\gamma}$. Therefore, we have the same conclusions as (\[partial f\], \[partial f 2\], \[line\], \[n1N\]) when $\gamma\leq n_1+1$, and all inequalities in the proof of Lemma \[lem3\] can be replaced by equalities. So $n_0=1$ by Lemma \[lem2\], $\check{S}=\{(1,4),(2,3)\}$, $\check{q}^{\beta}_{ii}=\check{q}^{\beta}_{ij}=0$ for any $(i,j)\in
S/\check{S}$, $\beta\in J$, which imply that $\tilde{Q}_{\beta}$ is a linear combination of $diag(D_2,0)$, $diag(D_3,0)$ for any $\beta\in J$. Hence, $1\leq n_1\leq 2$. But if $n_1=1$, it follows from (\[n1N\]) that $\|[\tilde{Q}_1,\tilde{Q}_{\beta}]\|^2=\frac{4}{3}>1$ for $\beta\in
J$ which contradicts with Lemma \[lem5\]. So we have $n_1=2$ and $2\leq\gamma\leq3$. If $\gamma=2$, then it follows from Lemma \[lem5\] and (\[line\]) the following contradiction: $$\frac{2}{3}=y_2\|[\tilde{Q}_1, \tilde{Q}_2]\|^2\leq \frac{1}{2}.$$ So we get $\gamma=3$. By (\[line\]) again, we have $y_1=y_2=y_3=\frac{1}{3}$ and $y_\alpha=0$ for $\alpha>3$, and $$\|[\tilde{Q}_1,\tilde{Q}_2]\|^2=\|[\tilde{Q}_1,\tilde{Q}_3]\|^2=\|[\tilde{Q}_2,\tilde{Q}_3]\|^2=1,$$ from which we can conclude the equality case of (ii) of Theorem \[thm1\] by Lemmas \[lem5\] and \[lem7\]. The proof of Theorem \[thm1\] is now completed.$\Box$
Simons-type inequality for Riemannian submersions {#simons-sec}
=================================================
Moving frame method for Riemannian submersions
----------------------------------------------
In this subsection we present a treatment of basic materials about Riemannian submersions by moving frame method.
Let $\pi: M^{n+m}\rightarrow B^n$ be a Riemannian submersion. We denote by $D$, $R$, $r$ (resp. $\hat{D}, \hat{R}, \hat{r}$; $\check{D},\check{R},\check{r}$) the Levi-Civita connection, the curvature operator and the Ricci operator on $M$ (resp. on the fibres; on $B$) respectively. Around each point $x\in M$, we can choose local orthonormal vertical vector fields $\{U_{n+1},\cdots,U_{n+m}\}$ and local orthonormal basic vector fields $\{X_1,\cdots,X_n\}$ which are horizontal and projectable such that $\{\check{X}_1:=\pi_*X_1,\cdots,\check{X}_n:=\pi_*X_n \}$ form a local orthonormal basis around $\pi(x)\in B$. Thus $\{X_1,\cdots,X_n,U_{n+1},\cdots,U_{n+m}\}$ form a local orthonormal basis of $TM$ around $x\in M$ and we denote by $\{\omega_1,\cdots,\omega_n,\omega_{n+1},\cdots,\omega_{n+m}\}$ the dual $1$-forms on $M$ with respect to this basis, *i.e.*, $$\omega_i(X_j)=\delta_{ij},\quad \omega_i(U_r)=\omega_r(X_i)=0,\quad \omega_r(U_s)=\delta_{rs},$$ where, from now on, we use the convention for indices as follows: $$h,i,j,k,l\in\{1,\cdots,n\};\quad
r,s,t,u,v\in\{n+1,\cdots,n+m\},\quad
\alpha,\beta,\gamma,\delta\in\{1,\cdots,n+m\}.$$ Also we denote by $\{\check{\omega}_1,\cdots,\check{\omega}_n\}$ the dual $1$-forms on $B$ with respect to the basis $\{\check{X}_1,\cdots,\check{X}_n\}$ and by $\{\hat{\omega}_{n+1},\cdots,\hat{\omega}_{n+m}\}$ the dual $1$-forms on the fibre(s) with respect to the basis $\{U_{n+1},\cdots,U_{n+m}\}$. Then the connection $1$-forms $\{\omega_{\alpha\beta}\}$ of $D$ on $M$, the connection $1$-forms $\{\hat{\omega}_{rs}\}$ of $\hat{D}$ on the fibre(s) and the connection $1$-forms $\{\check{\omega}_{ij}\}$ of $\check{D}$ can be defined as follows: $$\label{connection forms}
\begin{array}{ll}
\omega_{ij}=\langle DX_i,X_j\rangle,&
\omega_{ir}=\langle DX_i,U_r\rangle=-\langle DU_r,X_i\rangle=-\omega_{ri},\quad
\omega_{rs}=\langle DU_r,U_s\rangle;\\
\hat{\omega}_{rs}=\langle\hat{D}U_r,U_s\rangle,&
\check{\omega}_{ij}=\langle\check{D}\check{X}_i,\check{X}_j\rangle,
\end{array}$$ where without confusion we denote by bracket simultaneously the metrics on $M$, $B$ and the fibres. Let $\{\Omega_{\alpha\beta}\}$ (resp. $\{\hat{\Omega}_{rs}\}$; $\{\check{\Omega}_{ij}\}$) be the curvature $2$-forms on $M$ (resp. on the fibres; on $B$ ). Then we have the following structure equations: $$\label{str-eq1}
\Big\{
\begin{array}{ll}
d\omega_{\alpha}=\omega_{\alpha\beta}\wedge\omega_{\beta},\quad
\omega_{\alpha\beta}=-\omega_{\beta\alpha},\\
d\omega_{\alpha\beta}=\omega_{\alpha\gamma}\wedge\omega_{\gamma\beta}+\Omega_{\alpha\beta};
\end{array}$$ $$\label{str-eq2}
\Big\{\begin{array}{ll}
d\hat{\omega}_{r}=\hat{\omega}_{rs}\wedge\hat{\omega}_{s},\quad
\hat{\omega}_{rs}=-\hat{\omega}_{sr},&\\
d\hat{\omega}_{rs}=\hat{\omega}_{rt}\wedge\hat{\omega}_{ts}+\hat{\Omega}_{rs};&
\end{array}$$ $$\label{str-eq3}
\Big\{\begin{array}{ll}
d\check{\omega}_{i}=\check{\omega}_{ij}\wedge\check{\omega}_{j},\quad
\check{\omega}_{ij}=-\check{\omega}_{ji},&\\
d\check{\omega}_{ij}=\check{\omega}_{ik}\wedge\check{\omega}_{kj}+\check{\Omega}_{ij},&
\end{array}$$ where, from now on, repeated indices are implicitly summed over, and we will write the curvature forms as $\Omega_{\alpha\beta}=-\frac{1}{2}R_{\alpha\beta\gamma\delta}\omega_{\gamma}\wedge\omega_{\delta}$ and so the Ricci curvature $r=(R_{\alpha\beta})$ (resp. $\hat{r}=(\hat{R}_{rs})$; $\check{r}=(\check{R}_{ij})$) on $M$ (resp. on the fibre(s); on $B$) can be expressed as $R_{\alpha\beta}=R_{\alpha\gamma\beta\gamma}$ (resp. $\hat{R}_{rs}=\hat{R}_{rtst}$; $\check{R}_{ij}=\check{R}_{ikjk}$).
Now since $\pi_*[X_i,U_r]=[\check{X}_i,\pi_*U_r]=0$ and $\pi_*[U_r,U_s]=[\pi_*U_r,\pi_*U_s]=0$, $[X_i,U_r]$ and $[U_r,U_s]$ are vertical, thereby it follows from (\[connection forms\]) and the definitions of the tensors $T$ and $A$ in $(\ref{A T})$ that $$\label{ATcoeff}
\begin{array}{ll}
T^i_{rs}:=\omega_{ri}(U_s)=\langle T_{U_s}U_r,X_i\rangle=-\langle T_{U_s}X_i,U_r\rangle=T^i_{sr};&\\
A^r_{ij}:=\omega_{ir}(X_j)=\langle A_{X_j}X_i,U_r\rangle=-\langle A_{X_j}U_r,X_i\rangle=\omega_{ij}(U_r)=-A^r_{ji}.&
\end{array}$$ Hence one can see that the tensor $T$ (or its coefficients $\{T^i_{rs}\}$) is just the second fundamental form when it is restricted to vertical vector fields along the fibre(s). Meanwhile, we find that $$A_{X_i}X_j=-A_{X_j}X_i=\frac{1}{2}\mathscr{V}[X_i,X_j]$$ and thus $$A_{X}Y=\frac{1}{2}\mathscr{V}[X,Y],\quad for~~X,Y\in\mathscr{H},$$ which shows that $A$ measures the integrability of the horizontal distribution $\mathscr{H}$ and so it is usually called the *integrability tensor* of $\pi$. By (\[A norm\]) and (\[ATcoeff\]), we have $$\label{A norm2}
|A|^2=\sum_{r,i,j}(A^r_{ij})^2.$$ Moreover, formulas (\[ATcoeff\]) imply the following equations: $$\label{conn relation}
\begin{array}{ll}
\omega_{ir}=A^r_{ij}\omega_j-T^i_{rs}\omega_s,&\\
\omega_{ij}=\pi^*\check{\omega}_{ij}+A^r_{ij}\omega_r.&
\end{array}$$ Define the covariant derivatives of $T^i_{rs}$ and $A^r_{ij}$ by $$\label{cov-T-A}
\begin{array}{ll}
DT^i_{rs}:=dT^i_{rs}+T^i_{ts}\omega_{tr}+T^i_{rt}\omega_{ts}+T^j_{rs}\omega_{ji}
=:T^i_{rsj}\omega_j+T^i_{rst}\omega_t,&\\
DA^r_{ij}:=dA^r_{ij}+A^r_{kj}\omega_{ki}+A^r_{ik}\omega_{kj}+A^s_{ij}\omega_{sr}
=:A^r_{ijk}\omega_k+A^r_{ijs}\omega_s.
\end{array}$$ Then it is easily seen from (\[ATcoeff\]) and (\[cov-T-A\]) that $$\begin{array}{ll}\label{cov-T-A2}
T^i_{rsj}=\langle(D_{X_j}T)_{U_s}U_r,X_i\rangle=T^i_{srj},&T^i_{rst}=\langle(D_{U_t}T)_{U_s}U_r,X_i\rangle=T^i_{srt},\\
A^r_{ijk}=\langle(D_{X_k}A)_{X_j}X_i,U_r\rangle=-A^r_{jik},&A^r_{ijs}=\langle(D_{U_s}A)_{X_j}X_i,U_r\rangle=-A^r_{jis},
\end{array}$$ which are the only components of $DT$ and $DA$ that cannot be recovered from $T$ and $A$ at a point (cf. [@Be; @O]). Taking deferential of (\[conn relation\]) by using (\[cov-T-A\]) and the structure equations (\[str-eq1\], \[str-eq3\]) we get $$\label{d conn1}
(DA^r_{ij}+A^r_{ik}A^s_{jk}\omega_s+T^i_{rs}A^s_{jk}\omega_k)\wedge\omega_j
=(DT^i_{rs}-T^i_{rt}T^k_{ts}\omega_k)\wedge\omega_s+\Omega_{ir},$$ $$\begin{aligned}
\label{d conn2}
\Omega_{ij}&=&\pi^*\check{\Omega}_{ij}+(A^r_{ij}A^r_{kl}+A^r_{ik}A^r_{jl})\omega_k\wedge\omega_l\\
&&+(A^r_{ijk}-A^s_{ij}T^k_{sr}+A^s_{jk}T^i_{sr}+A^s_{ki}T^j_{sr})\omega_k\wedge\omega_r\nonumber\\
&&+(A^r_{ijs}+T^i_{ts}T^j_{tr}+A^s_{ik}A^r_{kj})\omega_s\wedge\omega_r.\nonumber\end{aligned}$$ Recall that the O’Neill’s formula $\{0\}$ in [@O] is just the Gauss equation on the fibre(s) derived from the structure equations (\[str-eq1\], \[str-eq2\]) and can be written as $$\label{o0}
R_{rstu}=\hat{R}_{rstu}-T^i_{rt}T^i_{su}+T^i_{st}T^i_{ru}.$$ Taking values of (\[d conn1\]) on $U_s\wedge U_t$, $X_j\wedge U_s$ and of (\[d conn2\]) on $U_s\wedge U_r$, $X_k\wedge U_r$ and $X_k\wedge X_l$, respectively, we can get the O’Neill’s formulas $\{1,2,2',3,4\}$ in [@O] as follows: $$\begin{aligned}
&&R_{irst}=T^i_{rts}-T^i_{rst},\label{o1}\\
&&R_{irjs}=T^i_{rsj}+A^r_{ijs}-T^i_{rt}T^j_{ts}+A^r_{ik}A^s_{jk},\label{o2}\\
&&R_{ijsr}=A^s_{ijr}-A^r_{ijs}+A^r_{ik}A^s_{kj}-A^s_{ik}A^r_{kj}+T^i_{tr}T^j_{ts}-T^i_{ts}T^j_{tr},\label{o2'}\\
&&R_{ijkr}=-A^r_{ijk}+A^s_{ij}T^k_{sr}-A^s_{jk}T^i_{sr}-A^s_{ki}T^j_{sr},\label{o3}\\
&&R_{ijkl}=\check{R}_{ijkl}\circ
\pi-2A^r_{ij}A^r_{kl}-A^r_{ik}A^r_{jl}+A^r_{il}A^r_{jk}.\label{o4}\end{aligned}$$ Taking value of (\[d conn1\]) on $X_j\wedge X_k$ we get $$R_{irjk}=A^r_{ijk}-A^r_{ikj}+2A^s_{jk}T^i_{rs},$$ which by combining with (\[cov-T-A2\], \[o3\]) implies $$\label{Arijk-identity}
A^r_{ijk}+A^r_{jki}+A^r_{kij}=A^s_{ji}T^k_{sr}+A^s_{kj}T^i_{sr}+A^s_{ik}T^j_{sr}.$$ Reversing $i$ and $j$, $r$ and $s$ in (\[o2\]) and using (\[cov-T-A2\]) and the symmetry of the curvature operator, we can get the following (cf. [@Be; @Gr]): $$\label{Arijs-identity}
A^r_{ijs}+A^s_{ijr}=T^j_{rsi}-T^i_{rsj}.$$
Let $\{K_{\alpha\beta}\}$ (resp. $\{\hat{K}_{rs}\}$; $\{\check{K}_{ij}\}$) be the sectional curvatures of $M$ (resp. of the fibre(s); of $B$). Then it follows from (\[o0\]-\[o4\]) that $$\label{sec-curv}
\begin{array}{lll}
K_{rs}=\hat{K}_{rs}+\sum_i\Big((T^i_{rs})^2-T^i_{rr}T^i_{ss}\Big),\\
K_{ir}=T^i_{rri}-\sum_s(T^i_{rs})^2+\sum_j(A^r_{ij})^2,\\
K_{ij}=\check{K}_{ij}\circ\pi-3\sum_r(A^r_{ij})^2,
\end{array}$$ where, unusually, repeated indices are not summed over. If the fibres are totally geodesic, *i.e.*, $T=0$, then by (\[o0\]-\[o4\]) we have the following identities about Ricci curvatures: $$\label{ricci-curv}
\begin{array}{lll}
R_{ir}=A^r_{ikk}=-\langle\check{\delta}A(X_i),U_r\rangle,\\
R_{rs}=\hat{R}_{rs}+A^r_{ij}A^s_{ij},\\
R_{ij}=\check{R}_{ij}\circ\pi-2A^r_{ik}A^r_{jk}.
\end{array}$$ Hence if $M$ is Einstein with totally geodesic fibres, then we have $$\label{einsteinM}
R_{ir}=A^r_{ikk}=-\langle\check{\delta}A(X_i),U_r\rangle=0,$$ which is equivalent to that the horizontal distribution $\mathscr{H}$ is Yang-Mills.
Laplacians of the integrability tensor
--------------------------------------
From now on, we assume that the Riemannian submersion $\pi: M^{n+m}\rightarrow B^n$ has totally geodesic fibres and Yang-Mills horizontal distribution, *i.e.*, $T=0$ and $A^r_{ikk}=0$ (by (\[einsteinM\])).
We define the covariant derivatives of $A^r_{ijk}$ and $A^r_{ijs}$ by $$\label{Arijkl-Arijst}
\begin{array}{ll}
DA^r_{ijk}:=dA^r_{ijk}+A^r_{ljk}\omega_{li}+A^r_{ilk}\omega_{lj}+
A^r_{ijl}\omega_{lk}+A^s_{ijk}\omega_{sr}=:A^r_{ijkl}\omega_l+A^r_{ijks}\omega_s,\\
DA^r_{ijs}:=dA^r_{ijs}+A^r_{kjs}\omega_{ki}+A^r_{iks}\omega_{kj}+A^r_{ijt}\omega_{ts}
+A^t_{ijs}\omega_{tr}=:A^r_{ijsk}\omega_k+A^r_{ijst}\omega_t.
\end{array}$$ The horizontal and vertical Laplacians of $A^r_{ij}$ are defined by $$\label{def-lap-A}
\triangle^{\mathscr{H}}A^r_{ij}:=A^r_{ijkk},\quad
\triangle^{\mathscr{V}}A^r_{ij}:=A^r_{ijss},$$ while the horizontal and vertical Laplacians of a function $f\in
C^{\infty}(M)$ are defined by $$\label{def-lap}
\triangle^{\mathscr{H}}f:=(X_iX_i-D_{X_i}X_i)f,\quad
\triangle^{\mathscr{V}}f:=(U_sU_s-D_{U_s}U_s)f.$$ It is easily seen that these Laplacians are well-defined and relate to the Laplace-Beltrami operator $\triangle$ of $M$ by $$\triangle=\triangle^{\mathscr{H}}+\triangle^{\mathscr{V}}.$$ Moreover, since the fibres are totally geodesic, $\triangle^{\mathscr{V}}$ is just the Laplace-Beltrami operator, also denoted by $\triangle$, along any fibre $F_b$ when restricted to actions on functions of $F_b$, *i.e.*, $$(\triangle^{\mathscr{V}}f)|_{F_b}=\triangle(f|_{F_b}),\quad
for~~any~~f\in C^{\infty}(M).$$ Therefore, if $M$ is closed, then for any function $f\in
C^{\infty}(M)$, we have $$\label{int-lap-0}
\int_M\triangle^{\mathscr{H}}f~dV_M=0,\quad
\int_M\triangle^{\mathscr{V}}f~dV_M=0.$$
Taking differential of the second equation of (\[cov-T-A\]) by using (\[cov-T-A\], \[Arijkl-Arijst\]) and the structure equations (\[str-eq1\]) we get $$\begin{aligned}
\label{dDArij}
&&DA^r_{ijk}\wedge\omega_k+DA^r_{ijs}\wedge\omega_s\\
&=&-(A^r_{hj}A^s_{hk}A^s_{il}+A^r_{ih}A^s_{hk}A^s_{jl}+A^r_{hl}A^s_{hk}A^s_{ij}+A^r_{ijs}A^s_{kl})\omega_k\wedge\omega_l\nonumber\\
&&-A^r_{ijl}A^s_{lk}\omega_k\wedge\omega_s+(A^r_{hj}\Omega_{hi}+A^r_{ih}\Omega_{hj}+A^s_{ij}\Omega_{sr}).\nonumber\end{aligned}$$ Evaluating (\[dDArij\]) on $X_k\wedge X_l$ and $U_s\wedge U_t$, respectively, we obtain $$\begin{aligned}
&&A^r_{ijlk}-A^r_{ijkl}\label{Arijkl-lk}\\
&=&-(A^r_{hj}A^s_{hk}A^s_{il}+A^r_{ih}A^s_{hk}A^s_{jl}+A^r_{hl}A^s_{hk}A^s_{ij}+2A^r_{ijs}A^s_{kl})\nonumber\\
&&+(A^r_{hj}A^s_{hl}A^s_{ik}+A^r_{ih}A^s_{hl}A^s_{jk}+A^r_{hk}A^s_{hl}A^s_{ij})\nonumber\\
&&-(A^r_{hj}R_{hikl}+A^r_{ih}R_{hjkl}+A^s_{ij}R_{srkl}),\nonumber\end{aligned}$$ $$\begin{aligned}
A^r_{ijts}-A^r_{ijst}=-(A^r_{hj}R_{hist}+A^r_{ih}R_{hjst}+A^u_{ij}R_{urst}).\label{Arijst-ts}\end{aligned}$$ Now since $T=0$ and $A^r_{ikk}=0$, by combining the identities (\[ATcoeff\], \[cov-T-A2\], \[o2’\], \[o4\], \[Arijk-identity\], \[Arijs-identity\], \[def-lap-A\]) with (\[Arijkl-lk\], \[Arijst-ts\]), we can calculate the Laplacians of the integrability tensor $A$ as follows: $$\begin{aligned}
\label{lap-h-A}
&&\langle A,\triangle^{\mathscr{H}}A\rangle:=A^r_{ij}(\triangle^{\mathscr{H}}A^r_{ij})=A^r_{ij}A^r_{ijkk}\\
&=&A^r_{ij}(-A^r_{jkik}-A^r_{kijk})=2A^r_{ij}A^r_{ikjk}=2A^r_{ij}(A^r_{ikjk}-A^r_{ikkj})\nonumber\\
&=&2A^r_{ij}\Big(-(A^r_{hk}A^s_{hk}A^s_{ij}+2A^r_{iks}A^s_{kj})+2A^r_{hk}A^s_{hj}A^s_{ik}\nonumber\\
&&\quad\quad\quad-(A^r_{hk}R_{hikj}+A^r_{ih}R_{hkkj}+A^s_{ik}R_{srkj})\Big)\nonumber\\
&=&2A^r_{ij}\Big(2A^r_{ih}A^s_{hk}A^s_{kj}+2A^r_{hk}A^s_{hj}A^s_{ik}\nonumber\\
&&\quad\quad-(A^r_{hk}\check{R}_{hikj}\circ\pi+A^r_{ih}\check{R}_{hkkj}\circ\pi)
-2A^s_{ik}R_{srkj}\Big)\nonumber\\
&=&-2\|[A^r,A^s]\|^2-A^r_{ij}A^r_{hk}\check{R}_{ijhk}\circ\pi+2A^r_{ij}A^r_{ih}\check{R}_{jh}\circ\pi-4A^r_{ij}A^s_{ik}R_{srkj},\nonumber\end{aligned}$$ $$\begin{aligned}
\label{lap-v-A}
&&\langle A,\triangle^{\mathscr{V}}A\rangle:=A^r_{ij}(\triangle^{\mathscr{V}}A^r_{ij})=A^r_{ij}A^r_{ijss}\\
&=&-A^r_{ij}A^s_{ijrs}=A^r_{ij}(A^s_{ijsr}-A^s_{ijrs})\nonumber\\
&=&-A^r_{ij}(A^s_{hj}R_{hirs}+A^s_{ih}R_{hjrs}+A^u_{ij}R_{usrs})\nonumber\\
&=&2A^r_{ij}A^s_{ik}R_{srkj}-A^r_{ij}A^s_{ij}\hat{R}_{rs},\nonumber\end{aligned}$$ where we denote by $A^r:=(A^r_{ij})$ the $(n\times n)$ skew-symmetric matrix corresponding to the operator $AU_r:~TM\rightarrow TM$ defined by $AU_r(X_i):=A_{X_i}U_r=A^r_{ij}X_j$, and the square norm of the Lie bracket in the last line of (\[lap-h-A\]) is implicitly summed over all the indices $r$ and $s$.
Simons-type inequality
----------------------
In this subsection we will derive the Simons-type inequality rendered in Theorem \[Thm-simonstype ineq\] for Riemannian submersions with totally geodesic fibres and Yang-Mills horizontal distributions.
We denote by $\nabla^{\mathscr{H}}$ (resp. $\nabla^\mathscr{V}$) the restriction to the horizontal (resp. vertical) distribution of the covariant derivative $D$ on $M$, *i.e.*, $$\nabla^{\mathscr{H}}W:=(DW)|_{\mathscr{H}},\quad \nabla^\mathscr{V}W:=(DW)|_{\mathscr{V}},\quad for~~ any~~ tensor~~ W~~
on~~ M.$$ From (\[A norm2\], \[def-lap-A\], \[def-lap\]) we can derive the following $$\label{lap-hv-Anorm}
\frac{1}{2}\triangle^{\mathscr{H}}|A|^2=\langle A,\triangle^{\mathscr{H}}A\rangle+|\nabla^{\mathscr{H}}A|^2,\quad
\frac{1}{2}\triangle^{\mathscr{V}}|A|^2=\langle A,\triangle^{\mathscr{V}}A\rangle+|\nabla^{\mathscr{V}}A|^2.
$$ Combining (\[o2’\], \[o3\], \[Arijs-identity\], \[Arijkl-lk\], \[Arijst-ts\], \[lap-hv-Anorm\]) we obtain $$\begin{aligned}
\label{laph-lapv}
&&(\frac{1}{2}\triangle^{\mathscr{H}}+2\triangle^{\mathscr{V}})|A|^2\\
&=&-2\|[A^r,A^s]\|^2-A^r_{ij}A^r_{hk}\check{R}_{ijhk}\circ\pi+2A^r_{ij}A^r_{ih}\check{R}_{jh}\circ\pi-4A^r_{ij}A^s_{ij}\hat{R}_{rs}\nonumber\\
&&+4A^r_{ij}A^s_{ik}R_{srkj}+|A^r_{ijk}|^2+4|A^r_{ijs}|^2\nonumber\\
&=&-\|[A^r,A^s]\|^2-A^r_{ij}A^r_{hk}\check{R}_{ijhk}\circ\pi+2A^r_{ij}A^r_{ih}\check{R}_{jh}\circ\pi-4A^r_{ij}A^s_{ij}\hat{R}_{rs}\nonumber\\
&&+|R_{ijkr}|^2+|R_{srij}|^2,\nonumber\end{aligned}$$ where, from now on, the indices within square norms are also implicitly summed over. If $M$ is closed, then by (\[int-lap-0\], \[laph-lapv\]) we get $$\label{int-ineq}
\int_M\Big(\|[A^r,A^s]\|^2+4A^r_{ij}A^s_{ij}\hat{R}_{rs}+A^r_{ij}A^r_{hk}\check{R}_{ijhk}\circ\pi-2A^r_{ij}A^r_{ih}\check{R}_{jh}\circ\pi\Big)dV_M\geq0.$$
As defined before Theorem \[Thm-simonstype ineq\] in Section \[sec1\], for $x\in M$, $\check{\kappa}(x)$ is the largest eigenvalue of the curvature operator $\check{R}$ of $B$ at $\pi(x)\in B$, $\check{\lambda}(x)$ is the lowest eigenvalue of the Ricci curvature $\check{r}$ of $B$ at $\pi(x)\in B$ and $\hat{\mu}(x)$ is the largest eigenvalue of the Ricci curvature $\hat{r}$ of the fibre at $x$. Then the inequality (\[int-ineq\]) induces the following: $$\label{int-ineq2}
\int_M\Big(\|[A^r,A^s]\|^2+4\hat{\mu}|A|^2+2\check{\kappa}|A|^2-2\check{\lambda}|A|^2\Big)dV_M\geq0.$$
When $n=2$, it is obvious that $[A^r,A^s]=0$ and $\check{\kappa}=\check{\lambda}$. Thus by (\[int-ineq2\]) we have $$\label{ineq-mu }
\int_M|A|^2\hat{\mu}~dV_M\geq0,$$ which verifies the first case (i) of Theorem \[Thm-simonstype ineq\].
When $m=1$, the first two terms of (\[int-ineq2\]) vanish and thus $$\label{ineq-kappa-lam}
\int_M|A|^2(\check{\kappa}-\check{\lambda})~dV_M\geq0,$$ which proves the second case (ii) of Theorem \[Thm-simonstype ineq\].
Now we are coming to discover the phenomenons of “duality" between symmetric matrices and skew-symmetric matrices, between submanifold geometry and Riemannian submersions, as well as their interactions. To do this, one needs only to apply the inequalities of Theorem \[thm1\] to (\[int-ineq2\]) with the skew-symmetric matrices $\{A^r\}$ instead of $\{B_r\}$, keeping in mind how the algebraic DDVV inequality (of symmetric matrices) applies to prove the Simons inequality in submanifold geometry (cf. [@Lu]). This completes the proof of the left two cases (iii, iv) of Theorem \[Thm-simonstype ineq\] immediately.
Equality conclusions
--------------------
In this subsection we will complete the proof of Theorem \[Thm-simonstype ineq\] by verifying the conclusions (a-d) for equality conditions of the Simons-type inequality case by case.
Firstly, it is a well-known fact that the total space $M$ of a Riemannian submersion with vanishing $T$ and $A$ is (at least locally) a Riemannian product $B\times F$, and vice versa. Henceforth, we assume that $A\neq0$. The proof of (a-d) of Theorem \[Thm-simonstype ineq\] goes on as follows:
**(a)** In each case of (i-iv) of Theorem \[Thm-simonstype ineq\], the equality assumption of the integral inequality compels (\[int-ineq\]) to attain its equality simultaneously, which then by (\[int-lap-0\], \[laph-lapv\]) shows immediately $$\label{Rijkr-Rsrij-0}
R_{ijkr}\equiv0,\quad R_{srij}\equiv0.$$ Now since the fibres are totally geodesic, the Ricci equation on any fibre $F_b$ shows that the normal curvature $\hat{R}^{\bot}_{srij}$ of $F_b$ equals $R_{srij}$ and thus vanishes. So each fibre has flat normal bundle in $M$. Moreover, it follows from (\[o2’\], \[o3\], \[Arijs-identity\], \[Rijkr-Rsrij-0\]) that $$\label{Arijk0-Arijs}
A^r_{ijk}=0,\quad A^r_{ijs}=\frac{1}{2}[A^r,A^s]_{ij}.$$ Noticing that the covariant derivative of $|A|^2$ can be calculated from (\[Arijk0-Arijs\]) as $$D|A|^2=2A^r_{ij}A^r_{ijk}\omega_k+2A^r_{ij}A^r_{ijs}\omega_s=0,$$ we arrive at the conclusion that $|A|^2\equiv Const=:C>0$. Then by (\[lap-h-A\]-\[lap-hv-Anorm\]) and (\[Rijkr-Rsrij-0\], \[Arijk0-Arijs\]), we have $$\begin{aligned}
&&\frac{1}{2}\triangle^{\mathscr{H}}|A|^2=
-2\|[A^r,A^s]\|^2-A^r_{ij}A^r_{hk}\check{R}_{ijhk}\circ\pi+2A^r_{ij}A^r_{ih}\check{R}_{jh}\circ\pi\equiv0,\label{laph-0}\\
&&\frac{1}{2}\triangle^{\mathscr{V}}|A|^2=
-A^r_{ij}A^s_{ij}\hat{R}_{rs}+\frac{1}{4}\|[A^r,A^s]\|^2\equiv0.\label{lapv-0}\end{aligned}$$ Now we come to prove the subcases (a1-a4) of (a) as follows.
- Now $n=2$ and $[A^r,A^s]\equiv0$. So by the definition of $\hat{\mu}$ and (\[lapv-0\]), we get $$|A|^2\hat{\mu}\geq A^r_{ij}A^s_{ij}\hat{R}_{rs}=0,$$ whereas $|A|^2\equiv C>0$ and $\int_M|A|^2\hat{\mu}dV_M=0$ by assumption.\
This proves that $\hat{\mu}\equiv0$.
- Now $m=1$ and $[A^r,A^s]\equiv0$. So by the definitions of $\check{\kappa}$, $\check{\lambda}$ and (\[laph-0\]), we get $$|A|^2(\check{\kappa}-\check{\lambda})\geq
\frac{1}{2}A^r_{ij}A^r_{hk}\check{R}_{ijhk}\circ\pi-A^r_{ij}A^r_{ih}\check{R}_{jh}\circ\pi=0,$$ whereas $|A|^2\equiv C>0$ and $\int_M|A|^2(\check{\kappa}-\check{\lambda})dV_M=0$ by assumption.\
This proves that $\check{\kappa}-\check{\lambda}\equiv0$.
- Now the equality assumption implies that the inequality in (i) of Theorem \[thm1\] (with $B_r=A^r$) attains its equality, *i.e.*, $$\label{equ-Lie}
\sum_{r,s}\|[A^r,A^s]\|^2=\frac{1}{3}\Big(\sum_r|A^r|^2\Big)^2=\frac{1}{3}|A|^4=\frac{1}{3}C^2.$$ Then by the definitions of $\hat{\mu},\check{\kappa},\check{\lambda}$ and (\[laph-0\], \[lapv-0\]), we have $$\begin{aligned}
&&|A|^2\hat{\mu}\geq
A^r_{ij}A^s_{ij}\hat{R}_{rs}=\frac{1}{4}\|[A^r,A^s]\|^2=\frac{1}{12}C^2,\nonumber\\
&&|A|^2(\check{\kappa}-\check{\lambda})\geq
\frac{1}{2}A^r_{ij}A^r_{hk}\check{R}_{ijhk}\circ\pi-A^r_{ij}A^r_{ih}\check{R}_{jh}\circ\pi=-\|[A^r,A^s]\|^2=-\frac{1}{3}C^2,\nonumber\end{aligned}$$ whereas $|A|^2\equiv C>0$ and $\int_M~|A|^2(\frac{1}{6}|A|^2+2\hat{\mu}+\check{\kappa}-\check{\lambda})~dV_M=0$ by assumption.\
This proves that $\hat{\mu}\equiv\frac{1}{12}C$, $\check{\kappa}-\check{\lambda}\equiv-\frac{1}{3}C$.
- The proof is almost the same with that of (a3) except for that the coefficient $\frac{1}{3}$ in (\[equ-Lie\]) would be substituted by $\frac{2}{3}$. So we omit it here.
**(b)** If the equality in (iii) (resp. (iv)) holds, as in the proof of (a3), the inequality in (i) (resp. (ii)) of Theorem \[thm1\] (with $B_r=A^r$) attains its equality, thereby, under some $K=O(n)\times O(m)$ action which can be realized by a choice of an orthonormal horizontal basis $\{X_1,\cdots,X_n\}$ and of an orthonormal vertical basis $\{U_{n+1},\cdots,U_{n+m}\}$, the matrices $A^r$’s are all equal to zero except $A^{n+1},A^{n+2},A^{n+3}$, which are in the forms of $C_1,C_2,C_3$ (resp. $diag(D_1, 0), diag(D_2, 0), diag(D_3, 0)$). Noticing that now we have $$|A|^2=|A^{n+1}|^2+|A^{n+2}|^2+|A^{n+3}|^2\equiv C>0,$$ we derive that $m\geq3$. Moreover, we can rewrite $A^{n+1},A^{n+2},A^{n+3}$ as follows: $$\label{case3A123}
\begin{array}{cc}
A^{n+1}=\sqrt{\frac{C}{6}}\left(\begin{array}{ccc}
0& 1 & 0 \\
-1& 0 & 0\\
0& 0& 0
\end{array}\right),& A^{n+2}=\sqrt{\frac{C}{6}}\left(\begin{array}{ccc}
0& 0 & 1 \\
0& 0 & 0\\
-1& 0& 0
\end{array}\right),\\
A^{n+3}=\sqrt{\frac{C}{6}}\left(\begin{array}{ccc}
0& 0 & 0 \\
0& 0 & 1\\
0& -1& 0
\end{array}\right)& \emph{for equality case of (iii);}
\end{array}$$
$$\label{case4A123}
\begin{array}{cc}
A^{n+1}=\sqrt{\frac{C}{12}}\left(\begin{array}{c|c}
\begin{smallmatrix}
0& 1 & 0&0 \\
-1& 0 & 0&0\\
0& 0& 0&1\\
0&0&-1&0
\end{smallmatrix}&0\\
\hline0&0
\end{array}\right),& A^{n+2}=\sqrt{\frac{C}{12}}\left(\begin{array}{c|c}
\begin{smallmatrix}
0& 0 & 1 &0\\
0& 0 & 0&-1\\
-1& 0& 0&0\\
0&1&0&0
\end{smallmatrix}&0\\
\hline0&0
\end{array}\right),\\
A^{n+3}=\sqrt{\frac{C}{12}}\left(\begin{array}{c|c}
\begin{smallmatrix}
0& 0 & 0 &1\\
0& 0 & 1&0\\
0& -1& 0&0\\
-1&0&0&0
\end{smallmatrix}&0\\
\hline0&0
\end{array}\right)& \emph{for
equality case of (iv),}
\end{array}$$
where $0$ in the diagonals of (\[case4A123\]) is a zero matrix of order $(n-4)$. As in the proof of (a3), we have the following equations if the equality in (iii) or (iv) holds: $$\label{case34-equs}
|A|^2\hat{\mu}= A^r_{ij}A^s_{ij}\hat{R}_{rs},\quad
|A|^2\check{\kappa}=
\frac{1}{2}A^r_{ij}A^r_{hk}\check{R}_{ijhk}\circ\pi,\quad
|A|^2\check{\lambda}=A^r_{ij}A^r_{ih}\check{R}_{jh}\circ\pi.$$
Using the formulas (\[case3A123\]) for equality case of (iii), the equations (\[case34-equs\]) can be turned to the following: $$\begin{array}{lll} \hat{\mu}=
\frac{1}{3}(\hat{R}_{n+1~n+1}+\hat{R}_{n+2~n+2}+\hat{R}_{n+3~n+3}),\\
\check{\kappa}=
\frac{1}{3}(\check{R}_{1212}\circ\pi+\check{R}_{1313}\circ\pi+\check{R}_{2323}\circ\pi),\\
\check{\lambda}=\frac{1}{3}(\check{R}_{11}\circ\pi+\check{R}_{22}\circ\pi+\check{R}_{33}\circ\pi).
\end{array}$$ Then recalling the definitions of $\hat{\mu},\check{\kappa},\check{\lambda}$, we obtain the following decompositions for $\hat{r},\check{R},\check{r}$ for equality case of (iii): $$\hat{r}=\hat{\mu}I_3\oplus \hat{r}',\quad
\check{R}\equiv\check{\kappa}I_3,\quad
\check{r}\equiv\check{\lambda}I_3,$$ where $\hat{r}'=\hat{r}|_{span\{U_{7},\cdots,U_{3+m}\}}$ if $m\geq4$ and $0$ if $m=3$, $\check{\lambda}=2\check{\kappa}$ because of $n=3$ now.
Similarly, using the formulas (\[case4A123\]) for equality case of (iv) and the first Bianchi identity, the equations (\[case34-equs\]) can be turned to the following: $$\begin{array}{lll} \hat{\mu}=
\frac{1}{3}(\hat{R}_{n+1~n+1}+\hat{R}_{n+2~n+2}+\hat{R}_{n+3~n+3}),\\
\check{\kappa}=
\frac{1}{6}(\check{R}_{1212}\circ\pi+\check{R}_{1313}\circ\pi+\check{R}_{1414}\circ\pi
+\check{R}_{2323}\circ\pi+\check{R}_{2424}\circ\pi+\check{R}_{3434}\circ\pi),\\
\check{\lambda}=\frac{1}{4}(\check{R}_{11}\circ\pi+\check{R}_{22}\circ\pi+\check{R}_{33}\circ\pi+\check{R}_{44}\circ\pi).
\end{array}$$ Then recalling the definitions of $\hat{\mu},\check{\kappa},\check{\lambda}$, we obtain the following decompositions for $\hat{r},\check{R},\check{r}$ for equality case of (iv): $$\hat{r}=\hat{\mu}I_3\oplus \hat{r}',\quad
\check{R}=\check{\kappa}I_6\oplus \check{R}',\quad
\check{r}\equiv\check{\lambda}I_4\oplus\check{r}',$$ where $\hat{r}'=\hat{r}|_{span\{U_{n+4},\cdots,U_{n+m}\}}$ if $m\geq4$ and $0$ if $m=3$, $\check{R}'=\check{R}|_{span\{X_i\wedge
X_j|1\leq i\leq n,~5\leq j\leq n\}}$ and $\check{r}'=\check{r}|_{span\{X_5,\cdots,X_n\}}$ if $n\geq5$ and $0$ if $n=4$.
From the decompositions, if $m=3$, then we can see that the $3$-dimensional fibres have constant Ricci curvature and thus have constant sectional curvature; if $n=3$ or $4$, then the base manifold $B^n$ has constant sectional curvature; if $n=5$, then by the definitions of $\check{\kappa},\check{\lambda}$ we have $$\begin{aligned}
&&\check{\lambda}\leq\check{R}_{55}=\check{R}_{1515}+\check{R}_{2525}+\check{R}_{3535}+\check{R}_{4545}\leq
3\check{\kappa}+\check{R}_{i5i5},\nonumber\\
&&\check{\lambda}=\check{R}_{ii}=\sum_{j=1}^5\check{R}_{ijij}=3\check{\kappa}+\check{R}_{i5i5},\quad
for~~ i=1,2,3,4.\nonumber\end{aligned}$$ These prove that $\check{R}_{i5i5}=\check{\kappa}$ for $i=1,2,3,4$, and so the base manifold $B^5$ has constant sectional curvature.
**(c)** Now $m=3,n=3$ and the equality in (iii) holds. In (b) we have proved that both of the fibres and the base manifold $B^3$ have constant sectional curvature. Due to a result of Hermann [@He] we see that the fibres are all isometric. Reset $|A|^2\equiv C=:24a>0$, then by (a3) and (b) we get $$\hat{\mu}=2a,\quad \check{\lambda}=2\check{\kappa}=16a,$$ which deduce the conclusions of (c1) and (c2).
The identities in (c3) can be calculated from the formulas (\[sec-curv\], \[ricci-curv\], \[case3A123\]). In fact, since we have $T=0$ and $A_{ikk}=0$, the formulas (\[sec-curv\], \[ricci-curv\]) turn into the following: $$\label{K_MR_M}
\begin{array}{ll}
K_{rs}=\hat{K}_{rs},\quad K_{ir}=\sum_j(A^r_{ij})^2,\quad K_{ij}=\check{K}_{ij}\circ\pi-3\sum_r(A^r_{ij})^2;\\
R_{ir}=0,\quad R_{rs}=\hat{R}_{rs}+A^r_{ij}A^s_{ij},\quad R_{ij}=\check{R}_{ij}\circ\pi-2A^r_{ik}A^r_{jk}.
\end{array}$$ Then using formulas (\[case3A123\], \[K\_MR\_M\]) and the known facts that $\hat{K}_{rs}=a$, $\check{K}_{ij}=8a$, we complete the proof. One should notice that the index range for $r$ in (c3) is $\{1,2,3\}$ rather than $\{n+1,n+2,n+3\}$ $(n=3)$ here.
**(d)** Based on results of (b) and formulas (\[case4A123\], \[K\_MR\_M\]), the proof of the assertions for (d2) and the heading paragraph of (d) are exactly the same with that of (c) despite that we reset $|A|^2\equiv C=:12a>0$ here in view of (a4). As for (d1), we first calculate the sectional curvatures of $B^4$ and $M^7$ respectively and find that $B$ has constant sectional curvature $4a$ and $M$ has constant sectional curvature $a$. In fact, by (a4), (b) and (\[case4A123\], \[K\_MR\_M\]) we know that $$\hat{\mu}=2a,\quad \check{\lambda}=3\check{\kappa}=12a,\quad K_{rs}=K_{ir}=K_{ij}=a.$$
Hence, $M^7$ is covered by $S^7(\frac{1}{\sqrt{a}})$, $B^4$ is covered by $S^4(\frac{1}{2\sqrt{a}})$ and we denote by $\pi_1, \pi_2$ the corresponding covering maps. Thus there is a Riemannian submersion $\pi_0: S^7(\frac{1}{\sqrt{a}})\rightarrow S^4(\frac{1}{2\sqrt{a}})$ (lift map of $\pi\circ\pi_1$ through $\pi_2$) such that $\pi_2\circ\pi_0=\pi\circ\pi_1$. Recall that Ranjan [@Ra] showed that $\pi_0: S^7(\frac{1}{\sqrt{a}})\rightarrow S^4(\frac{1}{2\sqrt{a}})$ is equivalent to the Hopf fibration (see also [@Es]). Without loss of generality, we can assume that $\pi_0$ is just the Hopf fibration, since otherwise we can alter $\pi_1, \pi_2$ by taking compositions with corresponding isometries (bundle isometry between $\pi_0$ and the Hopf fibration) of $S^7(\frac{1}{\sqrt{a}})$ and $S^4(\frac{1}{2\sqrt{a}})$ respectively. The proof of (d1) is now completed.
In conclusion, the proof of Theorem \[Thm-simonstype ineq\] is now completed.
The first part of this paper (*i.e.*, the skew-symmetric matrix inequality, Theorem \[thm1\]) was done during 2007-2008 when I was still a PhD student under the guidance of Professor Zizhou Tang who taught and helped me much all these years. By this opportunity I would like to express my deepest gratitude to him. Many thanks also to Professors Thomas E. Cecil, Qingming Chen, Xiuxiong Chen and Weiping Zhang for their kindly encouragements and supports.
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abstract: 'In boundary element methods, using double nodes at corners is a useful approach to uniquely define the normal direction of boundary elements. However, matrix equations constructed by conventional boundary integral equations (CBIEs) become singular under certain combinations of double node boundary conditions. In this paper, we analyze the singular conditions of the CBIE formulation for cases where the boundary conditions at the double node are imposed by combinations of Dirichlet, Neumann, Robin, and interface conditions. To address this singularity, we propose the use of hypersingular integral equations (HBIEs) for wave propagation problems that obey the Helmholtz equation. To demonstrate the applicability of HBIE, we compare three types of simultaneous equations: (i) CBIE, (ii) partial-HBIE where the HBIE is only applied to the double nodes at corners while the CBIE is applied to the other nodes, and (iii) full-HBIE where the HBIE is applied to all nodes. Based on our numerical results, we observe the following results. The singularity of the matrix equations for problems with any combination of boundary conditions can be resolved by both full-HBIEs and partial-HBIEs, and the partial-HBIE exhibits better accuracy than the full-HBIE. Furthermore, the computational cost of partial-HBIEs is smaller than that of full-HBIEs.'
address: 'Faculty of Engineering, Hokkaido University, Sapporo, 060-8628, Japan'
author:
- Satoshi Tomioka
- Shusuke Nishiyama
- Yutaka Matsumoto
- Naoki Miyamoto
bibliography:
- 'paper.bib'
title: Desingularization of matrix equations employing hypersingular integrals in boundary element methods using double nodes
---
Boundary element method, Hypersingular integral, Helmholtz equation, Double node, Corner, Boundary condition, Regularization of coefficient matrix, Rank deficiency
Introduction {#sec:intro}
============
The boundary element method (BEM), the finite difference method (FDM), and the finite element method (FEM) have been commonly used to solve boundary value problems. In the BEM, a set of simultaneous equations for determining unknown variables at nodes on the boundary is constructed in discretized boundary integral equations. The variables in simultaneous equations are nodal field values and normal derivatives only on individual boundary nodes, whereas the variables in the FDM or FEM are field values at domain nodes which are placed in the entire domain enclosed by the boundary. Therefore, the number of variables in the BEM is much smaller than that in the FDM or FEM, which is one of the advantages of BEM. Furthermore, the BEM can be easily applied to external problems, such as wave scattering problems, since it does not require the placement of nodes in a domain spreading to infinity and it does not require any other boundary conditions to represent radiation at a boundary enclosing the domain considered.
In most boundary problems, the field values, $u$, along the boundary are continuous; however, the normal derivatives, $q$, are discontinuous at corners since the normal directions at any corner point are different. By using a linear element or higher-order elements, the boundary elements share the nodes at both ends of the element with adjacent boundary elements. In this case, the normal direction, ${{{\boldsymbol{n}}}}$, at the corner node cannot be defined uniquely since the single node at any corner belongs to two boundary elements with different normal directions.
There are two approaches to addressing the definition problem of the normal direction. The first approach involves the use of non-conforming elements, which are also called discontinuous elements. In the non-conforming element, collocation nodes that represent $u$ and $q$ do not coincide with geometric nodes, but they do so in the conforming element. The non-conforming element has been applied to several problems; e.g., elastostatic problems [@Manolis:1986; @Parreira:1988; @Olukoko:1993; @Blazquez:1994; @Huesmann:1994; @Paris:1995; @Blazquez:1998], fluid flow problems [@Patterson:1982; @Dyka:1989], and acoustic problems [@Silva:1993]. Although the accuracy between the non-conforming element and the conforming element were compared by Manolis and Banerjee [@Manolis:1986], and Parreira [@Parreira:1988], they arrived at different conclusions.
The second approach includes a double node technique [@Brebbia:1984:Sec_5_2] or a multiple node technique for three-dimensional problems. In the double node technique, two normal derivatives, $q_{\alpha}={{{\boldsymbol{n}}}}_{\alpha}{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{\nabla u}$ and $q_{\beta}={{{\boldsymbol{n}}}}_{\beta}{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{\nabla u}$, and a field, $u$, are defined at the corner node, where ${{{\boldsymbol{n}}}}_{\alpha}$ and ${{{\boldsymbol{n}}}}_{\beta}$ denote the directions normal to the two boundary elements connected to the corner node. However, a set of simultaneous equations, called a matrix equation, becomes singular under certain boundary conditions; i.e., the rank of the matrix equation is reduced since some node equations are redundant. Further details will be presented in Sec. \[sec:CBIE-Rank\_deficient\].
To address the rank reduction problem caused by the double nodes, there are two categories of approaches. The first category involves the use of a local relation for each double node [@Walker:1989; @Yan:1994; @Kassab:1994; @Gao:2000]. By using a Taylor expansion around the corner node, this relation is described as a linear combination of the two normal derivatives and the field values at neighbor nodes of the corner node. The local relation is replaced by one of the redundant equations that reduces the rank; therefore, the matrix equation still includes a square matrix. In the second category, extra node equations are employed [@Mitra:1987; @Mitra:1993; @Subia:1995; @Deng:2013; @Zheng:2018]. Mitra and Ingber [@Mitra:1987] proposed a technique for replacing one of the redundant equations in each corner by an extra node equation with respect to an extra collocation node placed outside the domain considered. Following this study, the authors mentioned that “external collocation yields a coefficient matrix with a large number” [@Mitra:1993], and they improved the method using the extra node equations so that the location of the extra node on the boundary elements connects to the double node [@Mitra:1993; @Subia:1995]. Subia, Ingber, and Mitra demonstrated that there are no significant differences in accuracy between the method of the extra node equation and the non-conforming method [@Subia:1995]. The method that uses the extra node equations was extended to the problems of interface boundaries at which two or more domains are connected [@Deng:2013; @Zheng:2018]. Using these methods, the number of variables is not increased since the field at the extra node is known. Therefore, the coefficient matrix is the square matrix, which is similar to the matrix of the first category. If we simply add the local relations shown in the first category or the extra node equations in the second category instead of replacing them, the number of equations becomes larger than the number of variables, which is referred to as an overdetermined problem. We can solve this equation by using least-square methods, but the computational time required to solve this equation is much greater than solving the general simultaneous equation; therefore, the replacements are generally applied. The replacement of the equation should be performed individually while examining the types of boundary conditions at the corner. The individual examination increases the complexity of the programming of widely applicable BEM codes that include many types of boundary conditions.
In addition to the corner node problems, there are rank deficient or large condition number problems. We focus on two problems related to hypersingular boundary integral equations. The first is a non-uniqueness or a spurious solution problem in an external field for a wave scattering problem that obeys the Helmholtz equation, in which the domain considered is outside of the boundary enclosing a scatterer. In this situation, spurious solutions can be obtained at the eigenfrequency of the scatterer. To resolve this problem, two major approaches were proposed. Schenck employed equations related to additional nodes in the scatterer region and the method is called ‘Combined Helmholtz Integral Equation Formulation (CHIEF)’ [@Schenck:1968]. Chen et al. [@IL_Chen:2001] applied the additional node equations to interior problems in which additional points are placed outside of the boundary. These methods are similar to the extra node equation methods for the corner problems shown in the previous paragraph; however, they require a solver based on the least-square method since the set of simultaneous equations becomes an overdetermined equation. The other approach is referred to as the Burton-Miller method [@Burton-Miller:1971; @Benthien:1997; @Diwan:2013; @Langrenne:2015]. Burton and Miller [@Burton-Miller:1971] represented a boundary integral equation (BIE) for each node using a linear combination of two types of BIE called a conventional BIE (CBIE) and a hypersingular BIE (HBIE). In the CBIE, the field $u$ at the field point is denoted by integrals over the boundary on which sources are distributed. The HBIE is obtained by taking a gradient of the CBIE. The singularity of the integral in the HBIE is stronger than that of the CBIE; therefore, it is called a hypersingular integral. Bentihien and Schenck [@Benthien:1997] reviewed the non-uniqueness problem with comparisons of other methods, including the CHIEF and Burton-Miller methods.
The other rank deficient problem is found at a degenerate boundary which appears either at a crack in an elastostatic problem [@Portela:1992; @JT_Chen:1994; @Chyuan:2003; @Lu:2010] or at both surfaces of a thin metal with zero-thickness in an electromagnetic problem [@Chyuan:2003]. To resolve these problems, the CBIEs are applied to one side of the crack or the thin metal, and the HBIEs are applied to the other side. These methods are referred to as dual-BEM.
As shown in the previous two paragraphs, the use of HBIEs is effective for resolving rank deficient problems. In this paper, we will demonstrate that the corner singular problem for wave propagation problems can be solved by only using node equations based on HBIEs. In addition, to suppress the rank deficiency caused by the corners, we do not need to use HBIEs for every node. We will also illustrate that only the replacement of the node equations related to the corner node by HBIEs is sufficient, which is similar to the aforementioned dual-BEM.
In the application of HBIEs, the regularization of hypersingularity is a key issue. The authors developed an analytical regularization of the two-dimensional Helmholtz equation [@Tomioka:2010]; the other regularization methods are also found in the references of that study. In the regularization of HBIEs, we include the relations between two normal derivatives at the double nodes, similar to the local relation methods described above. Therefore, the method by HBIE can be considered an extension that uses the local relation. However, the contributions to the double node from all the nodes are considered in the method by HBIE; whereas the local relation method represents the relations between local nodal quantities only.
The method proposed in this paper to overcome the corner problem caused by the double nodes does not require local relations at corners, extra node equations, or least-square methods. Our method can also be applied to any kind of boundary conditions of boundary elements that include corner nodes. In addition, the proposed method can also be applied to interface boundary conditions. From these characteristics, no additional effort is required in prepossessing to prepare the input data for solving the boundary value problem.
The outline of this paper is as follows. In Sec. \[sec:CBIE-Rank\_deficient\], we demonstrate why the rank of the coefficient matrix of the CBIE is reduced in the case where the double node is employed. We also demonstrate the condition that results in rank deficiency based on the nature of the discretized node equations. In Sec. \[sec:HBIE\], we illustrate why the HBIE does not cause a rank deficiency. The numerical results and discussions for simple waveguide problems are presented in Sec. \[sec:results\] to demonstrate that the HBIE is applicable to any combination of boundary conditions. The advantages of the partial-HBIE method in which the HBIE is applied to only the double nodes and the CBIE is applied to other nodes are also shown. Finally, the summary is presented in Sec. \[sec:conclusion\].
Rank deficiency problem in CBIEs {#sec:CBIE-Rank_deficient}
================================
To illustrate rank deficient conditions for a set of discretized node equations of CBIEs, the discretization using linear elements is first shown; the double node technique that defines two sub-nodes for a double node are then shown; and lastly, the rank deficiency conditions are discussed based on comparisons between two equations for the two sub-nodes.
Discretization of CBIEs {#sec:CBIE-discretization}
-----------------------
A complex-valued time harmonic scalar wave satisfies the following Helmholtz equation with an assumed time factor ${\rm e}^{{\rm j}\omega t}$, where $\rm j$ is an imaginary unit and $\omega$ is the angular frequency: $$\begin{aligned}
\label{Helmholtz}
&&\nabla^2 u({{{\boldsymbol{r}}}})+k^2 u({{{\boldsymbol{r}}}})=0, \qquad {{{\boldsymbol{r}}}}\in\Omega,\end{aligned}$$ where $k$ indicates the wave number, which is a ratio of $\omega$ to the wave velocity, and $\Omega$ represents the spatial domain considered. A fundamental solution $u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})$, which represents the contribution to a field point ${{{\boldsymbol{r}}}_i}$ from a unit source placed at a source point ${{{\boldsymbol{r}}}}$ in free space satisfies $$\begin{aligned}
\label{funda1}
\nabla^2 u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})+k^2 u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})=-\delta({{{\boldsymbol{r}}}}-{{{\boldsymbol{r}}}_i}),\end{aligned}$$ where the differential operator $\nabla$ operates only on ${{{\boldsymbol{r}}}}$, but not on ${{{\boldsymbol{r}}}_i}$. This equation can be solved analytically. In two-dimensional problems, the outward propagating wave that obeys [Eq. (\[funda1\])]{} is $$\begin{aligned}
u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})=\frac{1}{4\rm j}H_0^{(2)}(kr),\qquad r=|{{{\boldsymbol{r}}}}-{{{\boldsymbol{r}}}_i}|,\end{aligned}$$ where the function $H_0^{(2)}(kr)$ is a second kind 0-th order Hankel function.
Using Green’s second identity and some integral operations for [Eqs. (\[Helmholtz\]) and (\[funda1\])]{}, we obtain the conventional boundary integral equation (CBIE), $$\begin{aligned}
\label{bie1}
c({{{\boldsymbol{r}}}_i})&\,u({{{\boldsymbol{r}}}_i})
={{\oint_{\Gamma_{}}^{} \left[ u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i}){\nabla u({{{\boldsymbol{r}}}})}{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}-u({{{\boldsymbol{r}}}}) {\nabla u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})}{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}\right]\,d{\Gamma}\,}},
\nonumber\\
$$ where $\Gamma$ denotes a boundary surrounding $\Omega$, ${{{\boldsymbol{r}}}}$ is the position of the source point on the boundary, ${{{\boldsymbol{r}}}_i}$ is the position of the field point, ${{{\boldsymbol{n}}}}$ is the outward-pointing normal unit vector; the contour integral is evaluated as a Cauchy principal value, and $c({{{\boldsymbol{r}}}_i})$ is the result of the following evaluation of Dirac’s delta function: $$\begin{aligned}
\label{def-c}
c({{{\boldsymbol{r}}}_i})u({{{\boldsymbol{r}}}_i})
{\triangleq}{{\int_{\Omega_{}}^{}\! \,\,u({{{\boldsymbol{r}}}})\delta({{{\boldsymbol{r}}}}-{{{\boldsymbol{r}}}_i})\,d{\Omega}\,}}
={{\int_{\Omega_{}}^{}\! \,\,\delta({{{\boldsymbol{r}}}}-{{{\boldsymbol{r}}}_i})\,d{\Omega}\,}}\,u({{{\boldsymbol{r}}}_i})\,.\end{aligned}$$ The coefficient $c({{{\boldsymbol{r}}}_i})$ depends on the shape of the boundary $\Gamma$ at the field point ${{{\boldsymbol{r}}}_i}$. Because of the nature of Dirac’s delta function, when ${{{\boldsymbol{r}}}_i}$ is located inside and outside the domain, $c({{{\boldsymbol{r}}}_i})$ evaluates to 1 and 0, respectively. In the case where ${{{\boldsymbol{r}}}_i}$ is located on $\Gamma$, $c({{{\boldsymbol{r}}}_i})$ is equal to the ratio of the interior angle ${\varDelta\theta}$ to the whole angle; e.g., ${\varDelta\theta}/2\pi$ for 2-dimensional problems. In addition, when we introduce a fundamental solution to a Laplace equation, $u_L^*$, which satisfies $$\begin{aligned}
\label{funda-laplace}
\nabla^2 u_L^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})=-\delta({{{\boldsymbol{r}}}}-{{{\boldsymbol{r}}}_i}),\end{aligned}$$ the coefficient $c({{{\boldsymbol{r}}}_i})$ can be expressed by a boundary integral: $$\begin{aligned}
\label{equi-pot}
c({{{\boldsymbol{r}}}_i})={{\int_{\Omega_{}}^{}\! \,\,\delta({{{\boldsymbol{r}}}}-{{{\boldsymbol{r}}}_i})\,d{\Omega}\,}}
=-{{\oint_{\Gamma_{}}^{} {\nabla u}_L^*{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}\,d{\Gamma}\,}},\end{aligned}$$ which is called an equipotential condition [@Brebbia:1980].
When ${{{\boldsymbol{r}}}_i}$ is located on $\Gamma$ and the boundary is partitioned into a number of boundary elements, [Eq. (\[bie1\])]{} can be written as a node equation for the node $i$ as a discrete algebraic expression: $$\begin{aligned}
\label{CBIE-point_i}
&
\sum_{j\in I}\left(c_i\delta_i^j+h_i^j\right)u_j^{}-\sum_{j\in I} g_i^j q_j^{}=0,\end{aligned}$$ where both the superscripts and the subscripts $i$ and $j$ denote the node numbers and not the element numbers; $I$ denotes a set of the node numbers, which includes $N$ members; $\delta_i^j$ denotes the Kronecker’s delta; and $q\equiv{\nabla u}{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}$. The factors $g_i^j$ and $h_i^j$ are results of boundary integrals in [Eq. (\[bie1\])]{} as shown below.
In [Eq. (\[CBIE-point\_i\])]{}, there are two variables, $u_j$ and $q_j$, at ${{{\boldsymbol{r}}}}_j$. One of them is specified by a boundary condition as a boundary value which is denoted by $\overline{x}_j$, and the other is an unknown variable denoted $x_j$; therefore, the number of variables is equal to that of the nodes, $N$. Since the node $i$ can be placed at every boundary node, we can obtain $N$ equations of [Eq. (\[CBIE-point\_i\])]{} as $$\begin{aligned}
\label{System_eq-point_i}
\sum_{j\in I} a_i^j x_j=\overline{b}_i, \quad \overline{b}_i=\sum_{j\in I} b_i^j \overline{x}_j
\qquad \mbox{ for each }i\in I,\end{aligned}$$ where both $a_i^j$ and $b_i^j$ are either $h_i^j$ or $g_i^j$ when simple boundary conditions are specified. The matrix composed of $a_i^j$ is called a coefficient matrix in the following discussions.
The coefficient $g_i^j$ represents the contribution to $u_i$ from the nodal quantity $q_j$ at the node $j$, and $h_i^j$ represents the contribution to $u_i$ from $u_j$. These coefficients are evaluated by boundary integrals, which depend on a method that interpolates $u({{{\boldsymbol{r}}}})$ and $q({{{\boldsymbol{r}}}})$ along the boundary element. To examine the nature of $g_i^j$ and $h_i^j$, we present evaluations using a shape function to interpolate them. Let us consider the discretization of the boundary integral of $u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})\,q({{{\boldsymbol{r}}}})$. By denoting the $k$-th boundary element as $\Gamma_k$ and a local node number as $l$, $q({{{\boldsymbol{r}}}})$ on $\Gamma_k$ is represented by a linear combination of shape functions $\phi^{(k,l)}({{{\boldsymbol{r}}}})$ and $q$ at the boundary nodes: $$\begin{aligned}
\label{linear-interporation}
q({{{\boldsymbol{r}}}})=\sum_{l=1}^{N_l} \phi^{(k,l)}({{{\boldsymbol{r}}}}) q_{(k,l)}^{}\quad \mbox{ on\ }\Gamma_k,\end{aligned}$$ where $N_l$ denotes the number of nodes in a boundary element; e.g., $N_l=2$ for the linear element, $N_l=3$ for the second-order element. A global node number $j$ can be mapped from the local node numbers $(k,l)$ by a permutation matrix $m_{(k,l)}^j$: $$\begin{aligned}
q_{(k,l)}^{}=\sum_j m^j_{(k,l)} q_j,\end{aligned}$$ where $m^j_{(k,l)}$ has the value 1 if $(k,l)$ and $j$ are associated, and 0 otherwise. Although the notation of $m^j_{(k,l)}$ is used here, it provides a symbolic meaning, and it is treated as a mapping function such as $j=m(k,l)$ in actual coding to avoid summation procedures with respect to $k$. By using these definitions, the boundary integral is discretized as $$\begin{aligned}
\label{use-permutation}
&
{{\oint_{\Gamma_{}}^{} u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})q({{{\boldsymbol{r}}}})\,d{\Gamma}\,}}
\nonumber\\&\quad
=\sum_k
{{\int_{\Gamma_{k}}^{}\! u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})\left(\sum_l\phi^{(k,l)}({{{\boldsymbol{r}}}}) \left(\sum_j m^j_{(k,l)}q_j\right)\right)\,d{\Gamma}\,}}
\nonumber\\&\quad
=\sum_j\left(\sum_k\sum_l m_{(k,l)}^j g^{(k,l)}_{i}\right) q_j
=\sum_j g_i^j q_j,\end{aligned}$$ where $g_i^{(k,l)}$ and $g_i^j$ are defined as $$\begin{aligned}
&
\label{g_int}
g^{(k,l)}_{i}={{\int_{\Gamma_{k}}^{}\! u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})\phi^{(k,l)}({{{\boldsymbol{r}}}})\,d{\Gamma}\,}},
\\
&
\label{g_int2}
g_i^j=\sum_k\sum_l m^j_{(k,l)} g^{(k,l)}_{i}.\end{aligned}$$ The integral on the right-hand side in [Eq. (\[g\_int\])]{} is evaluated by a numerical integral such as Gauss quadrature or by an analytical integral in the case of ${{{\boldsymbol{r}}}_i}\in\Gamma_k$, which is called a singular integral. Similarly, the other integral in [Eq. (\[bie1\])]{} is evaluated as $$\begin{aligned}
&
{{\oint_{\Gamma_{}}^{} q^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})u({{{\boldsymbol{r}}}})\,d{\Gamma}\,}}
=\sum_j h_i^j u_j,
\\
&
\label{h_int}
h_i^{(k,l)}={{\int_{\Gamma_{k}}^{}\! q^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})\phi^{(k,l)}({{{\boldsymbol{r}}}})\,d{\Gamma}\,}},
\\
&
\label{h_int2}
h_i^j=\sum_k\sum_l m^j_{(k,l)} h_i^{(k,l)},\end{aligned}$$ where $q^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})={\nabla u}^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i}){\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}})$, which is not the derivative at ${{{\boldsymbol{r}}}_i}$; i.e., $q^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})$ does not depend on ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}_i})$.
To discuss the rank of the matrix equation shown in [Eq. (\[System\_eq-point\_i\])]{}, the dependencies of both $g_i^j$ and $h_i^j$ on ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}_i})$ which is the normal unit vector at ${{{\boldsymbol{r}}}_i}$ are important. First, $g_i^j$ is independent of ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}_i})$ since [Eqs. (\[g\_int\]) and (\[g\_int2\])]{} do not include ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}_i})$ regardless of $j$ or $k$. In contrast, the case of $h_i^j$ is different from that of $g_i^j$. Although $h_i^j$ in [Eqs. (\[h\_int\]) and (\[h\_int2\])]{} looks independent of ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}_i})$ even when $q^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})$ is expressed using ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}})$, there is an exception at $i=j$ in which ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}})$ becomes ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}_i})$. The dependency of this exceptional case is presented in the next section.
![ Configuration of double node. Thick lines depict boundary elements. The two circles which are in contact express sub-nodes which represent a double node at a corner. The distance between the sub-nodes are zero; i.e., they are connected to a zero-sized element. Each sub-node belonging to the double node is connected to one non-zero-sized element. The outward-pointing normal vector is defined for each boundary element as ${{{\boldsymbol{n}}}}^{{\rm A}}$ or ${{{\boldsymbol{n}}}}^{{\rm B}}$. The normal vector at each sub-node, ${{{\boldsymbol{n}}}}_{\alpha}$ or ${{{\boldsymbol{n}}}}_{\beta}$, can be defined uniquely as ${{{\boldsymbol{n}}}}_{\alpha}={{{\boldsymbol{n}}}}^{{\rm A}}$ and ${{{\boldsymbol{n}}}}_{\beta}={{{\boldsymbol{n}}}}^{{\rm B}}$. []{data-label="fig:double_node"}](double_node-numbering-embed.eps){width="0.6\hsize"}
Double node technique {#sec:double_node}
---------------------
In the node equation for node $i$ shown in [Eq. (\[CBIE-point\_i\])]{}, both $u_j$ and $q_j$ are values of the nodes, which are located at the two ends of the boundary element in the case of the linear element. When the node ${{{\boldsymbol{r}}}}_i$ is located at a corner, the normal derivative $q_i$ of the linear element cannot be defined uniquely since the node belongs to two elements with different normal directions. Using a double node technique is one of the solutions.
Figure \[fig:double\_node\] illustrates a configuration of a double node. The node ${{{\boldsymbol{r}}}}_i$ is represented by two sub-nodes at the same position; ${{{\boldsymbol{r}}}}_{{\alpha}}={{{\boldsymbol{r}}}}_{{\beta}}$. The node at ${{{\boldsymbol{r}}}}_{{\alpha}}$ is connected to both $\Gamma_{{{\rm A}}}$ and a zero-sized element between ${{{\boldsymbol{r}}}}_{{\alpha}}$ and ${{{\boldsymbol{r}}}}_{{\beta}}$, and vice versa. The normal direction of the sub-nodes ${\alpha}$ and ${\beta}$ can be determined from the directions normal to $\Gamma_{{{\rm A}}}$ and $\Gamma_{{{\rm B}}}$, respectively; therefore, the direction of the derivatives of $q_{{\alpha}}$ and $q_{{\beta}}$ can be defined individually. Since the integral along the zero-sized element is identically zero regardless of the normal direction, we do not need to define the normal direction for the zero-sized element. Although the node ${\alpha}$ does not belong to $\Gamma_{{\rm B}}$, contributions of $g^{({{\rm B}},l)}_{{\alpha}}$ and $h^{({{\rm B}},l)}_{{\alpha}}$, which are evaluated by analytical integrals as singular integrals for accurate evaluation, are similar to $g^{({{\rm A}},l)}_{{\alpha}}$ and $h^{({{\rm A}},l)}_{{\alpha}}$.
Both $u$ and $q$ are defined at each sub-node in the same way as ordinary nodes, and the boundary condition is imposed for each sub-node. Either $u_{\alpha}$ or $u_{\beta}$ is unnecessary since $u_{\alpha}=u_{\beta}$; however, the same representation as the ordinary nodes simplifies the programming effort. In this case, the number of unknown variables increases by the number of double nodes, which equals the number of corners. Since one double node $i$ is replaced by the two sub-nodes ${\alpha}$ and ${\beta}$ for each corner, the number of node equations is also increased by the number of corners. Consequently, a set of CBIEs can be expressed by a matrix equation with a square coefficient matrix even when we apply double nodes.
Since the integral along the zero-sized elements is zero, the right-hand sides of [Eq. (\[equi-pot\])]{} for $i={\alpha}$ and $i={\beta}$ are the same. Therefore, the coefficients $c_{{\alpha}}$ and $c_{{\beta}}$ must have the same value: $$\begin{aligned}
\label{c-ia-ib}
c_{{\alpha}}^{}=c_{{\beta}}^{}.\end{aligned}$$
In the case of linear elements, the singular integral of $h_i^j$ for the corners ($i,j={\alpha},{\beta}$) becomes zero since the ${{{\boldsymbol{r}}}}-{{{\boldsymbol{r}}}_i}$ for $i={\alpha}$ or ${\beta}$ is perpendicular to ${{{\boldsymbol{n}}}}_j$ for $j={\alpha}$ or ${\beta}$: $$\begin{aligned}
\label{h-ia-ib-sing}
& h_{{\alpha}}^{{\alpha}}=h_{{\beta}}^{{\alpha}}
=h_{{\alpha}}^{{\beta}}=h_{{\beta}}^{{\beta}}
=0&& \mbox{ for linear elements}.\end{aligned}$$ Therefore, summarizing the discussions in the last paragraph in Sec. \[sec:CBIE-discretization\] and this result, we obtain the relations for the sub-nodes as $$\begin{aligned}
\label{g-ia-ib}
& g_{{\alpha}}^j=g_{{\beta}}^j&& \mbox{ for any $j$},
\\
\label{h-ia-ib-nosing}
& h_{{\alpha}}^j=h_{{\beta}}^j&& \mbox{ for any $j$}.\end{aligned}$$
Rank deficiency conditions in CBIEs {#sec:Rank_deficient_condition-CBIE}
-----------------------------------
By using the double node technique, the boundary nodes including sub-nodes are defined at the ends of boundary elements, and the boundary conditions are defined at each node. In problems with a single medium, there are three common types of boundary conditions; Dirichlet condition, Neumann condition, and Robin condition. These conditions are given, respectively, as $$\begin{aligned}
\label{Dirichlet}
&u_j=\overline{u}_j &&\mbox{ for } j\in I_D,
\\
\label{Neumann}
&q_j=\overline{q}_j &&\mbox{ for } j\in I_N,
\\
\label{Robin}
&\kappa_u u_j+\kappa_q q_j=\overline{\psi}_j&&\mbox{ for }j\in I_R,\end{aligned}$$ where the over-bars denote values imposed by the boundary conditions, $\kappa_u$ and $\kappa_q$ are the given constants determined by the problem considered, and the sets of the node numbers with corresponding boundary conditions are denoted by $I_D$, $I_N$, and $I_R$, respectively.
When we consider a multi-media problem, there exists a condition at the interface between the media. The interface condition is expressed by two continuous conditions for $u$ and $q$. The continuous conditions are given as $$\begin{aligned}
\label{Interface-cond}
&u_{j^{(2)}}=u_{j^{(1)}},
\quad
q_{j^{(2)}}=-\kappa_{21} q_{j^{(1)}}
&&\mbox{ for }j^{(1)},j^{(2)}\in I_I,\end{aligned}$$ where media numbers are denoted by (1) and (2); the nodes $j^{(1)}$ and $j^{(2)}$ are located at the same positions; $\kappa_{21}$ is determined by media constants of $\Omega^{(1)}$ and $\Omega^{(2)}$; and $I_I$ denotes the set of nodes with interface conditions.
In this section, we first present three simple examples when two sub-nodes of a double node at a corner point belong to $I_D$ or $I_N$, and discuss why the rank deficient problem arises. Then, the case in which the sub-nodes belong to $I_R$ or $I_I$ are presented.
The combinations of the boundary conditions where the two sub-nodes, ${\alpha}$ and ${\beta}$, belong to $I_D$ or $I_N$ are classified into the following three cases:
- both include the Dirichlet conditions: ${\alpha}\in I_D$ and ${\beta}\in I_D$,
- both include the Neumann conditions: ${\alpha}\in I_N$ and ${\beta}\in I_N$,
- one includes the Dirichlet condition and the other includes the Neumann condition:\
${\alpha}\in I_D$ and ${\beta}\in I_N$, or ${\alpha}\in I_N$ and ${\beta}\in I_D$.
### Case of two Dirichlet conditions {#sec:Dirichlet-case-CBIE}
Under these conditions (${\alpha}\in I_D$ and ${\beta}\in I_D$), the sub-node equations of [Eq. (\[CBIE-point\_i\])]{} for $i={{\alpha}}$ and ${{\beta}}$ can be arranged so that the terms including known values are moved to the right-hand side and the terms related to ${\alpha}$ and ${\beta}$ are moved out from the summation as $$\begin{aligned}
\label{Both-Dirichlet-ia}
&
-g_{{\alpha}}^{{\alpha}}q_{{\alpha}}^{}-g_{{\alpha}}^{{\beta}}q_{{\beta}}^{}
+\hspace*{-0.5em}\sum_{j\in I\setminus\{{\alpha},{\beta}\}}\hspace*{-0.5em}
a_{{\alpha}}^j x_{j}
=\overline{b}_{{\alpha}},
\\
\label{Both-Dirichlet-ib}
&
-g_{{\beta}}^{{\alpha}}q_{{\alpha}}^{}-g_{{\beta}}^{{\beta}}q_{{\beta}}^{}
+\hspace*{-0.5em}\sum_{j\in I\setminus\{{\alpha},{\beta}\}}\hspace*{-0.5em}
a_{{\beta}}^j x_{j}
=\overline{b}_{{\beta}},\end{aligned}$$ where $I=I_D\cup I_N$, i.e., all nodes, and $I{\setminus\{{\alpha},{\beta}\}}$ denotes a set of all node numbers except ${\alpha}$ and ${\beta}$; and the detailed descriptions of the third terms on the left-hand sides and the right-hand sides are written as $$\begin{aligned}
\label{Both-Dirichlet-ax}
&
\hspace*{-0.5em}\sum_{j\in I\setminus\{{\alpha},{\beta}\}}\hspace*{-0.5em}
a_{i}^j x_{j}
=\sum_{j\in I_N}h_{i}^j u_j
-\hspace*{-0.75em}\sum_{{j\in I_D},{j\ne{{\alpha},{\beta}}}}\hspace*{-0.75em} g_{i}^j q_j,
\\
\label{Both-Dirichlet-b}
&\overline{b}_{i}
=
-\sum_{j\in I_D}\left(c_{i}^{}\delta_{i}^j+h_{i}^j\right)\overline{u}_j
+\sum_{j\in I_N}g_{i}^j \overline{q}_j.\end{aligned}$$ Comparing the coefficients of the terms on the left-hand side of the two sub-node equations shown in [Eqs. (\[Both-Dirichlet-ia\]) and (\[Both-Dirichlet-ib\])]{}, we can evaluate whether the rank of the coefficient matrix is reduced or not. From [Eqs. (\[g-ia-ib\]) and (\[h-ia-ib-nosing\])]{}, the contributions $h_i^j$ and $g_i^j$ ($i={\alpha},{\beta}$) to the node $j$ are the same. The coefficients $c_\alpha$ and $c_\beta$ share the same value from [Eq. (\[c-ia-ib\])]{}; however, the multiplied terms, $u_j$, are different since $u_j$ is also multiplied by the Kronecker’s delta in [Eq. (\[CBIE-point\_i\])]{}. Therefore, we can examine rank reduction by examining which terms simply include the Kronecker’s delta. The Kronecker’s delta is not found in the first and the second terms on the left-hand sides in [Eqs. (\[Both-Dirichlet-ia\]) and (\[Both-Dirichlet-ib\])]{}. For the third terms, node $j$ does not include ${\alpha}$ and ${\beta}$; therefore, the $\delta_{\alpha}^j$ and $\delta_{\beta}^j$ are not included. They only appear in the right-hand sides, which are not related to rank reduction. Therefore, the left-hand sides of the two equations are identical, and the rank is always reduced by these two sub-node equations; i.e., the coefficient matrix becomes a singular matrix.
### Case of two Neumann conditions {#sec:Neumann-case-CBIE}
As in the previous sub-section, two sub-node equations of [Eq. (\[CBIE-point\_i\])]{} in the case of ${\alpha}\in I_N$ and ${\beta}\in I_N$ can be arranged as $$\begin{aligned}
\label{Both-Neumann-ia}
\left(c_{{\alpha}}^{}+h_{{\alpha}}^{{\alpha}}\right)u_{{\alpha}}^{}
+h_{{\alpha}}^{{\beta}}u_{{\beta}}^{}
&
+\hspace*{-0.5em}\sum_{j\in I\setminus\{{\alpha},{\beta}\}}\hspace*{-0.5em}
a_{{\alpha}}^j x_{j}
=\overline{b}_{{\alpha}},
\\
\label{Both-Neumann-ib}
h_{{\beta}}^{{\alpha}}u_{{\alpha}}^{}
+\left(c_{{\beta}}^{}+h_{{\beta}}^{{\beta}}\right)u_{{\beta}}^{}
&
+\hspace*{-0.5em}\sum_{j\in I\setminus\{{\alpha},{\beta}\}}\hspace*{-0.5em}
a_{{\beta}}^j x_{j}
=\overline{b}_{{\beta}}.\end{aligned}$$ Although the definitions of $a_i^j$ and $\overline{b}_i$ are different from [Eqs. (\[Both-Dirichlet-ax\]) and (\[Both-Dirichlet-b\])]{}, the third terms of [Eqs. (\[Both-Neumann-ia\]) and (\[Both-Neumann-ib\])]{} are identical since they do not include the Kronecker’s delta. In contrast, the first and second terms are different. By applying [Eqs. (\[c-ia-ib\]) and (\[h-ia-ib-sing\])]{}, the coefficients associated with $u_{\alpha}$ and $u_{\beta}$ in [Eq. (\[Both-Neumann-ia\])]{} are $c_{\alpha}$ and 0, respectively; while those in [Eq. (\[Both-Neumann-ib\])]{} are 0 and $c_{\alpha}$, respectively. Therefore, the rank of the matrix that includes the sub-node equations is not reduced in the case of ${\alpha}\in I_N$ and ${\beta}\in I_N$. In addition, the right-hand sides are the same since they do not include the Kronecker’s delta. This means that the results of these two sub-node equations involve the relation $u_{\alpha}=u_{\beta}$.
### Case of coupled Dirichlet and Neumann conditions {#sec:Coupled-case-CBIE}
In the case of ${\alpha}\in I_D$ and ${\beta}\in I_N$, two sub-node equations are $$\begin{aligned}
\label{Dirichlet-Neumann-ia}
-g_{{\alpha}}^{{\alpha}}q_{{\alpha}}^{}
&
+h_{{\alpha}}^{{\beta}}u_{{\beta}}^{}
+\hspace*{-0.5em}\sum_{j\in I\setminus\{{\alpha},{\beta}\}}\hspace*{-0.5em}
a_{{\alpha}}^j x_{j}
=\overline{b}_{{\alpha}},
\\
\label{Dirichlet-Neumann-ib}
-g_{{\beta}}^{{\alpha}}q_{{\alpha}}^{}
&
+\left(c_{{\beta}}^{}+h_{{\beta}}^{{\beta}}\right)u_{{\beta}}^{}
+\hspace*{-0.5em}\sum_{j\in I\setminus\{{\alpha},{\beta}\}}\hspace*{-0.5em}
a_{{\beta}}^j x_{j}
=\overline{b}_{{\beta}}.\end{aligned}$$ Since the second terms on the left-hand sides of the above equations are different, the rank of the coefficient matrix is not reduced in the case of ${\alpha}\in I_D$ and ${\beta}\in I_N$.
According to the examples in Secs. \[sec:Neumann-case-CBIE\] and \[sec:Coupled-case-CBIE\], we can understand that the two sub-node equations for ${\alpha}$ and ${\beta}$ are different when either $c_{{\alpha}}$ or $c_{{\beta}}$ is included in the coefficients of unknown variables in the two sub-node equations.
### Case of Robin condition {#Robin-case-CBIE}
We consider the case in which the Robin conditions are imposed on at least one of the two sub-nodes. Eliminating $u_j$ from [Eq. (\[CBIE-point\_i\])]{} using [Eq. (\[Robin\])]{}, and arranging the equation, we obtain: $$\begin{aligned}
\label{disc-Robin}
&
\sum_{j\in I_N}\left(c_i^{}\delta_i^j+h_i^j\right)u_j
-\sum_{j\in I_D} g_i^j q_j
-\sum_{j\in I_R}\left(\frac{\kappa_q}{\kappa_u}\left(c_i^{}\delta^j_i+h_i^j\right)+g_i^j\right)q_j
=\overline{b}_i.\end{aligned}$$ When $i={\alpha}\in I_R$, the term with the factor $c_{{\alpha}}$ for unknown $q_{{\alpha}}$ remains in the third terms of the left-hand side. This satisfies the condition described at the end of Sec. \[sec:Coupled-case-CBIE\], which ensures that the equations of the two sub-nodes are independent and not identical. It does not depend on the type of boundary condition of the node ${\beta}$.
![ Duplicated node belonging to boundary elements with interface condition. (a) Original positions of nodes. (b) Definition of sub-nodes. Two domains $\Omega^{(1)}$ and $\Omega^{(2)}$ are in contact at an interface. The domain $\overline{\Omega}$ is outside of the domains considered. The node [Q]{} in (a) is a general node belonging to the boundary element with the interface condition that can be separated by two nodes ${\rm c}^{(1)}$ and ${\rm c}^{(2)}$ in (b) where the positions are the same. Similar to [Q]{}, the node [P]{} is a node belonging to the interface; however, it is also the double node. Consequently, [P]{} is split into four sub-nodes; ${\alpha}^{(1)}$, ${\beta}^{(1)}\in\Omega_1$, and ${\alpha}^{(2)}$, ${\beta}^{(2)}\in\Omega_2$. For each pair of $\left({\beta}^{(1)},{\beta}^{(2)}\right)$ or $\left({\rm c}^{(1)},{\rm c}^{(2)}\right)$, the interface conditions are satisfied, and both $u$ and $q$ at these nodes are unknown. Either a Dirichlet, Neumann, or Robin condition is imposed at ${\alpha}^{(1)}$ and ${\alpha}^{(2)}$. []{data-label="fig:double_node-interface"}](double_node-numbering-interface.eps){width="0.9\hsize"}
![ Definition of sub-nodes at an intersection of the interface boundary elements. Between the pairs of sub-nodes, $({{{\alpha}(1)}},{{{\alpha}(2)}})$, $({{{\beta}(1)}},{{{\beta}(2)}})$, and $({{{\gamma}(1)}},{{{\gamma}(2)}})$, the interface boundary conditions are imposed. []{data-label="fig:double_node-interface-domain"}](double_node-numbering-interface-multi_domain.eps){width="0.9\hsize"}
### Case of interface condition {#sec:interface-case-CBIE}
Figure \[fig:double\_node-interface\] illustrates a configuration around a double node where two domains are in contact. The corner node labeled [P]{} is represented by the four sub-nodes at the corner; two sub-nodes labeled ${\alpha}^{(1)}$ and ${\alpha}^{(2)}$ do not belong to the interface boundary, and the other two sub-nodes labeled ${\beta}^{(1)}$ and ${\beta}^{(2)}$ belong to the interface boundary. There are eight quantities related to these four sub-nodes. Of the eight quantities, two quantities at the sub-nodes ${\alpha}^{(1)}$ and ${\alpha}^{(2)}$ are considered ordinary boundary conditions, and two variables at ${\beta}^{(1)}$ and ${\beta}^{(2)}$ can be eliminated by the interface conditions shown in [Eq. (\[Interface-cond\])]{}, which, for ${\beta}^{(1)}$ and ${\beta}^{(2)}$, are rewritten as $$\begin{aligned}
\label{u-cont}
&u_{{{{\beta}(2)}}}^{}=u_{{{{\beta}(1)}}}^{},
\quad
q_{{{{\beta}(2)}}}^{}=-\kappa_{21} q_{{{{\beta}(1)}}}^{},\end{aligned}$$ where the double suffixes, such as $u_{{\beta}^{(1)}}$, are denoted by $u_{{{{\beta}(1)}}}$ for simplicity. Consequently, there are four unknown variables and four equations with respect to the double node when we apply these conditions before constructing the simultaneous equations.
As mentioned in the previous sub-sections, the coefficients unrelated to the sub-nodes do not affect rank reduction; therefore, we only consider the nature of the sub-matrix composed of the coefficients for the four unknown variables.
In the case of ${\alpha}^{(1)}\in I_D$ and ${\alpha}^{(2)}\in I_D$, the given values are $\overline{u}_{{{{\alpha}(1)}}}$ and $\overline{u}_{{{{\alpha}(2)}}}$, and the unknown variables are $q_{{{{\alpha}(1)}}}$, $u_{{{{\beta}(1)}}}$, $q_{{{{\beta}(1)}}}$, $q_{{{{\alpha}(2)}}}$, $u_{{{{\beta}(2)}}}$, and $q_{{{{\beta}(2)}}}$. The independent variables are reduced by using [Eq. (\[u-cont\])]{} such as $\{q_{{{{\alpha}(1)}}}, q_{{{{\beta}(1)}}},q_{{{{\alpha}(2)}}},u_{{{{\beta}(1)}}}\}$. The terms associated with these independent variables in the individual sub-node equations for $i={\alpha}^{(1)}$, ${\beta}^{(1)}$, ${\alpha}^{(2)}$, and ${\beta}^{(2)}$, are written as $$\begin{aligned}
\label{submatrix_eq-Both_Dirichlet}
\left(
\begin{array}{cccc}
-g_{{{{\alpha}(1)}}}^{{{{\alpha}(1)}}}
&-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&0
&h_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
\\
-g_{{{{\beta}(1)}}}^{{{{\alpha}(1)}}}
&-g_{{{{\beta}(1)}}}^{{{{\beta}(1)}}}
&0
&c_{{{{\beta}(1)}}}^{}+h_{{{{\beta}(1)}}}^{{{{\beta}(1)}}}
\\
0
&\kappa_{21} g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&-g_{{{{\alpha}(2)}}}^{{{{\alpha}(2)}}}
&h_{{{{\alpha}(2)}}}^{{{{\beta}(1)}}}
\\
0
&\kappa_{21} g_{{{{\beta}(2)}}}^{{{{\beta}(2)}}}
&-g_{{{{\beta}(2)}}}^{{{{\alpha}(2)}}}
&c_{{{{\beta}(2)}}}^{}+h_{{{{\beta}(2)}}}^{{{{\beta}(1)}}}
\end{array}
\right)
\left(
\begin{array}{c}
q_{{{{\alpha}(1)}}}^{}\\q_{{{{\beta}(1)}}}^{}\\q_{{{{\alpha}(2)}}}^{}\\u_{{{{\beta}(1)}}}^{}
\end{array}
\right).\end{aligned}$$ By applying [Eqs. (\[c-ia-ib\]), (\[h-ia-ib-sing\]) and (\[g-ia-ib\])]{} to the sub-matrix to eliminate coefficients with the subscript ${\beta}^{(1)}$ and ${\beta}^{(2)}$, the sub-matrix is rewritten as $$\begin{aligned}
\label{submat1-Dirichlet-case}
\left(
\begin{array}{cccc}
-g_{{{{\alpha}(1)}}}^{{{{\alpha}(1)}}}
&-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&0
&0
\\
-g_{{{{\alpha}(1)}}}^{{{{\alpha}(1)}}}
&-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&0
&c_{{{{\alpha}(1)}}}^{}
\\
0
&\kappa_{21} g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&-g_{{{{\alpha}(2)}}}^{{{{\alpha}(2)}}}
&0
\\
0
&\kappa_{21} g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&-g_{{{{\alpha}(2)}}}^{{{{\alpha}(2)}}}
&c_{{{{\alpha}(2)}}}^{}
\end{array}
\right).\end{aligned}$$ Furthermore, by applying elementary matrix operations, the sub-matrix is transformed to $$\begin{aligned}
\left(
\begin{array}{cccc}
-g_{{{{\alpha}(1)}}}^{{{{\alpha}(1)}}}
&-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&0
&0
\\
0
&0
&0
&c_{{{{\alpha}(1)}}}^{}
\\
0
&\kappa_{21} g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&-g_{{{{\alpha}(2)}}}^{{{{\alpha}(2)}}}
&0
\\
0
&0
&0
&c_{{{{\alpha}(2)}}}^{}
\end{array}
\right),\end{aligned}$$ where the second row is the result of replacement with the difference between the first and the second row of the matrix in [Eq. (\[submat1-Dirichlet-case\])]{}, and the fourth row is obtained by similar operations. The non-zero components in both the second and the fourth row are only found in the fourth column; therefore, the rank of this sub-matrix is reduced by one.
In other cases, the product of the sub-matrices and the unknown vector after applying [Eqs. (\[c-ia-ib\]), (\[h-ia-ib-sing\]) and (\[g-ia-ib\])]{} are as follows. In the case of ${\alpha}^{(1)}\in I_N$ and ${\alpha}^{(2)}\in I_N$, $$\begin{aligned}
\label{submatrix_eq-Both_Neumann}
\left(
\begin{array}{cccc}
c_{{{{\alpha}(1)}}}^{}
&0
&-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&0
\\
0
&c_{{{{\alpha}(1)}}}^{}
&-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&0
\\
0
&0
&\kappa_{21} g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&c_{{{{\alpha}(2)}}}^{}
\\
0
&c_{{{{\alpha}(2)}}}^{}
&\kappa_{21} g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&0
\end{array}
\right)
\left(
\begin{array}{c}
u_{{{{\alpha}(1)}}}^{}\\u_{{{{\beta}(1)}}}^{}\\q_{{{{\beta}(1)}}}^{}\\u_{{{{\alpha}(2)}}}^{}
\end{array}
\right).\end{aligned}$$ In the case of ${\alpha}^{(1)}\in I_D$ and ${\alpha}^{(2)}\in N_N$, $$\begin{aligned}
\label{submatrix_eq-coupled_Dirichlet_Neumann}
\left(
\begin{array}{cccc}
-g_{{{{\alpha}(1)}}}^{{{{\alpha}(1)}}}
&-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&0
&0
\\
-g_{{{{\alpha}(1)}}}^{{{{\alpha}(1)}}}
&-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&c_{{{{\alpha}(1)}}}^{}
&0
\\
0
&\kappa_{21} g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&0
&c_{{{{\alpha}(2)}}}^{}
\\
0
&\kappa_{21} g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&c_{{{{\alpha}(2)}}}^{}
&0
\end{array}
\right)
\left(
\begin{array}{c}
q_{{{{\alpha}(1)}}}^{}\\q_{{{{\beta}(1)}}}^{}\\u_{{{{\beta}(1)}}}^{}\\u_{{{{\alpha}(2)}}}^{}
\end{array}
\right).\end{aligned}$$ Since $\kappa_{21}>0$ generally, [Eqs. (\[submatrix\_eq-Both\_Neumann\]) and (\[submatrix\_eq-coupled\_Dirichlet\_Neumann\])]{} do not become singular.
Furthermore, in the case of ${\alpha}^{(1)}\in I_R$ or ${\beta}^{(2)}\in I_R$, the coefficient matrix becomes more complex than [Eq. (\[submatrix\_eq-Both\_Neumann\])]{} or [Eq. (\[submatrix\_eq-coupled\_Dirichlet\_Neumann\])]{}; therefore, the rank is also not reduced.
### Case of internal interface condition {#sec:internal_interface-case-CBIE}
In the case where the node is located at a corner of two interface boundary elements as shown in [Fig.\[fig:double\_node-interface-domain\]]{}(a), there are four sub-nodes at the corner node. After applying [Eqs. (\[c-ia-ib\]), (\[h-ia-ib-sing\]) and (\[g-ia-ib\])]{}, and eliminating the quantities related to ${\alpha}^{(2)}$ and ${\beta}^{(2)}$, the terms related to the sub-nodes in the four sub-node equations are expressed as $$\begin{aligned}
\label{submatrix_eq-Interface}
\left(
\begin{array}{cccc}
-g_{{{{\alpha}(1)}}}^{{{{\alpha}(1)}}}
&
-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&
c_{{{{\alpha}(1)}}}
&
0
\\
-g_{{{{\alpha}(1)}}}^{{{{\alpha}(1)}}}
&
-g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&
0
&
c_{{{{\alpha}(1)}}}
\\
\kappa_{21}g_{{{{\alpha}(2)}}}^{{{{\alpha}(2)}}}
&
\kappa_{21}g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&
c_{{{{\alpha}(2)}}}
&
0
\\
\kappa_{21}g_{{{{\alpha}(2)}}}^{{{{\alpha}(2)}}}
&
\kappa_{21}g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&
0
&
c_{{{{\alpha}(2)}}}
\end{array}\right)
\left(
\begin{array}{cccc}
q_{{{{\alpha}(1)}}}\\q_{{{{\beta}(1)}}}\\u_{{{{\alpha}(1)}}}\\u_{{{{\beta}(1)}}}
\end{array}\right).\end{aligned}$$ By applying the elementary matrix operations to the coefficient matrix, the matrix is transformed to $$\begin{aligned}
\left(
\begin{array}{cccc}
- g_{{{{\alpha}(1)}}}^{{{{\alpha}(1)}}}
&
- g_{{{{\alpha}(1)}}}^{{{{\beta}(1)}}}
&
c_{{{{\alpha}(1)}}}
&
0
\\
0
&
0
&
c_{{{{\alpha}(1)}}}
&
-c_{{{{\alpha}(1)}}}
\\
\kappa_{21}g_{{{{\alpha}(2)}}}^{{{{\alpha}(2)}}}
&
\kappa_{21}g_{{{{\alpha}(2)}}}^{{{{\beta}(2)}}}
&
c_{{{{\alpha}(2)}}}
&
0
\\
0
&
0
&
c_{{{{\alpha}(2)}}}
&
-c_{{{{\alpha}(2)}}}
\end{array}\right).\end{aligned}$$ We can understand that the rank of this matrix is reduced by one since the determinant of a 2x2 sub-matrix composed of non-zero elements in the second and the fourth row in the above sub-matrix is zero. When we increase the number of the domains as shown in [Fig.\[fig:double\_node-interface-domain\]]{}(b), we can illustrate the singularity, although the details are not shown here. Therefore, the sub-node equations become singular when the node is located at an intersection of the interface boundaries which partitions the original single domain into two or more domains.
Summary of rank deficiency conditions for CBIEs {#sec:CBIE-summary}
-----------------------------------------------
In this section, we analyze the rank deficiency conditions for the sub-node equations of a double node for CBIEs. The rank of the coefficient matrix is reduced in the following cases:
- both sub-nodes of a double node are imposed by Dirichlet conditions,
- sub-nodes are located at an intersection of an interface boundary element and two boundary elements imposed by Dirichlet conditions,
- sub-nodes are located at intersections of interface boundary elements with different normal directions and they do not belong to any boundary with ordinary boundary conditions.
The original sub-node equations or the sub-matrix associated with the sub-nodes for these cases include a pair of [Eqs. (\[Both-Dirichlet-ia\]) and (\[Both-Dirichlet-ib\])]{} for the first case, [Eq. (\[submatrix\_eq-Both\_Dirichlet\])]{} for the second case, and [Eq. (\[submatrix\_eq-Interface\])]{} for the last case. These equations have common characteristics; all the equations include $q_{\alpha}$ and $q_{\beta}$ (or $q_{{{\alpha}(1)}}$ and $q_{{{\beta}(1)}}$ for the interface boundary) as the unknown variables. The coefficients associated with these unknowns are $g_i^j$ ($i,j\in\{{\alpha},{\beta}\}$), which possess the characteristics shown in [Eq. (\[g-ia-ib\])]{}. The terms with $q_{\alpha}$ and $q_{\beta}$ are canceled because of this characteristic during the transformations of the equations, and the ranks have been reduced.
If the coefficients $g_i^j$ did not satisfy [Eq. (\[g-ia-ib\])]{}; in other words, if they included information on ${{{\boldsymbol{n}}}}_i$, these singularities could be avoided.
Regularization of coefficient matrix using HBIEs {#sec:HBIE}
================================================
As described in Sec. \[sec:CBIE-summary\], the reason for a rank deficient matrix in CBIE is the coefficients $g_i^j$ do not carry information concerning the normal direction of the node $i$. To provide this information, we employ a hypersingular boundary integral equation (HBIE). The HBIE can be derived by taking a gradient of the CBIE shown in [Eq. (\[bie1\])]{} with respect to ${{{\boldsymbol{r}}}}_i$: $$\begin{aligned}
\label{grad-bie1}
{{\nabla\!_i}\left[c({{{\boldsymbol{r}}}_i})u({{{\boldsymbol{r}}}_i})\right]}
={{\oint_{\Gamma_{}}^{} &\left[({{\nabla\!_i}u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})})({\nabla u({{{\boldsymbol{r}}}})}){\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}\right.\nonumber\\&\left.
-u({{{\boldsymbol{r}}}}) ({{\nabla\!_i}{\nabla u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})}}){\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}\right]\,d{\Gamma}\,}},
$$ where ${\nabla\!_i}$ denotes the gradient with respect to ${{{\boldsymbol{r}}}_i}$. The gradient, ${{\nabla\!_i}{\nabla }} u^*$, has a stronger singularity than the CBIE, and this is known as hypersingularity. However, the integral of the hypersingular function can be regularized. In this study, we employ the regularization for the Helmholtz equation which uses the fundamental solution of Laplace’s equation [@Tomioka:2010].
Regularized gradient field
--------------------------
In this section, after presenting a regularization of the gradient at ${{{\boldsymbol{r}}}_i}$ according to the method shown in Ref. [@Tomioka:2010], a discretized form of the normal derivative, $q$, is derived.
The gradient is expressed by a 2x2 dyadic tensor ${{\overleftrightarrow{{\bf{C_{\mathit i}}}}}}$ and a vector ${{\boldsymbol{J}}}_i(u,q)$ as $$\begin{aligned}
\label{Reg-grad-bie}
&
{{\overleftrightarrow{{\bf{C_{\mathit i}}}}}}\,{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{\nabla u({{{\boldsymbol{r}}}_i})}+{{\boldsymbol{J}}}_i(u,q)={{\boldsymbol{0}}}.\end{aligned}$$ The second term can be evaluated by two types of boundary integrals as $$\begin{aligned}
&
\label{J_i-def}
{{\boldsymbol{J}}}_i(u,q)=
\sum_{\gamma={{\rm A}},{{\rm B}}}{{\boldsymbol{J}}}_i^{\gamma,\rm sing}(u,q)
+\sum_{k\ne {{\rm A}},{{\rm B}}}{{\boldsymbol{J}}}_{i}^{k,\rm reg}(u,q),\end{aligned}$$ where ${{\boldsymbol{J}}}_i^{\gamma, \rm sing}(u,q)$ is associated to the boundary integral along the boundary elements that include the node $i\in\{{\alpha},{\beta}\}$ and $\gamma\in\{{{\rm A}},{{\rm B}}\}$ ([Fig.\[fig:double\_node\]]{}); whereas ${{\boldsymbol{J}}}_i^{k,\rm reg}$ corresponds to integrals of the boundary elements not including the node $i$. Similar to the factor $c({{{\boldsymbol{r}}}_i})$ in CBIEs, the singular integral included in [Eq. (\[grad-bie1\])]{} contributes to ${{\overleftrightarrow{{\bf{C_{\mathit i}}}}}}$ on the left-hand side of [Eq. (\[Reg-grad-bie\])]{} and ${{\boldsymbol{J}}}_i^{\gamma,\rm sing}(u,q)$. The components of ${{\boldsymbol{J}}}_i^{\gamma,\rm sing}(u,q)$, ${{\boldsymbol{J}}}_{i}^{k,\rm reg}(u,q)$ and ${{\overleftrightarrow{{\bf{C_{\mathit i}}}}}}$ are written as $$\begin{aligned}
\label{J-AB-def}
&{{\boldsymbol{J}}}_i^{\gamma,\rm sing}(u,q)
=
-{{\int_{0}^{L_\gamma}\! \frac1r\left({{\frac{\partial{ u^*}}{\partial{r}}}}-{{\frac{\partial{u_L^*}}{\partial{r}}}}\right)\,d{r}\,}}\cdot u({{{\boldsymbol{r}}}_i}){{{\boldsymbol{n}}}_{{\gamma}}}\nonumber\\&\hspace*{3em}
+\left.{{\int_{0}^{L_\gamma}\! r{{\frac{\partial{ u^*}}{\partial{r}}}}\,d{r}\,}}\right.
\left(
-\frac12{{{\boldsymbol{n}}}_{{\gamma}}}{\left.{{\frac{\partial{^2u}}{\partial{{{\tau}_{{\gamma}}}^2}}}}\right|_{{{{\boldsymbol{r}}}_i}}}
+{{{\boldsymbol{{\tau}}}}_{{\gamma}}}{\left.{\frac{\partial^2u}{\partial{{\tau}_{{\gamma}}}\partial{n_{{\gamma}}}}}\right|_{{{{\boldsymbol{r}}}_i}}}
\right),
\\
\label{Jn-def}
&{{\boldsymbol{J}}}_i^{k,\rm reg}(u,q)
=
{{\int_{\Gamma_{k}}^{}\! \left\{({{\nabla\!_i}q}^*)u-({{\nabla\!_i}q}_L^*) u({{{\boldsymbol{r}}}_i}) -({{\nabla\!_i}u}^*)q \right\}\,d{\Gamma}\,}},
\\
\label{DC-def}
&{{\overleftrightarrow{{\bf{C_{\mathit i}}}}}}=
\left(\begin{array}{cc}
\displaystyle
{\underset{{\gamma}\,:\,{{\rm A}}-{{\rm B}}}{{\rm Diff}}\left[\mathstrut{\textstyle\frac{2\theta_\gamma+\sin{2\theta_\gamma}}{4\pi}}\right]}
&
\hspace*{-1.0em} \displaystyle
{\underset{{\gamma}\,:\,{{\rm A}}-{{\rm B}}}{{\rm Diff}}\left[\mathstrut{\textstyle\frac{-\cos{2\theta_\gamma}}{4\pi}+ u^*(L_\gamma)}\right]}
\\[1em]
\displaystyle
{\underset{{\gamma}\,:\,{{\rm A}}-{{\rm B}}}{{\rm Diff}}\left[\mathstrut{\textstyle\frac{-\cos{2\theta_\gamma}}{4\pi}- u^*(L_\gamma)}\right]}
&
\hspace*{-1.0em} \displaystyle
{\underset{{\gamma}\,:\,{{\rm A}}-{{\rm B}}}{{\rm Diff}}\left[\mathstrut{\textstyle\frac{2\theta_\gamma-\sin{2\theta_\gamma}}{4\pi}}\right]}
\end{array}\right)
,
\\
&{\underset{{\gamma}\,:\,{{\rm A}}-{{\rm B}}}{{\rm Diff}}\left[\mathstrut{f_\gamma}\right]}{\triangleq}f_{{{\rm A}}}-f_{{{\rm B}}},\end{aligned}$$ where $u_L^*$ denotes the fundamental solution to the Laplace equation that satisfies [Eq. (\[funda-laplace\])]{}, $q_L^*$ is its normal derivative with respect to ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}})$ at ${{{\boldsymbol{r}}}}$; and ${{\boldsymbol{\tau}}}_\gamma$, $\theta_\gamma$, and $L_\gamma$ are the tangential unit vectors along $\Gamma_\gamma$, the azimuth angle of $\Gamma_\gamma$ from ${{{\boldsymbol{r}}}_i}$, and the length of $\Gamma_\gamma$, respectively.
Similar to CBIEs, the non-singular integral in [Eq. (\[Jn-def\])]{} is expressed by the shape function as $$\begin{aligned}
\label{s_int-reg}
&
{{\boldsymbol{s}}}^{(k,l),\rm reg}_{i}={{\int_{\Gamma_{k}}^{}\! \nabla_i\nabla u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})\phi^{(k,l)}({{{\boldsymbol{r}}}}){\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}\,d{\Gamma}\,}},
\\
\label{sL_int-reg}
&
{{\boldsymbol{s}}}^{(k,l),\rm reg}_{L,i}={{\int_{\Gamma_{k}}^{}\! \nabla_i\nabla u_L^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})\phi^{(k,l)}({{{\boldsymbol{r}}}}){\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}\,d{\Gamma}\,}},
\\
\label{t_int-reg}
&
{{\boldsymbol{t}}}^{(k,l),\rm reg}_{i}={{\int_{\Gamma_{k}}^{}\! \nabla_i u^*({{{\boldsymbol{r}}}},{{{\boldsymbol{r}}}_i})\phi^{(k,l)}({{{\boldsymbol{r}}}})\,d{\Gamma}\,}},\end{aligned}$$ which are evaluated using numerical integrals. By using the permutation matrix, the second term on the left-hand side in [Eq. (\[J\_i-def\])]{} is denoted by a form of linear combinations as $$\begin{aligned}
\label{J-reg-lin}
\sum_{k\ne {{\rm A}},{{\rm B}}}{{\boldsymbol{J}}}_i^{k,\rm reg}(u,q)
&
=\sum_{j\ne {\alpha},{\beta}}{{\boldsymbol{s}}}_i^{j,\rm reg}u_j^{}
-\sum_{j\ne {\alpha},{\beta}}{{\boldsymbol{s}}}_{L,i}^{j,\rm reg}u_i^{}
-\sum_{j\ne {\alpha},{\beta}}{{\boldsymbol{t}}}_i^{j,\rm reg}q_j^{}.\end{aligned}$$
The two singular integrals on the right-hand side of [Eq. (\[J-AB-def\])]{} can be analytically obtained by using Taylor series expansions around ${{{\boldsymbol{r}}}_i}$ for the two second-order derivatives. The first term on the right-hand side of [Eq. (\[J\_i-def\])]{} is also expressed as linear combinations with respect to the nodes neighboring the sub-nodes ${\alpha}$ and ${\beta}$ as $$\begin{aligned}
\label{J-sing-lin}
&
\sum_{\gamma={{\rm A}},{{\rm B}}}{{\boldsymbol{J}}}_i^{\gamma,\rm sing}(u,q)
=\sum_{j\in \tilde{I}^{\rm sing}}{{\boldsymbol{s}}}_i^{j,\rm sing}u_j^{}
-\sum_{j\in \tilde{I}^{\rm sing}}{{\boldsymbol{t}}}_i^{j,\rm sing}q_j^{},
\nonumber\\
&\qquad
\tilde{I}^{\rm sing}=\{i|{{{\boldsymbol{r}}}_i}\in\Gamma_{{\rm A}}\cup\Gamma_{{\rm B}}\}=\{({{\rm A}},1),({{\rm A}},2),({{\rm B}},1),({{\rm B}},2)\}.\end{aligned}$$ By defining $$\begin{aligned}
{{\boldsymbol{s}}}_i^j={{\boldsymbol{s}}}_i^{j,\rm sing}+{{\boldsymbol{s}}}_i^{j,\rm reg}-{{\boldsymbol{s}}}_{L,i}^{j,\rm reg},
\quad
{{\boldsymbol{t}}}_i^j={{\boldsymbol{t}}}_i^{j,\rm sing}+{{\boldsymbol{t}}}_i^{j,\rm reg},\end{aligned}$$ the vector ${{\boldsymbol{J}}}_i(u,q)$ is expressed as the following form with a linear combination: $$\begin{aligned}
\label{J-sum_form}
{{\boldsymbol{J}}}_i(u,q)=\sum_j{{\boldsymbol{s}}}_i^{j}u_j-\sum_j{{\boldsymbol{t}}}_i^{j}q_j^{}.\end{aligned}$$ By substituting this equation into [Eq. (\[Reg-grad-bie\])]{} and by multiplying both sides by ${{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}$, the gradient field at ${{{\boldsymbol{r}}}_i}$ is expressed as $$\begin{aligned}
\label{grad-by-C}
{\nabla u({{{\boldsymbol{r}}}_i})}=-{{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}\,{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}\left(\sum_j{{\boldsymbol{s}}}_i^{j}u_j-\sum_j{{\boldsymbol{t}}}_i^{j}q_j^{}\right).\end{aligned}$$
Rank deficiency conditions in HBIEs {#sec:Rank_deficient_condition-HBIE}
-----------------------------------
Let us consider the relations of ${{\boldsymbol{s}}}_i^j$ and ${{\boldsymbol{t}}}_i^j$ when ${{{\boldsymbol{r}}}_i}$ is a double node; i.e., ${{{\boldsymbol{r}}}}_i={{{\boldsymbol{r}}}}_{\alpha}={{{\boldsymbol{r}}}}_{\beta}$. If both $u_j$ and $q_j$ at all nodes are determined as known values, the gradient in [Eq. (\[grad-by-C\])]{} must be expressed uniquely. Since the dyadic tensor ${{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}$ on the right-hand side of [Eq. (\[grad-by-C\])]{} is determined by a geometric configuration of the boundary elements connected to ${{{\boldsymbol{r}}}_i}$ as shown in [Eq. (\[DC-def\])]{}, ${\overleftrightarrow{{\bf{C_{\mathit{\alpha}}}}}}={\overleftrightarrow{{\bf{C_{\mathit{\beta}}}}}}$ is satisfied. Therefore, to uniquely determine the gradient, i.e., ${\nabla u({{{\boldsymbol{r}}}}_{\alpha})}={\nabla u({{{\boldsymbol{r}}}}_{\beta})}$, the coefficient vectors for ${\alpha}$ and ${\beta}$ must be the same. Strictly speaking, there is an exception in ${{\boldsymbol{s}}}_i^j$ for $j={\alpha},{\beta}$. In this case, the condition ${{\boldsymbol{s}}}_{{\alpha}}^j\ne{{\boldsymbol{s}}}_{{\beta}}^j$ is permitted when ${{\boldsymbol{s}}}_{{\alpha}}^{{\alpha}}+{{\boldsymbol{s}}}_{{\alpha}}^{{\beta}}={{\boldsymbol{s}}}_{{\beta}}^{{\alpha}}+{{\boldsymbol{s}}}_{{\beta}}^{{\beta}}$ is satisfied since $u_{{\alpha}}=u_{{\beta}}$. Therefore, $$\begin{aligned}
\label{s-cond-sing}
&{{\boldsymbol{s}}}_{{\alpha}}^j\ne{{\boldsymbol{s}}}_{{\beta}}^j && \mbox{ for $j\in\{{\alpha},{\beta}\}$},\\
\label{s-cond}
&{{\boldsymbol{s}}}_{{\alpha}}^j={{\boldsymbol{s}}}_{{\beta}}^j && \mbox{ for $j\not\in\{{\alpha},{\beta}\}$},\\
\label{t-cond}
&{{\boldsymbol{t}}}_{{\alpha}}^j={{\boldsymbol{t}}}_{{\beta}}^j && \mbox{ for any $j$}.\end{aligned}$$
Since $q_i={{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}_i})\cdot{\nabla u({{{\boldsymbol{r}}}_i})}$, by taking an inner product of ${{{\boldsymbol{n}}}}({{{\boldsymbol{r}}}_i})$ to [Eq. (\[grad-by-C\])]{}, the discretized equation for the HBIE is obtained as $$\begin{aligned}
\label{HBIE-point_i}
&
\sum_j v_i^j u_j^{} -\sum_j(-\delta_i^j+w_i^j)q_j^{}=0,\end{aligned}$$ where $$\begin{aligned}
\label{def-v_i^j}
&
v_i^j={{{\boldsymbol{n}}}}_i\,{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}\,{{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}\,{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}\,{{\boldsymbol{s}}}_i^j,
\\
\label{def-w_i^j}
&
w_i^j={{{\boldsymbol{n}}}}_i\,{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}\,{{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}\,{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}\,{{\boldsymbol{t}}}_i^j.\end{aligned}$$
When ${{{\boldsymbol{n}}}}_{\alpha}\ne{{{\boldsymbol{n}}}}_{\beta}$, we can derive the following conditions using [Eqs. (\[s-cond\]) and (\[t-cond\])]{}: $$\begin{aligned}
\label{v-cond}
& v_{{\alpha}}^j\ne v_{{\beta}}^j && \mbox{ when $j\ne{\alpha},{\beta}$},\\
\label{w-cond}
& w_{{\alpha}}^j\ne w_{{\beta}}^j && \mbox{ for any $j$}.\end{aligned}$$ These relations mean that the two sub-node equations of [Eq. (\[HBIE-point\_i\])]{} for $i={\alpha}$ and $i={\beta}$ are different. Even in the case of ${{{\boldsymbol{n}}}}_{\alpha}={{{\boldsymbol{n}}}}_{\beta}$, the two sub-node equations are also different since the coefficients for $u_{\alpha}$ of the two equations are always different because of [Eq. (\[s-cond-sing\])]{} and the coefficients for $q_{\alpha}$ are always different because of the nature of $\delta_i^j$.
The aforementioned difference in the two equations is not a sufficient condition to not reduce the rank since we have not proved that the determinant of the sub-matrix associated with the sub-nodes is not equal to zero for any problem. However, the HBIE is highly applicable to the problem in which the node equations constructed by the CBIE become singular.
![Analysis models. (A) Dirichlet conditions of $\overline{u}=0$ on $\Gamma_D$ and an incident wave condition with a mode proportional to $\cos(k_x x)$ on $\Gamma^{\rm Inc}$ are given. (B) Neumann conditions of $\overline{q}=0$ on $\Gamma_N$ and an incident condition with $\sin(k_x x)$ on $\Gamma^{\rm Inc}$. In both cases, either of the three termination conditions on $\Gamma^{\rm T}$ is given; (a) a short condition, (b) an open condition, or (c) a non-reflective condition. []{data-label="fig:model"}](WG_model.eps)
![Exact solutions for model (A-b) and model (B-b). In both models, the termination condition at $y=2$ is $R=+1$ (conditions for the open termination:(b)). The conditions $x=\pm1$ are Dirichlet conditions with $\overline{u}=0$ for (A) and Neumann conditions with $\overline{q}=0$ for (B). Although the solution is a complex valued function, this figure only illustrates the real part. The imaginary part obeys the same eigenfunction, but the amplitude is different from the real part. []{data-label="fig:exact_dist-open_end-1reg"}](exact_solution.eps){width="0.97\hsize"}
![Error distributions of numerical solutions: (i) CBIE, (ii) partial-HBIE, and (iii) full-HBIE. The boundary condition on $\Gamma^{\rm T}$ is an open termination condition (model (b)); and on $x=\pm1$, Dirichlet conditions (A), and Neumann conditions (B). The size of the boundary elements is 0.05. Each sub-figure illustrates the real part of the error, ${\rm Re}\left\{\Delta u\right\}$. The scales of the vertical axes are different between sub-figures. []{data-label="fig:error_dist-open_end-1reg"}](error_dist.eps){width="0.97\hsize"}
![ Element size dependence of error. (a) Short termination, (b) Open termination, and (c) Non-reflective condition. The horizontal axis, $N_\lambda$, denotes the number of elements in a wavelength; i.e., $N_\lambda=\lambda/\Delta_{\rm Ele}$ where $\Delta_{\rm Ele}$ is the boundary element size. In sub-figure (a), the results by CBIE (i) are not shown as the analyses failed because of the singularity of the coefficient matrix. []{data-label="fig:size-vs-error"}](size_vs_error.eps){width="0.7\hsize"}
![ Element size dependence of CPU time. The horizontal axis, $N_\lambda$, denotes the number of elements in a wavelength. The model is (A-b); i.e., Dirichlet conditions for the side boundaries, and open termination conditions for the top boundary. The CPU time for the other termination condition (a) and (c) are almost the same as this result; however, the result of the CBIE (i) is not obtained for the condition of the short termination (a). []{data-label="fig:size-vs-cpu"}](size_depend-cpu-open_end-comp-embed.eps){width="0.7\hsize"}
![ Analysis models including interfaces between media. The first letter in the label of each sub-figure presents the base model shown in [Fig.\[fig:model\]]{}, and the last symbol illustrates a shape of the interface boundaries drawn by the dot-dash lines. The media constants of each domain are the same. []{data-label="fig:model_with_4reg"}](WG_model_with_interface.eps){width="0.99\hsize"}
![ Comparisons of errors in the problems including the interface boundaries. The label at the top of each block presents the model shown in [Fig.\[fig:model\]]{} and [Fig.\[fig:model\_with\_4reg\]]{}. In each model, the error obtained by the three methods, (i) CBIE, (ii) partial-HBIE, and (iii) full-HBIE, are expressed as a triplet of band graphs. The case in which the band graph is not shown means that the set of simultaneous equations is singular. []{data-label="fig:error-interface_conditions"}](error_histo-embed.eps){width="0.9\hsize"}
Numerical results and discussions {#sec:results}
=================================
To demonstrate the validity of the method in regularizing the coefficient matrix using HBIEs, we analyzed simple models as shown in [Fig.\[fig:model\]]{}. The actual models correspond to electromagnetic wave propagation problems in a waveguide where the wall parallel to the propagation direction is made of metal. When we consider $u$ as the $z$-component of the electric field, $u=0$ at the side-walls of the waveguide since the component of the electric field parallel to the metal is zero. In the case of model-(A), the metal walls are located at $\Gamma_D$, which is called the TE${}_{10}$ mode for a waveguide with rectangular cross-section and width $W$. In the case of model-(B), virtual walls $\Gamma_N$ with $q=0$ are located at $x=\pm W/2$, which are equivalent to the placement of physical metal walls with $u=0$ at $x=\pm W$, which is called the TE${}_{20}$ mode. These incident conditions are given as functions of $x$ as $$\begin{aligned}
& \overline{u^{\rm inc}}(x)=\overline{u^{\rm inc}_0}\cos(k_x x) &&\mbox{for model-(A)},
\\
& \overline{u^{\rm inc}}(x)=\overline{u^{\rm inc}_0}\sin(k_x x) &&\mbox{for model-(B)},
\\
& k_x=\frac{\pi}{W}.\end{aligned}$$ In these modes, a propagation field to $\pm y$-directions are proportional to $\exp({\mp{\rm j}k_y})$, respectively, where $$\begin{aligned}
& k_y^2=k^2-k_x^2.\end{aligned}$$ By using this characteristic, the incident boundary condition on $\Gamma^{\rm inc}$ can be rewritten as a Robin condition: $$\begin{aligned}
\label{incident-condition}
{\rm j}k_y u+q=2{\rm j}k_y\overline{u^{\rm inc}}.\end{aligned}$$ On the boundary $\Gamma^{\rm T}$ at $y=L$, we analyzed the following three conditions for termination:
----- -------- -------------------- ----------- --
(a) Short: $u=0$ ($R=-1$),
(b) Open: $q=0$ ($R=+1$),
(c) ${\rm j}k_y u+q=0$ ($R=0$),
----- -------- -------------------- ----------- --
where $R$ denotes the reflection coefficient of the electric field on $\Gamma_T$; and the non-reflective condition in (c) is equivalent to the incident condition with $\overline{u^{\rm inc}}=0$ in [Eq. (\[incident-condition\])]{}. Among the above three, termination-(a) (short type) has two double nodes at $(\pm W/2,L)$, where both sub-nodes have two Dirichlet conditions; therefore, the rank of the coefficient matrix of CBIEs will be reduced as mentioned in Sec. \[sec:Dirichlet-case-CBIE\]. The exact solutions to these models are given by $$\begin{aligned}
\label{exact-TE10}
&
\hat{u}(x,y)=\overline{u^{\rm inc}_0}\cos(k_x x)
\left(e^{-{\rm j}k_y y}+R e^{{\rm j}k (y-2L)}\right)
&& \mbox{for model-(A)},
\\
\label{exact-TE20-central_half}
&
\hat{u}(x,y)=\overline{u^{\rm inc}_0}\sin(k_x x)
\left(e^{-{\rm j}k_y y}+R e^{{\rm j}k (y-2L)}\right)
&& \mbox{for model-(B)}.\end{aligned}$$ To simplify, we used $\overline{u^{\rm inc}_0}=1$, $W=2$, $L=2$, and $\lambda=2\pi/k=1$; under this condition, $\max|u|=2$ for the cases (a) and (b), and $\max|u|=1$ for the case (c), $\lambda_x=2\pi/k_x=4$ and $\lambda_y=2\pi/k_y\simeq 1.033$. As examples of $\hat{u}(x,y)$, the real part of $\hat{u}(x,y)$ for the models (A-b) and (B-b) are shown in [Fig.\[fig:exact\_dist-open\_end-1reg\]]{}.
We analyzed three types of the following simultaneous equations:
1. CBIE: BIEs for all nodes including sub-nodes are obtained from the CBIE,
2. partial-HBIE: BIEs for only sub-nodes related to double nodes are obtained from the HBIE, and BIEs for the other nodes are obtained from the CBIE.
3. full-HBIE: BIEs for all nodes including sub-nodes are obtained from the HBIE,
To solve the complex-valued simultaneous equations, we employed two subroutines based on an LU decomposition provided in Lapack [@Lapack], in which the subroutine names are ‘zgetrf’ and ‘zgetrs.’ After determining both $u$ and $q$ at all boundary nodes by solving the simultaneous equations, the internal field $u({{{\boldsymbol{r}}}}_{i\,'})$ can be evaluated by $$\begin{aligned}
\label{internal-field}
u({{{\boldsymbol{r}}}}_{i\,'})=\sum_{j\in I} g_{i\,'\,}^j q_j -\sum_{j\in I}h_{i\,'\,}^j u_j,\end{aligned}$$ which is derived from the discretized CBIE shown in [Eq. (\[bie1\])]{}. This process is the same for all types of simultaneous equations.
Figure \[fig:error\_dist-open\_end-1reg\] presents the distributions of errors based on the three types of simultaneous equations (i), (ii), and (iii) for the model-(A-b) and model-(B-b). In these models with termination-(b) (open type), the coefficient matrices, even in the case of the CBIE shown in [Fig.\[fig:error\_dist-open\_end-1reg\]]{}(i), are not singular as mentioned in Sec. \[sec:CBIE-summary\].
Based on the comparison between the sub-figures (A-b-i) and (A-b-iii) or between (B-b-i) and (B-b-iii) in [Fig.\[fig:error\_dist-open\_end-1reg\]]{}, we can observe that the error of the full-HBIE is several times larger than that the CBIE. This difference can be explained by two reasons. The first reason is the difference in singularity to evaluate the coefficients. The coefficients $w_i^j$ and $v_i^j$ in the HBIE are evaluated from ${{\boldsymbol{s}}}_i^j$ and ${{\boldsymbol{t}}}_i^j$ with a multiplication of dyadic ${{\overleftrightarrow{{\bf{C_{\mathit i}}}}}}$ as shown in [Eqs. (\[def-v\_i\^j\]) and (\[def-w\_i\^j\])]{}, respectively. The vectors ${{\boldsymbol{s}}}_i^j$ and ${{\boldsymbol{t}}}_i^j$ are evaluated by the boundary integral shown in [Eqs. (\[s\_int-reg\]), (\[sL\_int-reg\]) and (\[t\_int-reg\])]{}. Their integrands include the first or second order derivatives of the fundamental solution. The strongest singularity in the HBIE is $O(r^{-2})$, while the strongest singularity in the CBIE is $O(r^{-1})$. This error emerges significantly in the contributions between two nodes with short distances. The second reason is the multiplication by ${{\overleftrightarrow{{\bf{C_{\mathit i}}}}}}$. In the worst case, the errors of $v_i^j$ and $w_i^j$ are multiplied by the maximum norm, ${{ \settoheight{\dimen0}{$\mathstrut {{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}$} \advance\dimen0 .0pt \settodepth{\dimen1}{${{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}$} \advance\dimen1 4pt \,\hbox{
\vrule height\dimen0 depth\dimen1\, \vrule height\dimen0 depth\dimen1\, \hbox{${{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}$} \, \vrule height\dimen0 depth\dimen1\, \vrule height\dimen0 depth\dimen1\, }\,}}$, and the errors of ${{\boldsymbol{s}}}_i^j$ and ${{\boldsymbol{t}}}_i^j$, respectively. This amplification affects all coefficients regardless of the distances between nodes. According to Ref. [@Tomioka:2010], ${{ \settoheight{\dimen0}{$\mathstrut {{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}$} \advance\dimen0 .0pt \settodepth{\dimen1}{${{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}$} \advance\dimen1 4pt \,\hbox{
\vrule height\dimen0 depth\dimen1\, \vrule height\dimen0 depth\dimen1\, \hbox{${{\overleftrightarrow{{\bf{C_{\mathit i}^{\rm{-1}}}}}}}$} \, \vrule height\dimen0 depth\dimen1\, \vrule height\dimen0 depth\dimen1\, }\,}}\le 4\pi/(\pi-2)\simeq 11$ in the case of $\Delta\theta=\pi/2$ and $L_{{{\rm A}}}=L_{{{\rm B}}}$.
In contrast, the error in the case of the partial-HBIE shown in [Fig.\[fig:error\_dist-open\_end-1reg\]]{}(A-b-ii) is similar to the case of the CBIE in (A-b-i); and the relation between (B-b-ii) and (B-b-i) is also similar. The number of unknowns in the analyses presented in [Fig.\[fig:error\_dist-open\_end-1reg\]]{} was 164 including four double nodes; i.e., the number of HBIEs was only 8 and the number of CBIEs was 156 in the partial-HBIE. Since the number of HBIEs with a larger error is sufficiently smaller than that of CBIEs, the total error of partial-HBIEs does not increase so much as that of CBIEs.
Based on the comparisons between model-(A) and model-(B) in [Fig.\[fig:error\_dist-open\_end-1reg\]]{}, we can observe that the error of model-(B) is larger than that of model-(A) for each equation type. This reason is the same as the first reason for the difference between CBIEs and full-HBIEs. The coefficients $g_{i\,'}^j$ and $h_{i\,'}^j$ in [Eq. (\[internal-field\])]{} are results of the boundary integrals in which integrands include $u^*$ and $q^*$, respectively. Since the singularity of $q^*$ is stronger than that of $u^*$, the contribution of $h_{i\,'}^j$ to $u({{{\boldsymbol{r}}}}_{i\,'})$ is larger than that of $g_{i\,'}^j$, especially in the case where the distance between the field point and boundary elements is smaller than a wavelength. The error caused by $h_{i\,'}^j$ is also larger than $g_{i\,'}^j$. In model-(A), there are many nodes with $\overline{u}=0$. Therefore, the larger error caused by $h_{i\,'}^j$ does not appear, and the smaller error caused by $g_{i\,'}^j$ becomes dominant. In contrast, in model-(B), since there are no nodes with $\overline{u}=0$, the larger error caused by $h_{i\,'}^j$ remains.
We analyzed the errors of different boundary element sizes. The error is evaluated based on average sampling points as follows: $$\begin{aligned}
&
\langle|\Delta u|\rangle=\frac{1}{N_s}\sum_{i'=1}^{N_s}{|\Delta u_{i'}|},
\qquad
\Delta u_{i'}=u_{i'}-\hat{u}_{i'},\end{aligned}$$ where $N_s$ is the number of sampling points that are intersections of the grid in [Fig.\[fig:exact\_dist-open\_end-1reg\]]{} ($N_s=41^2$), and $N_s$ is unchanged for all results regardless of the element size. The errors are not normalized since the averaged intensities of the exact solutions, $\langle|\hat{u}|\rangle$, have almost the same order of magnitude; $\simeq$0.8 for the termination types (a) and (b), and $\simeq$0.6 for (c). Figure \[fig:size-vs-error\] illustrates the dependence between the error and the number of elements in a wavelength, $N_\lambda=\lambda/\Delta_{\rm Ele}$ where $\Delta_{\rm Ele}$ denotes the boundary element size. In the case of (a), there are no plots for the short termination type (i) since the simultaneous equations become singular. In all cases of (a), (b) and (c), the errors decrease as $N_\lambda$ increases with a decay proportional to $1/N_\lambda^2$, except for the points of $N_\lambda\gtrsim 200$ in (b). This property is reasonable if $h_i^j$, $g_i^j$, $v_i^j$, and $w_i^j$ have accuracies of $O(\Delta_{\rm Ele})$ in the case of the linear element, and the truncated error is proportional to $O(\Delta_{\rm Ele}^2)$. However, there is an error which does not show this characteristic [@Tomioka:2010]; the order of the error of ${{\boldsymbol{s}}}_i^{j,{\rm reg}}$ for a short distance between the nodes $i$ and $j$ obeys $O(\Delta_{\rm Ele})$. This error arises when $N_\lambda\gtrsim 200$. In the cases (b) and (c), the error of the full-HBIE (iii) is several times larger than the CBIE (i), and that of the partial-HBIE (ii) is almost the same as (i), which is similar to the result previously shown in [Fig.\[fig:error\_dist-open\_end-1reg\]]{}.
Figure \[fig:size-vs-cpu\] presents the computational time, which does not include the CPU time of the file input and output processes. The computation consists of several major steps; the computation of the components of the coefficient matrix $a_i^j$ (through $h_i^j$ and $g_i^j$ in [Eqs. (\[h\_int2\]) and (\[g\_int2\])]{}, respectively; or $v_i^j$ and $w_i^j$ in [Eqs. (\[def-v\_i\^j\]) and (\[def-w\_i\^j\])]{}, respectively); solving the matrix equation; and the evaluation of the internal field using [Eq. (\[internal-field\])]{}; for which individual computational costs are proportional to $N_\lambda^2$, $N_\lambda^3$, and $N_\lambda$, respectively. Higher-order terms appear with increasing $N_\lambda$. In the case where $N_\lambda\lesssim5$, most of the computation time is exhausted in minor common steps such as initializing tables for the Hankel functions. The computational cost for the full-HBIE (iii) is larger than that for the others. This is because of the difference between the evaluation time of $v_i^j$ and $w_i^j$ for the HBIE and that of $h_i^j$ and $g_i^j$ for the CBIE. In the HBIE, the cost of evaluating $v_i^j$ and $w_i^j$ is mainly exhausted in the numerical integrals of non-singular elements for the three vectors in [Eqs. (\[s\_int-reg\]), (\[sL\_int-reg\]) and (\[t\_int-reg\])]{}. In contrast, two scalar integrals in [Eqs. (\[h\_int\]) and (\[g\_int\])]{} are dominant in the CBIE. The cost of evaluating a coefficient with a vector is twice that of a scalar in two-dimensional problems, and the number of components in the HBIE is 3/2 times greater than the CBIE. Moreover, the operator ${{{\boldsymbol{n}}}}{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{\nabla _}i\Grad$ in [Eqs. (\[s\_int-reg\]) and (\[sL\_int-reg\])]{} has two vector components ${{{\boldsymbol{e}}}_r}{{{\boldsymbol{e}}}_r}{\hspace*{-0.0833em}\cdot\hspace*{-0.0833em}}{{{\boldsymbol{n}}}}$ and ${{{\boldsymbol{n}}}}$. Because some of the terms have common factors, the sum of costs was reduced from these estimations; however, the cost of evaluating the coefficient matrix component in the HBIE is almost four times larger than that in the CBIE. Even when the simultaneous equations are singular, it can be solved as a minimal-norm solution of underdetermined equations by using a solver based on a singular value decomposition (SVD). The details are not included in this paper because the authors do not understand whether the minimal-norm solution is always correct or not. By limiting the examples shown here, the accuracy of the CBIE by using a solver based on SVD was almost the same as in the case of the partial-HBIE. However, the computational time of SVD was much larger than in the case of LU decomposition; e.g., the time to solve 6,900 s for the CBIE using SVD called ‘zgelss’ in Lapack, 220 s for the partial-HBIE using the LU decomposition in the case of $N_\lambda=1,000$.
The above results can be summarized as follows. First, as mentioned in Sec. \[sec:Rank\_deficient\_condition-CBIE\], the set of simultaneous equations constructed by the CBIE for all nodes becomes singular when both boundary conditions of the double node are imposed by Dirichlet conditions. In contrast, when the node equations are constructed by the HBIE for all or a part of the nodes, the set of simultaneous equations does not become singular. Next, the accuracy of the HBIE is unfortunately worse than the CBIE when the set of equations is regular; however, in the case of the partial-HBIE where only the equations of the sub-nodes belonging to the double nodes are given by the HBIE and the others are given by the CBIE, the reduction in accuracy is negligibly small. Finally, more computational cost is required to compute the component of the coefficient matrix by the HBIE than the CBIE. The rise in computational cost can be suppressed by applying the HBIE to sub-nodes only. Therefore, we can conclude that the replacement of the CBIE by the HBIE only for the sub-nodes (partial-HBIE) is the best solution from the viewpoint of singularity, accuracy, and computational cost.
To demonstrate the applicability of the HBIE in the cases with interface boundaries, we evaluated the models shown in [Fig.\[fig:model\_with\_4reg\]]{}. In these models, the original model shown in [Fig.\[fig:model\]]{}(A) or (B) is partitioned into two or four sub-domains by interface boundaries, and the exact solutions are the same as [Eqs. (\[exact-TE10\]) and (\[exact-TE20-central\_half\])]{}. The number of sub-nodes for each double node is four at two intersections in the model-([A:]{}) and at four corners in ([A:]{}) and ([B:]{}). In addition, at the intersections of the four interface boundaries in ([A:]{}) and ([B:]{}), the number of sub-nodes for each double node is eight. The combinations of four types of boundary conditions (Dirichlet, Neumann, Robin, and interface conditions), can be examined by these models.
Figure \[fig:error-interface\_conditions\] presents comparisons of the errors. As predicted in Sec. \[sec:interface-case-CBIE\], when the simultaneous equations are constructed by the CBIE, the set of equations for each model including the interface boundary conditions is always singular. Similar to the above discussions on the single region problems, the error in the partial-HBIE (ii) is less than that in the full-HBIE (iii) in the case of multi-regions problems. Based on the comparison between model-(A) and either ([A:]{}) or ([A:]{}), we can observe that the errors in the multi-media problems with the interface boundaries (models-([A:]{}) or ([A:]{})) are larger than that in the single region problem (model-(A)). One of the reasons could be the same reason for which the model, including the boundary with $\overline{q}=0$ such as model-(B), has a larger error than that with $\overline{u}=0$ such as model-(A), and this is discussed in the description of [Fig.\[fig:error\_dist-open\_end-1reg\]]{}. The internal field is evaluated by [Eq. (\[internal-field\])]{} as the boundary integral where the boundary encloses the domain considered. In the case of the multi-regions partitioned by the interface boundaries, the boundary enclosing a single region must include the continuous boundary where $u\ne0$. Therefore, the error in $h_{i'}^j$ contributing from the interface boundary in [Eq. (\[internal-field\])]{} is added to the total error in the multi-region problems; whereas it is not added from the boundary with $\overline{u}=0$ in model-(A). The other reason is a difference in the distances between the internal points and their nearest boundary; an average of distances in the multi-region problem is shorter than that of the single region problem. Since the contributions $h_{i'}^j$ and $g_{i'}^j$ increase with decreasing distance, the error in the multi-region problems is larger than that in the single-region problem. The difference in error between the multi- and single-region problem can also be found in the comparison between model-(B) and ([B:]{}).
Consequently, we can demonstrate that the formulation based on the HBIE is applicable without rank deficiency even in the cases involving the interface boundaries, which is similar to the corner nodes in the single region problems.
One may question whether the partial-HBIE method can avoid spurious solutions of an external problem shown in Sec. \[sec:intro\] since the partial-HBIE method uses both CBIEs and HBIEs like the Burton-Miller method does to avoid spurious solutions. In the Burton-Miller method, a linear combination of CBIEs and HBIEs with an appropriate combination factor is used to prevent spurious solutions from arising when some of the sub-matrices become singular. Whereas, in the partial-HBIE method, the matrix equation consists of two sets of equations, CBIEs and HBIEs, without any modifications. Each set of equations has sub-matrices which may potentially produce spurious solutions. The partial-HBIE method, therefore, cannot avoid spurious solutions. To avoid them, we should use other methods, such as the Burton-Miller method [@Burton-Miller:1971], CHIEF [@Schenck:1968], or a virtual boundary method [@Tomioka:1993; @Tomioka:1994] which divides the external region into multiple regions by virtual boundaries to stop the external region from surrounding the internal region. If we use a virtual boundary method, the issue of multiple-duplicated nodes has to be solved; that is, however, not very difficult when using the method proposed in this paper.
Conclusion {#sec:conclusion}
==========
The method of using double nodes at corners is a useful approach to uniquely define the normal direction. However, a set of simultaneous equations in CBIE formulation produces rank deficient problems in the following cases: both sub-nodes belonging to any double node are imposed by Dirichlet conditions; an intersection of the interface boundary located between different media is not connected to the boundary imposed by ordinary boundary conditions; and an interface boundary is connected to the two boundaries imposed by Dirichlet conditions. This means that the applicable problem that uses the double nodes are limited in the CBIE formulation.
In contrast, when the coefficient matrix is constructed by HBIEs, the rank is not reduced for any combination of boundary conditions, including interface conditions. However, the contribution coefficients between nodes in HBIEs are less accurate than those in CBIEs for the problem without rank deficiency because of two reasons; a HBIE exhibits a stronger singularity of the integrand than a CBIE, and most of the coefficients are multiplied by the dyadic tensor with a large norm. Furthermore, the computational cost of evaluating the coefficients of HBIEs is higher than that of CBIEs.
To address the rank deficiency problem in CBIEs and the drawbacks in HBIEs, the coupling approach presented in this paper called partial-HBIE is the best choice. In the partial-HBIE, most node equations are constructed by CBIEs, and only the sub-node equations related to corners are constructed by HBIEs.
The method that uses HBIEs demonstrates the following advantages compared to other methods: it does not require any additional local relation between nodal points around double nodes, any extra boundary integral equation, and it does not require a least-square method, which can be computationally time-consuming. Furthermore, the partial-HBIE can be applied by only switching the sub-node equation for the double node from a CBIE to a HBIE; therefore, we can be relieved from the efforts involved in preparing input data and complex coding.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by JSPS KAKENHI Grant Number 18K04158.
|
---
abstract: 'We present a self-contained short proof of the seminal result of Dillencourt (SoCG 1987 and DCG 1990) that Delaunay triangulations, of planar point sets in general position, are 1-tough. An important implication of this result is that Delaunay triangulations have perfect matchings. Another implication of our result is a proof of the conjecture of Aichholzer [[et al.]{}]{} (2010) that at least $n$ points are required to block any $n$-vertex Delaunay triangulation.'
author:
- 'Ahmad Biniaz[^1]'
bibliography:
- 'Toughness-DT.bib'
title: A Short Proof of the Toughness of Delaunay Triangulations
---
Introduction
============
Let $P$ be a set of points in the plane that is in general position, i.e., no three points on a line and no four points on a circle. The [*Delaunay triangulation*]{} of $P$ is an embedded planar graph with vertex set $P$ that has a straight-line edge between two points $p,q\in P$ if and only if there exists a closed disk that has only $p$ and $q$ on its boundary and does not contain any other point of $P$. A graph is $1$-[*tough*]{} if for any $k$, the removal of $k$ vertices splits the graph into at most $k$ connected components. In 1987, Dillencourt proved the toughness of Delaunay triangulations.
\[Dillencourt-thr\] Let $T$ be the Delaunay triangulation of a set of points in the plane in general position, and let $S\subseteq V(T)$. Then $T\setminus S$ has at most $|S|$ components.
Dillencourt’s proof of Theorem \[Dillencourt-thr\] is nontrivial and employs a large set of combinatorial and structural properties of (Delaunay) triangulations. Using the same proof idea, he showed that if $T$ is a Delaunay triangulation of an arbitrary point set in the plane (not necessarily in general position) then $T\setminus S$ has at most $|S|+1$ components. Combining this with Tutte’s classical theorem that characterizes graphs with perfect matchings [@Tutte1947], implies the following well-known result.
\[matching-thr\] Every Delaunay triangulation has a perfect matching.
In this note we present a self-contained short proof of Theorem \[Dillencourt-thr\]. To that end, we first present an upper bound on the maximum size of an independent set of $T$. To facilitate comparisons we use the same definitions and notations as in [@Dillencourt1990]. The number of elements of a set $S$ is denoted by $|S|$. For a graph $G$, the vertex set of $G$ is denoted by $V(G)$, and $|G|=|V(G)|$.
Every interior face of $T$ is a triangle, and the boundary of $T$ is a convex polygon; see Figure \[DT-fig\](a). An edge is called a [*boundary edge*]{} if it is on the boundary of $T$, and is called an [*interior edge*]{} otherwise. For any interior edge $(p,q)\in T$ between two faces $pqr$ and $pqs$ it holds that $$\label{eq0}
\angle prq + \angle psq < 180.$$
$\begin{tabular}{ccc}
\multicolumn{1}{m{.24\columnwidth}}{\centering\includegraphics[width=.2\columnwidth]{fig/DT.pdf}}
&\multicolumn{1}{m{.52\columnwidth}}{\centering\includegraphics[width=.51\columnwidth]{fig/DT2.pdf}}&\multicolumn{1}{m{.24\columnwidth}}{\centering\includegraphics[width=.22\columnwidth]{fig/DT-path.pdf}}\\
(a)&(b)&(c)
\end{tabular}$
A combinatorial and a structural property {#property-section}
=========================================
A direct implication of Theorem \[Dillencourt-thr\] gives the upper bound $\lfloor(|T|+1)/2\rfloor$ on the size of any independent set of $T$; see e.g. [@Aichholzer2013]. We present a different self-contained proof for a slightly better bound.
\[independent-set-thr\] Let $T$ be the Delaunay triangulation of a set of points in the plane in general position, and let $I$ be an independent set of $T$. Then $|I|\leqslant \lfloor|T|/2\rfloor$, and this bound is tight.
This upper bound is tight as any maximum independent set in the $n$-vertex Delaunay triangulation of Figure \[sufficiency-fig\](b) has exactly $\lfloor n/2\rfloor$ vertices (regardless of the parity of $n$).
Now we prove the upper bound. Set $S:=V(T)\setminus I$, and let $u$ be a vertex of $S$ that is on the boundary of $T$ (observe that such a vertex exists). Let $v,w\notin V(T)$ be two points in the plane such that (i) $T$ lies in the triangle $(u,v,w)$ and (ii) neither of $v$ and $w$ lies in the disks that introduce edges of $T$; see Figure \[DT-fig\](b). Let $\mathcal{T}$ be the Delaunay triangulation of $V(T)\cup \{v,w\}$. Our choice of $v$ and $w$ ensures that any edge of $T$ is also an edge of $\mathcal{T}$, and thus $T\subset \mathcal{T}$. Set $\mathcal{S}:=S\cup \{v,w\}$. In the rest of the proof we show that $|I|\leqslant |\mathcal{S}|-2$. This implies that $|I|\leqslant |S|$ (because $|\mathcal{S}|=|S|+2$) which in turn implies that $|I|\leqslant \lfloor|T|/2\rfloor$ (because $|T|=|S|+|I|$, and $|I|$ and $|T|$ are integers).
To show that $|I|\leqslant |\mathcal{S}|-2$ we use a counting argument similar to that of [@Dillencourt1990 Lemma 3.8]. Let $\mathcal{T}[\mathcal{S}]$ be the subgraph of $\mathcal{T}$ that is induced by $\mathcal{S}$. In other words, $\mathcal{T}[\mathcal{S}]$ is the resulting graph after removing vertices of $I$ and their incident edges from $\mathcal{T}$. Since $\mathcal{T}$ is a triangulation and $I$ does not contain boundary vertices of $\mathcal{T}$, the removal of every vertex of $I$ creates a hole (a new face which is the union of original faces) whose boundary is a simple polygon. All edges of this polygon belong to $\mathcal{T}[\mathcal{S}]$ because $I$ is an independent set. Therefore, $\mathcal{T}[\mathcal{S}]$ is a connected plane graph, the boundaries of its interior faces are simple polygons, and the boundary of its outer face is the triangle $(u,v,w)$; see Figure \[DT-fig\](b). Each interior face of $\mathcal{T}[\mathcal{S}]$ contains either no point of $I$ or exactly one point of $I$. Interior faces that do not contain any point of $I$ are called [*good faces*]{}, and interior faces that contain a point of $I$ are called [*bad faces*]{}. Each good face is a triangle. Let $g$ and $b$ denote the number of good and bad faces, respectively. Thus the number of interior faces is $g+b$.
Since $|I|=b$, it suffices to show that $b\leqslant |\mathcal{S}|-2$. To do so, we assign to each edge $(p,q)\in \mathcal{T}[\mathcal{S}]$ certain [*distinguished angles*]{}. If $(p,q)$ is an interior edge then we distinguish the two angles of $\mathcal{T}$ that are opposite to $(p,q)$, and if $(p,q)$ is a boundary edge then we distinguish the unique angle of $\mathcal{T}$ that is opposite to $(p,q)$, as in Figure \[DT-fig\](b). Let $d$ be the total measure of all distinguished angles. We compute $d$ in two different ways: once with respect to the number of faces of $\mathcal{T}[\mathcal{S}]$ and once with respect to the number of edges of $\mathcal{T}[\mathcal{S}]$. Each good face contains three distinguished angles, their sum is $180^\circ$. The sum of the distinguished angles in each bad face is $360^\circ$ because these angles are anchored at the removed vertex in the face. Therefore $$\label{eq3}
d=180\cdot g+360\cdot b.$$
Now we compute $d$ with respect to the number of edges of $\mathcal{T}[\mathcal{S}]$ which we denote by $e$. By Euler’s formula, we have $e=|\mathcal{S}|+b+g-1$. By Inequality , the sum of (at most two) distinguished angles assigned to each edge is less than $180^\circ$. Therefore $$\label{eq2}
d< 180\cdot e= 180\cdot (|\mathcal{S}|+ b + g -1).$$
Combining and , we have $$180\cdot g + 360\cdot b < 180\cdot (|\mathcal{S}|+ b + g -1),$$ which simplifies to $b< |\mathcal{S}|-1$. Since $b$ and $|\mathcal{S}|$ are integers, $b\leqslant |\mathcal{S}|-2$.
Our proof of Theorem \[Dillencourt-thr\] employs Theorem \[independent-set-thr\] and the following structural property of Delaunay triangulations presented by the author [@Biniaz2019]. For the sake of completeness we repeat its proof.
\[Delaunay-thr\] Let $T$ be the Delaunay triangulation of a set of points in the plane in general position. Let $p$ and $q$ be two vertices of $T$ and let $D$ be any closed disk that has on its boundary only vertices $p$ and $q$. Then there exists a path, between $p$ and $q$ in $T$, that lies in $D$.
The proof is by induction on the number of vertices in $D$. If there is no vertex of $V(T)\setminus\{p,q\}$ in the interior of $D$, then $(p,q)$ is an edge of $T$, and so is a desired path. Assume that there exists a vertex $r\in V(T)\setminus\{p,q\}$ in the interior of $D$. Let $c$ be the center of $D$. Consider the ray ${\overrightarrow{pc}}$ emanating from $p$ and passing through $c$. Fix $D$ at $p$ and then shrink it along ${\overrightarrow{pc}}$ until $r$ lies on its boundary; see Figure \[DT-fig\](c). Denote the resulting disk $D_{pr}$, and notice that it lies fully in $D$. Compute the disk $D_{qr}$ in a similar fashion by shrinking $D$ along ${\overrightarrow{qc}}$. The disk $D_{pr}$ does not contain $q$ and the disk $D_{qr}$ does not contain $p$. By induction hypothesis there exists a path, between $p$ and $r$ in $T$, that lies in $D_{pr}$, and similarly there exists a path, between $q$ and $r$ in $T$, that lies in $D_{qr}$. The union of these two paths contains a path, between $p$ and $q$ in $T$, that lies in $D$.
Proof of Theorem \[Dillencourt-thr\] {#proof-section}
====================================
Recall $T$ and $S$. Pick an arbitrary representative vertex from each component of $T\setminus S$, and let $C$ be the set of these vertices. The number of components is $|C|$. Consider the Delaunay triangulation $T'$ of $S\cup C$. Observe that $C$ is an independent set of $T$. We prove by contradiction that $C$ is also an independent set of $T'$. Assume that there exists an edge $(c_1,c_2)\in T'$ such that $c_1,c_2\in C$. Since $T'$ is a Delaunay triangulation, by definition there exists a closed disk $D$ that has only $c_1$ and $c_2$ on its boundary and does not contain any other point of $S\cup C$. Now consider $T$ and $D$. By Theorem \[Delaunay-thr\] there exists a path between $c_1$ and $c_2$ in $T$, that lies in $D$. Since $D$ does not contain any point of $S$, all edges of this path belong to $T\setminus S$. This contradicts the fact that $c_1$ and $c_2$ belong to different components of $T\setminus S$. Therefore $C$ is an independent set of $T'$. By Theorem \[independent-set-thr\], we have $|C|\leqslant |T'|/2$. This and the fact that $|T'|=|S|+|C|$ imply that $|C|\leqslant |S|$.
Blocking Delaunay triangulations
================================
In this section, we use Theorem \[independent-set-thr\] and prove the conjecture of Aichholzer [[et al.]{}]{} [@Aichholzer2013] that at least $n$ points are required to block any $n$-vertex Delaunay triangulation. Let $P$ be a set of points in the plane and let $T$ be the Delaunay triangulation of $P$. A point set $B$ [*blocks*]{} or [*stabs*]{} $T$ if in the Delaunay triangulation of $P\cup B$ there is no edge between two points of $P$. In other words, every disk that introduces an edge in $T$ contains a point of $B$. Throughout this section we assume that $P\cup B$ is in general position.
In 2010, Aronov [[et al.]{}]{} [@Aronov2011] showed that $2n$ points are sufficient to block any $n$-vertex Delaunay triangulation, and if the vertices are in convex position then $4n/3$ points suffice. These bounds have been improved by Aichholzer [[et al.]{}]{} [@Aichholzer2013] (2010) to $3n/2$ and $5n/4$, respectively.
For the lower bound, Aronov [[et al.]{}]{} [@Aronov2011] showed the existence of $n$-vertex Delaunay triangulations that require $n$ points to be blocked, for example see Figure \[sufficiency-fig\](a) in which every disk (representing a Delaunay edge) requires a unique point to be blocked as the disks are interior disjoint. Aichholzer [[et al.]{}]{} [@Aichholzer2013] proved that at least $n-1$ points are necessary to block any $n$-vertex Delaunay triangulations, and stated the following conjecture.
\[Aichholzer-conj\] For any point set $P$ in the plane in convex position, $|P|$ points are necessary and sufficient to block the Delaunay triangulation of $P$.
An implication of Theorem \[independent-set-thr\] proves the necessity of $|P|$ blocking points in Conjecture \[Aichholzer-conj\] (even if $P$ is in general position); the sufficiency remains open.
$\begin{tabular}{cc}
\multicolumn{1}{m{.5\columnwidth}}{\centering\includegraphics[width=.25\columnwidth]{fig/coins.pdf}}
&\multicolumn{1}{m{.5\columnwidth}}{\centering\includegraphics[width=.27\columnwidth]{fig/sufficiency-fig2.pdf}}\\
(a)&(b)
\end{tabular}$
\[blocking-thr\] Let $P\cup B$ be any set of points in the plane in general position such that $B$ blocks the Delaunay triangulation of $P$. Then $|B|\geqslant |P|$, and this bound is tight.
Consider the Delaunay triangulation $T$ of $P\cup B$. Since $B$ blocks the Delaunay triangulation of $P$, the removal of $B$ from $T$ leaves exactly $|P|$ components each consisting of a single point of $P$. Thus $P$ is an independent set of $T$. By Theorem \[independent-set-thr\], we have $|P|\leqslant \lfloor|T|/2\rfloor\leqslant |T|/2$ which implies that $|B|\geqslant |P|$ (because $|T|=|P|+|B|$).
To verify the tightness of this bound, consider a set of $n$ points in convex position where $n-1$ points are at distances approximately 1 from one point, say $p$, so that no four points lie on a circle. In the Delaunay triangulation of this point set, $p$ is connected to all other points, as depicted in Figure \[sufficiency-fig\](b). This Delaunay triangulation can be blocked by $n$ points that are placed outside the convex hull: two points are placed very close to $p$ and $n-2$ points are placed very close to the $n-2$ convex hull edges that are not incident to $p$. A similar placement has also been used in [@Aichholzer2013] and [@Aronov2011].
[^1]: Part of this work has been done while the author was an NSERC postdoctoral fellow at University of Waterloo.
|
---
abstract: 'Tumor growth has long been a target of investigation within the context of mathematical and computer modelling. The objective of this study is to propose and analyze a two-dimensional probabilistic cellular automata model to describe avascular solid tumor growth, taking into account both the competition between cancer cells and normal cells for nutrients and/or space and a time-dependent proliferation of cancer cells. Gompertzian growth, characteristic of some tumors, is described and some of the features of the time-spatial pattern of solid tumors, such as compact morphology with irregular borders, are captured. The parameter space is studied in order to analyze the occurrence of necrosis and the response to therapy. Our findings suggest that transitions exist between necrotic and non-necrotic phases (no-therapy cases), and between the states of cure and non-cure (therapy cases). To analyze cure, the control and order parameters are, respectively, the highest probability of cancer cell proliferation and the probability of the therapeutic effect on cancer cells. With respect to patterns, it is possible to observe the inner necrotic core and the effect of the therapy destroying the tumor from its outer borders inwards.'
address: 'Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, Brazil'
author:
- 'E. A. Reis'
- 'L. B. L. Santos'
- 'S. T. R. Pinho'
title: A cellular automata model for avascular solid tumor growth under the effect of therapy
---
\[section\]
tumor growth, cellular automata, parameter space, necrosis, therapy
INTRODUCTION
============
Neoplastic diseases are the cause of 7 million deaths annually or 12of deaths worldwide [@Who]. Mathematical and computer modelling may lead to greater understanding of the dynamics of cancer progression in the patient [@Preszosi] [@MMC]. These techniques may also be useful in selecting better therapeutic strategies by subjecting available options to computer testing ([*in silico*]{}). Continuous models have been proposed to describe the stages of tumor growth since the middle of the 20th century [@Wheldon]. Initially, a tumor grows exponentially (linear rate). After this transient stage, the growth rate decreases and a steady state is attained, due to several factors including a lack of nutrients and hypoxia. This nonlinear behavior characterizes avascular tumor growth when neovascularization has not yet been triggered. The decelerating avascular growth may be guided by different rules such as, for example, Gompertzian and logistic functions. Gompertzian growth has been one of the most studied decelerating tumor growth over the past 60 years [@Laird], [@Demicheli] . It is found, for example, in some solid tumors such as breast carcinomas [@Clare] [@Spratt]. It is also observed in tumors [*in vitro*]{} [@Bellomo] [@Guiot].
Although continuous models are capable of describing the behavior of tumor growth, it would appear more reasonable to adopt a discrete approach when describing the prevascular stage. Due to the fact that the angiogenic process has not triggered early tumor growth, few cancer cells are present and growth depends predominantly on the interactions of these cells with adjacent cells and with the environment [@Patel]. In addition, in the discrete approach, it is easier to capture the time-spatial pattern generated by the model in order to compare it with actual patterns [@Castro]. Some cellular automata models [@Galam] [@Shen] and hybrid cellular automata [@Patel] [@Dormann] [@Gerlee] have been proposed to study tumor growth.
Another important topic that is analyzed in mathematical models is the response to therapy, including how tumor growth changes under the effect of a drug [@Rygaard]. The focus is directed towards identifying the optimal therapy to maximize the effect on cancer cells and minimize the effect on normal cells [@Martin] [@Swan]. Although various continuous chemotherapy models exist [@Murray], [@Costa], [@Matveev], to the best of our knowledge the majority of the discrete models cited in the literature have not yet been used to investigate this topic.
The objective of this work is to propose a two-dimensional cellular automata model consisting of 4 states (empty site, normal cell, cancer cell or necrotic tumor cell) to describe avascular tumor growth. It should be emphasized that in this study the term tumor growth is used to refer to the number of cancer cells rather than the volume of the tumor; in other words, it is assumed that the tumor volume is proportional to the number of cancer cells [@Wheldon]. Assuming that the angiogenic process has not yet been triggered, there is no increase in nutrients, which are uniformly distributed over the lattice. In this simple model, some relevant processes involved in the prevascular phase of tumor growth are assessed: a dynamic proliferation of cancer cells and the competition between normal cells and cancer cells for nutrients and/or space. Since necrosis is often present in the prevascular stage of tumor growth [@Bellomo] [@Adam], the possibility of necrosis in the model must also be taken into consideration. Finally, the effect of therapy is included in order to investigate whether the system evolves to a state of cure.
This paper is organized as follows. In section \[sec2\], the model is presented, together with its local rules, parameters and the scope of the algorithm. Section \[sec3\] describes the simulated time series of cell density in the presence or absence of treatment, and shows the features of the time-spatial pattern of simulated solid tumors. In section \[sec4\], the parameter related to the process of necrosis and the effect of therapy is analyzed. Finally, in section \[sec5\], our results are discussed from the point of view of the phenomenon and some concluding remarks are made.
THE MODEL {#sec2}
=========
We propose a two-dimensional ($L \times L$) cellular automata model [@Wolfram] under periodic boundary conditions, using a Moore neighborhood with a radius of 1. At the initial condition ($t=0$), there is only one cancer cell (to ensure better visualization, this was taken from the center of the lattice). Since the intention it to model a non-viral tumor, normal cells would not be transformed into cancer cells with the exception of the cancer cell that triggers tumor growth at $t = 0$ [@Thecell]. The lattice represents a tissue sample; there is a cell in each site that may be in one of four states: normal cell (NoC), cancer cell (CC), necrotic cell (NeC) or empty site (ES). We assume that the nutrients are uniformly available over the lattice. In this respect, lack of space is identified with lack of nutrients in our model.
There is a growth potential $P_c$ value associated with each normal or cancer cell. Although the two-dimensional character of the model mimics the [*in vitro*]{} situation, the growth potential of cancer cells simulates the three-dimensional tumor [*in vivo*]{} in the sense that it represents the total number of cancer cells, i.e., in addition to the cancer cells on the lattice, the cancer cells generated by these cells.
The local rules are such that:
- \(i) The initial value of the mitotic probability of cancer cells is represented by a parameter $p_0$ that measures the available resources at the beginning of the tumor. After that, it decreases by a factor $\Delta
p_{mitot}$ until reaching the null value: $$\label{pmitot}
\Delta p_{mitot} = \exp \left[-
\left(\frac{n_{noc}(t)}{n_{cc}(t)}\right)^2 \right ]$$
in which $n_{noc}$ and $n_{cc}$ are the number of normal and cancer cells at time $t$, respectively. As shown, $\Delta
p_{mitot}(t)$ depends only on the dynamics, setting up a feedback inhibition mechanism [@Castro]: as the tumor grows, $\Delta
p_{mitot}(t)$ decreases because of the combined effect of the decrease in the number of normal cells and the increase in the number of cancer cells. In order to intensify the effect of this mechanism (see [@Lobato]), an exponent 2 in equation (\[pmitot\]) is considered. Since there is no new available source of nutrients and/or space, it decreases as the density of cancer cells increases because the available nutrients and/or space are reduced. The effect of the proliferation of cancer cells is that their growth potential increases by a unit at each time step.
- \(ii) The cancer cells compete with normal cells for the empty sites, depending on the potential growth of neighboring cells. According to the majority rule, a normal cell is displaced by a cancer cell following local battles occurring between healthy and cancerous cells \[15\].
- \(iii) If the growth potential of a cancer cell reaches a threshold value that is a fraction $f$ of the lattice size $L$, it becomes necrotic and its growth potential falls to zero.
- \(iv) Both normal and cancer cells may die, with probabilities $p_{drugn}$ and $p_{drugc}$, respectively, due to the continuous infusion of a drug that is applied after tap time steps; in this case, the site becomes empty.
- \(v) if there are no cancer cells in the neighborhood of a dead cell (empty site), regeneration of normal cells occurs; if the cancer cells in the neighborhood of a necrotic cell die as a result of the therapy, the necrotic cell is eliminated.
The algorithm was computationally implemented in FORTRAN 77 in accordance with the following steps: input data; calculate $\Delta
p_{mitot}(t)$; identify the state of the cell (choose one of the subroutines: normal cell (NoC), cancer cell(CC), necrotic cell(NeC), empty site (ES); update the cells of the lattice; after N iterations, output data.
The input data are the following cellular automata (CA) parameters:
- 1\) The spatial parameters: lattice size L; necrosis threshold fraction $f$ of lattice size;
- 2\) The temporal parameters: the length of the time series t final and the initial time of therapy infusion $t_{ap}$;
- 3\) The probabilities of: the initial proliferation of cancer cells $p_0$; the effect of therapy on normal cells and cancer cells ($p_{drugn}$ and $p_{drugc}$).
The output data are the time series of the density of each type of cell and the final configuration of the lattice at any time step. In addition, the time-spatial configurations, controlled by a package denominated g2 [@g2_manual] whose commands are inserted into the computer program in FORTRAN, are generated in “real time”. This package may be used in C, PYTHON and PERL.
The following is a description of each subroutine based on the local rules:
- a\) [**Normal Cell (NoC)**]{} - a random number $y$ is compared to $p_{drugn}$. If $y < p_{drugn}$, the growth potential $P_{noc}$ is confirmed: if $P_{noc} = 0$, the cell dies and the site becomes empty; otherwise, it remains occupied by a normal cell but $P_{noc}
= 0$. If $y \ge p_{drugn}$ and if there is at least one neighboring cancer cell, then the normal cell is ’dislocated’, $P_{noc} = 0$ and the site becomes empty; otherwise it remains a normal cell.
- b\) [**Cancer Cell (CC)**]{} - if all of its neighbors are cancer cells and its growth potential reaches a fraction $f$ of lattice size $L$, the cancer cell becomes necrotic. Otherwise, a number $y$ is randomly chosen. If $y < p_{drugc}$ and $t > t_{ap}$, the potential $P_{cc}$ is confirmed: if $P_{cc} > 0$, it is reduced by a unit and the cell remains a cancer cell; otherwise the cell dies and the site becomes empty. Finally, if $y \ge p_{drugc}$, the cell remains a cancer cell; however, its growth potential $P_{cc}$ increases by a unit.
- c\) [**Necrotic Cell (NeC)**]{} - if at least one of its neighbors is neither a cancer cell nor a necrotic cell, it is eliminated and the site becomes empty; otherwise, it continues necrotic.
- d\) [**Empty Site (ES)**]{} - if there are cancer and normal cells in its neighborhood, the local battle between cancer cells and normal cells is such that if the sum of the potential growth of its neighboring cancer cells is greater or equal to the sum of the potential growth of its normal cell neighbors, then the empty site is occupied by a cancer cell that diffuses from one of the randomly chosen neighbors; otherwise, it is occupied by one of the randomly chosen normal cells that were previously dislocated. If there are only normal cells in its neighborhood, it becomes a normal cell through a process of regeneration. Finally, if none of its neighbors are cancer cells or normal cells, it remains empty.
RESULTS: SIMULATED TIME SERIES AND TIME-SPATIAL PATTERNS {#sec3}
========================================================
Computational simulations of the model were performed in order to analyze two classes of behavior: cases in which no treatment was given and treatment cases. In each subsection, the time series of the simulated tumor as well as time-spatial patterns are shown.
NO TREATMENT CASE {#subsec31}
-----------------
In the case of no treatment, the following parameters are considered: $p_{drugn} = p_{drugc} = t_{ap} = 0$. For fixed values of $L$, $t_{final}$ and $p_0$, but different values of $f$, in Figures \[Figure1\] and \[Figure3\], the time series of the density of cells for nonnecrotic and necrotic tumors, respectively, are shown. In both cases, the cell densities reach saturated values due to the effects of the competition between normal cells and cancer cells, and the time-dependent mitotic probability. Comparing Figures \[Figure1\]b and \[Figure3\]b, the stationary value of cancer cell density is clearly greater in necrotic tumors than in nonnecrotic ones. This is a consequence of the fact that $\Delta
p_{mitot}(t)$ assumes smaller values in necrotic tumors compared to nonnecrotic ones because it does not depend on the density of necrotic cells. In both cases, the average of the different samples is considered, corresponding to different seeds of random numbers.
![\[Figure1\] Nonnecrotic tumor: the average of simulated time series of the density of: a) normal cells, b) cancer cells. We consider $M_{samples} = 200$ and the following parameter values: $L
= 251$, $p_{0} = 0.95$, $f = 0.6$, $p_{drugn} = p_{drugc} =
0.0$.](ReisFig1a.jpg "fig:"){width="6.0cm"} ![\[Figure1\] Nonnecrotic tumor: the average of simulated time series of the density of: a) normal cells, b) cancer cells. We consider $M_{samples} = 200$ and the following parameter values: $L
= 251$, $p_{0} = 0.95$, $f = 0.6$, $p_{drugn} = p_{drugc} =
0.0$.](ReisFig1b.jpg "fig:"){width="6.0cm"}
![\[Figure2\] Non-necrotic case: assuming $M_{samples}=200$ and the same parameters values of Figure \[Figure1\], the average of simulated time series of: a) the number of cancer cells (grey color); b) the growth potential of cancer cells (black color). The Gompertzian fitting is applied to (a) (black color) those in which parameters of time series are $\alpha_0=(6.58 \pm 0.09) \times 10^{-2}$, $\beta=(1.631 \pm 0.008)
\times 10^{-2}$ and $n_0=(1.6 \pm 0.06) \times 10^{2}$.](ReisFig2a.jpg "fig:"){width="6.0cm"} ![\[Figure2\] Non-necrotic case: assuming $M_{samples}=200$ and the same parameters values of Figure \[Figure1\], the average of simulated time series of: a) the number of cancer cells (grey color); b) the growth potential of cancer cells (black color). The Gompertzian fitting is applied to (a) (black color) those in which parameters of time series are $\alpha_0=(6.58 \pm 0.09) \times 10^{-2}$, $\beta=(1.631 \pm 0.008)
\times 10^{-2}$ and $n_0=(1.6 \pm 0.06) \times 10^{2}$.](ReisFig2b.jpg "fig:"){width="6.0cm"}
![\[Figure3\] Necrotic case: the average of simulated time series of density of a) normal cells, b) cancer cells. We consider $M_{samples}=200$ and the following parameter values: $L=251$, $p_0=0.95$, $f=0.2$, $p_{drugn}=0.0$, $p_{drugc}=0.0$.](ReisFig3a.jpg "fig:"){width="6.0cm"} ![\[Figure3\] Necrotic case: the average of simulated time series of density of a) normal cells, b) cancer cells. We consider $M_{samples}=200$ and the following parameter values: $L=251$, $p_0=0.95$, $f=0.2$, $p_{drugn}=0.0$, $p_{drugc}=0.0$.](ReisFig3b.jpg "fig:"){width="6.0cm"}
![\[Figure4\] Assuming $M_{samples}=200$ and the same parameters values of Figure \[Figure3\], the average of simulated time series of: a) the number of cancer cells (grey color); b) the growth potential of cancer cells (black color). The Gompertzian fitting is applied on (a) (black color) those in which parameters of time series are $\alpha_0=(5.86 \pm 0.06) \times 10^{-2}$, $\beta=(1.450 \pm 0.006) \times 10^{-2}$ and $n_0=(1.926 \pm 0.005)
\times 10^{2}$.](ReisFig4a.jpg "fig:"){width="6.0cm"} ![\[Figure4\] Assuming $M_{samples}=200$ and the same parameters values of Figure \[Figure3\], the average of simulated time series of: a) the number of cancer cells (grey color); b) the growth potential of cancer cells (black color). The Gompertzian fitting is applied on (a) (black color) those in which parameters of time series are $\alpha_0=(5.86 \pm 0.06) \times 10^{-2}$, $\beta=(1.450 \pm 0.006) \times 10^{-2}$ and $n_0=(1.926 \pm 0.005)
\times 10^{2}$.](ReisFig4b.jpg "fig:"){width="6.0cm"}
Figures \[Figure2\]a and \[Figure4\]a show that the tumor growth obeys the Gompertzian function both in nonnecrotic and necrotic cases. This behavior is observed with respect to the number of cancer cells for a range of values of $p_0$. In Gompertzian growth, the specific growth rate of the number of cancer cells decreases logarithmically:
$$\label{gompertzrate}
\frac{1}{n_{cc}}\frac{dn_{cc}}{dt} = \alpha_0
- \beta \log \left(\frac{n_{cc}}{n_0}\right)$$
where
- a\) $n_0$ is the initial population of cancer cells;
- b\) $\alpha_0$ is the specific growth rate of $n_0$ cells at $t=0$;
- c\) $\beta$ measures how rapidly the curve departs from a singular exponential and curves over, assuming its characteristic shape.
The solution of (\[gompertzrate\]) is
$$\label{gompertzsolution}
n_{cc}(t)=n_0 \exp \left\{\frac{\alpha_0}{\beta}[1-\exp(-\beta t)]\right\}.$$
It is evident that the stationary value of $n_{cc}$ is $n_{{cc}_{\infty}}=n_0 \exp(\alpha_0/\beta)$. According to the Gompertzian fitting represented by equation \[gompertzsolution\], the results of the simulations shown in Figures \[Figure2\] and \[Figure4\] correspond respectively to the following parameters:
- I\) The nonnecrotic tumor: $\alpha_0=(6.58 \pm 0.09) \times 10^{-2}$, $\beta=(1.631 \pm
0.008) \times 10^{-2}$ and $n_0=(1.60 \pm 0.06) \times 10^{2}$
- II\) The necrotic tumor: $\alpha_0=(5.86 \pm 0.06) \times
10^{-2}$, $\beta=(1.450 \pm 0.006) \times 10^{-2}$ and $n_0=(1.926
\pm 0.005) \times 10^{2}$
Comparing the above parameters of (I) and (II), the behavior of nonnecrotic and necrotic tumors was found to be very similar. It was also found that the number of cancer cells obeys the Gompertzian fitting.
An important confirmation of our model is shown by comparing the Gompertzian fitting parameters of simulated tumors with the corresponding parameters of actual tumors [@Demicheli] [@Brunton]. For instance, in the case of the testicular tumors shown in Table 3 of reference [@Demicheli], the $\beta$ values are in the range of $[0.005; 0.016]$ $day^{-1}$. Our simulated $\beta$ values are within the above range if we consider the time step of our simulations to be one day.
In relation to the parameters $\alpha_{0}$ and $n_0$, the simulated values are not comparable to the actual values, since no information on the initial size of the tumor was included in our model. Both $\alpha_0$ and $n_0$ are strongly dependent on that information.
![\[Figure5\] The spatial distribution of states at consequent time steps (see the time step bar) using the same parameters values of Figure \[Figure3\] except $L=81$ and $p_0=0.8$. The final time step is $t_{final}=600$. Dark colors (light grey, grey, and dark grey) correspond, respectively, to normal, cancer, and necrotic cells.](S1.jpg "fig:"){width="40.00000%"} ![\[Figure5\] The spatial distribution of states at consequent time steps (see the time step bar) using the same parameters values of Figure \[Figure3\] except $L=81$ and $p_0=0.8$. The final time step is $t_{final}=600$. Dark colors (light grey, grey, and dark grey) correspond, respectively, to normal, cancer, and necrotic cells.](S2.jpg "fig:"){width="40.00000%"} ![\[Figure5\] The spatial distribution of states at consequent time steps (see the time step bar) using the same parameters values of Figure \[Figure3\] except $L=81$ and $p_0=0.8$. The final time step is $t_{final}=600$. Dark colors (light grey, grey, and dark grey) correspond, respectively, to normal, cancer, and necrotic cells.](S3.jpg "fig:"){width="40.00000%"} ![\[Figure5\] The spatial distribution of states at consequent time steps (see the time step bar) using the same parameters values of Figure \[Figure3\] except $L=81$ and $p_0=0.8$. The final time step is $t_{final}=600$. Dark colors (light grey, grey, and dark grey) correspond, respectively, to normal, cancer, and necrotic cells.](S4.jpg "fig:"){width="40.00000%"}
Finally, it would be very interesting to discover whether it is possible to relate the cellular automata no-therapy parameters $L$, $f$ and $p_0$ with the Gompertzian parameters for the range of values of $p_0$ and to assess whether the number of cancer cells obeys a Gompertzian growth pattern. In this case, some preliminary conclusions may be drawn with respect to $n_\infty$: it decreases in accordance with the necrotic parameter $f$ but it increases linearly as a function of lattice size $L$ and, exponentially with $p_0$. Concerning the other Gompertzian parameters, the answer to this question is not so simple. Following these conclusions leads us to a much more interesting quandary if we want to make the model more realistic: to estimate realistic ranges of the CA no-therapy parameters $L$, $f$ and $p_0$ based on the Gompertzian parameters of actual solid tumors.
With respect to the time-spatial patterns of the simulated tumors, Figure \[Figure5\] shows the lattice configuration at some time during the steady state in the necrotic (Figure \[Figure5\]b) cases. Using the g2 package, it was found that, although tumor growth leads to the compact shape that is characteristic of solid tumors, the growth process is such that its boundary is irregular for any time $t$, as, for example, for the time steps represented in Figure \[Figure5\]. A similar type of behavior is observed for the time-spatial patterns in the nonnecrotic tumor. In relation to the process of necrosis, the necrotic region was found to be inside the tumor at any time $t$ [@Kansal] [@Bellomo], as shown in Figure \[Figure5\].
TREATMENT CASE {#subsec32}
--------------
Figures \[Figure6\]a and \[Figure6\]b show the simulated time series of the number of cancer cells and normal cells for the different values of $p_{drugc} > p_0$ and $p_{drugc} < p_0$ corresponding to successful treatment (cure) and unsuccessful treatment (non-cure), respectively. In each figure, two values of $p_{drugn}$ are considered. It can be clearly seen that, for both values of $p_{drugn}$, the success of the treatment does not change. It is also clear that the tumor size increases until tap (see figures \[Figure6\]a and \[Figure6\]b). However when $p_{drugc}
> p_0$, the reduction in tumor size starts, as expected, after $t_{ap}$ time steps.
When treatment is successful ($p_{drugc} > p_0$), it was found that, for an upper value of $p_{drugn}$ ($p_{drugn} = 10^{-2}$; light grey), more normal cells are eliminated compared with a lower value of $p_{drugn}$ - $p_{drugn} = 10^{-4}$ as shown in Figure \[Figure6\]a (dark grey). Therefore, we may conclude that very small values of $p_{drugn}$ correspond to optimal therapy. However, in the case of unsuccessful treatment ($p_{drugc} < p_0$), when $p_{drugn}$ is increased, both a decrease in the number of normal cells and a slower rate of increase of cancer cells is found (see fig \[Figure6\]b).
A systematic analysis of the parameter space presented in the next section will provide a more precise conclusion about the role of $p_{drugn}$ parameter.
![\[Figure6\]a) Successful treatment: the average of simulated time series of the number of normal cells (-) and cancer cells (circle) assuming $M_{samples}=200$ and the following parameters values: $L=251$, $p_0=0.8$, $f=0.2$, $p_{drugc}= 0.9$, $t_{ap}= 500$, and two values for $p_{drugn}$: $10^{-4}$ (dark grey), and $10^{-2}$ (light grey); b) Unsuccessful treatment: the time series of the number of normal cells (-) and cancer cells (circle) assuming $M_{samples}=200$ and the following parameters values: $L=251$, $p_0=0.8$, $f=0.2$, $p_{drugc}= 0.7$, $t_{ap}= 500$, and two values for $p_{drugn}$: $10^{-4}$ (dark grey), and $10^{-2}$ (light grey).](ReisFig6a.jpg "fig:"){width="6.0cm"} ![\[Figure6\]a) Successful treatment: the average of simulated time series of the number of normal cells (-) and cancer cells (circle) assuming $M_{samples}=200$ and the following parameters values: $L=251$, $p_0=0.8$, $f=0.2$, $p_{drugc}= 0.9$, $t_{ap}= 500$, and two values for $p_{drugn}$: $10^{-4}$ (dark grey), and $10^{-2}$ (light grey); b) Unsuccessful treatment: the time series of the number of normal cells (-) and cancer cells (circle) assuming $M_{samples}=200$ and the following parameters values: $L=251$, $p_0=0.8$, $f=0.2$, $p_{drugc}= 0.7$, $t_{ap}= 500$, and two values for $p_{drugn}$: $10^{-4}$ (dark grey), and $10^{-2}$ (light grey).](ReisFig6b.jpg "fig:"){width="6.0cm"}
In order to analyze the effect of therapy on cancer cells, the time-spatial distribution of the simulated tumors was followed using the g2 package. Figure \[Figure7\] shows the configuration of the lattice at different time steps from the beginning of therapy until the tumor is eliminated.
![\[Figure7\] The spatial distribution of states at consequent time steps (see the time step bar) using the same parameters values of Figure \[Figure6\]a except $L=81$. The final time step is $t_{final}=1400$. Dark colors (light grey, grey, dark grey, and black) correspond, respectively, to normal, cancer, necrotic cells, and empty sites.](C1.jpg "fig:"){width="40.00000%"} ![\[Figure7\] The spatial distribution of states at consequent time steps (see the time step bar) using the same parameters values of Figure \[Figure6\]a except $L=81$. The final time step is $t_{final}=1400$. Dark colors (light grey, grey, dark grey, and black) correspond, respectively, to normal, cancer, necrotic cells, and empty sites.](C2.jpg "fig:"){width="40.00000%"} ![\[Figure7\] The spatial distribution of states at consequent time steps (see the time step bar) using the same parameters values of Figure \[Figure6\]a except $L=81$. The final time step is $t_{final}=1400$. Dark colors (light grey, grey, dark grey, and black) correspond, respectively, to normal, cancer, necrotic cells, and empty sites.](C3.jpg "fig:"){width="40.00000%"} ![\[Figure7\] The spatial distribution of states at consequent time steps (see the time step bar) using the same parameters values of Figure \[Figure6\]a except $L=81$. The final time step is $t_{final}=1400$. Dark colors (light grey, grey, dark grey, and black) correspond, respectively, to normal, cancer, necrotic cells, and empty sites.](C4.jpg "fig:"){width="40.00000%"}
RESULTS: PARAMETER SPACE {#sec4}
========================
Analysis of the parameter space is important in order to confirm the robustness of the model and to identify the most relevant parameters for the dynamics of the model.
According to the relevance of some CA parameters to the features of the phenomenon, this analysis of parameter space was divided into two parts: the occurrence of necrosis (no treatment cases) and reaching a state of cure (treatment cases).
THE OCCURRENCE OF NECROSIS {#subsec41}
--------------------------
In the first part, the parameters are again established as: $p_{drugn} = p_{drugc} = t_{ap} = 0$ with the aim of evaluating the effect of necrosis, and the minimum value of f is obtained for each pair of values $(L; p_0)$. The values of $L$ are presumed to range from 101 to 251, increasing the interval by $\Delta L = 25$. With respect to $p_0$, the whole interval from 0.1 to 0.9 is taken into account, increasing $\Delta p_{mitot}(t) = 0.1$. Figure \[Figure8\] shows that a transition exists between the nonnecrotic (lower) and necrotic (upper) regions of parameter space.
It was found that:
$$\label{fmin} f_{min}=a(L) p_0 + b(L)$$
where the linear and the angular coefficients, $b(L)$ and $a(L)$ are different for distinct values of the lattice size.
![\[Figure8\] a) Parameter space in the case of no treatment: $f^{min} \times p_0 \times L$; the following parameters values are fixed: $p_{drugn}=0$, $p_{drugc}=0$; $t_{final}=5000$. b) the dependence of the angular coefficient of $f_{min}$, $a(L)$, with $L$.](ReisFig8a.jpg "fig:"){width="6.0cm"} ![\[Figure8\] a) Parameter space in the case of no treatment: $f^{min} \times p_0 \times L$; the following parameters values are fixed: $p_{drugn}=0$, $p_{drugc}=0$; $t_{final}=5000$. b) the dependence of the angular coefficient of $f_{min}$, $a(L)$, with $L$.](ReisFig8b.jpg "fig:"){width="6.0cm"}
The angular coefficient $a(L)= a_0 L^\gamma$ where $a_0=1.12$ and $\gamma=0.29$ (see figure \[Figure8\]b).
Since the lattice size is an intrinsic parameter of the model, that transition is such that the the maximal growth probability $p_0$ is the control parameter, while the necrotic parameter $f$ is the order parameter.
REACHING THE CURE STATE {#subsec42}
-----------------------
If the tumor is submitted to systemic treatment represented by a probability $p_{drugc} \ne 0$ starting at $t_{ap} \ne 0$ that affects normal cells with a probability $p_{drugn} < p_{drugc}$, our aim is to establish the minimum value of $p_{drugc}$ in order to achieve a state of cure, i.e., no cancer cells.
The analysis is now more complex than the one performed in subsection \[subsec41\] in the sense that there are 4 important parameters that control the behavior of the therapeutic effect: $p_{drugc}$, $p_{drugn}$, $t_{ap}$, and $p_{0}$. The relevance of $p_0$ is evident in subsection \[subsec32\]. This analysis is divided into two parts.
In the first part, motivated by the behavior observed in figure \[Figure6\]a, the parameter is defined as $p_{drugn}$, assuming very small values ($p_{drugn} = 10^{-4})$ to simulate optimal therapy. Lattice size is also established as $L = 251$ and the necrotic parameter $f = 0.6$. Analogously to the method applied in subsection \[subsec41\], the minimum value of $p_{drugc}$ was obtained for each pair of values $(p_0; t_{ap})$. The range of values for $p_0$ and tap are, respectively, $[0.1; 0.9]$ and $[0; t_{final}]$. Figure \[Figure9\] shows the relevant role of $p_0$ dividing the parameter subspace into two regions: the lower is the non-cure state while the upper corresponds to the state of cure. In this case $p_0$ is again the control parameter but $p_{drugc}$ is the order parameter.
![\[Figure9\] Parameter subspace of treatment case: $p_{drugc}^{min} \times p_0 \times t_{ap}$; the following parameters values are fixed: $L=251$, $f=0.6$, $p_{drugn}=10^{-4}$; $t_{final}=5000$.](ReisFig9.jpg){width="6.0cm"}
In the second part, a value of $p_0 = 0.8$ remains fixed, while the pair varies $(p_{drugn}; t_{ap})$, again in order to obtain the minimum value of $p_{drugc}$ that would be sufficient to effectively eliminate the tumor, now related to the effect of the drug on the normal cells. The whole interval of $p_{drugn}$ was taken into account from very small values $10^{-4}$ until $0.9$. It would be necessary to extend $t_{final}$ in order to maintain the duration of application, since in this analysis the parameter $t_{ap}$ varied. Since this behavior is similar for any value of $p_{drugn}$, in figure \[Figure10\] the minimum value of $p_{drugc}$ is shown to increase as a function of $t_{ap}$ in accordance with a Lorentzian function: $$p_{drugc}^{min} = A_0 + \frac{2A_1}{\pi}\,\frac{A_3}{4(t_{ap}-A_2)^2 + A_3^2}$$
This result means that the minimal rate of infusion of the drug to eliminate the tumor has to be greater if the treatment begins later. It emphasizes how important it is to initiate treatment as early as possible in order to reduce the infusion rate of the drug.
![\[Figure10\] Parameter subspace of treatment case: $p_{drugc}^{min} \times t_{ap}$; the following parameters values are fixed: $L=251$, $f=0.6$, $p_0=0.6$; $p_{drugn}=10^{-4}$; $\delta_t=t_{final}-t_{ap}=500$. The Lorentzian fitting (grey color) parameters are $A_0=0.60$, $A_1=186.61$, $A_2=4829.92$, $A_3=776.29$.](ReisFig10.jpg){width="6.0cm"}
Finally in the cases in which the tumor is eliminated ($p_{drugc} >
p_0$), for fixed values of $L$, $f$, $p_0$ and $p_{drugn}$, the cure time ($t_{cure}$) is estimated in relation to the initial time application $t_{ap}$, and $p_{drugc}$ (see Figure \[Figure11\]). Note that $t_{cure}$ is not a parameter but a consequence of the time evolution of the system. Figure ref[Figure11]{} shows that $t_{cure}$ increases linearly with $t_{ap}$ with an angular coefficient equal to 1. Figure \[Figure11\] also shows that, for a fixed $t_{ap}$, $t_{cure}$ decreases with $p_{drugc}$. This result also shows how important it is to start treatment as early as possible to reduce the amount of time required to reach the state of cure.
![\[Figure11\] Analysis of cure time in the treatment case: $t_{cure} \times p_{drugc}^{min} \times t_{ap}$; the following parameters values are fixed: $L=251$, $f=0.6$, $p_0=0.6$, $p_{drugn}=10^{-4}$, $t_{final}=500$.](ReisFig11.jpg){width="6.0cm"}
DISCUSSION AND CONCLUDING REMARKS {#sec5}
=================================
The model proposed in this study is capable of capturing the Gompertzian behavior of avascular tumor growth. The competition between normal and cancer cells and the dynamic character of the mitotic probability are the relevant components of the success of this model.
The number of cancer cells simulates tumors [*in vitro*]{} due to the two-dimensional character of the model, and their potential growth simulates a tumor in vivo due to its three-dimensional nature. Figures \[Figure2\] and \[Figure4\] show that the model is able to capture the dynamics of both [*in vivo*]{} and [*in vitro*]{} avascular tumor growth. The simulated values for the most important Gompertzian parameter $\beta$, which characterizes the Gompertzian shape, are compatible with the parameter values of some tumors [@Demicheli] [@Brunton].
The model is also able to capture necrotic and nonnecrotic tumors depending on the values of the parameter $f$. It is well-known that necrosis, unlike apoptosis, is a typical phenomenon found in a group of cells that is simulated in our model by the changing of the potential growth of the cells.
The time-spatial patterns reveal a tumor with a compact shape and irregular boundaries, as occurs in some solid tumors [@Patel], [@Ferreira]. Evidence was also found to confirm the three stages of avascular tumor growth [@Adam]:
- a\) Stage I - when the tumor grows exponentially due to available resources (nutrients and oxygen) - see first stage in Figure \[Figure5\];
- b\) Stage II - when the stabilization of $\Delta p_{mitot}$ starts but there are still enough resources to ensure that necrosis does not occur - see second stage in Figure \[Figure5\]b;
- c\) Stage III - when necrosis may occur depending on the value of the parameter $f$ because the resources are insufficient to provide for tumor growth - see third stage in Figure \[Figure5\]c in the case of necrosis.
The next stage, the angiogenic phase of the tumor [@Folkman], which has not yet been dealt with in our model, corresponds to vascular growth. To be able to evaluate this stage, the process of angiogenesis would have to be taken into account [@Adam].
In the case of no treatment ($p_{drugc} = p_{drugn} = 0$), the minimum value of $f$ for necrosis to occur is governed by the equation (\[fmin\]) that was obtained from the investigation of the parameter space. This means that, for a simulated tissue of dimension $L$, the occurrence of necrosis depends linearly on the maximum value of mitotic probability of the specific tumor.
In relation to the case in which therapy was implemented, a continuous strategy of systemic therapy, i.e. chemotherapy, was selected. Although the schedule of chemotherapy is usually periodic, in this case it was decided to simulate this less realistic situation so that the effect of the parameters $p_0$ and $p_{drugc}$ could be compared in a simple fashion. It was thus found that, with respect to avascular tumor growth, when the values of the parameter $p_{drugn}$ are very small and when $p_{drugc}$ is slightly greater than $p_0$, the tumor is completely eliminated (see Figure \[Figure7\]). Since neovascularization has not yet been triggered, a state of cure may be expected to occur with a periodic schedule.
The time-spatial patterns of the cases in which therapy was implemented (see Figure \[Figure8\]) show that the drug acts from the borders of the simulated solid tumor inwards, as would be expected in the case of solid tumors.
With respect to the effect of parameters on the state of cure, for fixed values of lattice size $L$ and necrotic parameter $f$, and for a range of values of $t_{ap}$ and $p_{drugn}$, the minimum value of $p_{drugc}$ again coincides with $p_0$ (see Figure \[Figure9\]). This reinforces the importance of the higher value of $p_0$ in the response of the tumor to therapy. It retains the memory of cancer cells with respect to the onset of mitosis.
Finally, Figures \[Figure10\] and \[Figure11\] illustrate a very relevant finding from the phenomenological point of view: the importance of initiating therapy as early as possible in order to reach the state of cure. The cure time was found to be proportional to $t_{ap}$ that measures the instant when the infusion starts (see Figure \[Figure11\]) and the minimum value of the therapeutic infusion to eliminate the tumor increases nonlinearly as a function of the starting point of therapy.
Future studies should be carried out to generalize the model with the objective of including the angiogenic process and the periodic schedule of systemic therapy. However, the most important perspective of this line of investigation is to compare the model with the in vitro tumor growth of cells from specific tissue samples and to compare parameter values. It would then be possible to relate the Gompertzian fitting parameters with the parameters of the model.
[**Acknowledgements:**]{} The authors would like to thank Ramon El-Bachá for his very useful discussions on the process of tumor growth and Nelson Alves Jr. for his valuable collaboration at the beginning of this study and his help in manipulating the time spatial patterns. This work is partially supported by CNPq – Conselho Nacional de Desenvolvimento Científico e Tecnológico (Brazilian Agency).
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|
---
author:
- Name
title: WMW Power Paper
---
Katie R. Mollan, Ilana M. Trumble, Sarah A. Reifeis, Orlando Ferrer, Camden P. Bay, Pedro L. Baldoni, and Michael G. Hudgens
**Department of Biostatistics and Center for AIDS Research, The University of North Carolina, Chapel Hill, NC**
Corresponding Author’s Footnote:
Katie R. Mollan is a Senior Biostatistician at The University of North Carolina at Chapel Hill, 3126 McGavran-Greenberg Hall, CB \#7420, Chapel Hill, NC 27599 (email: kmollan@unc.edu)
ABSTRACT {#abstract .unnumbered}
========
Accurate power calculations are essential in small studies containing expensive experimental units or high-stakes exposures. Herein, exact power of the Wilcoxon Mann-Whitney rank-sum test is formulated using a Monte Carlo approach and defining $P(X<Y)\equiv p$ of effect size, where $X$ and $Y$ denote random observations from two distributions hypothesized to be equal under the null. This approach is feasible even without background data. Simulations were conducted comparing the exact power approach to existing approaches by Rosner & Glynn (2009), Shieh et al. (2006), Noether (1987), and O’Brien-Castelloe (2006). Approximations by Noether and O’Brien-Castelloe are for small sample sizes. The Rosner & Glynn and Shieh et al. approaches performed well in many small sample scenarios, though both are restricted to location-shift alternatives and neither approach is theoretically justified for small samples. The exact method is recommended and available in the R package `wmwpow`.
KEYWORDS: Mann-Whitney test, Monte Carlo simulation, non-parametric, power analysis, Wilcoxon rank-sum test
1. Introduction {#introduction .unnumbered}
===============
Despite the current era of big data, there remains a practical need for power calculations of small preclinical, first-in-human, and basic science studies involving two independent samples. Accurate power calculations are critical when each experimental unit is expensive (e.g., macaques for preclinical HIV vaccine experiments) or the study is high stakes (e.g., novel HIV cure strategies where toxicity risks are unknown). Anti-conservative power approximations can result in an underpowered study and conservative approximations can lead to using more experimental units than necessary. Owing to small sample sizes, determining power in this setting is challenging because asymptotic approximations may not be reliable. An additional challenge common in many studies, such as preclinical or first-in-human trials, is the absence of relevant background data to inform power calculations.
In small studies with a continuous outcome (e.g., Kulkarni et al. 2011; Archin et al. 2014; Denton et al. 2014), the Wilcoxon Mann-Whitney (WMW) rank-sum test is often utilized to test for differences between groups (Wilcoxon 1945; Mann and Whitney 1947). Thus it is of interest to compute power of the WMW test against different alternatives. Previous work on calculating power of the WMW test for a continuous outcome includes Haynam & Govindarajulu (1966), Noether (1987), Collings & Hamilton (1988), Lehmann (1998), Shieh et al. (2006), Zhao et al. (2008), Rosner & Glynn (2009), and Divine et al. (2010). Power of the WMW test for ordered categorical outcomes was considered previously by Hilton & Mehta (1993), Kolassa (1995), and Tang (2011, 2016).
In this paper, an exact approach for determining the power of the WMW test is formulated using Monte Carlo simulation. The approach is exact in that no asymptotic approximation is employed, and the amount of Monte Carlo error can be controlled by the user. In addition to being exact, an appealing aspect of this approach is that it can be implemented with or without background data. Effect size is defined by $p=P(X<Y)$, where $X$ and $Y$ denote random observations from the two distributions being compared. Equivalently, the effect size can be expressed by the odds $p/(1-p)$ (O’Brien & Castelloe 2006; Divine et al. 2013, 2017). Under a location-shift alternative, the WMW test null hypothesis is $p=0.5$, analogous to a fair coin toss. This effect size can be easily understood by collaborative investigators. Moreover, when background data are lacking, it can be more productive to discuss plausible values for $p$ with collaborators than to elicit parameterizations for each distribution or to quantify effect size using standard deviation units. Further, as shown here and by Rosner & Glynn (2009), in many design scenarios the underlying distributions have minimal impact on power for a fixed effect size $p$.
The outline of the remainder of this paper is as follows. Section 2 presents several approaches to calculating power of the WMW test (with details in the Appendices). Section 3 presents simulation results comparing WMW test power calculations. Section 4 provides a motivating example, and Section 5 concludes with a discussion.
2. Methods {#methods .unnumbered}
==========
Suppose $X_1,..., X_m$ and $Y_1,..., Y_n$ are independent identically distributed (iid) random variables with continuous cumulative distribution functions $F$ and $G$, respectively. It is of interest to test the null hypothesis $H_0: F=G$ versus the two-sided alternative hypothesis $H_A: F\neq G$. The WMW test statistic is $W= \sum_{i=1}^{m} \sum_{j=1}^{n} \varphi(Y_j - X_i)$ where $\varphi(Y_j - X_i)=1$ when $Y_j > X_i$, and 0 otherwise; i.e., the WMW statistic counts the number of times a $Y_j$ is larger than a $X_i$. Under $H_0$, the WMW statistic has mean $\mu_0=mn/2$ and variance $\sigma^2_0=mn(N+1)/12$ where $N=n+m$; as $m$ and $n$ tend to infinity, $(W-\mu_0)/\sigma_0$ has a limiting standard normal distribution under $H_0$ (Mann & Whitney 1947).
Shieh et al. (2006) derived a large-sample approximation for power of the WMW test using the exact variance of $W$ under the alternative hypothesis $H_A$, and demonstrated that their approach was more accurate than the Noether (1987) and Lehmann (1998) approximations. Effect size in Shieh et al. was defined in terms of $G(x) = F(x-\theta)$, where $\theta$ is the location shift in the cumulative distribution function (CDF) and $H_0:\theta=0$. The Shieh et al. method is reformulated here using effect size $p$ (Appendix A) to facilitate interpretation and comparison to other approaches to estimating power of the WMW test. For large $m$ and $n$, power for the two-sided WMW test against a specific alternative hypothesis can be approximated by: $$\label{eq:pow}
P\bigg\{\bigg|\frac{W-\mu_0}{\sigma_0} \bigg| > z_{\alpha/2} \biggm| H_A\bigg\} \approx \Phi\Big(\frac{\mu - \mu_{0}-z_{\alpha/2}\sigma_{0}}{\sigma}\Big) + \Phi\Big(\frac{\mu_0 -\mu - z_{\alpha/2}\sigma_{0} }{\sigma}\Big) $$ where $\alpha$ is the significance level, $\Phi(\cdot)$ is the CDF of a standard normal distribution, $z_{\alpha/2} = \Phi^{-1}(1-\alpha/2)$, and $\mu$ and $\sigma$ are the mean and standard deviation of the WMW statistic under $H_A$, respectively. The mean under $H_A$, $\mu =mn/p$, depends upon effect size $p$, and the variance under $H_A$ can be expressed as: $$\label{eq:ssig}
\sigma^2=mn\{p(1-p) + (n-1)(p_{2}-p^2) + (m-1)(p_{3}-p^2)\}$$ where $\sigma^2$ depends upon effect size $p$ and underlying distributions $F$ and $G$ through $p_{2}$ and $p_{3}$ (Lehmann 1998; Shieh et al. 2006).
Noether (1987) provides an approximation to the power of the WMW test which also relies on the normal approximation in Equation \[eq:pow\], but does not require selecting parametric models for $F$ and $G$. Instead, two additional assumptions are supposed: (i) $\sigma^2 = \sigma^2_0$, i.e., the variance of $W$ under $H_A$ is equal to the variance under $H_0$; and (ii) $N/(N+1) \approx 1$ (Appendix B). Assumptions (i) and (ii) may be dubious for small sample sizes. Clearly the approximation $N/(N+1) \approx 1$ only holds for large $N$. In addition, a study with small $m$ and $n$ will have adequate power only for large effect sizes, in which case $\sigma$ will not, in general, equal $\sigma_0$ (Shieh et al. 2006).
Rosner and Glynn (2009) also provide a method for estimating the power of the WMW test which relies on the normal approximation in Equation \[eq:pow\] but does not require selecting parametric models for $F$ and $G$. Rosner and Glynn derive a closed-form estimate of power for location-shift alternatives defined after first applying a probit transformation to $F$ and $G$.
With modern computing, empirical (Monte Carlo) power calculation for the WMW test is feasible and accurate, particularly for small studies. As described below, empirical power computation entails repeated sampling from $F$ and $G$. Options for selecting $F$ and $G$ include: (i) specifying parametric distributions for both $F$ and $G$; (ii) specifying a parametric distribution for $F$ and choosing a value for $p$, which in turn imply a distribution for $G$; or (iii) resampling from a sufficient amount of background data (Collings & Hamilton 1988; Hamilton & Collings 1991). For studies where background data are unavailable or sparse, the resampling approach (iii) is not feasible. While approach (i) is feasible for small studies, it can be harder to interpret (e.g., presenting a mean difference in standard deviation units) compared to (ii) where one selects $p$ or odds. Options (i) and (ii) are available in the R package described below.
The empirical method can provide power estimates that are effectively exact in practice. The general approach entails simulating multiple datasets from $F$ and $G$, and computing the proportion of simulated datasets where the WMW test rejects the null. As the number of simulated datasets approaches $\infty$, empirical power converges in probability to the exact power of the WMW test. For a finite number of simulated data sets, the Monte Carlo error can be quantified, such that the number of simulations may be chosen to ensure this error is within an acceptable tolerance. In particular, let $Q$ be the number of rejections of $H_0$ among $S$ simulated datasets and let $p_q$ be the probability of rejecting $H_0$ with $Q \sim Binomial(S,p_q)$. For simulations under $H_0$, $p_q$ equals the type I error rate, and for simulations under a particular alternative hypothesis $H_A$, $p_q$ equals power. The power (or type I error) is estimated empirically by $\hat p_q = Q/S$. By the central limit theorem, for large $S$, $\hat p_q$ will be approximately normal with mean $p_q$ and the standard error of $\hat p_q$ will be no larger than $1/\sqrt[]{4S}$, which is $\approx 0.0016$ for $S=100,000$. This implies that $S=100,000$ simulated datasets will provide a precise power estimate to two decimal places. E.g., suppose $S=100,000$ and $Q=80,000$; then $\hat p_q=0.8$ and the corresponding 99% Wald confidence interval (CI) for $p_q$ rounded to two decimal places is (0.80, 0.80). With $S=10,000$ the standard error of $\hat p_q$ is no larger than $\approx 0.005$, and for $Q/S=8,000/10,000$ the 99% CI for $p_q$ is (0.79, 0.81).
The `wmwpow` R package provides three functions for estimating power: `wmwpowp`, `wmwpowd`, and `shiehpow`. For all three functions, the user inputs the sample sizes ($m,n$) and the significance level ($\alpha$). The function `wmwpowp` also takes inputs of the distribution for $F$ and the effect size $p$, and returns empirical power. For example, suppose the user inputs an exponential distribution with rate parameter $\mu$ for $F$ and a particular value for $p$; then `wmwpowp` solves for $G$. Available choices in `wmwpowp` for $F$ are the exponential, normal, and double exponential (Laplace) distributions, corresponding to the derivations in Appendix C. In each case, $F$ and $G$ are assumed to be in the same family or class of distributions; e.g., if $F$ is specified to be normal with mean $\mu_x$ and variance $\sigma_x^2$, then $G$ is assumed to be normal as well. If $F$ is exponential with rate $\mu$ and $p$ is fixed, then $G$ is completely specified. On the other hand, if $F$ is normal or double exponential and $p$ is fixed, then $G$ is not completely specified without additional assumptions. Therefore, for the normal and double exponential distributions, the function `wmwpowp` also takes as an input the scalar $k$ which specifies the ratio of standard deviations for $F$ and $G$. For $k=1$, choosing $p \neq 0.5$ corresponds to a location-shift alternative. Choosing $k \neq 1$ allows for unequal variances and thus a wider class of alternative hypotheses.
If specifying parametric distributions for both $F$ and $G$ is preferred, the function `wmwpowd` can be used to compute empirical power. `wmwpowd` allows the user to select from many standard continuous parametric distributions, including beta, exponential, normal, and Weibull. The function `wmwpowd` outputs the empirical power as well as the effect size $p$ and the equivalent odds corresponding to the $F$ and $G$ specified by the user.
The `wmwpow` package also includes the function `shiehpow`, which implements the Shieh et al. method for location-shift alternatives assuming normal, shifted exponential, or double exponential distributions. The function `shiehpow` uses a shifted exponential distribution, whereas the exponential distribution in `wmwpowp` uses one rate parameter that defines both shape and location such that a common support \[$0, \infty$) is maintained for $F$ and $G$.
3. Empirical Comparisons {#empirical-comparisons .unnumbered}
========================
The performance of methods by Noether (1987), O’Brien-Castelloe (2006), Shieh et al. (2006), and Rosner & Glynn (2009) were compared to empirical power results. Each method was formulated such that $\alpha$, $m$, $n$, and $p$ were the inputs, as well as an assumed probability distribution, when required. Power was estimated for effect size $p$ ranging from 0.50 to 0.95 by 0.05 (odds ranging from 1 to 19).
The approach of Shieh et al. was implemented using the R package `wmwpow`, function `shiehpow` with the formulae shown in Appendix A. The Noether approach (Appendix B) was also implemented in R. The O’Brien-Castelloe approach was applied using the SAS Power procedure (*twosamplewilcoxon*, SAS/STAT v14.2); default settings were used and distributional assumptions were $X \sim N(0,1)$ and $Y\sim N(\mu_y,1)$, solving for $\mu_y$ by inputting values of $p$ into the equation shown in Appendix C.2. Rosner & Glynn (2009) provided a SAS macro (*%WilcxPowerContinuousNties*) for their approach. Empirical power was computed as the proportion of rejections of $H_0$ under a specific alternative hypothesis over $S$ simulated datasets; $S=100,000$ simulated datasets were used for $n,m<20$, and $S=10,000$ simulated datasets for $n,m\geq20$. Computations were conducted in R version 3.4.3 and SAS version 9.4 (Cary, NC).
Comparisons between the empirical power calculations and results from Shieh et al., Rosner-Glynn, Noether, and the O’Brien-Castelloe methods are shown in Table \[tab:tab1\] and Figure \[fig:f1a\]-\[fig:f50\]. For $m=n=6$ per group, the Shieh et al. and Rosner-Glynn methods provided very similar results (Figure \[fig:f1a\]). For a given $p$, varying the distributions for $F$ and $G$ had negligible effect on the power. The O’Brien-Castelloe approximation was typically anti-conservative for small $m$ and $n$ (e.g., $m=n=6$). The Noether approximation was both anti-conservative or over-conservative depending upon effect size $p$ and sample sizes (Figures \[fig:f1a\] and \[fig:f1b\]). As $m$ and $n$ increase, power results from the methods evaluated here became increasingly similar, as expected. For $m,n \geq 50$, all of the methods yielded similar results (Figure \[fig:f50\]).
Generally, the Shieh et al. and Rosner-Glynn approaches tended to well approximate exact (empirical) power. However, for small unequal sample sizes (e.g., $m=6, n=12$), the Shieh et al. and Rosner-Glynn power estimates can differ, as demonstrated in the bottom of Table \[tab:tab1\]. Note the Rosner-Glynn approach gives the same power estimate when $m=6,n=12$ and $m=12,n=6$ for a fixed effect size $p$. In contrast, Shieh et al. power estimates need not be the same when the values of $m$ and $n$ are interchanged as can be seen from Equation \[eq:ssig\] and Appendix A ($p_{2}$ and $p_{3}$ are unequal for non-symmetric distributions).
Empirical power for alternative hypotheses where $F$ and $G$ are normal with unequal variances is shown in Figure \[fig:fknorm\]. For $m=n=6$, power decreases as the degree of variance heterogeneity increases (i.e., as $k$ increases). Varying $k$ had less impact for $m=n=15$. Note that if $k \neq 1$, then the null hypothesis $H_0: F = G$ does not hold even if $p=0.5$. Hence, in Figure \[fig:fknorm15\] the empirical power is above $\alpha=0.05$ for $p=0.5$ and $k=3,4$.
------------ --------------------- ------- ------- -------- ------- -------- --------
p=0.5 p=0.7 p=0.75 p=0.8 p=0.85 p=0.9
n=6, m=6 Noether 3 22 32 44 56 67
O’Brien-Castelloe n/a 26 39 54 69 82
Rosner-Glynn 5 19 27 38 53 74
Empirical - Normal 4 18 28 40 56 75
Empirical - Exp 4 18 28 40 56 74
Empirical - Laplace 4 18 28 39 55 72
Shieh - Normal 5 18 27 38 53 74
Shieh - Shifted Exp 5 19 28 39 53 72
Shieh - Laplace 5 19 27 38 53 72
n=15, m=15 Noether 3 48 66 81 91 97
O’Brien-Castelloe n/a 52 72 87 96 99
Rosner-Glynn 5 46 67 87 98 >99
Empirical - Normal 5 47 67 85 96 >99
Empirical - Exp 5 46 68 86 96 >99
Empirical - Laplace 5 46 68 85 95 99
Shieh - Normal 5 46 67 86 98 >99
Shieh - Shifted Exp 5 46 67 85 97 >99
Shieh - Laplace 5 46 67 86 97 >99
n=6, m=12 Rosner-Glynn 5 25 37 53 73 92
Empirical - Exp 4 24 37 54 73 90
Shieh - Shifted Exp 5 23 36 54 74 93
n=12, m=6 Rosner-Glynn 5 25 37 53 73 92
Empirical - Exp 4 26 39 55 72 86
Shieh - Shifted Exp 5 27 39 53 69 86
------------ --------------------- ------- ------- -------- ------- -------- --------
4. Motivating Example {#motivating-example .unnumbered}
=====================
Consider a proposed study of $m=n=15$ per group with the sample size limited by ethical (e.g., safety), recruitment, or budgetary constraints. Given the limited feasible sample size, an accurate assessment of power is crucial for deciding whether the study should proceed as planned. In this study, background data on the outcome are dearth, and yet power calculations are still needed if null hypothesis significance testing is planned. In some cases, the study team should change the study design to focus on estimation and collection of pilot data without hypothesis testing. Here we proceed assuming that group comparisons are essential to the study objectives.
For example, in early phase clinical trials evaluating potential cures for HIV, sample sizes are typically limited to mitigate potential risks to participants. An outcome of interest, HIV replication index, is a relatively new measure used in HIV cure research with limited background data.
Suppose the investigators choose a 0.05 significance level and decide $p=0.8$ or larger is a meaningful effect size, i.e., an 80% or larger true probability that the HIV replication index for any given individual in the placebo group is higher than for any given individual in the treatment group. Assuming $p=0.8$ (or equivalently, a true odds of 4 or larger) and using an empirical power approach, thirty individuals ($m=n=15$ per group) will provide 85% power to detect a difference between two independent groups (placebo versus treatment). Rosner-Glynn, and O’Brien-Castelloe power estimates were both 87% and the Shieh et al. estimate was 86%, whereas the Noether approximation was conservative in this example (81% power). Empirical power for the exact 2-sided WMW test was conducted assuming a normal distribution for $\log_{10}$ replication index and 100,000 datasets of size $m=n=15$ were generated.
5. Discussion {#discussion .unnumbered}
=============
Empirical power calculation is accurate and feasible for many power scenarios including small sample settings, unequal variance, and unequal group sample sizes. The power approximations of Noether and O’Brien-Castelloe are not reliably accurate for small sample sizes. The Rosner & Glynn and Shieh et al. approaches performed well in many small sample scenarios, though both are restricted to location-shift alternatives and neither approach is theoretically justified to provide accurate power estimates for small samples. In contrast, the empirical power approach can evaluate a wider class of alternative hypotheses and is valid for any sample size.
In some settings, it may be anticipated that ties will occur in the observed data. Ties can arise when the underlying variable is continuous, but the variable is measured or recorded with limited granularity such that two or more individuals may have the same recorded value. Estimating power of the WMW test when ties may or may not be present was not considered here; it is often not practical to ascertain the *a priori* probability of a tie occurring. Zhao et al. (2008) generalized the Noether (1987) method to handle ties, making the assumption that the variance of the test statistic $W$ under the alternative $H_A$ is the same as under $H_0$; this assumption may be dubious when group sample sizes are small. If adequate background information is available regarding ties, one can simulate data accordingly (e.g., resample from the background data), and proceed with empirical power calculation. Ordered categorical data can be thought of as an extreme case of ties, and can be simulated directly using category probabilities (e.g., the tabled distribution within the SAS function `RAND`). WMW test power calculation for ordered categorical data is also available in StatXact software (Hilton & Mehta 1993).
Exact power calculation via Monte Carlo simulation is recommended whenever computationally feasible. Empirical power calculation for the rank-sum test is available in the commercial software PASS by inputting parametric distributions for $F$ and $G$. However, PASS version 16 does not yet provide $p$ or odds as an input or output value. The R package `wmwpow` can be used to compute empirical power with either $p$ or odds as an input (or alternatively $F$ and $G$), and is free and publicly available on CRAN.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Genevieve Clutton, Kristina De Paris, and J. Victor Garcia-Martinez and the UNC HIV research community for requesting power calculations that motivated this work. We also thank the Editor, Associate Editor, two reviewers, and Nader Gemayel for their helpful comments and suggestions, .
Funding {#funding .unnumbered}
=======
This research was supported by the University of North Carolina at Chapel Hill Center for AIDS Research (CFAR), an NIH funded program P30 AI50410.
Appendices {#appendices .unnumbered}
==========
Appendix A {#appendix-a .unnumbered}
==========
As shown in Lehmann (1998) and Shieh et al. (2006), the variance $\sigma^2$ of the WMW statistic under $H_A$ depends on $F$ and $G$; this dependence can be formulated using $p_{2}$ and $p_{3}$ for a location-shift alternative, with $p_{2}=p_{3}$ when distributions $F$ and $G$ are symmetric. When the underlying distributions of $F$ and $G$ are exponential , $\theta = -\ln[2(1-p)]$ for $p$ in (0.5,1), $p_{2}=1-2/3e^{-\theta}$, and $p_{3}= 1-e^{-\theta} + 1/3e^{-2\theta}$. When the distributions of $F$ and $G$ are double exponential (Laplace), $\theta = -L(4(p - 1)/e^{2})$ where $L$ is the Lambert-W function used to solve for $x$ when $y=xe^{x}$ and $p_{2}=p_{3} = 1-(7/12 + \theta/2)e^{-\theta} -1/12e^{-2\theta}$. Lastly, for the normal case, $F \sim N(0,1)$, $\theta = \sqrt[]{2}\Phi^{-1}(p)$ and $p_{2}=p_{3}= E[\{\Phi(Z+\theta)\}^2], \textrm{where} \space \ Z \sim N(0,1)$.
Appendix B {#appendix-b .unnumbered}
==========
Noether (1987) provided an approximation to the power of the WMW test assuming $\sigma = \sigma_0$, and $N/(N+1) \approx 1$, where $N=m+n$. Consider a one-sided WMW test, in which case the power equals: $$1-\beta = P\bigg(Z > \frac{\mu_0 - \mu}{\sigma} + \frac{z_\alpha \sigma_0}{\sigma}\bigg).$$
Let $c=m/N$ and $z_{\beta} = \Phi^{-1}(1-\beta)$. Then under the assumption $\sigma = \sigma_0$, it follows that $$\Big(\frac{\mu_0 - \mu}{\sigma_0}\Big)^2=\frac{12N^2 c(1-c)(p- 0.5)^2}{N+1} = (z_\alpha + z_\beta)^2$$ or equivalently $$\frac{N^2}{N+1} = \frac{(z_\alpha + z_\beta)^2}{12c(1-c)(p-0.5)^2}.$$ Assuming $N/(N+1) \approx 1$, it follows that $$N\approx \frac{(z_\alpha + z_\beta)^2}{12c(1-c)(p-0.5)^2}$$ and therefore power of the WMW test is approximated by $$1-\beta \approx \Phi\Big[\sqrt{12Nc(1-c)(p-0.5)^2} - z_{\alpha} \Big].$$
Appendix C {#appendix-c .unnumbered}
==========
Consider the general form for $p = P(X<Y) = \int_{-\infty}^{\infty} \int_{-\infty}^{y} f_X(x) g_Y(y)dxdy=\int_{-\infty}^{\infty} g_Y(y) F_X(y)dy$, where $f_X(x)$ and $g_Y(y)$ are probability density functions for $X$ and $Y$, respectively. The following three distributions are implemented in the R package `wmwpow`, function `wmwpowp`.
C.1: Exponential {#c.1-exponential .unnumbered}
----------------
Let $X\sim Exp(\mu)$ and $Y\sim Exp(\lambda)$, where $\mu$ and $\lambda$ are exponential rate parameters. Then $p = P(X<Y) = \int_{0}^{\infty} \int_{x}^{\infty} \mu \lambda e^{-\mu x} e^{-\lambda y} dydx = \mu/(\lambda+\mu)$, and therefore $\lambda = \mu(1-p)/p$.
C.2: Normal {#c.2-normal .unnumbered}
-----------
Let $X\sim N(\mu_x, \sigma^2_x)$ and $Y\sim N(\mu_y, \sigma^2_y)$ such that $X-Y \sim N(\mu_x - \mu_y, \sigma^2_x+\sigma^2_y)$. This implies $p = P(X-Y<0) = \Phi\Big(\frac{\mu_y - \mu_x}{\sqrt[]{\sigma^2_x+\sigma^2_y}}\Big)$, and therefore $\mu_y = \mu_x + \Phi^{-1}(p)\sqrt[]{\sigma^2_x+\sigma^2_y}$.
C.3: Double Exponential {#c.3-double-exponential .unnumbered}
-----------------------
Let $X\sim Laplace(\mu_x, \sigma_x)$ and $Y\sim Laplace(\mu_y, \sigma_y)$. Then $\mu_y$ can be found as follows. Recall the cumulative distribution function of a Laplace random variable is$$F_X(x) =
\begin{cases}
\frac{1}{2} e^{\frac{x-\mu_x}{\sigma_x}} & \text{if $x \leq \mu_x$} \\
1-\frac{1}{2} e^{-\frac{x-\mu_x}{\sigma_x}} & \text{if $x > \mu_x$.} \\
\end{cases}$$ This implies $p=P(X<Y)$ $$= \int_{-\infty}^{\mu_x} \bigg(\frac{1}{2} e^{\frac{y-\mu_x}{\sigma_x}}\bigg) \bigg(\frac{1}{2\sigma_y} e^{-\frac{|y-\mu_y|}{\sigma_y}}\bigg)dy + \int_{\mu_x}^{\infty} \bigg(1-\frac{1}{2} e^{-\frac{y-\mu_x}{\sigma_x}}\bigg) \bigg(\frac{1}{2\sigma_y} e^{-\frac{|y-\mu_y|}{\sigma_y}}\bigg)dy.$$
Thus $\mu_y$ can be found by solving numerically $$\Bigg[\int_{-\infty}^{\mu_x} \bigg(\frac{1}{2} e^{\frac{y-\mu_x}{\sigma_x}}\bigg) \bigg(\frac{1}{2\sigma_y} e^{-\frac{|y-\mu_y|}{\sigma_y}}\bigg)dy + \int_{\mu_x}^{\infty} \bigg(1-\frac{1}{2} e^{-\frac{y-\mu_x}{\sigma_x}}\bigg) \bigg(\frac{1}{2\sigma_y} e^{-\frac{|y-\mu_y|}{\sigma_y}}\bigg)dy\Bigg] - p = 0$$
using any standard one-dimensional root finding method given $p$, $\mu_x$, $\sigma_x$, and $\sigma_y$.
References {#references .unnumbered}
==========
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|
---
abstract: 'We discuss properties of random fractals by means of a set of numbers that characterize their universal properties. This set is the generalized singularity spectrum that consists of the usual spectrum of multifractal dimensions and the associated complex analogs. Furthermore, non-universal properties are recovered from the study of a series of functions which are generalizations of the so-called energy integral.'
address:
- 'The Isaac Newton Institute of Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 0EH, United Kingdom'
- 'The James Franck Institute, University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA'
- 'Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom'
author:
- 'Francisco J. Solis'
- Louis Tao
title: Lacunarity of Random Fractals
---
and
Fractals, stochastic processes
A number of the properties of fractals are associated with their Hausdorff, box-counting, or fractal dimensions. But further information of a universal character is also encoded in secondary (singular) dimensions, some of which may be complex. For example, a recurrent theme in the study of fractals is that of asymptotic or logarithmic periodicity [@Badii; @Smith]. Most fractal objects and mathematical constructions thereof are not exactly scale invariant. Rather, they obey simple recurrence relations that relate an infinite but discrete set of scales. Recent physical examples include the appearance of complex exponents in diffusion-limited aggregation [@Saleur], crack propagation in two dimensions [@Blumenfeld], and Boolean delay equations in the modelling of climate dynamics [@Mullhaupt].
For instance, instead of the full scale invariance of a function of local variables, F(x)=\^[D]{}F(x/), \[eq:scaling\] we have the logarithmic analog of Bloch’s theorem, F(x)=\^[D]{}F(x/) G(), \[eq:bloch\] where $G$ may be a periodic function.
This is a rather well-known relation but we feel that it has been under exploited. The period of $G$ is independent of the scaling dimension, $D$, and gives further information of the properties of the fractal object.
The ways in which the fractal dimensions of the object manifest itself are manifold. Consider, in particular, a set of analytical quantities calculated from the real space distribution of a fractal object, namely, the set of correlation integrals: C\_q(r) = d(x) (d(y) (r - |x - y|))\^[q-1]{}, \[eq:correlation\] where $q \ge 2$, $\theta$ is the Heaviside step function, and $d\mu$ is the measure of the object in question. The scaling properties of these correlation integrals with respect to the distance $r$ define a countable set of dimensions forming part of the multifractal dimension spectrum [@GP; @Halsey; @PV].
The study of the scaling properties of such correlations is facilitated by considering the corresponding energy integrals: I\_q() = r\^[-]{} d C\_q(r), < d, \[eq:energy\] where $\tau$ is restricted to be less than the spatial dimension, $d$. Note that $I_q(\tau)$ is related to the Mellin transform of $C_q(r)$ [@Mellin].
At this point the spatial information of the fractal has been encoded in these energy integrals, which are typically, but not always, meromorphic. For particular cases of deterministic Cantor sets, it is shown in [@Bessis; @Fournier] that the complex structure of these functions (\[eq:energy\]) reveal singularities that correspond to the relevant scaling dimensions of the theory. One usually one keeps only the most relevant, , the one with the smallest real part, but the rest of the spectrum is important in studying finite-size effects.
More precisely, the most relevant singularity is a pole on the real axis and has a numerical value that is a lower bound to the Hausdorff dimension. This was first proved by Frostman for $q = 2$ [@Frostman] (see also Falconer [@FalconerBk]). In some cases, the Hausdorff dimension also corresponds to the box-counting dimension [@FalconerBk; @Mainieri].
Typically, the rest of the singularities are also poles, and appear as pairs of complex conjugates with real parts not smaller than the Hausdorff dimension. The imaginary parts of these poles correspond obviously to the logarithmic wavelength of the fractal, while the residues appear as the amplitudes of oscillations observed in the asymptotic scaling of various correlation integrals (\[eq:correlation\]). This program of the analysis of a fractal object has been carried out, albeit in a somewhat scattered way, for the middle-third Cantor set and some of its deterministic generalizations [@Bessis; @Fournier; @Orlandini]. This complex singularity spectrum has been called the [*multilacunarity*]{} spectrum [@Fournier]. It is the goal of our work to show that it is well-defined for classes of random fractals, and we explicitly compute the lacunarity of a particular example.
It turns out that for some simple but important examples the Hausdorff dimension is easy to calculate with the use of a little ingenuity. Just as simply, the satellite dimensions can be calculated in the same way without resorting to the explicit computation of the correlation integrals (\[eq:correlation\]) or the energy integrals (\[eq:energy\]).
For the middle-third Cantor set, and for many other objects with simple recursive descriptions, we consider an equation that relates the relative scales, $\ell_i$, and the relative (normalized) measures, $p_i$, at successive levels of approximation, \_i = 1. \[eq:partition\]
In the case of the middle-third Cantor set, $\ell_{i=1,2} = 1/3$ and $p_{i=1,2} = 1/2$, and (\[eq:partition\]) becomes $2^{q-1}/{3^\tau} = 1$. The unique real solution, $\tau(q) = (q-1) \ln 2 / \ln 3$, gives the Hausdorff dimension. In general, $\tau=\tau(q)$ is not linear in $q-1$ and gives one the desired multifractal dimension spectrum via the relation $D_q = \tau(q)/(q-1)$. In this way, the universal properties of the fractal, namely, its generalized dimensions, are rather easily computed [@Halsey; @Feigenbaum].
However, as noticed in [@Fournier], a study of the middle-third Cantor set (and some deterministic generalizations), (\[eq:partition\]) also has complex roots. For instance, in the case of the middle-third Cantor set, (q) = (q-1) + i , j = 0, 1, 2, …\[eq:complex\] The imaginary parts of these complex roots correspond to the period of $G$ in (\[eq:bloch\]) and the logarithmic period observed in the correlation integrals [@Smith].
The reason for the surprising success of this approach, which reduces the sometimes formidable calculation of the energy integrals (\[eq:energy\]) to the computation of a partition function (\[eq:partition\]), is that one has implicitly utilized the recursive structure of the fractal distribution \[encoded as (\[eq:bloch\])\]. In doing so, one successfully captures the fact that the fractal has a very well defined set of real-space singularities, and equates the determination of the spectrum of the many-body problem to the solution of a relatively simple transcendental equation.
So far we have examined results for deterministic fractals, where the scale invariance of (\[eq:bloch\]) is exactly satisfied. Consider now the case in which the fractal is not exactly self-similar, but is only statistically self-similar, , scaling functions obey (\[eq:bloch\]) only on average. We shall make precise what this averaging procedure entails (for related issues of averaging of stochastic hierarchical processes, see [@Hentschel]).
For a process that generates a generalized Cantor set by replacing each segment at level $l$ by $m$ segments at level $l+1$, we let the length of the segments, $\ell_i (i=1, \ldots, m)$, be random variables with random probability measures $p_i (i=1, \ldots, m)$. For this and other random fractals, we define a new set of correlation integrals as the expectations of (\[eq:correlation\]), given a probability distribution of the $p_i$’s and the $\ell_i$’s: \_q(r) = E, where $E$ denotes expectation and $C_q(r)$ is the value of the correlation integral for a single realization of the random fractal. A new energy integral may be defined in precisely the same fashion: \_q() = E, < d.
It has been shown by Falconer [@FalconerDq] that the probabilistic version of equation (\[eq:partition\]) still gives the relevant dimension spectrum, i.e, one has to solve the expectation equation, E= 1, \[eq:expectation\] to obtain the multifractal dimension spectrum.
The conditions for the existence of a unique and meaningful real solution of (\[eq:expectation\]) have been studied [@FalconerDq; @Graf; @cuttreesums], and simple extensions of these considerations lead to the existence of well-defined complex solutions in complete analogy with the deterministic case.
Consider then the following example of a randomized Cantor set. At level $l$ of the recursive construction, we divide each segment into $n$ equal segments and pick $m$ of them at random with uniform probability. We assign to each of the $m$ smaller segments a measure $1/m$-th of the original segment. To simplify the presentation, we examine only the case for $q=2$ and do not consider the more general model of Falconer [@FalconerDq], which involves possibly non-uniform probability distributions $p_i$ and $\ell_i$. However, the computation may be generalized for higher-order correlation integrals and for non-uniform probability distributions satisfying the restrictions outlined in [@FalconerDq].
At level $l$ of this process, the energy integral is simply related to the energy integral of the previous level: \^[(l)]{}() = m n\^[-]{} \^[(l-1)]{}() + (m,n,), \[eq:recur\] where the superscripts denote the finite-level approximation of the energy integral. The function $\R$ is given by (m,n,) = . \[eq:residue\] Explicit derivation of $\R$ for this example is given as an appendix. Note that in (\[eq:recur\]), the prefactor $m n^{-\tau}$ is the expected value of the partition function for this model. So that the energy integral of the limiting distribution is simply () = . \[eq:2energy\]
All of the singularities of $\I(\tau)$ are given by the zeros of the denominator on the right-hand side of (\[eq:2energy\]), m n\^[-]{} = 1, \[eq:model\] as expected. It is easily checked that (\[eq:residue\]) is not singular at $\tau = 1$. The roots of (\[eq:model\]) are at \_j = + i , j = 0,1,2,…\[eq:poles\] Thus, we expect the correlation integral to exhibit oscillations of period $\ln n$. By using properties of inverse Mellin transforms [@Mellin], the correlation integral can be written as (r) = r\^[D]{} ( + 2 \_[j=1]{}\^ || (2j - \_j)), \[eq:c2\] where $D = \ln m/\ln n$ is the (second-order) correlation dimension. $\alpha_j$ and $\phi_j$ are real and are determined by the residues at $\tau_j$ (for $j$ non-negative): The residue at the $j$-th pole is of the form, $\alpha_j \exp(i\phi_j)$. We propose to call the modulus of $\alpha_j e^{i\phi}/\tau_j$ (and the higher-order analogs) the lacunary amplitudes. In Fig. 1, we compare the expected scaling of the correlation integral with an ensemble average of the scaling for the case $n = 3$ and $m = 2$. The average is performed over several (in this case, twelve) numerical realizations of the random fractal (approximated at level $l = 15$). We plot the residuals, $r^{-\ln m / \ln n} \C(r)$, versus $r$. The dashed line is the ensemble average, and the solid line exhibits the expected oscillations of (\[eq:c2\]).
While it may seem superfluous to obtain the correlation integral from numerical computations once it has been calculated analytically, this exercise was interesting since we have not performed the average over a large ensemble. Rather we took the spatial average of a few instances of a random process. That both results were essentially the same over a number of logarithmic periods is a rather natural self-averaging property of many fractal objects. The fluctuation of the ensemble average about the expectation may be described by higher-order correlation integrals and will be discussed in a forthcoming publication [@ST]. In the present case, the small size of the ensemble produces disagreement with predictions at $r$ of the order of the system size. Furthermore, numerical resolution affects the correlation for $r$ smaller than $e^{-13}$.
We note that the measurement of the lacunarity spectrum is accessible using a variety of correlations. For instance, we may identify these complex dimensions in the logarithmically periodic oscillation of the Fourier transform [@Fourier] and the diffraction spectrum [@Fourier; @diffraction]. However, while the same logarithmic periods are observed, the shape and phase of the oscillations vary. The periods arise from the additional discrete scale invariance of the underlying model and the numerical value of these periods are determined by the solutions of (\[eq:expectation\]). The different measures of correlation (Fourier or Mellin transforms) reveal differently-valued residues located at the roots of (\[eq:expectation\]).
Previous studies of the inverse fractal problem, , the extraction of the (possibly stochastic) hierarchical process from the observed multifractal dimension spectrum, revealed ambiguities in the standard procedure: Namely, many models can be made to fit a given multifractal dimension spectrum [@Feigenbaum; @Chhabra]. The approach described above provides the [*maximal*]{} characterization of the underlying multiplicative process without additional dynamical information. In this way, we may further distinguish between different hierarchical processes that give rise to fractals with similar dimension spectra (see also [@Hentschel]).
We have to stress that the lacunarity spectrum does not resolve all the inherent ambiguities. Many models, random or deterministic, can be made to fit a given lacunarity spectrum. However, using the lacunarity spectrum, we may distinguish between processes that have the same multifractal dimension spectrum. What the lacunarity spectrum reveals is the possible additional discrete scale invariance which are not furnished by the previous attempts to characterize fractal systems. Furthermore, non-universal information is recovered from studying correlations (\[eq:correlation\]) and energies (\[eq:energy\]).
How does this work bear on fractal sets generated by low-order (deterministic or stochastic) dynamical systems? Preliminary work [@ST] suggests that the complex solutions of the Lyapunov partition function (a dynamical analog of (\[eq:partition\]) and (\[eq:expectation\]), see [@Lyap]) describes the anomalous scaling observed in simulations (see, for instance, [@Badii]). In addition, the fact that the lacunarity spectrum can be calculated from a partition function (\[eq:partition\]) immediately implies that periodic orbit expansions [@periodic] can be used to calculate the spectrum.
As for fractals generated by systems governed by large numbers of degrees of freedom (as featured most prominently in phenomena modeled by diffusion limited aggregation and in inhomogeneities of highly turbulent flows), this generalized multifractal description complements the traditional views. Studies thus far have taken the position that the deviations from strict power-law scaling are anomalous and, hence, have focused on establishing possible causes for this apparent deviation (an example being inertial range intermittency [@PV; @intermittent] in turbulent fluids). In our approach, log-periodic deviations may be accommodated rather naturally (see also [@Smith; @Novikov]). Future efforts will be directed towards the description of turbulent intermittency using the analysis followed in this paper.
We gratefully acknowledge J. Fournier, R. Rosner, A. Sornborger, and E. Spiegel for helpful conversations. We thank R. Ball for bringing his work with R. Blumenfeld to our attention. We also wish to thank the anonymous referee for useful suggestions. This work was completed while F. J. S. was a Rosenbaum fellow at the Newton Institute. L. T. is supported by the U. K. Particle Physics and Astronomy Research Council.
In this Appendix we present the details leading to the explicit formulas of the energy integral as given by (\[eq:residue\]) and (\[eq:2energy\]).
We need to consider first the behavior of the measure upon averaging. Since we have assigned an equal probability to every possible case of segmentation, the expectation value for the density, $\rho(x)dx=d\mu(x)$, is uniform, i.e., $$E\left[\rho(x)\right]=1.$$
To be able to perform the required multiple integrals, we need to evaluate the expectation of products of densities. Sufficient information about the joint distribution of these densities is obtained by considering one step in the recursive construction. After one such step, the interval $L^{0}=[0,1]$ is divided into $n$ subintervals $L^{1}_{i}, i=1, 2,\ldots, n$ from which $m$ subintervals will be chosen randomly. The set of all points $(x,y)$ that belong to the subintervals $L^{1}_{i}$ and $L^{1}_{j}$, respectively, form a square of size $n^{-1}\times n^{-1}$ in the $x$-$y$ plane. Such a square will be labelled $(i,j)$, as shown in Figure 2. The shaded region $R$ corresponds to those cases in which $x$ and $y$ are in disjoint intervals, i.e., $i \ne j$.
[30 mm]{}
Consider now one of the shaded squares, say $(i,j)$. The probability that the intervals $L^{1}_{i}$ and $L^{1}_{j}$ are indeed chosen is simply ${n-2\choose m-2}/{n\choose m}$. In this case, each of the intervals will support a measure of total mass $1/m$. Furthermore, since segmentation for each of these intervals proceeds uncorrelated, we have E& = & E\_[i]{}E\_[j]{}\
& = & , (x,y)(i,j), where $E_{k}$ denote expectation conditioned to the event that segment $k$ is indeed chosen.
Next, we consider the diagonal squares. These squares will contribute to the energy integral with probability ${n-1\choose m-1}/
{n\choose m}=m/n$. The process of segmentation for a subinterval is identical to that of the original interval. Therefore, if $L^{1}_{i}$ is chosen, the average of the product of densities $\rho(x)\rho(y)$ restricted to this interval is identical to that of the original interval up to rescaling. The rescaling matches $x$ from the $L^{1}_{i}$ segment with the point $x'=nx-i-1$ in $L^{0}$. We have, for $x,y \in L^{1}_{i}$, E& = & (m/n) E\_[i]{}\
& = & (m/n)(1/m)\^[2]{} ENote also that $|x-y|=(1/n)|x'-y'|$. Thus the contribution to the energy integral from each of the diagonal squares is proportional to the overall expectation of the energy integral $$E\left[\int_{(i,i)}\frac{d\mu(x)d\mu(y)}{|x-y|^{\tau}}\right]=
(m/n)(1/m)^{2} n^{\tau}\, {\I}_2(\tau)$$
Summing over all squares we obtain relation (\[eq:recur\]) where ${\R}(m,n,\tau)$ can now be identified with the expectation value of the energy integral restricted to the shaded region.
To simplify the calculation of ${\R}(m,n,\tau)$ we note that $$E\left[ \int_{R}\frac{d\mu(x)d\mu(y)}{|x-y|^{\tau}}\right] =
\frac{m(m-1)}{n(n-1)}\left(\frac{n}{m}\right)^{2}
\int_{R}\, \frac{dx dy}{|x-y|^{\tau}}$$ Furthermore, the last integral satifies $$\int_{0}^{1}\int_{0}^{1}\, \frac{dx dy}{|x-y|^{\tau}}=
\int_{R}\, \frac{dx dy}{|x-y|^{\tau}} +
n^{s-1}\int_{0}^{1}\int_{0}^{1}\, \frac{dx dy}{|x-y|^{\tau}}$$ which readily gives the final result $$R(m,n,\tau) = \frac{2 n (m-1)(1-n^{\tau-2})}
{m (n-1) (\tau-1)(\tau-2)}.$$
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---
abstract: 'Machine learning (ML) is increasingly deployed in real world contexts, supplying “actionable insights” and forming the basis of automated decision-making systems. While issues resulting from biases *pre-existing* in training data have been at the center of the fairness debate, these systems are also affected by *technical and emergent biases*, which often arise as context-specific artifacts of implementation. This position paper interprets technical bias as an epistemological problem and emergent bias as a dynamical feedback phenomenon. In order to stimulate debate on how to change machine learning practice to effectively address these issues, we explore this broader view on bias, stress the need to reflect on epistemology, and point to value-sensitive design methodologies to revisit the design and implementation process of automated decision-making systems.'
bibliography:
- 'references.bib'
---
Introduction
============
Data-driven decision-making is rapidly being introduced in high-stakes social domains such as medical clinics, criminal justice, and public infrastructure. The proliferation of biases in these systems leads to new forms of erroneous decision-making, causing disparate treatment or outcomes across populations [@barocas_big_2016]. The ML community is working hard to understand and mitigate the unintended and harmful behavior that may emerge from poor design of real-world automated decision-making systems [@amodei_concrete_2016]. While many technical tools are being proposed to mitigate these errors, there is insufficient understanding of *how the machine learning design and deployment practice* can safeguard critical human values such as safety or fairness. The AI Now Institute identifies “a deep need for interdisciplinary, socially aware work that integrates the long history of bias research from the social sciences and humanities into the field of AI research” [@campolo_ai_2017].
How can ML practitioners, often lacking consistent language to go beyond technical descriptions and solutions to “well-defined problems,” engage with fundamentally human aspects in manner that is *constructive rather than dismissive or reductive*? And how may other disciplines help to enrich the practice? In this paper, we argue that practitioners and researchers need to take a step back and adopt a broader and more holistic view on bias than currently advocated in many classrooms and professional fora. Our discussion emphasizes the need to reflect on questions of epistemology and underlines the importance of dynamical behavior in data-driven decision-making. We do not provide full-fledged answers to the problems presented, but point to methodologies in value-sensitive design and self-reflection to contend more effectively with issues of fairness, accountability, and transparency throughout the design and implementation process of automated decision-making systems.
A Broader View On Bias
======================
Most literature addressing issues of fairness in ML has focused on the ways in which models can inherit *pre-existing biases* from training data. Limiting ourselves to these biases is problematic in two ways. Firstly, it narrows us to look at how these biases lead to *allocative harm*; a primarily economic view of how systems allocate or withhold an opportunity or resource, such as being granted a loan or held in prison. In her NIPS 2017 keynote, Kate Crawford made the case that at the root of all forms of allocative harm are biases that cause *representational harm*. This perspective requires us to move beyond biases in the data set and “think about the role of ML in harmful representations of human identity,” and how these biases “reinforce the subordination of groups along the lines of identity” and “affect how groups or individuals are understood socially,” thereby also contributing to harmful attitudes and cultural beliefs in the longer term [@crawford_keynote:_2017]. It is fair to say that representation issues have been largely neglected by the ML community, potentially because they are hard to formalize and track.
*Responsible representation* requires analyses beyond scrutinizing a training set, including questioning how sensitive attributes might be represented by different features and classes of models and what governance is needed to complement the model. Additionally, while ML systems are increasingly implemented to provide “actionable insights” and guide decisions in the real world, the core methods still fail to effectively address the inherent *dynamic nature* of interactions between the automated decision making process and the environment or individuals that are acted upon. This is particularly true in contexts where observations or human responses (such as clicks and likes) are *fed back* along the way to update the algorithm’s parameters, allowing biases to be further reinforced and amplified.
The tendency of ML-based decision-making systems to formalize and reinforce socially sensitive phenomena necessitates a broader taxonomy of biases that includes risks beyond those pre-existing in the data. As argued by Friedman and Nissenbaum in the nascent days of value-sensitive design methodologies, two other sources of bias naturally occur when designing and employing computer systems, namely *technical bias* and *emergent bias* [@friedman_value-sensitive_1996; @friedman_bias_1996].
While understanding pre-existing bias has lent itself reasonably well to statistical approaches for understanding a given data set, technical and emergent bias require engaging with the domain of application and the ways in which the algorithm is used and integrated. For automated decision-making tools to be responsibly integrated in any context, it is critical that designers (1) assess technical bias by reflecting on their *epistemology* and understanding the values of users and stakeholders, and (2) assess emergent bias by studying the *feedback mechanisms* that create intimate, ever-evolving coupling between algorithms and the environment they act upon.
Technical Bias Is About Epistemology
====================================
Friedman describes a source of technical bias as “the attempt to make human constructs amenable to computers - when, for example, we quantify the qualitative, make discrete the continuous, or formalize the nonformal” [@friedman_value-sensitive_1996]. This form of bias originates from all the tools used in the process of turning data into a model that can make predictions. While technical bias is domain-specific, we identify four sources in the machine learning pipeline.
Firstly, both collected and existing data $X$ are at some point measured and transformed into a computer readable scale. Depending on the objects measured, each variable may have a different scale, such as nominal, ordinal, interval, or ratio. Consider for example Netflix’s decision to let viewers rate movies with “likes” instead of a 1-5 star rating. As such, movie ratings moved from an ordinal scale (a number score in which order matters, but the interval between scores does not) to a nominal scale (mutually exclusive labels: you like a movie or you don’t). While the nominal scale might make it easier for viewers to rate movies, it affects how viewers are *represented* and what content gets recommended by the ML system. As such, these choices can produce *measurement bias*, so careful consideration is necessary to understand its effects on system outcomes [@hardt_nips_2017].
Secondly, based on gathered data $X$ and available domain knowledge, practitioners *engineer features* and *select model classes*. Features $\varphi(X)$ can be the available data attributes, transformations thereof based on knowledge and hypotheses, or generated in an automated fashion. Since each feature can be regarded as a model of attributes of the system or population under study, it is relevant to ask how representative it is as a proxy and why it may be predictive of the outcome. Models are used to make predictions based on features. A model class $f(\cdot;\theta)$, with parameters $\theta$, should be selected based on the complexity of the phenomenon in question and the amount and quality of the available data. Is the individual or object that is subject to the decision easily reduced to numbers or equations to begin with? What information in the data is inherently lost by virtue of the mapping $f(\varphi(X);\theta)$ having a limited complexity? The process of representation, abstraction and compression can be collectively described as inducing *modeling bias*. ML can be seen as a *compression* problem in which complex phenomena are stored as a pattern in a finite-dimensional parameter space. From an information theoretic perspective, modeling bias influences the extent to which distortion can be minimized when *reconstructing* a phenomenon from a compressed or sampled version of the original [@cover_elements_2012; @dobbe_fully_2017].
Thirdly, label data $Y$ is used to represent the output of the model. Training labels may be the actual outcome for historical cases, or some discretized or proxy version in cases where the actual outcomes cannot be measured or exactly quantified. Consider for example the use of records of arrest to predict crime rather than the facts of whether the crime was actually committed. How *representative* are such records of real crime across all subpopulations? What core information do they miss for representing the intended classes? And what bias lies hidden in them? We propose to refer to such issues as *label bias*.
Lastly, given a certain parameterization $(\varphi(\cdot),f(\cdot,\theta))$ and training data $(X,Y)$, a model is trained and *tuned to optimize certain objectives*. At this stage, various metrics may inform the model builder on where to tweak the model. Do we minimize the number of false positives or false negatives? In recidivism prediction, a false positive may be someone who incorrectly gets sentenced to prison, whereas a false negative poses a threat to safety by failing to recognize a high-threat individual. There are inherent trade-offs between prioritizing for equal prediction accuracy across groups versus for an equal likelihood of false positives and negatives across groups [@chouldechova_fair_2016; @kleinberg_inherent_2017]. Technical definitions of fairness are motivated by different metrics, illuminating the inherent ambiguity and context-dependence of such issues. For a given context, what is the right balance? And who gets to decide? We coin the effects of these trade-offs *optimization bias*.
The many questions posed above illuminate the range of places in the machine learning design process where issues of *epistemology* arise: they require *justification* and often *value judgment*. Our theory of knowledge and the way we formalize and solve problems determines how we represent and understand sensitive phenomena. How do we represent phenomena in ways that are deemed correct? What evidence is needed in order to justify an action or decision? What are legitimate classes or outcomes of a model? And how do we deal with inherent trade-offs of fairness? These challenges are deeply context-specific, often ethical, and challenge us to understand our epistemology and that of the domain we are working in.
The detrimental effects of overlooking these questions in practice are obvious in high-stakes domains, such as predictive policing and sentencing, where the decision to treat crime as a prediction problem reduces the perceived autonomy of individuals, fated to either commit a crime or act within the law. Barabas et al. argue that rather than prediction, “machine learning models should be used to surface covariates that are fed into a *causal model* for understanding the social, structural and psychological drivers of crime” [@barabas_interventions_2018]. This is a strong message with many challenges, but it points in the right direction: in these contexts, machine learning models should *facilitate rather than replace* the critical eye of the human expert. It forces practitioners and researchers to be humble and reflect on how *our own skills and tools may benefit or hurt an existing decision-making process*.
Emergent Bias Is About Dynamics
===============================
Complementing pre-existing and technical biases, “emergent bias arises only in a context of use by real users \[...\] as a result of a change in societal knowledge, user population, or cultural values.” [@friedman_value-sensitive_1996]. Recently, convincing examples of emergent bias have surfaced in contexts where ML is used to automate or mediate human decisions. In predictive policing, where discovered crime data (e.g., arrest records) are used to predict the location of new crimes and determine police deployment, runaway feedback loops can cause increasing surveillance of particular neighborhoods regardless of the true crime rate [@ensign_runaway_2018], leading to over-policing of “high-risk” individuals [@stroud_chicagos_2016]. In optimizing for attention, recommendation systems may have a tendency to turn towards the extreme and radical [@tufekci_opinion_2018]. When machine learning systems are unleashed in feedback with society, they may be more accurately described as *reinforcement learning* systems, performing *feedback control* [@recht_ethics_2018]. Therefore, a decision-making system has its own *dynamics*, which can be modified by feedback, potentially causing bias to accrue over time. To conceptualize these ideas at a high level, we adopt the system formulation depicted in Figure \[fig:my\_label\].
![A Simple Feedback Model[]{data-label="fig:my_label"}](decision_maker_v2.pdf){width="45.00000%"}
The machine learning system acts on the environment through decisions, control actions, or interventions. From the environment, the decision maker considers observations, historical data, measurements and responses, conceivably updating its model in order to steer the environment in a beneficial direction. For example, in the case of predictive policing, ‘the environment’ describes a city and its citizens, and ‘the decision maker’ is the police department, which determines where to send police patrols or invest in social interventions.
The dynamical perspective offered by the conception of a feedback model allows for a focus on interactions, which can add clarity to debates over key issues like fairness and algorithmic accountability. Situations with completely different fairness interpretations may have identical *static* observational metrics (properties of the joint distribution of input, model and output), and thus a causal or dynamic model is necessary to distinguish them [@hardt_equality_2016]. On the other hand, a one-step feedback model, incorporating temporal indicators of well-being for individuals affected by decisions, offers a way of comparing competing definitions of fairness [@liu_delayed_2018]. Similarly, calls for “interpretability” and proposed solutions often omit key operative words – Interpretable to whom? And for what purpose? [@kohli_translation_2018]. The dynamic viewpoint adds clarity to these questions by focusing on causes and effects of decision making systems, and situating interpretability in context.
Beyond providing a more realistic and workable frame of thinking about bias and related issues, the feedback system perspective may also allow inspiration to be drawn from areas of *Systems Theory* that have traditionally studied feedback and dynamics. For instance, the field of *System Identification* uses statistical methods to build mathematical models of dynamical systems from measured data, often to be employed to control *dynamical systems* with strict safety requirements, such as airplanes or electric power systems [@ljung_system_1998; @astrom_feedback_2010]. Inspiration may be drawn from the rich literature on *closed-loop identification*, which considers the identification of models with data gathered *during* operation, while the same model is also used to safeguard a system [@van_den_hof_closed-loop_1998]. That said, modeling socio-technical systems is more challenging than engineered systems. The complexity of modeled phenomena, the role of unmodeled phenomena such as external economic factors, and slower temporal dynamics all pose barriers to directly applying existing engineering principles.
Our Positionality Shapes Our Epistemology
=========================================
As ML practitioners and researchers, we are wired to analyze challenges in ways that *abstract, formalize and reduce complexity*. It is natural for us to think rigorously about technical roots of biases in the systems we design, and propose and techno-fixes to prevent negative impact from their proliferation. However, it is of crucial importance to acknowledge that the methods and approaches we use to reduce, formalize, and gather feedback from experiments are *themselves* inherent sources of bias. Epistemologies differ tremendously from application to application and ultimately shape the way a decision-maker justifies decisions and affects individuals. Technology *intimately touches and embodies values* deemed critical in employing the intended decision-making system. As such, we need to go beyond our formal tools and analyses to engage with others and reflect on our epistemology. In doing so, we aim to determine *responsible ways* in which technology can help put values into practice, and understand its fundamental limitations.
With a plethora of issues surfacing, it is easy to either consider banning ML altogether, or otherwise dismiss requests to fundamentally revisit its role in enabling data-driven decision-making in sensitive environments. Instead, we propose three principles to nourish debate on the middle ground:
**1**: *Do fairness forensics* [@crawford_keynote:_2017]. Keep track of biases in an open and transparent way and engage in constructive dialogue with domain experts, to understand proven ways of formalizing complex phenomena and to breed awareness about how bias works and when/where users should be cautious.
**2**: *Acknowledge that your positionality shapes your epistemology* [@takacs_how_2003]. Our personal backgrounds, the training we received, the people we represent or interact with all have an impact on how we look at and formalize problems. As ML practitioners, we should set aside time and energy for critical self-reflection, to identify our own biases and blind spots, to harbor communication with the groups affected by the systems we design, and to understand where we should enrich our epistemology with other viewpoints.
**3**: *Perform value-sensitive design*. Determine what values are relevant in building a decision-making system and how they might be embodied or challenged in the design and implementation by engaging with users and other affected stakeholders [@van_den_hoven_ict_2007; @friedman_value_2013].
As Takacs describes it, the benefits of self-reflection go well beyond arriving at the “best solution” to a complex problem [@takacs_how_2003; @takacs_positionality_2002]. “This means learning to listen with open minds and hearts, learning to respect different ways of knowing the world borne of different identities and experiences, and learning to examine and re-examine one’s own worldviews. \[...\] When we constantly engage to understand how our positionality biases our epistemology, we greet the world with respect, interact with others to explore and cherish their differences, and live life with a fuller sense of self as part of a web of community.” As machine learning systems rapidly change our information gathering and shape our decisions and worldviews in ways we cannot fully anticipate, self-reflection and awareness of our epistemology becomes ever more important for machine learning practitioners and researchers to ensure that automated decision-making systems contribute in beneficial and sustainable ways.
#### Acknowledgements
We thank Moritz Hardt and Ben Recht for helpful comments and suggestions. This work is funded by a Tech for Social Good Grant from CITRIS and the Banatao Institute at UC Berkeley.
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author:
- |
\
Institut für Theoretische Physik, ETH Zürich, CH-8093 Zürich, Switzerland\
CERN, Theory Division, CH-1211 Geneva 23, Switzerland\
E-mail:
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Massimo D’Elia\
INFN - Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy\
Dipartimento di Fisica dell’Università di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy\
E-mail:
title: |
`CERN-PH-TH-2017-022`\
Continuum limit and universality\
of the Columbia plot
---
Introduction
============
![([*left*]{}) Columbia plot: we focus on the $N_f=3$ chiral critical point (arrow), and its $N_f=4$ analogue; ([*right*]{}): adding a vertical axis for the chemical potential $\mu$, a possible QCD chiral critical point occurs when the surface swept by the $\mu=0$ chiral critical line intersects the physical quark masses’ vertical line.](columbia_plot_arrow.png){width="45.00000%"}
The QCD phase diagram summarizes the various behaviors of QCD as a function of temperature $T$ and matter density, or equivalently quark chemical potential $\mu$. Since the chiral and the center symmetry, which play crucial roles in the phase diagram, are both explicitly broken in QCD by the quark masses, it is useful to consider these masses as QCD parameters: $m_{u,d}$ for the two light quark masses considered degenerate for simplicity, and $m_s$ for the strange quark mass. Our expectations for the $\mu=0$ phase diagram, projected along the $T$-direction, are contained in the “Columbia plot” Fig. 1 (left).
The upper-right and lower-left corners of the Columbia plot are simpler to analyze:\
- In the first, all quarks are infinitely massive. They decouple, and the resulting $SU(N_c=3)$ Yang-Mills theory obeys the global $Z(3)$ center symmetry, which is spontaneously broken at high temperature via a first-order transition.\
- In the second, all quarks are massless, and the theory obeys the global $SU(N_f=3)$ chiral symmetry, which is spontaneously broken at low temperature and restored at high temperature. For 3 massless flavors or more, one expects symmetry restoration to occur via a first-order transition [@Pisarski_Wilczek], because no $3d$ $SU(N_f), N_f \geq 3$, second-order universality class is known [@Vicari].\
In the middle of the Columbia plot, where both symmetries are badly broken explicitly, Monte Carlo simulations indicate an analytic crossover as $T$ is raised. Thus, there must exist two critical, second-order lines separating the two first-order regions above from the central crossover region. Because no particular symmetry is at play along these critical lines, their universality class should be that of a $3d$ $\phi^4$ theory, i.e. that of the $3d$ Ising model.
A simple way to pin down the location of these two critical lines is to consider the $N_f=3$ case, with all quark masses equal, shown as the diagonal of the Columbia plot. Two critical quark masses should be observed, to be determined with high precision via Monte Carlo simulations. In practice, it is difficult to adopt a reference scale, since an $N_f=3$ theory is a distortion of real-world QCD. So, it is convenient to trade the critical quark mass for the ratio of the corresponding $T=0$ “pion” mass $m_\pi^c$ over the transition temperature $T_c$: $m_\pi^c / T_c$ is of order 1 for real-world QCD, and allows to separate the regime of “light” and that of “heavy” quarks.
Besides increasing our fundamental knowledge, the quantitative determination of the Columbia plot is useful when considering the effect of a chemical potential $\mu$, which can be viewed as an additional vertical axis. The two $\mu=0$ critical lines discussed above will sweep critical surfaces as $\mu$ is turned on. The chiral critical surface, near the lower-left corner, may bend away from the origin, and reach the physical quark mass values for a sufficiently large $\mu$: this signals the presence of a QCD chiral critical point, as in Fig. 1 (right). Or this critical surface may bend the other way, and there may be no chiral critical point [@PdF_OP]. Even in the first case, reaching the critical point will require more bending if the $\mu=0$ critical line corresponds to smaller quark masses. Thus, an accurate determination of this critical line is an important ingredient to shape our understanding of the finite-density properties of QCD. This can all be studied at $\mu=0$, without having to face the “sign problem” present at non-zero chemical potential.
The importance of pinning down the $\mu=0, N_f=3$ critical points has been recognized. The technical difficulty is to control the approach to the continuum limit. For the heavy quark case, the masses are $\gtrsim {\cal O}(1)$ GeV, which requires fine lattices to avoid UV cutoff effects. Numerical work so far has focused on coarse lattices with $N_t=4$ time-slices [@heavy]. Such difficulties are absent in the light quark case, so that one would expect reasonable accuracy for lattice spacings ${\cal O}(0.1)$ fm, i.e. $N_t \gtrsim 8$. However, one observes $(i)$ large cutoff effects ($\sim 30\%$) for $N_t=4,6,8$ [@Ukawa1], and $(ii)$ enormous discrepancies (a factor of $\sim$ four!) between staggered and Wilson fermions at these values of $N_t$.
Action $N_t$ $m_\pi^c$ Ref. Year
-------------------- ------- ------------------- -------------- ------
standard staggered 4 $\sim 290$ MeV [@KS_Nt4] 2001
p4 staggered 4 $\sim 67$ MeV [@p4_Nt4] 2004
standard staggered 6 $\sim 150$ MeV [@KS_Nt6] 2007
HISQ staggered 6 $\lesssim 45$ MeV [@HISQ] 2011
stout staggered 4-6 could be zero [@Varnhorst] 2014
Wilson-clover 6-8 $\sim 300$ MeV [@Ukawa1] 2014
Wilson-clover 4-10 $\sim 100$ MeV [@Ukawa2] 2016
: Summary of previous studies of the $N_f=3$, $\mu=0$ chiral critical point – adapted from [@Varnhorst]. The general trend is: finer lattices and/or improved actions drive the critical ”pion” mass down; Wilson fermions favor much heavier values. The last line (Wilson, $N_t=10$) was presented at the Lattice conference [@Ukawa2]. Other, related studies have kept fixed to their physical value the strange quark mass [@ms_fixed], or the ratio $m_s/m_{u,d}$ [@Endrodi].
Table I, adapted from [@Varnhorst], summarizes the results of past studies, and adds some results presented at the Lattice conference [@Ukawa2]. The general trend is: the more one approaches the continuum limit, by decreasing the lattice spacing or by improving the action, the softer the transition, and the lighter the critical “pion” is. The most remarkable case is that of [@Varnhorst]: using a stout-improved action with staggered fermions on an $N_t=4$ lattice, no sign of a first-order transition was found, down to arbitrarily small quark masses! In contrast, with Wilson fermions Ref. [@Ukawa1] finds a critical “pion” mass of about 300 MeV after extrapolating from $N_t=6$ and $8$.
Thus, one is led to mistrusting the staggered simulations for two reasons: they indicate a very small critical “pion” mass, consistent with zero; and they disagree strongly with Wilson fermion results, both at finite lattice spacing and after continuum extrapolation.
One plausible culprit for these two puzzles might be rooting. With staggered fermions, the Dirac determinant is raised to the power $3/4$ to mimic $N_f=3$ degenerate flavors. The danger of this procedure has been pointed out [@rooting]: the consensus is that danger appears when the chiral limit is approached first, before the continuum limit is taken. This is potentially the case here, since the quark masses needed for criticality quickly approach zero as $N_t$ is increased.
To eliminate a possible issue with rooting, we have studied the case of $N_f=4$ degenerate flavors: no rooting is required, and the thermal transition in the chiral limit is expected to be first-order as for $N_f=3$. Actually, a naive counting of the degrees of freedom suggests that the first-order transition will be stronger for $N_f=4$ than for $N_f=3$, so that the critical “pion” mass will be heavier, thus reducing the computing cost of the simulations.
![$N_f=3$ comparison of $m_\pi^c/T_c$ as a function of the lattice spacing $a^2$, between standard staggered fermions [@KS_Nt6] and non-perturbatively improved Wilson fermions [@Ukawa1]. The $N_t=4$ and $6$ Wilson fermion data would mistakenly suggest small lattice artifacts. The $N_t=10$ Wilson point was presented at the Lattice conference [@Ukawa2]. The numbers along the $y$-axis indicate the results of linear extrapolations in $a^2$.](m_pi_over_Tc_Nf3_procs.pdf){width="75.00000%"}
Results: $N_f=4$
================
As argued above, we have simulated standard staggered fermions with $N_f=4$ flavors and Wilson plaquette action, in order to bypass potential harmful effects of rooting and to keep computing costs down. The numerical simulations have been performed using a code running on GPUs [@gpupaper].
For successive values of $N_t=4, 6, 8$ and $10$, we have simulated lattices of spatial size $N_s \geq 2 N_t$, and determined the light bare quark mass $m_q^c$ for which the finite-temperature chiral transition is second-order. Following [@PdF_OP], the order of the phase transition was established by monitoring the Binder cumulant $B_4(\bar\psi \psi)$, where $B_4(X) \equiv \frac{\langle (\delta X)^4 \rangle}{\langle (\delta X)^2 \rangle^2}$ and $\delta X \equiv X - \langle X \rangle$. Near criticality, $B_4$ should be a function of the ratio $L_s/\xi$ of the spatial lattice size over the spatial correlation length, which diverges as $|m_q - m_q^c|^{-\nu}$. The critical value 1.604.. and the critical exponent $\nu \approx 0.63$ are known from the $3d$ Ising universality class. Thus, one can expand $B_4(m_q)$ near $m_q^c$ as $$B_4(m_q) = 1.604 + c_1 (m_q - m_q^c) N_s^{1/\nu} + {\cal O}((m_q - m_q^c)^3)$$ An illustrative fit (including cubic terms) is shown in Fig. 3 (left) for $N_t=4$.
For a given value of $N_t$, this procedure determines the bare parameters $a m_q^c$ and $\beta$ required for criticality. A zero-temperature simulation is then performed, at these parameters, to determine the $T=0$ pion mass $(a m_\pi^c)(N_t)$. Finally, one obtains the physically meaningful ratio $m_\pi^c/T_c = N_t a m_\pi^c$, and repeats this procedure for successive values of $N_t$. These successive ratios are shown in Fig. 3 (right).
The zero-temperature simulations require large lattices $(N_s^0)^3 \times N_t^0$, both spatially (to maintain $(m_\pi L_s) \gg 1$) and temporally (to achieve $T\approx 0$). To reduce the computing effort, we chose $N_s^0 \geq 2 N_t$ and $N_t^0 \geq 4 N_t$, i.e. we used a $20^3 \times 40$ “zero-temperature” lattice in combination with a $20^3 \times 10$ finite-temperature lattice. We are aware that our choice is only marginally satisfactory, and causes systematic errors in the extracted pion mass. However, Fig. 3 (right) shows variations of order 100% as $N_t$ is varied, which makes our systematic errors negligible in comparison.
![([*left*]{}): finite-size scaling of the Binder cumulant $B_4(\bar\psi \psi)$ with the bare quark mass, here for $N_t=4$. ([*right*]{}): $N_f=4$ variation of $m_\pi^c/T_c$ as a function of the lattice spacing $a^2$, with linear and quadratic extrapolations. $N_f=3$ staggered results are shown for comparison: the critical “pion” mass is smaller, as expected.](Nf4_Nt4_B4.pdf "fig:"){width="55.00000%"} ![([*left*]{}): finite-size scaling of the Binder cumulant $B_4(\bar\psi \psi)$ with the bare quark mass, here for $N_t=4$. ([*right*]{}): $N_f=4$ variation of $m_\pi^c/T_c$ as a function of the lattice spacing $a^2$, with linear and quadratic extrapolations. $N_f=3$ staggered results are shown for comparison: the critical “pion” mass is smaller, as expected.](m_pi_over_Tc_Nf4_procs.pdf "fig:"){width="55.00000%"}
Discussion
==========
Our $N_f=4$ results Fig. 3 (right) are qualitatively similar to previous $N_f=3$ results: the dramatic reduction or $m_\pi^c/T_c$ as the lattice spacing is reduced is still present, and thus not related to rooting. Actually, a similar reduction is now visible in the Wilson data as well, with the new $N_t=10$ point [@Ukawa2] (see Fig. 2). Therefore, there is no reason to doubt the universality of the continuum limit: the Wilson and the staggered values should converge as $N_t$ is increased.
What is remarkable, however, is how slow this convergence is, even with a non-perturbatively improved action as in the Wilson case! Note that $N_t=10$, from which the continuum value of $m_\pi^c/T_c$ will probably differ by a factor 2 or more, corresponds to a lattice spacing $a \sim 0.13$ fm. State-of-the-art thermodynamic studies use a maximum number of $N_t=16$ time-slices.
Let us speculate on the reason for such large cutoff effects. Taste-breaking could be the explanation: $N_f=4$ staggered fermions possess 16 ”pions”, but only one of them is really light, and the 15 others become degenerate with it in the continuum limit only. Thus, as the lattice spacing is reduced, the number of light pions effectively increases. But this should make the transition stronger, not weaker as observed. Moreover, the opposite occurs with Wilson fermions: there, the doublers become heavier toward the continuum limit. Nevertheless, Wilson and staggered fermions both lead to the transition becoming softer in the continuum limit. Thus, the explanation must reside elsewhere.
Perhaps cutting off all but the $N_t$ lowest Matsubara frequencies has larger than expected consequences. One simple exercise consists of calculating the pressure of a free massless boson on the lattice, and comparing it with the continuum Stefan-Boltzmann law [@Karsch_SB]. An instructive figure can be found in Fig. 2 of [@Helvio]. It shows that the lattice pressure can easily differ from the continuum one by factors well beyond 10. The pressure deficit due to the Matsubara cutoff is less pronounced if the boson is more massive, so that the cutoff will extend the parameter regime of the confined phase, and push the critical ”pion” mass upward.
If this guess is correct, then it is essential to reduce the [*temporal*]{} lattice spacing, not so much the spatial one. Lattice actions with anisotropic couplings would afford an economical approach to the continuum limit. For the measurement of $m_\pi^c/T_c$, the non-perturbative tuning of the (gauge and fermion) anisotropy coefficients is not needed, as long as the continuum limit is consistent. Alternatively, one could improve the action so as to approach the Stefan-Boltzmann law better, in the spirit of the p4-improved action [@p4].
Controlling the continuum extrapolation of the $N_f=4$ finite temperature transition will serve us for the $N_f=2+1$ case as well. There, unexpected results have been obtained for the upper-left corner of the Columbia plot: the thermal transition appears to be first-order – on a coarse, $N_t=4$ lattice [@Nf2]. Lattice corrections should also be carefully considered in the search for a conformal window in the number of flavors: it has been proposed to identify the lower edge of this window as the number of massless flavors for which the critical temperature of the chiral phase transition reaches zero [@Pallante] – there too, lattice corrections may well be very significant.
Finally, it is clear that the continuum value of $m_\pi^c/T_c$, for $N_f=4$ and even more so for $N_f=3$, is going to be extremely small. At present, continuum extrapolations as in Fig. 3 (right) are compatible with a zero value. Could it actually be exactly zero, in contradiction with the predictions of [@Pisarski_Wilczek] ?
Numerical simulations have been performed on a GPU farm located at the INFN Computer Center in Pisa and on the QUONG cluster in Rome.
[99]{}
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|
---
abstract: |
We argue that storage rings can be used for the detection of low–frequency gravitational–wave background. Proceeding from the measurements by Schin Dat${\acute{\rm e}}$ and Noritaka Kumagai (Nucl. Instrum. Meth. [**A421**]{}, 417 (1999)) and Masaru Takao and Taihei Shimada (Proceedings of EPAC 2000, Vienna, 2000, p.1572) of variations of the machine circumference of the SPring–8 storage ring we explain the systematic shrinkage of the machine circumference by the influence of the relic gravitational–wave background. We give arguments against a possibility to explain the observed shrinkage of the machine circumference of the SPring–8 storage ring by diastrophic tectonic forces. We show that the forces, related to the [*stiffness*]{} of the physical structures, governing the path of the beam, can be neglected for the analysis of the shrinkage of the machine circumference caused by the relic gravitational–wave background. We show the shrinkage of the machine circumference can be explained by a relic gravitational–wave background even if it is treated as a stochastic system incoming on the plane of the machine circumference from all quarters of the Universe. We show that the rate of the shrinkage of the machine circumference does not depend on the radius of the storage ring and it should be universal for storage rings with any radii.\
PACS: 04.30.Nk, 04.80.Nn, 02.50.Ey
author:
- 'A. N. Ivanov[^1] [^2] and A. P. Kobushkin[^3] [^4]'
title: 'Storage rings as detectors for relic gravitational–wave background ?'
---
[*Atominstitut der Österreichischen Universitäten, Arbeitsbereich Kernphysik und Nukleare Astrophysik, Technische Universität Wien,\
Wiedner Hauptstr. 8-10, A-1040 Wien, Österreich\
und\
Institut für Mittelenergiephysik Österreichische Akademie der Wissenschaften,\
Boltzmanngasse 3, A-1090, Wien, Österreich*]{}
Introduction
============
The existence of gravitational waves has been predicted by Einstein’s general theory of relativity [@JW73]–[@JP99] . Starting with the pioneering work by Weber [@JW60] one of the most challenging problems of experimental physics is the detection of gravitational radiation. In the seventies of the last century the existence of gravitational waves has been confirmed indirectly in a set of accurate measurements of secular orbital period changes in the Hulse–Taylor binary pulsar [@JT82]. An attempt for the observation of the cosmic low–frequency gravitational–wave background has been undertaken by Stinebring, Ryba, Taylor and Romani [@DS90].
An interesting influence of gravity on the parameters of storage rings of the Large Electron Positron Collider (LEP) at CERN and the SPring–8 in Japan has been found by Arnaudon [*et al.*]{} [@LA95] and Dat${\acute{\rm e}}$ and Kumagai [@SD99], and Takao and Shimada [@MT00], respectively. Below we will discuss only the measurements for the SPring–8 storage ring [@SD99; @MT00], though our results should be applicable also to other storage rings.
In the analysis of the influence of gravity on the SPring–8 electron storage ring Dat${\acute{\rm e}}$, Kumagai, Takao and Shimada have considered the changes of the machine circumference $C_0 \simeq
1436\,{\rm m}$ in dependence of gravitational coupling of the storage ring to the Moon and the Sun. According to [@SD99; @MT00], the change $\Delta C$ of the reference value of the machine circumference is defined by the gravitational interaction of the electron storage ring with the Moon and the Sun due to the tidal and seasonal forces. A total rate of a change of the machine circumference can be written as $$\begin{aligned}
\label{label1.1}
\frac{\Delta C}{\Delta t} = \Bigg(\frac{\Delta C}{\Delta t}\Bigg)_{\rm
m} + \Bigg(\frac{\Delta C}{\Delta t}\Bigg)_{\rm s} +
\Bigg(\frac{\Delta C}{\Delta t}\Bigg)_{\rm us},\end{aligned}$$ where first two terms are caused by the tidal (m) and seasonal (s) forces, but the third term describes a rate of a change of the machine circumference due to unknown sources (us).
The theoretical predictions for $(\Delta C/\Delta t)_{\rm m} + (\Delta
C/\Delta t)_{\rm s}$ caused by the tidal and seasonal forces have been fully confirmed experimentally [@SD99; @MT00]. Nevertheless, measuring the rate $\Delta C/\Delta t$ of the changes of the machine circumference of the storage ring there has been found a systematic shrinkage of the machine circumference with the rate of about $2\times
10^{-4}\,{\rm m/yr}$ [@MT00], which cannot be explained by the tidal and seasonal forces induced by the Moon and the Sun. In (\[label1.1\]) this shrinkage is described by the third term $(\Delta C/\Delta t)_{\rm us}$. In this letter we give arguments that this phenomenon can be understood as an influence of a cosmic very low–frequency gravitational–wave background. Therefore, below we denote the third term as $(\Delta C/\Delta t)_{\rm gw}$.
The paper is organized as follows. In Section 2 we estimate the rate of the shrinkage of the machine circumference due to the gravitational strain. In Section 3 we solve the equations of motion of the storage ring in the field of the cylindrical relic gravitational wave. We show that the solution of the equations of motion gives the same result obtained within the hypothesis of the [*gravitational strain*]{}. We show that the rate of the shrinkage of the machine circumference does not depend on the radius of the storage ring and should be universal for storage rings with any radii. In Section 4 we give arguments against a possibility to explain the observed shrinkage of the machine circumference of the SPring–8 storage ring by diastrophic tectonic forces. In Section 5 we discuss the influence of the [*stiffness*]{} of the physical structures of the storage ring, governing the path of the beam. We argue that the forces, induced by the [*stiffness*]{} of the physical structures of the storage ring, governing the path of the beam, can be neglected for the analysis of the shrinkage of the machine circumference caused by the relic gravitational–wave background. In Section 6 we investigate the shrinkage of the machine circumference induced by a stochastic spherical relic gravitational–wave background incoming on the plane of the machine circumference from all quarters of the Universe. We show that the stochastic relic gravitational–wave background, incoming on the plane of the machine circumference from all quarters of the Universe, does not destroy the shrinkage of the machine circumference. The former is due to the fact that the effect of the shrinkage of the machine circumference is of the second order in gravitational wave interactions. We show that the independence of the rate of the shrinkage of the machine circumference on the radius of the storage ring retains in the case of the interaction of the storage ring with the stochastic relic gravitational–wave background. In the Conclusion we discuss the obtained results.
Gravitational strain and shrinkage of machine circumference
===========================================================
It is well–known that on the Earth one of the main fundamental effects of gravitational waves is the [*gravitational strain*]{}: a fractional distortion in the length of the object induced by the gravitational field [@JP99].
In this connection we assume that the storage ring is sensitive to the influence of low–frequency gravitational waves, which produce a variation $\delta C_{\rm gw}(t)$ of the machine circumference $C_0$. Following [@JW73]–[@JP99] we treat low–frequency gravitational waves as perturbations of the metric.
For the calculation of $\delta C_{\rm gw}(t)$ we define a perturbation of the metric $h_{ab}(t,z)\,(a,b = x,y,z)$ as a monochromatic plane wave traveling along the $z$–axis with frequency $\omega$ and wave number $k = \omega/c$ [@JW73]–[@JP99] $$\begin{aligned}
\label{label2.1}
h_{ab}(t,z) = \left(\begin{array}{llcl} h_{xx}(t,z) & h_{xy}(t,z) &
0\\ h_{yx}(t,z) & h_{yy}(t,z) & 0\\ ~~0 & ~~0 & 0
\end{array}\right) = \left(\begin{array}{llcl}
\Delta_+ & ~\Delta_\times & 0\\
\Delta_\times & - \Delta_+ & 0\\
~0 & ~~0 & 0
\end{array}\right)\,\cos(\omega t - k z + \delta),\end{aligned}$$ where $\Delta_+$ and $\Delta_\times$ are constant amplitudes of the diagonal and non–diagonal components of the monochromatic plane wave, $h_{tt}(t,z) = h_{ta}(t,z) = h_{at}(t,z) = 0$ [@JW73]–[@JP99] and $\delta$ is an arbitrary phase. We define the monochromatic plane wave in the so–called [*transverse traceless*]{} gauge $h_{aa}(t,z) =
h_{xx}(t,z) + h_{yy}(t,z) = 0$ (see pp.946–948 of Ref.[@JW73]).
Placing the storage ring in the $xy$–plane at $z = 0$ the variation $\delta C_{\rm gw}(t)$ can be defined by the contour integral $$\begin{aligned}
\label{label2.2}
&&\delta C_{\rm gw}(t) = \oint_{C_0}\sqrt{dx^2 + dy^2 +
h_{xx}(t,0)dx^2 + h_{yy}(t,0)dy^2 + 2h_{xy}(t,0)dxdy} -
C_0=\nonumber\\ &&= \frac{C_0}{2\pi}\int^{2\pi}_0(\sqrt{1 -
h_{xy}(t,0)\sin2\varphi - h_{xx}(t,0)\cos2\varphi} - 1)\,d\varphi =
\oint_{C_0}\delta {\ell}_{\rm gw},\end{aligned}$$ where we have used polar coordinates $x = (C_0/2\pi)\cos\varphi$ and $y = (C_0/2\pi)\sin\varphi$ and the relation $h_{yy}(t,0) = -
h_{xx}(t,0)$ (\[label2.1\]). A change of the length of a segment between two adjacent points of the machine circumference of the storage ring due to the [*gravitational strain*]{} caused by the monochromatic plane wave $h_{ab}(t,z)$ we denote as $\delta
{\ell}_{\rm gw}$. Expanding the square root in powers of $h_{ab}(t,0)$ we represent $\delta {\ell}_{\rm gw}$ in the following form $$\begin{aligned}
\label{label2.3}
\delta {\ell}_{\rm gw} = \frac{C_0}{2\pi}\,\Big( -
\frac{1}{2}(h_{xx}\cos 2\varphi + h_{xy}\sin2\varphi)-
\frac{1}{8}(h_{xx}\cos 2\varphi +h_{xy}\sin2\varphi)^2 + \ldots\Big)
d\varphi.\end{aligned}$$ Keeping the first non–vanishing contribution we get $$\begin{aligned}
\label{label2.4}
\delta C_{\rm gw}(t) = \oint_{C_0}\delta {\ell}_{\rm gw} \simeq -
\frac{1}{16}\,C_0\,(h^2_{xx}(t,0) + h^2_{xy}(t,0)) = -
\frac{1}{16}\,C_0\,h^2_0\,\cos^2(\omega t + \delta),\end{aligned}$$ where the amplitude $h_0$ is equal to $h_0 = \sqrt{\Delta^2_+ +
\Delta^2_\times}$.
We would like to accentuate that the amplitude $h_0$ of the monochromatic plane wave is not the real amplitude of the relic gravitational–wave background. The relation of the amplitude $h_0$ to the amplitude $h^{\rm gw}_0$ of the relic gravitational–wave background can be found, for example, in the following way.
Notice that a real relic gravitational–wave background should be treated as a perturbation of the Friedmann–Robertson–Walker metric [@JW73]–[@JP99]. In terms of a perturbation of the Friedmann–Robertson–Walker metric, caused by the relic gravitational–wave background $h^{\rm gw}_{ab}(t,z)$, a change of the length of a segment between two adjacent points of the machine circumference of the storage ring can be determined by [@JW73]–[@JP99] $$\begin{aligned}
\label{label2.5}
\delta s_{\rm gw} = R_{\rm U}\,\Big( - \frac{1}{2}(h^{\rm gw}_{xx}\cos
2\varphi +h^{\rm gw}_{xy}\sin2\varphi) - \frac{1}{8}(h^{\rm
gw}_{xx}\cos 2\varphi +h^{\rm gw}_{xy}\sin2\varphi)^2 + \ldots\Big)
d\varphi,\end{aligned}$$ where $R_{\rm U}$ is the radius of the Universe at the present time. According to [@JW73]–[@JP99], the radius of the Universe is equal to $R_{\rm U} =(c/H_0)\sqrt{k/(\Omega - 1)}$, where $c =
9.45\times 10^{15}\,{\rm m\cdot yr^{-1}}$, $H_0 = (7.63 \pm
0.75)\times 10^{-11}\,{\rm yr}^{-1}$ [@DG00], $\Omega$ is a [*density parameter*]{} as the ratio of the energy density in the Universe to the critical energy density [@JW73]–[@JP99], and $k = 0,
\pm 1$ for flat, closed and open Universe, respectively [@JW73]–[@JP99]: (1) $k = 0$ with $\Omega = 1$, (2) $k = 1$ with $\Omega > 1$ and (3) $k = - 1$ with $\Omega < 1$. For our estimate we will use $R_{\rm U} \sim c/H_0 = 1.25\times 10^{26}\,{\rm
m}$. This agrees with the value of the [*Volume today*]{} equal to $V
= 2\pi^2 R^3_{\rm U} = 3.83\times 10^{79}\,{\rm m}^3$ (see [@JW73], p.738, Box 27.4).
It is obvious that the contour integral of $\delta s_{\rm gw}$ over the machine circumference of the storage ring should give the same variation of the length of the machine circumference as Eq.(\[label2.4\]): $$\begin{aligned}
\label{label2.6}
\delta C_{\rm gw}(t) = \oint_{C_0}\delta {\ell}_{\rm gw} =
\oint_{C_0}\delta s_{\rm gw}.\end{aligned}$$ Keeping the first non–vanishing contributions we obtain the relation between $h_0$ and $h^{\rm gw}_0$ equal to $$\begin{aligned}
\label{label2.7}
h^{\rm gw}_0 = \sqrt{\frac{C_0}{2\pi R_{\rm U}}}\,h_0 \sim 1.4\times
10^{-12}\,h_0,\end{aligned}$$ where $h^{\rm gw}_0 = \sqrt{(\Delta^{\rm gw}_+)^2 + (\Delta^{\rm
gw}_{\times})^2}$.
Since by definition of a perturbation, $h_0 \ll 1$, the relation (\[label2.7\]) gives a correct upper limit on the value of the real amplitude of the gravitational–wave background $h^{\rm gw}_0 \ll
1.4\times 10^{-12}$ [@JW73]–[@JP99]. A more detailed estimate for $h^{\rm gw}_0$, related to the experimental shrinkage of the machine circumference of the storage ring [@MT00], we derive below.
Notice that it is rather clear that the contribution of the gravitational waves to the variation of the machine circumference, $\delta C_{\rm gw}(t) \sim O((h^2_0)$, is of the second order. In fact, the mass quadrupole moment of the storage ring, located in the $xy$–plane at $z = 0$, has only two equal components $D_{xx} = D_{yy}
= D$. Due to this, the interaction of this mass quadrupole moment with gravitational waves is proportional to $D\,(h_{xx} + h_{yy})$, which is zero by definition for gravitational waves in the [*transverse traceless gauge*]{} $h_{xx} = - h_{yy}$ [@JW73]–[@JP99].
The change of the storage ring of the machine circumference $\Delta
C_{\rm gw}$ induced by the gravitational waves (\[label2.1\]) for the time interval $\Delta t = t_2 - t_1$ is equal to $$\begin{aligned}
\label{label2.8}
\Delta C_{\rm gw} = \delta C_{\rm gw}(t_2) - \delta C_{\rm gw}(t_1) =
\frac{1}{16}\,C_0\,h^2_0\,\sin(\omega \Delta t)\,\sin(\omega(t_2 +
t_1) + 2\delta).\end{aligned}$$ For the rate of the change of the machine circumference at $\Delta t
\to 0$ we get $$\begin{aligned}
\label{label2.9}
\frac{\Delta C_{\rm gw}(t)}{\Delta t} =
\frac{1}{16}\,C_0\,h^2_0\,\omega\,\sin(2\omega t + 2\delta).\end{aligned}$$ For the comparison with the experimental rate we have to average the theoretical rate (\[label2.9\]) over the data–taking period $\tau$. This gives $$\begin{aligned}
\label{label2.10}
\hspace{-0.3in}\Big\langle\frac{\Delta C_{\rm gw}(t)}{\Delta
t}\Big\rangle_{\tau} =
\frac{1}{16}\,C_0\,h^2_0\,\omega\,\frac{1}{\tau}
\int^{+\tau/2}_{-\tau/2}dt\,\sin(2\omega t + 2\delta) =
\frac{1}{16}\,C_0\,h^2_0\,\sin 2\delta\;\frac{\sin \omega \tau}{\tau}.\end{aligned}$$ In the low–frequency limit $\omega \tau \ll 1$, corresponding to the case of the relic gravitational–wave background, the relation (\[label2.8\]) can be transcribed into the form $$\begin{aligned}
\label{label2.11}
\frac{1}{C_0}\Big\langle\frac{\Delta C_{\rm gw}(t)}{\Delta
t}\Big\rangle_{\tau} = \frac{1}{16}\,h^2_0\,\omega\,\sin 2\delta .\end{aligned}$$ Since the experimental rate of the change of the machine circumference is equal to $$\begin{aligned}
\label{label2.12}
\frac{1}{C_0}\Big(\frac{\Delta C}{\Delta t}\Big)_{\exp} \simeq -
1.4\times 10^{-7}\,{\rm yr^{-1}},\end{aligned}$$ a comparison of theoretical and experimental rates leads to the relation $$\begin{aligned}
\label{label2.13}
\omega\, h^2_0\,\sin 2\delta = \frac{16}{C_0}\,\Big(\frac{\Delta
C}{\Delta t}\Big)_{\exp} \simeq -\,2.2 \times 10^{-6}\,{\rm yr}^{-1}.\end{aligned}$$ The experimentally observed shrinkage of the machine circumference of the storage ring [@MT00] imposes a constraint on the phase of the gravitational wave, i.e. $-\sin 2\delta > 0$. For further estimates we set $|\sin 2\delta|\sim 1$.
Then, since $h_0 \ll 1$, we get the lower limit on the frequency, $\omega \gg 2\times 10^{-6}{\rm yr}^{-1}$. This corresponds to an oscillation period $T \ll 3\times 10^{-3}\,{\rm Gyr}$ of the shrinkage of the machine circumference, which is smaller compared with the age of the Universe $T \simeq 15\,{\rm Gyr}$ [@GB88]. Since the oscillation period exceeds greatly any reasonable interval of experimental measurements, the rate of the shrinkage of the machine circumference, induced by the relic gravitational–wave background, should be constant in time during any data–taking period. This agrees with Eq.(\[label2.11\]).
We can also give a lower limit on the amplitude $h_0$. According to the experimental data by Takao and Shimada [@MT00], the oscillation period of the rate of the machine circumference (\[label2.12\]) should be much greater than 5 years, $T \gg 5\,{\rm yr}$. This gives $\omega \ll 1\,{\rm yr}^{-1}$ and $h_0 \gg 10^{-3}$ and according to (\[label2.7\]) we get $h^{\rm gw}_0 \gg 10^{-15}$, the lower limit on the amplitude of the relic gravitational–wave background imposed by the experimental shrinkage of the machine circumference of the storage ring (\[label2.12\]).
The rate of the shrinkage of the machine circumference, represented in terms of the relic gravitational–wave perturbations of the Friedmann–Robertson–Walker metric (\[label2.5\]), reads $$\begin{aligned}
\label{label2.14}
\Big\langle\frac{\Delta C_{\rm gw}(t)}{\Delta t}\Big\rangle_{\tau} =
\frac{\pi}{8}\,R_{\rm U}\,(h^{\rm gw}_0)^2\,\omega\,\sin 2\delta.\end{aligned}$$ This shows that the rate of the shrinkage of the machine circumference does not depend on the length of the machine circumference of the storage ring.
Thus, in our interpretation of the shrinkage of the machine circumference as induced by the relic gravitational–wave background, the value of the rate of the shrinkage of the machine circumference should be universal and equal to $(\Delta C_{\rm gw}(t)/\Delta
t)_{\exp} = -\,2\times 10^{-4}\,{\rm m/yr}$ [@MT00] for storage rings with any radii both for the SPring–8 with radius $R_0 \simeq
229\,{\rm m}$ and for the DAPHNE with radius $R_0 \simeq 15\,{\rm m}$.
Another important quantity characterizing the relic gravitational–wave background is the density parameter $\Omega_{\rm
gw}$ defined by [@JW73]–[@JP99] $$\begin{aligned}
\label{label2.15}
\Omega_{\rm gw} = \frac{\omega^2 (h^{\rm gw}_0)^2}{12H^2_0}.\end{aligned}$$ For the frequency $\omega \ll 1\,{\rm yr}^{-1}$ we get the upper limit $\Omega_{\rm gw} \ll 10^{-10}$, where we have used that $\omega h_0
\sim 1.5\times 10^{-3}\sqrt{\omega} \ll 1.5\times 10^{-3}\,{\rm
yr^{-1}}$ for $\omega \ll 1\,{\rm yr}^{-1}$ giving due to (\[label2.7\]) the relation $\omega h^{\rm gw}_0 \sim 2\times
10^{-15}\sqrt{\omega} \ll 2\times 10^{-15}\,{\rm yr^{-1}}$. The estimate $\Omega_{\rm gw} \ll 10^{-10}$ does not contradict contemporary cosmological models [@JP99].
Equations of motion and shrinkage of machine circumference
==========================================================
In this Section we show that the analysis of the influence of the relic gravitational–wave background through the solution of equations of motion for the storage ring in the cylindrical relic gravitational–wave field gives the same result that we have obtained in Section 2.
According to [@JW73] (see pp.1004–1011 of Ref.[@JW73]) the non–relativistic motion of a massive particle in the $xy$–plane at $z = 0$ induced by the cylindrical gravitational–wave background can be described the equations of motion $$\begin{aligned}
\label{label3.1}
\frac{d^2x}{dt^2}&=& - R^x_{0x0}\,x - R^x_{0y0}\,y,\nonumber\\
\frac{d^2y}{dt^2}&=& - R^y_{0x0}\,x - R^y_{0y0}\,y,\end{aligned}$$ where $R^{\alpha}_{\beta\gamma\delta}$ is the Riemann tensor defined by [@JW73] $$\begin{aligned}
\label{label3.2}
R^{\alpha}_{\beta\gamma\delta} = \frac{\partial
\Gamma^{\alpha}_{\beta\delta}}{\partial x^{\gamma}} - \frac{\partial
\Gamma^{\alpha}_{\beta\gamma}}{\partial x^{\delta}} +
\Gamma^{\alpha}_{\mu\gamma}\,\Gamma^{\mu}_{\beta\delta} -
\Gamma^{\alpha}_{\mu\delta}\,\Gamma^{\mu}_{\beta\gamma}.\end{aligned}$$ The Christoffel symbols or differently the “covariant connection coefficients” $\Gamma^{\alpha}_{\lambda\mu}$ are determined in terms of the metric tensor [@JW73] (see also [@DG00]) $$\begin{aligned}
\label{label3.3}
\Gamma^{\alpha}_{\lambda\mu} =
\frac{1}{2}\,g^{\alpha\nu}\,\Big(\frac{\partial g_{\mu\nu}}{\partial
x^{\lambda}} + \frac{\partial g_{\lambda\nu}}{\partial x^{\mu}} -
\frac{\partial g_{\mu\lambda}}{\partial x^{\nu}}\Big).\end{aligned}$$ For the calculation of the Christoffel symbols we use the following metric tensor [@JW73] (see also [@DG00]) $$\begin{aligned}
\label{label3.4}
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}(t - z),\end{aligned}$$ where $\eta_{\mu\nu} = {\rm diag}(- 1, 1, 1, 1)$ and $h_{\mu\nu}(t -
z)$ is a symmetric tensor defined in the [*transverse traceless gauge*]{} with non–zero components $h_{xx}(t - z) = - h_{yy}(t - z)$ and $h_{xy}(t - z) = h_{yx}(t - z)$ [@JW73]. In terms of $h_{\mu\nu}(t - z)$ the Christoffel symbols are given by $$\begin{aligned}
\label{label3.5}
\Gamma^{\alpha}_{\lambda\mu} =
\frac{1}{2}\,\eta^{\alpha\nu}\,\Big(\frac{\partial
h_{\mu\nu}}{\partial x^{\lambda}} + \frac{\partial
h_{\lambda\nu}}{\partial x^{\mu}} - \frac{\partial
h_{\mu\lambda}}{\partial x^{\nu}}\Big) -
\frac{1}{2}\,h^{\alpha\nu}\,\Big(\frac{\partial h_{\mu\nu}}{\partial
x^{\lambda}} + \frac{\partial h_{\lambda\nu}}{\partial x^{\mu}} -
\frac{\partial h_{\mu\lambda}}{\partial x^{\nu}}\Big).\end{aligned}$$ The components of the Riemann tensor contributing to the equations of motion (\[label3.1\]) $$\begin{aligned}
\label{label3.6}
R^x_{0x0} &=& - \frac{d\Gamma^x_{0x}}{dt} - \Gamma^x_{x0}\Gamma^x_{0x}
- \Gamma^x_{y0}\Gamma^y_{0x}\quad,\quad R^x_{0y0} = -
\frac{d\Gamma^x_{0y}}{dt} - \Gamma^x_{x0}\Gamma^x_{0y} -
\Gamma^x_{y0}\Gamma^y_{0y},\nonumber\\ R^y_{0x0} &=& -
\frac{d\Gamma^y_{0x}}{dt} - \Gamma^y_{x0}\Gamma^x_{0x} -
\Gamma^y_{y0}\Gamma^y_{0x}\quad,\quad R^y_{0y0} = -
\frac{d\Gamma^y_{0y}}{dt} - \Gamma^y_{x0}\Gamma^x_{0y} -
\Gamma^y_{y0}\Gamma^y_{0y}.\end{aligned}$$ The Christoffel symbols read $$\begin{aligned}
\label{label3.7}
\hspace{-0.3in}\Gamma^x_{0x}&=& + \frac{1}{2}\,\frac{dh_{xx}}{dt} -
\frac{1}{4}\, \frac{d}{dt}(h^2_{xx} + h^2_{xy})\quad\quad\;,\;
\Gamma^y_{0y} = - \frac{1}{2}\,\frac{dh_{xx}}{dt} - \frac{1}{4}\,
\frac{d}{dt}(h^2_{xx} + h^2_{xy}),\nonumber\\
\hspace{-0.3in}\Gamma^x_{0y}&=& + \frac{1}{2}\,\frac{dh_{xy}}{dt} -
\frac{1}{2}\,h_{xx} h_{xy}\,\frac{d}{dt}{\ell
n}\Big(\frac{h_{xy}}{h_{xx}}\Big)\;,\; \Gamma^y_{0x} = +
\frac{1}{2}\,\frac{dh_{xy}}{dt} + \frac{1}{2}\,h_{xx}
h_{xy}\,\frac{d}{dt}{\ell n}\Big(\frac{h_{xy}}{h_{xx}}\Big).\end{aligned}$$ For the monochromatic gravitational waves the ratio $h_{xy}/h_{xx} =
\Delta_{\times}/\Delta_+$ is constant and the Christoffel symbols $\Gamma^x_{0y}$ and $\Gamma^y_{0x}$ are linear in $h_{ab}$.
For the calculation of the components of the Riemann tensor, defining the equations of motion (\[label3.1\]), we keep also the terms of order $O(h^2_{ab})$ inclusively and obtain $$\begin{aligned}
\label{label3.8}
R^x_{0x0} &=& - \frac{1}{2}\,\frac{d^2h_{xx}}{dt^2} +
\frac{1}{4}\,\frac{d^2}{dt^2}(h^2_{xx} + h^2_{xy}) -
\frac{1}{4}\,\Big[\Big(\frac{dh_{xx}}{dt}\Big)^2 +
\Big(\frac{dh_{xy}}{dt}\Big)^2\Big],\nonumber\\ R^x_{0y0} &=&
R^y_{0x0} = - \frac{1}{2}\,\frac{d^2h_{xy}}{dt^2} ,\nonumber\\
R^y_{0y0} &=&+ \frac{1}{2}\,\frac{d^2h_{xx}}{dt^2} +
\frac{1}{4}\,\frac{d^2}{dt^2}(h^2_{xx} + h^2_{xy}) -
\frac{1}{4}\,\Big[\Big(\frac{dh_{xx}}{dt}\Big)^2 +
\Big(\frac{dh_{xy}}{dt}\Big)^2\Big].\end{aligned}$$ Substituting (\[label3.8\]) in the equations of motion (\[label3.1\]) we get $$\begin{aligned}
\label{label3.9}
\ddot{x}&=&\frac{1}{2}\,\ddot{h}_{xx}(t)\,x +
\frac{1}{2}\,\ddot{h}_{xy}(t)\,y - \frac{1}{4}\,(\ddot{h^2}(t) -
\dot{h}^2(t))\,x, \nonumber\\ \ddot{y}&=&
\frac{1}{2}\,\ddot{h}_{xy}(t)\,x - \frac{1}{2}\,\ddot{h}_{xx}(t)\,y -
\frac{1}{4}\,(\ddot{h^2}(t) - \dot{h}^2(t))\,y,\end{aligned}$$ where overdots stand for the derivative with respect to time. We have denoted $h^2 = h^2_{xx} + h^2_{xy}$ and $\dot{h}^2 = \dot{h}^2_{xx} +
\dot{h}^2_{xy}$.
The equations of motion (\[label3.9\]) can be treated as the Lagrange equations derived from the Lagrange function $$\begin{aligned}
\label{label3.10}
L(t,x,y,\dot{x},\dot{y}) &=& \frac{1}{2}\,(\dot{x}^2 + \dot{y}^2) -
\frac{1}{8}\,(\ddot{h^2}(t) - \dot{h}^2(t))\,(x^2 + y^2)\nonumber\\
&&+ \frac{1}{4}\,\ddot{h}_{xx}(t)\,(x^2 - y^2) +
\frac{1}{2}\,\ddot{h}_{xy}(t)\,xy.\end{aligned}$$ In the polar coordinates $x = r\cos\Phi$ and $y = r\sin \Phi$ we get $$\begin{aligned}
\label{label3.11}
L(t, r, \varphi, \dot{r}, \dot{\Phi}) &=& \frac{1}{2}\,\dot{r}^2 -
\frac{1}{8}\,(\ddot{h^2}(t) - \dot{h}^2(t))\,r^2\nonumber\\ &&+
\frac{1}{2}\,r^2\dot{\Phi}^2 +
\frac{1}{4}\,r^2\,(\ddot{h}_{xx}(t)\,\cos 2\Phi +
\ddot{h}_{xy}(t)\,\sin 2\Phi).\end{aligned}$$ Assuming that the radius $r$ is almost constant we can factorize radial and angular degrees of freedom. $$\begin{aligned}
\label{label3.12}
L(t, r, \varphi, \dot{r}, \dot{\Phi}) &=& \frac{1}{2}\,\dot{r}^2 -
\frac{1}{8}\,(\ddot{h^2}(t) - \dot{h}^2(t))\,r^2\nonumber\\ &&+
R^2_0\,\Big[\frac{1}{2}\,\dot{\varphi}^2 +
\frac{1}{4}\,(\ddot{h}_{xx}(t)\,\cos 2\Phi + \ddot{h}_{xy}(t)\,\sin
2\Phi)\Big].\end{aligned}$$ where $R_0$ is the radius of the machine circumference, $R_0 =
C_0/2\pi$.
The equations of motion for the radius $r(t)$ and the azimuthal angle $\Phi(t)$ are equal to $$\begin{aligned}
\label{label3.13}
\ddot{r}(t) &=& - \frac{1}{4}\,(\ddot{h^2}(t) -
\dot{h}^2(t))\,r(t),\nonumber\\ \ddot{\Phi}(t)&=&-
\frac{1}{2}\,(\ddot{h}_{xx}(t)\,\sin 2\Phi(t) - \ddot{h}_{xy}(t)\,\cos
2\Phi(t)).\end{aligned}$$ Since $\Phi(t) \ll 1$, the solution reads $$\begin{aligned}
\label{label3.14}
\Phi(t) &=& \frac{1}{2}\,h_{xy}(t) =
\frac{1}{2}\,\Delta_{\times}\,\cos(\omega t + \delta),\nonumber\\
\dot{\Phi}(t) &=& \frac{1}{2}\,\dot{h}_{xy}(t) = -
\frac{1}{2}\,\Delta_{\times}\,\omega\,\sin(\omega t + \delta).\end{aligned}$$ For the frequencies of the gravitational wave background corresponding to the low–frequency limit $\omega \to 0$ we get $$\begin{aligned}
\label{label3.15}
\dot{\Phi}(t) = - \frac{1}{2}\,\Delta_{\times}\,\omega\,\sin\delta.\end{aligned}$$ This predicts the rotation of the machine circumference with a practically constant velocity in dependence on the polarization and phase of the gravitational wave background.
Assuming that $r(t)$ is a smooth function of $t$ and replacing the radius $r(t)$ by $R_0 = C_0/2\pi$ in the r.h.s. of (\[label3.13\]) we get $$\begin{aligned}
\label{label3.16}
\frac{1}{R_0}\,\frac{dr(t)}{dt} = - \frac{1}{4}\,\int^t_0
d\tau\,(\ddot{h^2}(\tau) - \dot{h}^2(\tau)) + D\end{aligned}$$ where a constant $D$ cancels all constant contributions to the r.h.s. of (\[label3.16\]).
For the relic monochromatic cylindrical gravitational wave $h_{xx} =
\Delta_+\,\cos(\omega t + \delta)$ and $h_{xy} =
\Delta_\times\,\cos(\omega t + \delta)$ the r.h.s. of (\[label3.15\]) is equal to $$\begin{aligned}
\label{label3.17}
\frac{1}{R_0}\,\frac{dr(t)}{dt} = \frac{1}{16}\,h^2_0\,\omega\,
\sin(2\omega t + 2\delta) - \frac{1}{8}\,h^2_0\omega^2 t.\end{aligned}$$ At leading order in the low–frequency limit $\omega \to 0$ we get $$\begin{aligned}
\label{label3.18}
\frac{1}{C_0}\frac{dC(t)}{dt} = \frac{1}{16}\,h^2_0\,\omega\,\sin
2\delta.\end{aligned}$$ This agrees fully with our result (\[label2.11\]) obtained within the hypothesis of the [*gravitational strain*]{}.
Some similar formulas calculated in this section one can find in the paper by van Holten [@JH99] devoted to the analysis of the cyclotron motion in a gravitational–wave background.
The fluctuations of the Friedmann–Roberson–Walker metric $h_{ab}(t -
z)$ we define as [@VR82]–[@MS00] $$\begin{aligned}
\label{label3.19}
ds^2 = a^2(t)( - dt^2 + dz^2 + dx^2 + dy^2 + h^{\rm gw}_{ab}(t -
z)dx^a dx^b),\end{aligned}$$ where $a(t)$ is a scale factor [@JW73]–[@JP99]. From (\[label3.19\]) we obtain [@BA88] $$\begin{aligned}
\label{label3.20}
h_{ab}(t - z) = a^2(t)h^{\rm gw}_{ab}(t - z).\end{aligned}$$ The explicit expression for the scale factor $a(t)$ depends on the epoch of the evolution of the Universe [@JW73]–[@JP99] (see also [@BA88]–[@BA96]). The relation (\[label3.20\]) does not contradict our estimate (\[label2.7\]).
Shrinkage of the machine circumference as diastrophism of the Earth crust
=========================================================================
In this Section we give arguments against a possibility to explain the observed shrinkage of the machine circumference of the SPring–8 storage ring by diastrophic tectonic forces or [*diastrophism of the Earth crust*]{}.
There are major forces acting within the Earth crust. They can be forces of [*compression*]{}, [*tension*]{} or [*shearing*]{}. They may be directly due to plate tectonics or caused by more localized or regionalized stresses. When these forces actually deform parts of the crust, the resulting landforms produced are said to have been formed by [*diastrophism*]{}. Diastrophism can cause [*uplifting*]{}, [*rifting*]{}, [*doming*]{}, and [*tilting*]{} of regions of the Earth surface. However, the major forms of diastrophism are associated with either [*folding*]{} or [*faulting*]{}. When these forces actually deform parts of the crust, the resulting landforms produced are said to have been formed by [*diastrophism*]{} [@D1].
Since the shrinkage of the machine circumference can be identified to some extent with the deformation of the part of the Earth crust, an alternative source of the systematic shrinkage could be, in principle, caused by [*diastrophism*]{}. First let us estimate the value of the [*part of the Earth crust*]{} which is undergone by [*diastrophism*]{}. The area occupied by the storage ring is equal to $S_0
= C^2_0/4\pi = 0.164\,{\rm km}^2$ with a radius $R_0 = C_0/2\pi =
0.229\,{\rm km}$. The deformation of this part of the Earth crust leads to the systematic shrinkage of the crust to the center of the machine circumference of the storage ring. Since neither [*faulting*]{} nor [*tilting*]{} can contribute to this shrinkage, so only [*folding*]{} of the Earth crust can be attracted to the explanation of this phenomenon.
Indeed, [*folding*]{} occurs when rocks buckle or fold due to horizontal or vertical pressure. They are shaped into an arch (called [*anticline*]{}) or a trough (called [*syncline*]{}), or they may override an adjacent fold [@D1].
However, the linear scale of the machine circumference of the storage ring $R_0$ is smaller compared with linear scales $L$ of tectonic forces providing [*moldings*]{} of the Earth crust, which are of order of a few kilometers, and, correspondingly $L \gg R_0$. Of course, the Global Positioning System (GPS) admits a measuring of motion of some parts of the Earth crust with a velocity comeasurable with the rate of the shrinkage of the machine circumference of the storage ring which is of order $2\times 10^{-4}\,{\rm m/yr}$ [@D1]. However, it is very unlikely that in such a seismic active country as Japan a center of tectonic forces, leading to the shrinkage of the machine circumference, would be localized with a great precision at the center of the machine circumference during more than 5 years.
Therefore, we can conclude that the observed shrinkage of the machine circumference of the SPring–8 storage ring, located in Japan, can be hardly caused by diastrophic tectonic forces. It seems extremely incredible that the diastrophic tectonic forces, discussed above, would really be able to produce a longer then few–year lasting folding of the Earth crust with the scale $D \simeq 0.458\,{\rm km}$ to the center of the machine circumference of the storage ring.
Thus, one can believe that the mechanism of the shrinkage of the machine circumference related to the gravitational–wave background seems to be more credible and probable with respect to any one caused by tectonic forces.
Shrinkage of the machine circumference and stiffness of the physical structures, governing the path of the beam
===============================================================================================================
In this section we discuss the influence of the forces, related to the [*stiffness*]{} of the physical structures of the storage ring, governing the path of the beam (mounts of magnets, for instance). In fact, one can imagine that the forces, induced by the [*stiffness*]{} of the physical structures of the storage ring, can prevent the machine circumference of the storage ring from the shrinkage caused by the relic gravitational–wave background. If it is so this should mean that the observed shrinkage of the machine circumference of the storage ring cannot be explained by the influence of the relic gravitational–wave background. Below we show that the forces, related to the [*stiffness*]{} of the physical structures of the storage ring, can be neglected for the analysis of the shrinkage of the machine circumference, caused by the relic gravitational–wave background.
Let us denote the forces, caused by the [*stiffness*]{} of the physical structures of the storage ring, as $\vec{F}_{\rm stiff}$. The observation of the fluctuations of the machine circumference, induced by the tidal and seasonal forces [@SD99; @MT00], assumes that the forces, produced by the [*stiffness*]{} of the physical structures of the storage ring, are smaller compared with the tidal and seasonal forces.
Since the seasonal forces are smaller compared with the tidal forces but have been measured experimentally by the change of the machine circumference, it is obvious that the forces, induced by the [*stiffness*]{} of the physical structures of the storage ring, should be smaller compared with the seasonal forces.
This can be written in the form of the inequality $$\begin{aligned}
\label{label5.1}
|\vec{F}_{\rm stiff}| \ll | \vec{F}_{\rm s}(\vec{r}\,)|,\end{aligned}$$ where the force $\vec{F}_{\rm s}(\vec{r}\,)$ is defined by [@SD99] $$\begin{aligned}
\label{label5.2}
\vec{F}_{\rm s}(\vec{r}\,) = - \bigtriangledown\,U_{\rm s}(\vec{r}\,).\end{aligned}$$ The potential $U_{\rm s}(\vec{r}\,)$, produced by the Sun, is given by [@SD99] $$\begin{aligned}
\label{label5.3}
U_{\rm s}(\vec{r}\,) = G_N M_{\odot}\,\Bigg(\frac{1}{|\vec{R}_s -
\vec{r}\,|} - \frac{1}{R_s} - \frac{\vec{r}\cdot
\vec{R}_s}{R^3_s}\Bigg),\end{aligned}$$ where $G_N = 6.636 \times 10^4\,{\rm m^3\,kg^{-1}\,yr^{-2}}$ [@DG00], $M_{\odot} = 1.989\times 10^{30}\,{\rm kg}$ is the mass of the Sun, $R_s = 1.496\times 10^{11}\,{\rm m}$ is the distance between centers of the Sun and the Earth, $|\vec{r}| = R_{\oplus} =
6.378\times 10^6\,{\rm m}$ is the radius of the Earth.
The rate of the change of the machine circumference, caused by the tidal and seasonal forces is of order of $|\Delta C/\Delta t| =
4\times 10^{-4}\,{\rm m/yr}$ [@SD99; @MT00]. The the experimental rate of the shrinkage of the machine circumference, $|\Delta C/\Delta
t| = 2\times 10^{-4}\,{\rm m/yr}$ [@MT00], is of the same order of magnitude. This implies that the forces, leading to the shrinkage of the machine circumference, can be of gravitational nature. Moreover, the forces, induced by the [*stiffness*]{} of the physical structures of the storage ring, governing the path of the beam, should be smaller compared with the forces responsible for the shrinkage.
In order to get a quantitative confirmation of this assertion we suggest to compare the energy densities of the seasonal forces and the relic gravitational–wave background. Following [@JW73]–[@JP99] we define the energy density of the seasonal forces and the gravitational–wave background as $$\begin{aligned}
\label{label5.4}
T^{\rm s}_{00} &\sim& \frac{1}{32\pi c^2 G_N}\,\Big\langle \Big(
\frac{2\pi}{T_s}\,U_{\rm s}(\vec{r}\,)\Big)^2\Big\rangle = \frac{\pi
G_N M^2_{\odot} R^4_{\otimes}}{40 c^2 T^2_s R^6_s},\nonumber\\ T^{\rm
gw}_{00} &=& \frac{c^2\omega^2(h^{\rm gw}_0)^2}{32\pi G_N},\end{aligned}$$ where $T_s = 0.5\,{\rm yr}$ is a period of the seasonal forces[^5].
Assuming that the energy density of the gravitational–wave background $T^{\rm gw}_{00}$ should be of the same order of magnitude as the energy density of the seasonal forces $T^{\rm s}_{00}$, $T^{\rm
gw}_{00} \sim T^{\rm s}_{00}$, we can get a constraint on the amplitude and frequency of the gravitational–wave background responsible for the observed shrinkage of the machine circumference [@MT00]. Setting $T^{\rm gw}_{00} \sim T^{\rm s}_{00}$ we obtain $$\begin{aligned}
\label{label5.5}
\omega h^{\rm gw}_0 \sim \frac{2\pi}{\sqrt{5}}\,\frac{G_N M_{\odot}
R^2_{\otimes}}{c^2 T_s R^3_s}\simeq 10^{-16}\,{\rm yr^{-1}}.\end{aligned}$$ For $\omega \ll 1\,{\rm yr^{-1}}$ the relation (\[label5.5\]) gives $h^{\rm gw}_0 \gg 10^{-16}$ that agrees with our estimate $h^{\rm
gw}_0 \gg 10^{-15}$ given in section 2.
This should testify that the [*stiffness*]{} of the physical structures of the storage ring, governing the path of the beam, can be neglected for the analysis of the shrinkage of the machine circumference, caused by the relic gravitational–wave background.
Stochastic relic gravitational–wave background
==============================================
In this section we analyse the shrinkage of the machine circumference [@MT00] coupled to the relic gravitational–wave background treated as a stochastic system [@BA88; @BA96; @SWB1]–[@SWB3]. We show that the suggested explanation of the shrinkage of the machine circumference, measured at the SPring–8 [@MT00], by the relic gravitational–wave background survives even if the storage ring interacts with the stochastic relic gravitational waves coming from all quarters of the Universe. This is related to the fact that the observed shrinkage of the machine circumference of the storage ring is an effect of the second order of the interaction of the gravitational waves with the machine circumference.
Below we consider spherical relic gravitational waves [@SWB3; @SGW1] converging to the center of the machine circumference[^6]. The relic gravitational waves are polarized in the $(\varphi_s\vartheta_s)$ plane, defined by unit vectors $\vec{e}_{\varphi_s}$ and $\vec{e}_{\vartheta_s}$ as it is shown in Fig.1, perpendicular to the direction of the propagation, which is anti–parallel to the unit vector $\vec{e}_r$. The polarization tensor, determined in the [*transverse traceless gauge*]{}, has the following non–vanishing components: $\Delta_{\varphi_s\varphi_s} = -
\Delta_{\vartheta_s\vartheta_s} = \Delta_+$ and $\Delta_{\varphi_s\vartheta_s} = \Delta_{\vartheta_s\varphi_s} =
\Delta_\times$.
It is convenient to analyse the influence of the stochastic relic gravitational–wave background on the shrinkage of the machine circumference in terms of the [*gravitational strain*]{} as it is done in Section 2.
The [*gravitational strain*]{} of the machine circumference, induced by the stochastic relic gravitational waves incoming from all quarters of the Universe, can be defined by $$\begin{aligned}
\label{label6.1}
\hspace{-0.3in}\delta C_{\rm gw}(t) = - \frac{C_0}{16\pi}\int
d\Omega_s \int^{2\pi}_0\langle( h_{xx}(t)\,\cos 2\varphi +
h_{xy}(t)\sin2\varphi)^2\rangle d\varphi,\end{aligned}$$ where $h_{xx}(t)$ and $h_{xy}(t)$ are given by $$\begin{aligned}
\label{label6.2}
h_{xx}(t) &=& \cos\vartheta_s\,(\Delta_+\,\cos 2\varphi_s +
\Delta_\times\,\sin 2\varphi_s)\,\cos\Big(\omega \Big(t -
\frac{R_0}{c}\Big) + \delta\Big),\nonumber\\ h_{xy}(t)&=&
\cos\vartheta_s\,(- \Delta_+\,\sin 2\varphi_s + \Delta_\times\,\cos
2\varphi_s)\,\cos\Big(\omega \Big(t - \frac{R_0}{c}\Big) +
\delta\Big)\end{aligned}$$ and $h_{yy}(t) = - h_{xx}(t)$. Below we neglect $R_0/c = 7.6\times
10^{-7}\,{\rm s}$, where $R_0 \simeq 229\,{\rm m}$ is the radius of the machine circumference, relative to the data–taking period $\tau$, which is about a few years [@MT00]. The quantities $h_{xx}(t)$, $h_{yy}(t)$ and $h_{xy}(t)$ are the projections of the components of the polarization tensor of the spherical relic gravitational wave on the plane of the machine circumference (see Fig.1). They depend on the angles $\vartheta_s$ and $\varphi_s$, which are the angle of the slope of the polarization plane of the gravitational wave relative to the plane of the machine circumference and the azimuthal angle, respectively (see Fig.1). At $\vartheta_s = \varphi_s = 0$ we get a gravitational wave equivalent to the cylindrical gravitational wave defined by (\[label2.1\]).
Integration over the angles $\vartheta_s$ and $\varphi_s$, where $d\Omega_s = \sin\vartheta_sd\vartheta_sd\varphi_s$, takes into account the contribution of the stochastic relic gravitational waves incoming on the plane of the machine circumference from all quarters of the Universe[^7]. Following [@BA96; @SWB1]–[@SWB3] we assume that the stochastic relic gravitational–wave background is isotropic.
{height="0.30\textheight"}
In Eq.(\[label6.1\]) the brackets $\langle \ldots\rangle$ mean $$\begin{aligned}
\label{label6.3}
\langle f\rangle = \int^{\infty}_0 d\omega\,S_h(\omega)\,f(\omega),\end{aligned}$$ where $S_h(\omega)$ is a spectral density, caused by the averaging over stochastic degrees of freedom of the relic gravitational–wave background [@SWB1]–[@SWB2]. We suppose that the spectral density $S_h(\omega)$ is normalized to unity.
The properties of the spectral density $S_h(\omega)$ depend on the theoretical model of the stochastic relic gravitational–wave background. We do not suggest any theoretic model of a stochastic gravitational–wave background and our approach to the description of the stochastic relic gravitational–wave background is phenomenological to full extent. The properties of the spectral density $S_h(\omega)$, such as a localization in the region of very low frequencies, $\omega \ll 1\,{\rm yr}^{-1}$, and so, we specify in terms of constraints on the averaged frequencies $\langle \omega
\rangle$ and $\langle \omega^2 \rangle$. These constraints come from the comparison of the experimental and theoretical rates of the shrinkage of the machine circumference, where the theoretical rate is defined by the interaction of the storage ring with the stochastic relic gravitational–wave background.
Substituting (\[label6.2\]) into (\[label6.1\]) we get $$\begin{aligned}
\label{label6.4}
\hspace{-0.3in}\delta C_{\rm gw}(t) &=& -
\frac{C_0}{16\pi}\int^{\infty}_0d\omega\,S_h(\omega)\,\cos^2(\omega t
+ \delta)\int d\Omega_s\,\cos^2\vartheta_s\nonumber\\
\hspace{-0.3in}&\times& \int^{2\pi}_0\,\Big[\Delta_+\,\cos 2(\varphi_s
+ \varphi) + \Delta_\times\,\sin 2(\varphi_s +
\varphi)\Big]^2\,d\varphi,\end{aligned}$$ Integrating over the angular variables we obtain the [*gravitational strain*]{}, induced by the stochastic spherical relic gravitational–wave background incoming on the plane of the machine circumference from all quarters of the Universe. It reads $$\begin{aligned}
\label{label6.5}
\hspace{-0.3in}\delta C_{\rm gw}(t) &=& -
\frac{4\pi}{3}\,\frac{1}{16}\,C_0h^2_0\int^{\infty}_0
d\omega\,S_h(\omega)\,\cos^2(\omega t + \delta),\end{aligned}$$ where $h_0 = \sqrt{\Delta^2_+ + \Delta^2_\times}$.
The relative rate of the shrinkage of the machine circumference can be defined by $$\begin{aligned}
\label{label6.6}
\frac{1}{C_0}\,\frac{\Delta C_{\rm gw}}{\Delta t} =
\frac{4\pi}{3}\,\frac{1}{16}\,h^2_0 \int^{\infty}_0
d\omega\,\omega\,S_h(\omega)\,\sin(2\omega t + 2\delta).\end{aligned}$$ In such a form the rate of the shrinkage of the machine circumference resembles two–point correlation functions of the operators of the gravitational waves appearing in the description of the relic gravitational–wave background as a stochastic system [@SWB1]–[@SWB3].
For the comparison of the theoretical rate of the change of the machine circumference (\[label6.6\]) with the experimental data one has to average the theoretical rate over the data–taking period $\tau$. This gives $$\begin{aligned}
\label{label6.7}
\frac{1}{C_0}\,\Big\langle \frac{\Delta C_{\rm gw}}{\Delta
t}\Big\rangle_{\tau} &=& \frac{4\pi}{3}\,\frac{1}{16}\,h^2_0
\int^{\infty}_0 d\omega\,\omega\,S_h(\omega)\frac{1}{\tau}\int^{+
\tau/2}_{-\tau/2}dt\,\sin(2\omega t + 2\delta) =\nonumber\\
&=&\frac{4\pi}{3}\,\frac{1}{16}\,h^2_0\,\sin2\delta \int^{\infty}_0
d\omega\,\omega\,S_h(\omega)\,\frac{\sin \omega \tau}{\omega \tau}.\end{aligned}$$ Since we deal with a relic gravitational–wave background, we suppose that the frequencies of the relic gravitational waves satisfy the constraint $\omega \tau \ll 1$.
For the validity of this constraint we have to assume that the spectral density $S_h(\omega)$ is localized in the region of frequencies of order of $\omega \ll 1\,{\rm yr}^{-1}$.
In the case of the dominance of the region $\omega \tau \ll 1$ in the integrand of the integral over $\omega$ in the r.h.s. of (\[label6.7\]), we can transcribe Eq.(\[label6.7\]) into the form $$\begin{aligned}
\label{label6.8}
\frac{1}{C_0}\,\Big\langle \frac{\Delta C_{\rm gw}}{\Delta
t}\Big\rangle_{\tau} = \frac{4\pi}{3}\,\frac{1}{16}\,h^2_0\, \langle
\omega \rangle\,\sin 2\delta,\end{aligned}$$ where $\langle \omega \rangle$ we determine as an averaged frequency of the stochastic relic gravitational–wave background weighted with the spectral density $S_h(\omega)$. It reads $$\begin{aligned}
\label{label6.9}
\langle \omega\rangle = \int^{\infty}_0 d\omega\,\omega\,S_h(\omega).\end{aligned}$$ The expression (\[label6.8\]) differs by a factor $4\pi/3$ from the rate of the shrinkage of the machine circumference given by Eq.(\[label2.11\]). Such a factor is caused by the summation over all directions of the relic gravitational waves coupled to the storage ring. The appearance of the factor $4\pi/3$ does not change significantly our estimates made below Eq.(\[label2.13\]).
Now from the comparison of the theoretical rate (\[label6.8\]) with the experimental one (\[label2.12\]) we get $\langle \omega \rangle
\gg 5\times 10^{-7}\,{\rm yr}^{-1}$. For the averaged period of oscillations of the machine circumference this gives $\langle T\rangle
\ll 10^{-2}\,{\rm Gyr}$.
The upper limit on the density parameter $\Omega_{\rm gw} \ll
10^{-10}$, given by (\[label2.15\]), is left unchanged for $\sqrt{\langle \omega^2\rangle} \ll 1\,{\rm yr}^{-1}$, which is not related to the factor $4\pi/3$. Remind that the constraint $\sqrt{\langle \omega^2\rangle} \ll 1\,{\rm yr}^{-1}$ is caused by the experimental fact that the period of the observed shrinkage of the machine circumference of the SPring–8 storage ring should be greater than 5 years [@MT00].
Of course, the constraints $\langle \omega \rangle \gg 5\times
10^{-7}\,{\rm yr}^{-1}$ and $\sqrt{\langle \omega^2\rangle} \ll
1\,{\rm yr}^{-1}$ can be justified only by the properties of the spectral density $S_h(\omega)$ within a certain theoretical model of the stochastic relic gravitational–wave background.
For the rate of the shrinkage of the machine circumference, given in terms of the stochastic relic gravitational–wave perturbations of the Friedmann–Robertson–Walker metric (\[label2.5\]), we obtain $$\begin{aligned}
\label{label6.10}
\Big\langle \frac{\Delta C_{\rm gw}}{\Delta t}\Big\rangle_{\tau} =
\frac{4\pi}{3}\,\frac{\pi}{8}\,R_{\rm U}\,(h^{\rm gw}_0)^2\, \langle
\omega \rangle\,\sin 2\delta,\end{aligned}$$ where $h^{\rm gw}_0 = \sqrt{(\Delta^{\rm gw}_+)^2 + (\Delta^{\rm
gw}_{\times})^2}$.
The r.h.s. of (\[label6.10\]) does not depend on the length of the the machine circumference. Therefore, the rate of the shrinkage of the machine circumference should be universal for all storage rings with any radii.
From the comparison of (\[label6.10\]) with (\[label6.8\]) one can conclude that the relation between the amplitudes $h_0$ and $h^{\rm
gw}_0$, given by Eq.(\[label2.7\]), is retained for the stochastic relic gravitational–wave background incoming on the plane of the machine circumference from all quarters of the Universe.
Thus, within our phenomenological approach to the description of the stochastic relic gravitational–wave background the interaction of the stochastic relic gravitational–wave background, incoming on the plane of the machine circumference from all quarters of the Universe, with the storage ring does not destroy the shrinkage of the machine circumference of the storage ring, observed in [@MT00]. Formally, this is due to the phenomenon of the shrinkage of the machine circumference is of the second order in gravitational wave interactions.
Conclusion
==========
The results obtained above should be understood as a hint that experimental analysis of fine variations of the machine circumferences of the storage rings can, in principle, contain an information about the relic gravitational–wave background on the same footing as the storage rings are sensitive to the tidal and seasonal forces [@LA95]–[@MT00].
We argue that if the systematic shrinkage of the machine circumference of the storage ring, observed at the SPring–8 [@MT00], is caused by the influence of the relic gravitational–wave background, the same effect should be measured for the machine circumference of the storage ring of any accelerator, for example, the LEP at CERN [@LA95], the ELSA at University of Bonn, the DAPHNE at Frascati, the VEPP–4 at Novosibirsk and others.
We have shown that the rate of the shrinkage of the machine circumference, represented in terms of the relic gravitational–wave perturbations of the Friedmann–Robertson–Walker metric, does not depend on the length of the machine circumference and should be universal for any storage ring with any radius (see Eqs.(\[label2.14\]) and (\[label6.10\])).
This makes very simple the experimental analysis of the validity of our hypothesis of the influence of the relic gravitational–wave background on the shrinkage of the machine circumference of the SPring–8 storage ring. Indeed, it is sufficient to measure the rates of the shrinkage of the machine circumferences of the storage rings of the LEP at CERN, the DAPHNE at Frascati, the VEPP–4 at Novosibirsk or of any other accelerators. If the rates of the shrinkage of the machine circumferences of the storage rings would have been found comeasurable with the value $ (\Delta C(t)/\Delta t)_{\exp} =
-\,2\times 10^{-4}\,{\rm m/yr}$, obtained for the SPring–8 storage ring [@MT00], this should testify the detection of the relic gravitational–wave background. Any negative result should bury the hypothesis.
We argue that the shrinkage of the machine circumference of the storage ring cannot be related to diastrophic tectonic forces. Then, since the value of the rate of the shrinkage of the machine circumference is comeasurable with the change of the machine circumference, induced by the seasonal forces, the influence of the [*stiffness*]{} of the physical structures of the storage ring, governing the path of the beam (mounts of magnets, for instance), can be neglected. In fact, as has been measured by Dat${\acute {\rm e}}$ and Kumagai [@SD99] and Takao and Shimada [@MT00] the forces, related to the [*stiffness*]{} of the physical structures of the storage ring, governing the path of the beam (mounts of magnets, for instance), are smaller compared with the seasonal forces.
We have solved Einstein’s equations of motion for the storage ring in the field of the cylindrical relic gravitational wave and computed the rate of the shrinkage of the machine circumference. We have shown that the rate of the shrinkage of the machine circumference, obtained from the solution of Einstein’s equations of motion for the storage ring in the field of the cylindrical relic gravitational wave, coincides with the rate obtained in terms of the [*gravitational strain*]{}. In addition to the shrinkage of the machine circumference we have found a slow rotation of the storage ring defined by the non–diagonal component of the polarization tensor of the relic gravitational wave in the [*transverse traceless gauge*]{}, $\Phi(t) = h_{xy}(t)/2$,
Finally we have discussed the interaction of the storage ring with a stochastic relic spherical gravitational–wave background. We have shown that, since the shrinkage of the machine circumference is a phenomenon of the second order in gravitational wave interactions, it cannot be destroyed even if one takes into account the contribution of the relic gravitational waves incoming on the plane of the machine circumference from all quarters of the Universe. We have obtained an additional factor $4\pi/3$ relative to the rate of the shrinkage of the machine circumference, induced by the cylindrical relic gravitational–wave background (\[label2.11\])[^8]. This changes only the lower limit of the frequencies of the gravitational waves responsible for the observed shrinkage.
Indeed, we get $\langle \omega \rangle \gg 5\times 10^{-7}\,{\rm
yr}^{-1}$ instead of $\langle \omega \rangle\gg 2\times 10^{-6}\,{\rm
yr}^{-1}$. We would like to emphasize that the factor $4\pi/3$ does not influence on the upper limit on the density parameter $\Omega_{\rm
gw} \ll 10^{-10}$, given by (\[label2.15\]) and agreeing well with predictions of all cosmological models [@JW73]–[@JP99] (see also [@BA96; @SWB1]–[@SWB3]).
The upper limit on the density parameter $\Omega_{\rm gw} \ll
10^{-10}$ is retained if $\sqrt{\langle \omega^2\rangle} \ll 1\,{\rm
yr}^{-1}$, which is caused by the experimental fact that the shrinkage of the machine circumference lasts longer than 5 years [@MT00].
Of course, the constraints $\langle \omega \rangle \gg 5\times
10^{-7}\,{\rm yr}^{-1}$ and $\sqrt{\langle \omega^2\rangle} \ll
1\,{\rm yr}^{-1}$ can be justified by the properties of the spectral density $S_h(\omega)$, defined by the theoretical model of the stochastic relic gravitational–wave background.
One can suppose that in the case of the validity of our explanation of the shrinkage of the machine circumference by the relic gravitational–wave background, the constraints on the averaged frequencies of the relic gravitational waves can be likely used to set “limits on a low–frequency cosmological spectrum”.
For the better understanding of the mechanism of the shrinkage of the machine circumference of the storage ring, caused by the relic gravitational–wave background, we recommend readers to consult the paper by Schin Dat${\acute {\rm e}}$ and Noritaka Kumagai [@SD99], suggested a nice physical explanation of fine variations of the machine circumference of the SPring–8 storage ring induced by the tidal forces.
Acknowledgement {#acknowledgement .unnumbered}
===============
We are grateful to Manfried Faber and Heinz Oberhummer for fruitful and stimulating discussions and reading the manuscript. We thank Schin Dat${\acute{\rm e}}$ for calling our attention to the data \[8,9\] and encouraging discussions. Discussions with Igor N. Toptygin and his comments on the results obtained in the manuscript are greatly appreciated. We thank Berthold Schoch and Wolfgang Hillert from University of Bonn for fruitful discussions. We thank Natalia Troitskaya for interesting comments on our results and discussions.
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[^1]: E–mail: ivanov@kph.tuwien.ac.at, Tel.: +43–1–58801–14261, Fax: +43–1–58801–14299
[^2]: Permanent Address: State Polytechnic University, Department of Nuclear Physics, 195251 St. Petersburg, Russian Federation
[^3]: E–mail: kobushkin@kph.tuwien.ac.at and kobushkin@bitp.kiev.ua
[^4]: Permanent Address: Bogoliubov Institute for Theoretical Physics, 03143, Kiev and Physical and Technical National University KPI, Prospect Pobedy 37, 03056, Kiev, Ukraine
[^5]: We have used for the estimate of the time–derivative of the potential $U_s(\vec{r}\,)$ the relation $|\dot{U}_{\rm s}(\vec{r}\,)| \sim \omega_s|U_{\rm s}(\vec{r}\,)|$, where $\omega_s = 2\pi/T_s$ is a characteristic frequency of the time–variations of the seasonal forces [@AM77].
[^6]: The spherical gravitational waves, converging to the center of the machine circumference, we define as [@SGW1]: $h_{ab}(t,|\vec{r} -
\vec{R}_0|) \sim \cos(\omega(t - |\vec{r} - \vec{R}_0|/c) +
\delta)/|\vec{r} - \vec{R}_0|$, where the vector $\vec{R}_0$ is the radius–vector of the machine circumference, located in the plane of the machine circumference $|\vec{R}_0| = R_0 = C_0/2\pi$, and $\vec{r}$ is the radius–vector of the observer. It is zero , $\vec{r}
= 0$, at the center of the machine circumference.
[^7]: It is assumed that the Earth is transparent for the relic gravitational waves.
[^8]: Of course, the stochastic relic gravitational–wave background would not produce a rotation of the machine circumference, which is linear in the gravitational wave $\Phi(t) = h_{xy}(t)/2$. Such a rotation has been obtained in Section 3 by solving Einstein’s equations of motion of the storage ring in the field of the cylindrical relic gravitational wave.
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abstract: 'In previous work we found that the spectral state switch and other spectral properties of both neutron star (NS) and galactic black hole candidates (GBHC), in low mass x-ray binary systems could be explained by a magnetic propeller effect that requires an intrinsically magnetic central compact object. In later work we showed that intrinsically magnetic GBHC could be easily accommodated by general relativity in terms of magnetospheric eternally collapsing objects (MECO), with lifetimes greater than a Hubble time, and examined some of their spectral properties. In this work we show how a standard thin accretion disk and corona can interact with the central magnetic field in atoll class NS, and GBHC and active galactic nuclei (AGN) modeled as MECO, to produce jets that emit radio through infrared luminosity $L_R$ that is correlated with mass and x-ray luminosity as $L_R \propto M^{0.75 - 0.92}L_x^{2/3}$ up to a mass scale invariant cutoff at the spectral state switch. Comparing the MECO-GBHC/AGN model to observations, we find that the correlation exponent, the mass scale invariant cutoff, and the radio luminosity ratios of AGN, GBHC and atoll class NS are correctly predicted, which strongly implies that GBHC and AGN have observable intrinsic magnetic moments and hence do not have event horizons.'
author:
- |
Stanley L. Robertson$^1$ and Darryl J. Leiter$^2$\
$^1$Physics Dept.,Southwestern Oklahoma State University, Weatherford, OK 73096, USA (stan.robertson@swosu.edu)\
$^2$FSTC, Charlottesville, VA 22901, USA (dleiter@aol.com)\
title: 'On the Origin of the Universal Radio-X-Ray Luminosity Correlation In Black Hole Candidates'
---
accretion, accretion disks–black hole physics–magnetic fields–X-rays: binaries– jets and outflows–Radio continuum
Introduction
============
In earlier work (Robertson & Leiter 2002, hereafter RL02) we extended analyses of magnetic propeller effects (Campana et al. 1998, Zhang, Yu & Zhang 1998) of neutron stars (NS) in low mass x-ray binaries (LMXB) to the domain of GBHC. From the luminosities at the low/high spectral state transitions, accurate rates of spin were found for NS and accurate quiescent luminosities were calculated for *both* NS and GBHC. The NS magnetic moments were found to be consistent with $\sim 10^{8 - 9}$G magnetic fields, in good agreement with those others have found (e.g. Bhattacharya 1995) from spin-down rates for similarly spinning 200 - 600 Hz millisecond pulsars. GBHC spins were found to be typically 10 - 50 Hz. Their magnetic moments of $\sim 10^{29}$ gauss cm$^3$ are $\sim 200$ times larger than those of ‘atoll’ class NS (e.g. Burderi et al. 2002, DiSalvo & Burderi 2003). The implied magnetic fields of GBHC are in good agreement with fields of $\sim 10^8$G that have been found at the base of the jets of GRS 1915+105 (Gliozzi, Bodo & Ghisellini 1999, Vadawale, Rao & Chakrabarti 2001) and in the accretion disk of Cygnus X-1 (Gnedin et al. 2003). At accretion disk inner radii corresponding to the low/high spectral state switch, the magnetic fields of both ‘atoll’ class NS and GBHC are $\sim 5\times10^7$G, which may account for some of their strong similarities (e.g. Yu et al. 2003, Tanaka & Shibazaki 1996, van der Klis 1994).
In later work (Leiter & Robertson 2003, Robertson & Leiter 2003, hereafter RL03) we have described how the Einstein field equations of General Relativity applied to compact plasmas with equipartition magnetic fields permit the existence of magnetic, eternally collapsing objects (MECO) that can have lifetimes in excess of a Hubble time. These highly redshifted, faintly (as distantly observed in quiescence) radiating objects can produce ‘ultrasoft’ thermal spectral peaks and the magnetic propeller effects found in RL02. Here we examine the accretion disk - magnetosphere interaction and show how the magnetosphere can drive jets. Our model should be applicable for any jet producing objects with sufficiently large magnetic moments, whether T-Tauri stars or NS, or GBHC and active galactic nuclei (AGN) modeled as MECO. In this context, the scaling of the magnetic moments of MECO with mass will be an important consideration. As shown in RL03, the MECO is dominated by a photon-photon collision generated pair plasma which is stabilized at high redshift deep inside the photon orbit by an Eddington limit radiation pressure generated by an equipartition magnetic field intrinsic to the MECO. The surface value of the MECO intrinsic magnetic field is calculated by equating the synchrotron generated photon pressure ($\propto B^4$) to the gravitational force per unit area, which is proportional to the density. Since the density is inversely proportional to the square of the MECO mass, $M$, the internal magnetic field scales as $M^{-1/2}$ and the MECO magnetic moment, $\mu$, scales as $M^{-1/2}(2GM/c^2)^3
\propto M^{5/2}$.
In the following, for NS, GBHC and AGN, we will assume the existence of a gas pressure dominated, geometrically thin accretion disk (Shakura & Sunyaev 1973). For gas pressure dominance, it has been shown (e.g., Merloni & Fabian 2002) that the hard x-ray spectral tail and reflection features of the low state spectrum can be adequately explained by reprocessing of the soft thermal disk photons in an accretion disk corona (ADC). The physical size of a corona is consistent with limits found for the source of the power-law x-ray emissions of LMXB (Church & Balucińska-Church 2003).
It has been suggested, however, (Markoff, Falcke & Fender 2001, Falcke, Körding & Markoff 2003) that the power-law x-ray emissions might originate in a jet. Flat or inverted spectrum synchrotron radio- infrared emissions are generally believed to originate in jets and low state jets have been resolved (Stirling et al. 2001) and studied over a wide range of luminosity variation (Corbel et al. 2000, 2003). As a result of these outflows, it has been pointed out (Fender, Gallo & Jonker 2003) that the low quiescent luminosities of GBHC cannot be taken as evidence of advective accretion flows (ADAF) through event horizons and as noted by Abramowicz, Kluzniak and Lasota (2002) there is presently no other observational evidence of event horizons.
Whether or not the x-rays originate in the jet, there is a strong coupling between x-ray and radio emissions that must be related to the accretion flow and jet structure. A universal low state radio / X-ray correlation ($L_R \propto L_x^{0.7}$) (Gallo, Fender & Pooley 2003) with a cutoff at the low/high state transition (Fender et al. 1999, Tannenbaum et al. 1972, Corbel et al. 2003) has been found for GBHC and NS ( Migliari et al. 2003). A similar radio / x-ray correlation (Merloni, Heinz & Di Matteo 2003, Falcke, Körding & Markoff, 2003) and its suppression at the transition to the high/soft state (Maccarone, Gallo & Fender 2003) have been shown to hold for AGN as well. These radio / X-ray luminosity correlations have been examined for scale invariant jets (Heinz & Sunyaev 2003, hereafter HS03), yielding constraints on the accretion processes. In the context of HS03, Merloni, Heinz & Di Matteo (2003) (Hereafter MHD03) have examined their correlation for compatibility with various accretion flow models and found better consistency with an ADAF / jet model than with radiatively efficient disk / jet or pure jet models.
An ADAF/jet model (Meier 2001) can also account for the low/high spectral state transition as a transition from an ADAF to a standard thin disk. It relies on a rapid black hole spin to provide energy to drive the jet. The model predicts that stable high/soft states would not exist for AGN more massive than $7 \times 10^4 M_\odot$ (Meier 2001) or, with more generous allowance for hysteresis effects, $\sim 4\times 10^6 M_\odot$ (Maccarone, Gallo & Fender 2003). The theoretical mass limit occurs because the Eddington scaled luminosity at which a thin disk (constrained to match a radiatively inefficient ADAF accretion rate) becomes radiation dominated is mass dependent. Since the high/soft state nevertheless appears to exist in AGN more massive than $6 \times 10^7 M_\odot$, the ADAF transition model cannot be regarded as established. Understanding the origin of the mass limit error of the model remains an open question (Maccarone, Gallo & Fender 2003).
Black hole models that rely entirely on the jet to produce the power-law x-ray emissions may have difficulties with constraints on the physical size of a jet. For dipping sources the size of the region of the low state power-law production has been found (Church & Balucińska-Church 2003) to be $\leq 10^9$ cm. In addition, there is the enigma of the size of the hard spectral producing region increasing while the jet dies in the high state. It is also unclear how black hole and NS behaviours could be so similar with the magnetic fields of even the weakly magnetized atoll class NS being capable of disrupting the inner accretion disk. On the other hand, we will show that our MECO model, with a radiatively efficient disk, will provide a superior fit to the radio / X-ray correlations and provide a mass scale invariant cutoff at the high/soft state transition while permitting radio-infrared and some of the x-ray luminosity to originate in a jet.
The Disk - Intrinsic Magnetic Moment Interaction
================================================
In the magnetic propeller model, the inner disk and magnetosphere radius, $r_m$, determines the spectral state. Very low to quiescent states correspond to an inner accretion disk radius outside the light cylinder. In the low/hard/radio-loud/jet-producing state of the active propeller regime, the inner disk radius lies between light cylinder and Keplerian co-rotation radii. Most, and perhaps all, of the accretion flow is ejected in the low/hard state. The high/soft state corresponds to an inner disk inside the co-rotation radius with the flow of accreting matter able to reach the central object where it produces an ultrasoft thermal spectral component. The cooling of the accretion disk corona and the former base of the jet by the soft photons also contributes to a softening of the x-ray spectrum. The whole complex of spectral state switch phenomena is related to the cessation or regeneration of magnetospherically driven outflow and presence or absence of dominant soft emissions from a central source.
The inner disk temperature is generally high enough to produce a very diamagnetic plasma at the magnetopause. Surface currents on the inner disk distort the magnetopause and they also substantially shield the trailing disk such that the region of strong disk-magnetosphere interaction is mostly confined to a ring or torus, of width $\delta r$ and half height $H$. This shielding leaves most of the disk under the influence of its own internal shear dynamo fields, (e.g. Balbus & Hawley 1998, Balbus 2003). At the inner disk radius the magnetic field of the central MECO is much stronger than the shear dynamo field generated within the inner accretion disk. In MHD approximation, the force density on the inner ring is $F_v = (\nabla \times B) \times B / 4\pi$. For simplicity, we assume coincident magnetic and spin axes of the central object and take this axis as the $z$ axis of cylindrical coordinates $(r,\phi,z)$.
The magnetic torque per unit volume of plasma in the inner ring of the disk that is threaded by the intrinsic magnetic field of the central object, can be approximated by $\tau_v=rF_{v\phi} =
r \frac{B_z}{4\pi} \frac{\partial B_{\phi}}{\partial z}
\sim r \frac{B_zB_{\phi}}{4\pi H}$, where $B_{\phi}$ is the average azimuthal magnetic field component. We stress that $B_{\phi}$, as used here, is an average toroidal magnetic field component. The toroidal component likely varies episodically between reconnection events (Goodson & Winglee 1999, Matt et al. 2002, Kato, Hayashi & Matsumato 2004, Uzdensky 2002).
The average flow of disk angular momentum entering the inner ring is $\dot{M}r v_k$, where $\dot{M}$ is mass accretion rate and $v_k$ is the Keplerian speed in the disk. This angular momentum must be extracted by the magnetic torque, $\tau$, hence: $$\tau = \dot{M}r v_k = r\frac{B_zB_{\phi}}{4\pi H}(4\pi r H \delta r).$$ In order to proceed further, we assume that $B_{\phi} = \lambda B_z$, $B_z=\mu/r^3$, and use $v_k=\sqrt{GM/r}$, where $\lambda$ is a constant, presumed to be of order unity, $\mu$ is the magnetic dipole moment of the central object $M$, its mass, and $G$, the Newtonian gravitational force constant. With these assumptions we obtain $$\dot{M} = (\frac{\lambda \delta r}{r}) \frac {\mu^2}{r^5 \omega_k}$$ where $\omega_k = v_k/r$ and the magnetopause radius, $r_m$ is given by $$r_m = (\frac{\lambda \delta r}{r})^{2/7}(\frac{\mu^4}{GM\dot{M}^2})^{1/7}$$ We scale the accretion rate to that needed to produce luminosity at the Eddington limit for a central object of mass $M$, and define $$\dot{m}= \frac{\dot{M}}{\dot{M}_{Edd}} ~\propto~ \frac{\dot{M}}{M}$$ and using $r_g=GM/c^2$ and eq. 3, we define $$\chi = r/r_g ~\propto~ \dot{m}^{-2/7}.$$
In order to estimate the size of the boundary region, $(\delta r /r)$, we normalize this disk-magnetosphere model to an average atoll NS (Table 1, RL02) of mass $M = 1.4 M_{\odot}$. The average rate of spin is $\sim 450$ Hz, the co-rotation radius is $\sim 30$ km, and the maximum luminosity for the low state is $GM\dot{M}/2r ~\sim 2 \times 10^{36}$ erg s$^{-1}$ From this we find $\dot{M} = 6.4 \times 10^{16} g s^{-1}$. Then for an average magnetic moment of $\sim 1.5 \times 10^{27}$ gauss cm$^3$, we find that ($\frac{\lambda \delta r}{r})^{2/7} \sim 0.3$. Thus $\xi = \frac{\lambda \delta r}{r} \sim 0.015$; i.e., the boundary region is suitably small, though likely larger than the scale height of the trailing disk. For later convenience, we define parameters $\beta$ and $\xi$ as $$\beta = \mu/M^3 ~~~~~~ \xi = \lambda \delta r /r$$ Then in terms of the variables defined so far, we can express the (reprocessed) disk luminosity as $$L_d = \frac{GM\dot{M}}{2r}=\xi \frac{\sqrt{GM} \mu^2}{2 r^{9/2}}=\frac{\xi \mu^2 \omega_k}{2r^3}
~\propto~ \beta^2 M^2 \dot{m}^{9/7}$$ At the co-rotation radius we reach the maximum luminosity, $L_c$, of the low/hard state, with Keplerian angular speed $\omega_k = \omega_s$, the angular speed of the magnetosphere, and $r_m=(GM/\omega_s^2)^{1/3}$. Thus $$L_c=\xi\frac{\mu^2 \omega_s^3}{2GM}$$ Two additional quantities needed for the analysis of the flow into the base of a jet are the scaling parameters for the poloidal magnetic field and the inner disk density. The magnetic field at the base of the jet is simply $$B_m = \frac{\mu}{r_m^3} ~\propto ~ \frac{\beta}{\chi_m^3}$$ For $\rho$, the density in the disk, we assume a standard ‘alpha’ disk, for which $\dot{M} = \rho 4 \pi r H v_r$. The radial inflow speed, $v_r$ is proportional to $v_s H/r$, where $v_s$ is sound speed in the disk. Using $H/r \propto v_s/v_k$, taking $v_s^2 \propto B^2/\rho$ and solving the mass flow rate equation for $\rho$ yields: $$\rho_m ~\propto~ \frac{\mu^2}{M r_m^5} ~\propto~ \frac{\beta^2}{\chi_m^5}$$
Mass Ejection and Radio Emission
--------------------------------
The radio luminosity of a jet is a function of the rate at which the magnetosphere can do work on the inner ring of the disk. This depends on the relative speed between the magnetosphere and the inner disk; i.e., $\dot{E} =\tau (\omega_s - \omega_k)$, or $$\dot{E} = \xi\frac{\mu^2 \omega_s (1 - \frac{\omega_k}{\omega_s})}{r^3}
~\propto~ \mu^2 M^{-3}\dot{m}^{6/7}\omega_s(1 - \frac{\omega_k}{\omega_s})$$
Disk mass, spiraling in quasi-Keplerian orbits from negligible speed at radial infinity must regain at least as much energy as was radiated away in order to escape. For this to be provided by the magnetosphere requires $\dot{E} \geq GM\dot{M}/2r$, from which $\omega_k \leq 2\omega_s/3$. Thus the magnetosphere alone is incapable of completely ejecting all of the accreting matter once the inner disk reaches this limit and the radio luminosity will be commensurately reduced and ultimately cut off. [^1]
The radio flux, $F_{\nu}$, of jet sources has a power law dependence on frequency of the form $$F_{\nu}~ \propto~ \nu^{-\alpha}$$ The spectral energy distributions of GBHC and AGN in radio-infrared show very little, if any, evolution in the low state during outbursts; i.e., $\alpha$ is essentially constant ($\alpha \approx -0.5$, radio; $-0.15$, IR, see e.g., Chaty et al. 2003). To determine the dependence of radio flux on $\mu$, $M$ and $\dot{m}$, we use the model and methods of HS03. Their analysis was based on a radiation transfer equation (Rybicki & Lightman 1979) which gives the radio flux from a jet viewed at right angle to the jet axis as $$F_{\nu}~\propto~ \int R(z)^2 j_{\nu}(z)[1-\exp{(-\tau_{\nu}(z))}]/\tau_{\nu}(z)dz$$ Here $z$ is a coordinate along the conical jet axis of symmetry, $R(z)$ is the radius of the jet, $j_{\nu}(z)$ is the optically thin synchrotron emissivity for a power law distribution of electrons over energy and $\tau_{\nu}(z)$ is the optical depth for a viewing angle perpendicular to the jet axis. Noting that $\tau_{\nu}(z)$ becomes huge below the shock at the base of the jet, the integral can be taken from $z \approx 0$ to $z \rightarrow \infty$. Both $j_{\nu}(z)$ and $\tau_{\nu}(z)$ depend on the magnetic field and density distributions along the jet. We assume that the magnetic field will be proportional to $B_m$ of eq. (9). The density after passage through the jet shock will remain proportional to $\rho_m$ of eq. (10). Thus we assume that $B(z)=\beta f(z)/\chi_m^3$ and $\rho(z) = g(z)\beta^2/\chi_m^5$, where f(z) and g(z) are distribution functions along the jet.
In order to evaluate the integral for $F_{\nu}$, it is helpful to scale $z$ and $R(z)$ to match the disk radius at the base of the jet nozzle. For this purpose, we define variables scaled the same as $\chi$; $\zeta = z \dot{m}^{-2/7}/r_g$ and $R_{\zeta}(\zeta)= R(z)\dot{m}^{-2/7}/r_g$. With this scaling, $R_{\zeta}$ can automatically always match $\chi_m$ at the base of the jet and: $$F_{\nu} \propto M^3\dot{m}^{6/7}\int_0^{\infty}
R_{\zeta}(\zeta)^2 j_{\nu}(\zeta)[1-\exp{(-\tau_{\nu}(\zeta))}]/\tau_{\nu}(\zeta)d\zeta$$ The integral above has magnetic field and density dependence only via $\beta = \mu/M^3$ in Equations (9) and (10). For given inner disk radius in units of $r_g$, having $B_m$ determined predominantly by the central object represents a case that was not considered in HS03. Nevertheless, using their method (and with notation adapted from their eq. (8) to the present case, $\phi_B= \beta$ and $\phi_C=\beta^2$) we obtain $\alpha$ from a differentiation of the logarithm of the integral with respect to $\ln(\nu)$. A second differentiation of $\ln{(\alpha)}$ with respect to $\ln{(\dot{m})}$ yields a zero because the MECO magnetic field, and thus $\beta$, is independent of $\dot{m}$, thus assuring that there is no low state spectral evolution as $\dot{m}$ changes. Further, following HS03, we assume scale invariance of the jet morphology. For given $\chi_m$, the integral is invariant with respect to $\dot{m}$.
The scaling of $B$ and $\rho$ satisfies the conditions for applicability of the method used by HS03 to obtain their eq. (10a). Then by similar method we obtain the dependence of $F_\nu$ on M and $F_{\nu} \propto \beta^q$, where $$q=\frac{13+2p+\alpha(p+6)}{p+4}.$$ Taking the canonical value of $p=2$, we obtain $q=(17+8\alpha)/6$ and for the accretion disk-intrinsic magnetic moment interaction and spectral index described by equations (1) through (14) we find $$F_{\nu} \propto M^{(17+2\alpha)/6} \dot{m}^{6/7} \beta^{(17+8\alpha)/6}\nu^{-\alpha}$$ If $\beta \propto M^{-1/2}$, as would apply for MECO AGN/GBHC this recovers the HS03 dependence of $F_{\nu} \propto M^{(17/12-\alpha /3)}$ from their eq. (10a), but for strict scale invariance of the integral, there is no further dependence on $\dot{m}$ here. This differs from the $\dot{m}$ dependence found by HS03 because the dominant magnetic field of the jet originates in the MECO rather than being generated in the accretion flow of the disk.
With $\mu$ in eq. (11) written in terms of $\beta$, a comparison with eq. (16) shows the radio flux to be proportional to $\dot{E}$. Thus we can take the integrated radio flux as luminosity, $L_R$, to be given by $$L_R = C' \dot{E} = C_o M^{(2\alpha -1)/6}\beta^{(5+8\alpha)/6}\dot{E}/\omega_s$$ where $C_o$ is a constant dependent on the radio bandwidth.
As noted by HS03, there will also be optically thin x-ray emission from the jet. In this case, taking $\tau_{\nu} << 1$ in eq. 14, we obtain what is essentially an integral over the jet source volume for the optically thin x-ray emission of the jet. Since $F_{\nu,x}$ depends on $\dot{m}$ in the same way as before, both radio-infrared and the jet part of the x-ray fluxes are proportional to $\dot{E}$. While the base of the jet contributes to the x-ray flux, its radiating volume is likely much smaller than that of the ADC, which produces most of the x-ray flux. The cutoff of this part of the x-ray flux and the onset of soft thermal emissions as the inner disk radius pushes inside corotation and the accretion flow reaches the central object marks the spectral state transition. The ADC actually grows in the high state (while producing a declining fraction of the x-ray luminosity) as it is cooled by photons from the central object.
Finally, we note that the degree of collimation of a jet actually appears to depend on the scale height and pressure of the corona (Kato, Minishige & Shibata 2004), but $F_\nu$ can still be calculated for a largely uncollimated outflow; for example, a large angle flow spreading out from the inner rings of the disk. In this case, we would obtain an integral similar to eq. 14. with $R_\zeta^2 d\zeta$ replaced by a column length parallel to the line of sight looking into the plasma and integrated over the area of the flow, projected perpendicular to the line of sight. Though there would be differences of numerical factors depending on viewing angle, scale invariance for given $r_m/r_g = \chi_m$ would still require all coordinates to be scaled in terms of $r_g$ and $\dot{m}^{2/7}$ in the same way as in eq. 14 and we would still obtain $F_\nu \propto \dot{E}$. Thus the scaling results we have obtained for magnetospherically driven outflows are very robust, even though there may be considerable uncertainty about the geometric details of the flow.
Radio - X-ray Correlation
-------------------------
Since $\dot{E} \propto r^{-3}$ and $L_d \propto r^{-9/2}$, it is apparent that we should expect radio luminosity, $L_R \propto L_d^{2/3}$. In particular we find $$L_R = C(M,\beta, \omega_s)2L_c^{1/3} L_d^{2/3}(1-\omega_k/\omega_s)$$ where $$C(M,\beta, \omega_s)=C_o M^{\frac{(2\alpha -1)}{6}}\beta^{\frac{(5+8\alpha)}{6}}/\omega_s$$ Strictly speaking, $L_d$ should be the bolometric luminosity of the disk, however, the x-ray luminosity over a large energy band is a very substantial fraction of the disk luminosity. To compare with the correlation exponent of 2/3 obtained here, recent studies, including noisy data for both GBHC and AGN have yielded $0.71 \pm 0.01$ (Gallo, Fender & Pooley 2003), 0.72 (Markoff et al. 2003, Falcke, Körding & Markoff 2003), $0.60 \pm 0.11$ (MHD03) and $0.64 \pm 0.09$ (Maccarone, Gallo & Fender 2003). For $\alpha$ in the range 0, -0.5, $\beta \propto M^{-1/2}$, $\omega_s \propto M^{-1}$ and $L_c \propto M$, the MECO model yields $C(M) \propto M^{(9-4\alpha)/12}$ and $$log L_R = (2/3)log L_x + (0.75 - 0.92) log M + const.$$ which is a better fit to the “fundamental plane” of MHD03 than any of the ADAF, disk/corona or disk/jet models they considered (see their Figure 5 for a $\chi^2$ density plot). In terms of Eddington scaled luminosities $L_{R,Edd} = L_R/L_{Edd}$ and $L_{d,Edd} = L_d/L_{Edd}$ eq. (18) can be written as $$L_{R,Edd} = C(M,\beta, \omega_s) 2L_{c,Edd}^{1/3}
L_{x,Edd}^{2/3}(1-\omega_k/\omega_s)$$
Another outstanding physical property of the MECO model for GBHC and AGN is that the Eddington scaled x-ray luminosity, $L_x/L_{Edd}$, which is $\propto L_c/M$ at the spectral state switch, is mass scale invariant. With $\mu$ scaling as $M^{2.5}$ and $\omega_s$ as $M^{-1}$, eq. (8) shows that $L_c/M$ is constant. $L_c/L_{Edd} \approx 0.02$ has been found for GBHC and AGN (Ghisellini & Celotti 2001, Maccarone, Gallo & Fender 2003), and remarkably, a similar value is found for atoll class NS (Tanaka & Shibazaki 1996, RL02). With this common ratio, the Eddington scaled radio luminosities of MECO objects will correlate with x-ray emissions at the low/high spectral state switch in proportion to $C(M,\beta, \omega_s)$. For $\alpha$ in the range 0, -0.5, $C(M, \beta, \omega_s) \propto M^{0.42 - 0.58}$ for MECO AGN/GBHC, in good agreement with the MHD03 correlation.
Even though they are not MECO the behavior of NS x-ray binaries can also be described by our model since they contain a central object with an intrinsic magnetic moment. According to eq. 18, the ratio of peak radio luminosity to $L_c$ is also just proportional to $C(M,\beta,\omega_s)
\sim \mu^b/(M^{3b-a} \omega_s)$, where $a=(2\alpha-1)/6$ and $b=(5+8\alpha)/6$. Using average values of $\mu, M$ and $\omega_s$ for the limited sample from Table 1 of RL02, and with $\alpha$ in the range 0, -0.5, we predict that GBHC should have peak radio / x-ray luminosity ratios that are 10 - 13 times larger than for atoll NS. Fender and Kuulkers (2001) compared the ratio of radio to x-ray luminosities at the radio peak for GBHC and NS. Omitting two GBHC and one NS that are abnormally radio loud the comparison between GBHC and atoll class NS ratio averages is $\sim 31/2.4 = 13$, from data in their Table 1.
Both $L_c$ and peak radio emissions for Z class NS are larger because their magnetic moments are about 10X larger than those of atolls, but $L_c$ increases by less than $10^2$, because on average they likely spin more slowly than atolls. These Z sources and abnormally loud radio sources, such as Cygnus X-3 can be explained with the magnetic model. It has been suggested that Cygnus X-3 is a NS (Brazier et al. 1990, Mitra 1998). Based on its apparent 79 Hz spin and spin down rate, it should have a magnetic moment about 13X that of an average MECO-GBHC and spin about 2.6X faster than an average MECO-GBHC (RL02). Given an adequate accretion supply, its low state steady radio emissions (not bubble events) could be $13^2 2.6^3 \sim 3000$ times stronger than an average MECO-GBHC.
If we let $x=L_d/L_c$, then for $x < 1$, corresponding to the low state, eq. (18) takes the form: $$L_R =C(M,\beta, \omega_s) 2L_c(x^{2/3} - x)$$ Neglecting $C(M,\beta, \omega_s)$, the function has a maximum value of $(0.3 L_c)$ at $x=0.3$. Scaling the function for $L_c$ at the cutoff or $0.3 L_c$ at the maximum is quite easy. An ‘eyeball’ fit to the data of Figure 1 provides a sense of the sharpness of the cutoff associated with $L_c$. Since $\omega_s$ is known for several NS from burst oscillations, eq. (8) has been used (RL02) to constrain their magnetic moments.
$L_c$, in eqs. (8) and (22), is an individual cutoff luminosity that depends on $\omega_s$, which varies, depending upon the average accretion rate produced by a binary companion or an AGN environment. Consequently the mass scale invariant cutoff ratio $L_c/ L_{Edd}$ has a random variability, but apparently within only a narrow range. At any epoch, it is likely that most GBHC and NS are near spin equilibrium; being neither spun up nor spun down by accretion on an average over a few outburst cycles. In high states they can be spun up by accretion while in low states they are spun down by magnetospherically driven outflows. If the magnetic fields of NS weaken with age, for whatever reason, then they would be spun up commensurately by accretion disks that can then more easily penetrate to smaller radii and drive the magnetosphere to higher Keplerian angular speeds. On the other hand the highly redshifted, slow collapse of a MECO stabilizes its intrinsic equipartition magnetic field and leads us to expect very little field decay for the MECO-GBHC. Since the MECO intrinsic magnetic field is determined solely by its mass, the correct mass scaling found here for MECO-GBHC/AGN, which was based on $\omega_s \propto M^{-1}$ without additional multipliers, suggests that MECO/AGN most likely are in a state of slow spin equilibrium.
Conclusions
===========
In a previous paper (RL02) we found that the spectral state switch and other spectral properties of low mass x-ray binaries, including both NS and GBHC, could be explained by a magnetic propeller effect that requires an intrinsically magnetized central object. Subsequently (RL03) we applied the Einstein field equations of General Relativity to the case of a highly compact, Eddington limited, pair dominated plasma with an intrinsic equipartition magnetic field. We found that the Einstein equations permit the existence of intrinsically magnetic, highly red shifted, extremely long lived, collapsing, radiating MECO objects that can produce the required propeller effects. In addition to accounting for the strong spectral similarities of NS and GBHC, the magnetosphere-accretion disk interaction associated with the MECO model has provided explanations for radio / x-ray luminosity correlations, the mass scale invariant spectral state switch phenomenon with its suppression of the radio jet outflow in the high/ soft state, the “ultrasoft” thermal peak and hard spectral tail of the high state, and, finally, the quiescient luminosities described as spin-down driven radiations.
In conclusion, we have shown here how a standard, thin, gas pressure dominated accretion disk and corona can interact with the central intrinsic magnetic moments of MECO-GBHC/AGN and NS in x-ray biniaries to drive low state jets. In the case of the MECO-GBHC/AGN the radio-infrared emissions of the jets have been found to correlate with the x-ray luminosity up to a mass scale invariant cutoff $L_c / L_{Edd}$ at the spectral state switch. In this context we obtained radio-infrared luminosities for MECO that vary as $M^{0.75-0.92}L_x^{2/3}$, consistent with observations of GBHC and AGN, and correctly predicted the observed relative radio luminosities of NS, GBHC, and AGN. While much detailed work remains to be done, the successful comparison of the MECO model predictions with observations strongly suggests that GBHC and AGN may have observable intrinsic magnetic moments anchored within them and hence they do not have event horizons.\
[**Acknowledgements**]{}\
We thank the anonymous referee for many comments and suggestions that have substantially improved this paper. We thank Elena Gallo for providing data for Figure 1. Useful information has been generously provided by Mike Church, Heino Falcke and Thomas Maccarone. We are very grateful to Abhas Mitra for many helpful discussions of gravitational collapse and pertinent astrophysical observations.
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[^1]: For very rapid inner disk transit through the co-rotation radius, fast relative motion between inner disk and magnetosphere can heat the inner disk plasma and strong bursts of radiation pressure from the central object may drive large outflows while an extended jet structure is still largely intact. This process has been calculated for inner disk radii inside the marginally stable orbit (Chou & Tajima 1999) using pressures and poloidal magnetic fields of unspecified origins. A MECO is obviously capable of suppling both the field and a radiation pressure. The hysteresis of the low/high and high/low state transitions may be associated with the need for the inner disk to be completely beyond the corotation radius before a jet can be regenerated after it has subsided.
|
---
abstract: 'Centuries of development in natural sciences and mathematical modeling provide valuable domain expert knowledge that has yet to be explored for the development of machine learning models. When modeling complex physical systems, both domain knowledge and data contribute important information about the system. In this paper, we present a data-driven model that takes advantage of partial domain knowledge in order to improve generalization and interpretability. The presented model, which we call EVGP (Explicit Variational Gaussian Process), uses an explicit linear prior to incorporate partial domain knowledge while using data to fill in the gaps in knowledge. Variational inference was used to obtain a sparse approximation that scales well to large datasets. The advantages include: 1) using partial domain knowledge to improve inductive bias (assumptions of the model), 2) scalability to large datasets, 3) improved interpretability. We show how the EVGP model can be used to learn system dynamics using basic Newtonian mechanics as prior knowledge. We demonstrate that using simple priors from partially defined physics models considerably improves performance when compared to fully data-driven models. simulate the systems.'
author:
- |
Daniel L. Marino, Milos Manic\
Department of Computer Science\
Virginia Commonwealth University\
Richmond, Virginia\
`marinodl@vcu.edu, misko@ieee.org`
title: 'Combining Physics-Based Domain Knowledge and Machine Learning using Variational Gaussian Processes with Explicit Linear Prior'
---
Introduction
============
For centuries, scientists and engineers have worked on creating mathematical abstractions of real world systems. This principled modeling approach provides a powerful toolbox to derive white-box models that we can use to understand and analyze physical systems. However, as the complexity of physical systems grow, deriving detailed principled models becomes an expensive and tedious task that requires highly experienced scientists and engineers. Moreover, incorrect assumptions leads to inaccurate models that are unable to represent the real system.
Data-driven black-box models provide an appealing alternative modeling approach that requires little to none domain knowledge. These models are fit to data extracted from the real system, minimizing the problems derived from incorrect assumptions. However, using data-driven models while completely ignoring domain knowledge may lead to models that do not generalize well and are hard to understand. Completely black-box approaches ignore the structure of the problem, wasting resources [@szegedy2015going] and making the model less explainable [@adadi2018peeking].
Gray-box models combine domain knowledge and data as both provide important and complementary information about the system. Domain knowledge can be used to construct a set of basic assumptions about the system, giving the data-driven model a baseline to build upon. Data can be used to fill the gaps in knowledge and model complex relations that were not considered by the domain expert.
In this paper, we explore an approach for embedding domain knowledge into a data-driven model in order to improve generalization and interpretability. The presented gray-box model, which we called EVGP (Explicit Variational Gaussian Process), is a scalable approximation of a sparse Gaussian Process (GP) that uses domain knowledge to define the prior distribution of the GP. In this paper domain knowledge is extracted from physics-based knowledge, however the EVGP can be applied to any domain.
The work on this paper has three corner stones (Fig. \[figure:overview\]): 1) Gaussian processes are used for learning complex non-linear behavior from data and model uncertainty, 2) Partial domain knowledge is used as prior in order to improve inductive bias, 3) Variational Inference is used to find an approximation that scales well to large datasets. Inductive bias refers to the assumptions made by the model when doing predictions over inputs that have not been observed. The presented approach provides uncertainty estimations which are fundamental in order to avoid the risk associated with overconfidence in unexplored areas [@frigola2014variational] and warns the user of possible incorrect estimations [@marino2017data].
![Illustration of the EVGP model. []{data-label="figure:model_intuition"}](overview.pdf)
![Illustration of the EVGP model. []{data-label="figure:model_intuition"}](model_intuition_v2.pdf)
The work in this paper is highly applicable when: 1) modeling physical systems with uncertainty estimations, 2) partial domain knowledge of the system is available, 3) large quantities of data are available. The aim is to help the engineer and take advantage of available knowledge without requiring the derivation of complex and detailed models. Instead, an engineer only has to provide simple, partially formed, models and the EVGP takes care of filling the gaps in knowledge. We show how the EVGP model can be used to learn system dynamics using basic physics laws as prior knowledge.
Variational GP using explicit features {#section:evgp}
======================================
The EVGP model is designed to solve regression problems under uncertainty. Given a dataset ${{\cal D}}= ({{ {\boldsymbol{x}} }}^{(i)} , {{ {\boldsymbol{y}} }}^{(i)} ) $ composed of input/output pairs of samples $ { \left( {{ {\boldsymbol{x}} }}^{(i)} , {{ {\boldsymbol{y}} }}^{(i)} \right) }$, we would like to obtain a predictive distribution $p({{ {\boldsymbol{y}} }}|{{ {\boldsymbol{x}} }}, {{\cal D}})$ that estimates the value of the output ${{ {\boldsymbol{y}} }}$ for a given input ${{ {\boldsymbol{x}} }}$. The EVGP model approximates $p({{ {\boldsymbol{y}} }}|{{ {\boldsymbol{x}} }}, {{\cal D}})$ using Variational Inference. The EVGP is defined as a distribution $p({{ {\boldsymbol{y}} }}|{{ {\boldsymbol{x}} }}, w)$ where $w$ are a set of parameters with prior distribution $p(w)$.
In the following sections we describe in detail: *A)* the EVGP model, *B)* the variational loss function used to train the model, *C)* the predictive distribution that approximates $p({{ {\boldsymbol{y}} }}|{{ {\boldsymbol{x}} }}, {{\cal D}})$.
Model Definition
----------------
\[section:model\_definition\] The EVGP model takes the following form: $$\begin{aligned}
y = g({{ {\boldsymbol{x}} }}) + \epsilon_y ; \;\;\;
{ { g { \left( {{ {\boldsymbol{x}} }}\right) } } } = { { h { \left( {{ {\boldsymbol{x}} }}\right) } } }^T {{{ {\boldsymbol{\beta}} }}}+ { { f { \left( {{ {\boldsymbol{x}} }}\right) } } } \label{eq:evgp}
\end{aligned}$$ where ${ { f { \left( {{ {\boldsymbol{x}} }}\right) } } } \sim { {\cal GP} { \left( 0, { { k { \left( {{ {\boldsymbol{x}} }}, {{ {\boldsymbol{x}} }}' \right) } } } \right) } }$ is a Gaussian process with kernel $k$, $\epsilon_y \sim { {\cal N} \left( 0, {{ {\boldsymbol{\Sigma}} }}_y \right) }$ is the observation noise and $g({{ {\boldsymbol{x}} }})$ is the denoised prediction for the input ${{ {\boldsymbol{x}} }}$. Figure \[figure:model\_intuition\] offers a visual representation of the model. The following is the description of the main components of the EVGP model:
- Domain knowledge is embedded in the explicit function ${ { h { \left( {{ {\boldsymbol{x}} }}\right) } } }^T {{{ {\boldsymbol{\beta}} }}}$, parameterized by ${{{ {\boldsymbol{\beta}} }}}$. The function ${ { h { \left( {{ {\boldsymbol{x}} }}\right) } } }$ describes a set of explicit features (hence the name of our method) provided by the domain expert. ${{{ {\boldsymbol{\beta}} }}}$ is modeled using a normal distribution with a prior that is also extracted from domain knowledge. In this paper, $h({{ {\boldsymbol{x}} }})^T {{{ {\boldsymbol{\beta}} }}}$ is derived from partially defined Neutonian mechanics.
- The Gaussian Process $f({{ {\boldsymbol{x}} }})$ adds the ability to learn complex non-linear relations that ${ { h { \left( {{ {\boldsymbol{x}} }}\right) } } }^T {{{ {\boldsymbol{\beta}} }}}$ is unable to capture.
Given a dataset ${{\cal D}}$, the exact predictive distribution $p(y|x, {{\cal D}})$ for the model in Eq. [(\[eq:evgp\])]{} is described in [@rasmussen2004gaussian]. For the rest of the paper, we refer to the exact distribution as EGP. Computing the EGP predictive distribution has a large computational cost and does not scale well for large datasets. To alleviate this problem, sparse approximation methods [@quinonero2005unifying] use a small set of $m$ inducing points $({{ {\boldsymbol{f}} }}_m, {{ {\boldsymbol{X}} }}_m)$ instead of the entire dataset to approximate the predictive distribution.
In order to construct a sparse approximation for the model in Eq. [(\[eq:evgp\])]{}, we use a set of $m$ inducing points $({{ {\boldsymbol{f}} }}_m, {{ {\boldsymbol{X}} }}_m)$ as parameters that will be learned from data. Given $({{ {\boldsymbol{f}} }}_m, {{ {\boldsymbol{X}} }}_m)$ and a set of test points $({{ {\boldsymbol{g}} }}, {{ {\boldsymbol{X}} }})$, the prior distribution of the model can be expressed as follows:
-- --
-- --
where ${{ {\boldsymbol{X}} }}$ denotes the data matrix, where each row represents an individual sample. The rows of ${{ {\boldsymbol{H}} }}_x$ represent the value of the function $h()$ applied to the real samples ${{ {\boldsymbol{X}} }}$. The rows of ${{ {\boldsymbol{H}} }}_m$ represent the value of the function $h()$ applied to the inducing (learned) points ${{ {\boldsymbol{X}} }}_m$. Using the conditional rule for multivariate Gaussian distributions, we obtain the definition of the denoised sparse EVGP model: $$\begin{aligned}
{ { p { \left( {{ {\boldsymbol{g}} }}\mid {{ {\boldsymbol{X}} }}, {\omega}\right) } } } \sim &
{ {\cal N} \left( {{ {\boldsymbol{H}} }}_x {{{ {\boldsymbol{\beta}} }}}+ {{ {\boldsymbol{\mu}} }}_{{{ {\boldsymbol{f}} }}\mid {\omega}},
{{ {\boldsymbol{\Sigma}} }}_{{{ {\boldsymbol{f}} }}\mid {\omega}} \right) } \label{eq:g_given_w}\end{aligned}$$ where ${\omega}= {{\left\{{{ {\boldsymbol{f}} }}_m, {{{ {\boldsymbol{\beta}} }}}\right\}}}$ are the parameters of our model, ${{ {\boldsymbol{\mu}} }}_{{{ {\boldsymbol{f}} }}\mid {\omega}}
= {{ {\boldsymbol{K}} }}_{{{ {\boldsymbol{x}} }}m}{{ {\boldsymbol{K}} }}_{mm}^{-1}{{ {\boldsymbol{f}} }}_m$, and ${{ {\boldsymbol{\Sigma}} }}_{{{ {\boldsymbol{f}} }}\mid {\omega}}
= {{ {\boldsymbol{K}} }}_{{{ {\boldsymbol{x}} }}{{ {\boldsymbol{x}} }}} - {{ {\boldsymbol{K}} }}_{{{ {\boldsymbol{x}} }}m} {{ {\boldsymbol{K}} }}_{mm}^{-1} {{ {\boldsymbol{K}} }}_{mx}$. Equation [(\[eq:g\_given\_w\])]{} defines our scalable EVGP model. In following sections we give prior distributions to the parameters ${\omega}$ and perform approximate Bayesian inference. Note that Eq. [(\[eq:g\_given\_w\])]{} is also conditioned on ${{ {\boldsymbol{X}} }}_m$, however we do not indicate this explicitly in order to improve readability.
Variational Loss {#variational-lower-bound}
----------------
In this section we present the loss function that we use to fit our model. In this paper we follow a Variational Bayesian approach (see Appendix \[section:variational\_inference\] for a brief overview). Given a training dataset ${{\cal D}}$ and a prior distribution $p({\omega})$, we wish to approximate the posterior distribution $p({\omega}|{{\cal D}})$. The posterior of ${\omega}$ is approximated using a variational distribution ${ { p_\phi { \left( w \right) } } } \approx p({\omega}|{{\cal D}})$ parameterized by $\phi$. For the EVGP parameters ${\omega}= {{\left\{{{ {\boldsymbol{f}} }}_m, {{{ {\boldsymbol{\beta}} }}}\right\}}}$, we use the following variational posterior distributions: $$\begin{aligned}
{ { p_\phi { \left( {{ {\boldsymbol{f}} }}_m \right) } } } = { {\cal N} \left( {{ {\boldsymbol{f}} }}_m | {{ {\boldsymbol{a}} }}, {{ {\boldsymbol{A}} }}\right) } ; \;\;\;
{ { p_\phi { \left( {{{ {\boldsymbol{\beta}} }}}\right) } } } = { {\cal N} \left( {{{ {\boldsymbol{\beta}} }}}| {{ {\boldsymbol{b}} }}, {{ {\boldsymbol{B}} }}\right) }\end{aligned}$$
The prior-distributions for ${\omega}$ are also defined as multivariate normal distributions: $$\begin{aligned}
{ { p { \left( {{ {\boldsymbol{f}} }}_m \right) } } } = { {\cal N} \left( {{ {\boldsymbol{f}} }}_m \mid 0, {{ {\boldsymbol{K}} }}_{mm} \right) } ; \;\;\;
{ { p { \left( {{{ {\boldsymbol{\beta}} }}}\right) } } } = { {\cal N} \left( {{{ {\boldsymbol{\beta}} }}}\mid {{ {\boldsymbol{\mu}} }}_\beta, {{ {\boldsymbol{\Sigma}} }}_{\beta} \right) }\end{aligned}$$ these prior distributions represent our prior knowledge, i.e. our knowledge before looking at the data.
Given the training dataset ${{\cal D}}= ({{ {\boldsymbol{y}} }}, {{ {\boldsymbol{X}} }})$, the parameters $\phi$ of ${ { p_\phi { \left( {\omega}\right) } } }$ are learned by minimizing the negative Evidence Lower Bound (ELBO). For the EVGP, the negative ELBO takes the following form: $$\begin{aligned}
{ { {\mathcal{L}}{ \left( \phi \right) } } }
= & - \log { {\cal N} \left( {{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{H}} }}_x {{ {\boldsymbol{b}} }}+ {{ {\boldsymbol{K}} }}_{xm} {{ {\boldsymbol{K}} }}_{mm}^{-1} {{ {\boldsymbol{a}} }}, {{ {\boldsymbol{\Sigma}} }}_y \right) }
\nonumber \\
& + \dfrac{1}{2} { \left[
{\text{Tr}{ \left( {{ {\boldsymbol{M}} }}_1 {{ {\boldsymbol{A}} }}\right) }}
+ {\text{Tr}{ \left( {{ {\boldsymbol{M}} }}_2 {{ {\boldsymbol{B}} }}\right) }}
+ {\text{Tr}{ \left( {{ {\boldsymbol{\Sigma}} }}_y^{-1} {{ {\boldsymbol{\Sigma}} }}_{{{ {\boldsymbol{f}} }}\mid {\omega}} \right) }}
\right] } \nonumber \\
& + {\mathcal{L}}_{KL}
\label{eq:elbo}\end{aligned}$$ where ${{ {\boldsymbol{M}} }}_1 = { \left( {{ {\boldsymbol{K}} }}_{mm}^{-1} {{ {\boldsymbol{K}} }}_{mx} \right) } {{ {\boldsymbol{\Sigma}} }}_y^{-1} { \left( {{ {\boldsymbol{K}} }}_{xm} {{ {\boldsymbol{K}} }}_{mm}^{-1} \right) }$, and ${{ {\boldsymbol{M}} }}_2 = {{ {\boldsymbol{H}} }}_x^T {{ {\boldsymbol{\Sigma}} }}_y^{-1} {{ {\boldsymbol{H}} }}_x$. The term ${\mathcal{L}}_{KL}$ is the KL-divergence between the posterior and prior distributions for the parameters: $$\begin{aligned}
{\mathcal{L}}_{KL} = &
D_{KL}{ \left( { {\cal N} \left( {{ {\boldsymbol{a}} }}, {{ {\boldsymbol{A}} }}\right) } \mid \mid { {\cal N} \left( 0, {{ {\boldsymbol{K}} }}_{mm} \right) } \right) }
+ D_{KL}{ \left( { {\cal N} \left( {{ {\boldsymbol{b}} }}, {{ {\boldsymbol{B}} }}\right) } \mid\mid { {\cal N} \left( {{ {\boldsymbol{\mu}} }}_{\beta}, {{ {\boldsymbol{\Sigma}} }}_{\beta} \right) } \right) }
\end{aligned}$$
A detailed derivation of the variational loss in Eq. [(\[eq:elbo\])]{} is presented in Appendix \[appendix:elbo\]. The negative ELBO (Eq. \[eq:elbo\]) serves as our loss function to learn the parameters $\phi$ given the training dataset ${{ {\boldsymbol{y}} }}, {{ {\boldsymbol{X}} }}$. In order to scale to very large datasets, the ELBO is optimized using mini-batches (see Appendix \[appendix:minibatches\]). In our case, the parameters of the variational approximation are: $\phi={{{ {\boldsymbol{a}} }}, {{ {\boldsymbol{A}} }}, {{ {\boldsymbol{b}} }}, {{ {\boldsymbol{B}} }}, {{ {\boldsymbol{X}} }}_m}$.
Predictive distribution {#section:evgp-prediction}
-----------------------
After learning the parameters $\phi$, we would like to provide estimations using the approximated variational distribution ${ { p_\phi { \left( {\omega}\right) } } }$. Given a set of test points ${\hat{{{ {\boldsymbol{X}} }}}}$, the estimated denoised predictive distribution is computed as an expectation of Eq. [(\[eq:g\_given\_w\])]{} w.r.t. ${ { p_\phi { \left( {\omega}\right) } } }$:
[cc]{}
Note that ${\hat{{{ {\boldsymbol{g}} }}}}$ is just a denoised version of ${\hat{{{ {\boldsymbol{y}} }}}}$. Eq. [(\[eq:prediction\])]{} approximates $p({\hat{{{ {\boldsymbol{g}} }}}}| {\hat{{{ {\boldsymbol{x}} }}}}, {{\cal D}})$, using the learned distribution ${ { p_\phi { \left( {\omega}\right) } } }$ (see Appendix \[appendix:elbo\]). The result is equivalent to [@titsias2009variational] with the addition of ${{ {\boldsymbol{H}} }}_x {{ {\boldsymbol{b}} }}$ for the mean and ${{ {\boldsymbol{H}} }}_{{\hat{x}}} {{ {\boldsymbol{B}} }}{{ {\boldsymbol{H}} }}_{{\hat{x}}}^T$ for the covariance. These additional terms include the information provided by the prior function ${{ {\boldsymbol{H}} }}_x$ with the parameters ${{ {\boldsymbol{b}} }}$ and ${{ {\boldsymbol{B}} }}$ that were learned from data.
The approximated predictive distribution with observation noise is the following: $$\begin{aligned}
{ { p_\phi { \left( {\hat{{{ {\boldsymbol{y}} }}}}\mid {\hat{{{ {\boldsymbol{X}} }}}}\right) } } } =
{ {\cal N} \left( {\hat{{{ {\boldsymbol{g}} }}}}\middle\vert {{ {\boldsymbol{\mu}} }}_{{\hat{{{ {\boldsymbol{g}} }}}}|{\hat{{{ {\boldsymbol{x}} }}}}},
{{ {\boldsymbol{\Sigma}} }}_{{\hat{{{ {\boldsymbol{g}} }}}}|{\hat{{{ {\boldsymbol{x}} }}}}} + {{ {\boldsymbol{\Sigma}} }}_y \right) } \nonumber
\end{aligned}$$ where ${ { p_\phi { \left( {\hat{{{ {\boldsymbol{y}} }}}}\mid {\hat{{{ {\boldsymbol{X}} }}}}\right) } } } \approx p({\hat{{{ {\boldsymbol{y}} }}}}\mid {\hat{{{ {\boldsymbol{x}} }}}}, {{\cal D}})$. In the next section, we show how Eq. [(\[eq:prediction\])]{} can be used to model system dynamics and predict the next state of a physical system given the control input and current state.
Embedding physics-based knowledge {#section:physics-evgp}
=================================
In this paper, we apply the EVGP model to learn the dynamics of a physical system. The state ${{{ {\boldsymbol{z}} }}}_{[t]}$ of the physical system can be modeled as follows: $$\begin{aligned}
{{{ {\boldsymbol{z}} }}}_{[t+1]} &\sim g({{ {\boldsymbol{z}} }}_{[t]} \oplus {{ {\boldsymbol{u}} }}_{[t]}) \label{eq:dynamic_evgp}\\
{{{ {\boldsymbol{y}} }}}_{[t]} &\sim {{ {\boldsymbol{z}} }}_{[t]} + \epsilon_y \nonumber\end{aligned}$$ where ${{{ {\boldsymbol{u}} }}}_{[t]}$ is the control input and ${{{ {\boldsymbol{y}} }}}_{[t]}$ is the measured output of the system. The symbol $\oplus$ denotes concatenation and ${{ {\boldsymbol{x}} }}_{[t]} = {{{ {\boldsymbol{z}} }}}_{[t]} \oplus {{{ {\boldsymbol{u}} }}}_{[t]}$ is the input to the EVGP model. For example, in the case of a mechatronic system: ${{{ {\boldsymbol{u}} }}}_{[t]}$ are the forces applied by the actuators (e.g. electric motors); ${{{ {\boldsymbol{z}} }}}_{[t]}$ is the position and velocity of the joints; ${{{ {\boldsymbol{y}} }}}_{[t]}$ is the output from the sensors.
We assume independent EVGP models for each output ${{{ {\boldsymbol{y}} }}}_{[t]}$ in the equation [(\[eq:dynamic\_evgp\])]{}. The function $g()$ in Eq. [(\[eq:dynamic\_evgp\])]{} is modeled using the EVGP model from Eq. [(\[eq:evgp\])]{} and Eq. [(\[eq:prediction\])]{}. In the following sections we present how we can use simple Newtonian mechanics to define useful priors $h({{ {\boldsymbol{x}} }})^T {{ {\boldsymbol{\beta}} }}$ for the EVGP model.
Priors from simple Newtonian dynamics {#multi-body-dynamics}
-------------------------------------
Figure \[figure:pendulum\] shows a simple example of a single rigid-body link. The simplest model that we can use for this system comes from Newton’s second law $u = J {\ddot{ q_1 }}$, where $u$ is the torque applied to the system, $J$ is the moment of inertia, and $q_1$ is the angle of the pendulum. Using Euler discretization method, we obtain the following state-space representation that serves as the prior $h({{ {\boldsymbol{x}} }})^T {{ {\boldsymbol{\beta}} }}$ for our EVGP model: $$\begin{aligned}
{ \begin{bmatrix} q_1[t+1] \\ {\dot{ q_1 }}[t+1] \end{bmatrix} } =
{ \begin{bmatrix} q_1[t] \\ {\dot{ q_1 }}[t] \end{bmatrix} } +
\Delta{t} { \begin{bmatrix} {\dot{ q_1 }}[t] \\ \dfrac{1}{J}u[t] \end{bmatrix} }
=
\underbrace{
{ \begin{bmatrix} 1 & \Delta{t} & 0 \\
0 & 1 & \Delta{t}/J \end{bmatrix} }}_{{{ {\boldsymbol{\beta}} }}\text{ prior mean } {{{ {\boldsymbol{\mu}} }}_\beta}}
\underbrace{
{ \begin{bmatrix} q_1[t] \\ {\dot{ q_1 }}[t] \\ u[t] \end{bmatrix} }}_{h({{ {\boldsymbol{x}} }}_{[t]})}
\label{eq:inertia_prior}
\end{aligned}$$ We refer to this prior as IF (inertia+force). $\Delta {t}$ is the discretization time and ${ \begin{bmatrix} q_1[t] & {\dot{ q_1 }}[t] \end{bmatrix} }^T$ is the state ${{ {\boldsymbol{z}} }}_{[t]}$ of the system. The IF prior in Eq. [(\[eq:inertia\_prior\])]{} does not include gravitational effects. Gravity pulls the link to the downward position with a force proportional to $\sin{q_1}$. Hence, a prior that considers gravitational forces can be constructed by including $\sin(q_1[t])$: $$\begin{aligned}
\text{IFG}_{[t+1]} =
{ \begin{bmatrix} 1 & \Delta{t} & 0 & 0\\
0 & 1 & \Delta{t}/J & -\gamma \end{bmatrix} }
{ \begin{bmatrix} q_1[t] \\ {\dot{ q_1 }}[t] \\ u[t] \\ \sin(q_1[t]) \end{bmatrix} }
\label{eq:gravity_prior}\end{aligned}$$ we call this prior IFG (inertia+force+gravity). We purposely did not define $J$ and $\gamma$. One of the advantages of the presented approach is that the algorithm can learn the parameters from data if they are not available. If the user does not know the value of $J$ and $\gamma$, a prior with large standard deviation can be provided for these parameters (large $\Sigma_{{ {\boldsymbol{\beta}} }}$). Although parameters like $J$ and $\gamma$ are easy to compute for a simple pendulum, for more complex systems they may be hard and tedious to obtain. Our objective is to take advantage of domain knowledge and allow the model to fill in the gaps in knowledge.
For the rest of the paper, priors derived from Eq. [(\[eq:inertia\_prior\])]{} are referred as IF priors, while Eq. [(\[eq:gravity\_prior\])]{} priors are referred as IFG. In the experiments (section \[section:experiments\]) we compare the performance for both priors in order to illustrate how performance can be progressively improved with more detailed priors.
Simplified priors for Acrobot and Cartpole {#section:acrobot_cartpole_priors}
------------------------------------------
In addition to the pendulum, we consider the Acrobot and Cartpole systems in our analysis. For these systems, we consider much simpler priors than the exact dynamic models derived from multi-body dynamics. We use the same principles shown in the previous section in order to get simple priors for the Acrobot and Cartpole.
Figure \[figure:acrobot\] shows a diagram of the Acrobot system. A simple prior for this system can be constructed using the prior in Eq. [(\[eq:inertia\_prior\])]{} for each one of the links of the Acrobot: $$\begin{aligned}
\text{IF} =
{ \begin{bmatrix} 1 & 0 & {\Delta{t}}& 0 & 0 \\
0 & 1 & 0 & {\Delta{t}}& 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & \gamma_1 \\ \end{bmatrix} }
{ \begin{bmatrix} q_1[t] \\ q_2[t] \\ {\dot{ q_1 }}[t] \\ {\dot{ q_2 }}[t] \\ u[t] \end{bmatrix} }
; \;\;
\text{IFG} =
{ \begin{bmatrix} 1 & 0 & {\Delta{t}}& 0 & 0 & 0 & 0 \\
0 & 1 & 0 & {\Delta{t}}& 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & -\gamma_2 & -\gamma_3 \\
0 & 0 & 0 & 1 & \gamma_1 & 0 & -\gamma_4 \\ \end{bmatrix} }
{ \begin{bmatrix} q_1[t] \\ q_2[t] \\ {\dot{ q_1 }}[t] \\ {\dot{ q_2 }}[t] \\
u[t] \\
\sin_1 \\ \sin_{12} \end{bmatrix} }
\label{eq:acrobot_priors}
\end{aligned}$$ where $\sin_1 = \sin(q_1[t])$, and $\sin_{12} = \sin(q_1[t] + q_2[t])$. In this case, the input $u[t]$ only drives the second link. The idea with these priors is to construct an intuitive and simple model from “noisy” Newtonian dynamics. These priors are extremely simple as they do not consider friction or coriolis/centrifugal forces. However, they provide important information about the mechanics of the system. These priors convey the following information:
- The position should increase proportional to the velocity by a factor of ${\Delta{t}}$.
- The position should stay the same if the velocity is zero.
- The velocity should stay the same if no external forces are applied.
- For the IFG prior, gravity pulls the links to the downward position proportional to the sine of the angle w.r.t. the horizontal plane. Gravity has no effect when the links are completely down/up.
The objective with these priors is to demonstrate how extremely simplified priors extracted with simple physics can be used to improve performance of data-driven models. The IF and IFG priors for the Cartpole (Figure \[figure:cartpole\]) are constructed using the same principles: $$\begin{aligned}
\text{IF} =
{ \begin{bmatrix} 1 & 0 & {\Delta{t}}& 0 & 0 \\
0 & 1 & 0 & {\Delta{t}}& 0 \\
0 & 0 & 1 & 0 & \gamma_1 \\
0 & 0 & 0 & 1 & 0 \\ \end{bmatrix} }
{ \begin{bmatrix} q_1[t] \\ q_2[t] \\ {\dot{ q_1 }}[t] \\ {\dot{ q_2 }}[t] \\ u[t] \end{bmatrix} } ; \;\;
\text{IFG} =
{ \begin{bmatrix} 1 & 0 & {\Delta{t}}& 0 & 0 & 0 \\
0 & 1 & 0 & {\Delta{t}}& 0 & 0 \\
0 & 0 & 1 & 0 & \gamma_1 & 0 \\
0 & 0 & 0 & 1 & 0 & -\gamma_2 \\ \end{bmatrix} }
{ \begin{bmatrix} q_1[t] \\ q_2[t] \\ {\dot{ q_1 }}[t] \\ {\dot{ q_2 }}[t] \\
u[t] \\
\sin(q_2[t]) \end{bmatrix} }\end{aligned}$$
Experiments {#section:experiments}
===========
In order to evaluate the performance of the presented model, we performed experiments on a set of simulated systems: Pendulum, Cartpole and Acrobot. We also performed qualitative tests on a toy-dataset to visualize the performance of the EVGP model.
We used Drake [@drake] to simulate the Pendulum, Cartpole and Acrobot systems and obtain the control/sensor data used to train and evaluate the EVGP models. We used the squared exponential function for the covariance kernels. The reason for this choice is that all the experiments involve continuous systems. The EVGP model was implemented using Tensorflow and the minimization of the negative ELBO loss was done using the ADAM optimizer [@kingma2014adam]. The experiments were run in a computer with a single GPU (Nvidia Quadro P5000) with an Intel(R) Xeon(R) CPU (E3-1505M at 3.00GHz).
Experiments on Toy Dataset {#section:toy_dataset}
--------------------------
The toy dataset is intended to serve as an illustration of the behavior of the EVGP model and visualize the qualitative differences between several GP models. We use a modified version of the toy dataset used in [@snelson2006sparse] [@titsias2009variational]. The dataset[^1] is modified as follows: $$\begin{aligned}
(y, x) \leftarrow ( 6 y + 3 x, x)
\end{aligned}$$ The modification is intended to provide a global linear trend to the data (See Figure \[fig:toy\_dataset\]). Figure \[fig:toy\_dataset\] shows the distribution learned using different versions of a Gaussian Process. Figures \[fig:gp\_estimate\] and Figure \[fig:egp\_estimate\] show the exact posterior distributions for a GP and EGP [@rasmussen2004gaussian] model, respectively. Figures \[fig:vgp\_estimate\] and \[fig:evgp\_estimate\] show the variational approximations obtained with a VGP [@hensman2013gaussian] and EVGP model. The standard deviation (black line) is used to visualize the uncertainty. The figures show how the uncertainty grows as we move away from the training dataset.
The original dataset is composed of 200 samples, Figure \[fig:toy\_dataset\] shows that the variational approximations are able to successfully approximate their exact counterparts with as few inducing points as m=10. The position of the inducing points are shown with green crosses.
In this case, the prior-knowledge that we provide to the EVGP is a simple linear function $h(x, {{ {\boldsymbol{\beta}} }}) = x \beta_1 + \beta_2$. Figure \[fig:evgp\_estimate\] shows how we can use the prior in order to control the global shape of the function. The figure shows how the EGP and EVGP models use the prior knowledge to fit the global behavior of the data (linear) while using the kernels to model the local non-linear behavior.
\
Learning system dynamics
------------------------
We evaluated the performance of the EVGP model in learning system dynamics using data obtained from simulations of the Pendulum, Cartpole and Acrobot systems. Concretely, we evaluated the accuracy of the EVGP model with IF and IFG priors in predicting the next state of the system given the current control inputs and state.
**Data:** to evaluate the difference in generalization, we sampled two different datasets for each system: one for training and one for testing. The datasets were sampled by simulating the system using random initial states ${{{ {\boldsymbol{z}} }}}_{[0]} \sim \alpha \; {{ { {\cal U} { \left( -1, 1 \right) } } }}$ and random control inputs ${{ {\boldsymbol{u}} }}_{[t]} \sim \eta \; { {\cal N} \left( 0, 1 \right) }$ drawn from uniform and normal distributions, respectively. Table \[table:sample\_scales\] shows the values of the scales ($\alpha, \eta$) that were used to sample the trajectories. These values were chosen in order to cover at least the range ($-\pi, \pi$) for the angles on the systems. In Table \[table:sample\_scales\], $H$ refers to the number of sampled trajectories, ${\left\lvert {{\cal D}}\right\rvert}$ refers to the total number of samples. All trajectories were sampled for 100 time steps and ${\Delta{t}}=0.03 s$.
**Baseline:** we compare the EVGP model with a standard VGP model, a residual VGP (RES-VGP), and a residual Deep Bayesian Neural Network (RES-DBNN). The VGP model is based on [@hensman2013gaussian] and uses a zero mean prior. The residual VGP and DBNN models assume the system can be approximated as ${{{ {\boldsymbol{z}} }}}_{[t+1]} = {{{ {\boldsymbol{z}} }}}_{[t]} + g_r({{{ {\boldsymbol{z}} }}}_{[t]} \oplus {{{ {\boldsymbol{u}} }}}_{[t]})$. Approximating residuals is a common approach used to simplify the work for GP and DBNN models [@deisenroth2011pilco] [@gal2016improving] [@marino2019modeling]. The RES-DBNN model is trained using the local reparameterization trick presented in [@kingma2015variational] with Normal variational distributions for the weights. For these set of experiments, we did not consider the exact GP and EGP models given the large number of samples in the training datasets.
[r]{}[0.35]{}
Table \[table:model\_parameters\] shows the number of inducing points (m) used for the VGP and EVGP models. The table also shows the number of hidden units for the DBNN model, where \[15, 15\] means a two-layer network with 15 units in each layer. We used the LeakyRelu activation function.
Table \[table:complexity\] shows the space and time complexity of the models considered in this paper. In this table, $m$ is the number of inducing points, $o$ is the number of outputs, $L$ is the number of hidden layers, and $n$ is the number of hidden units in each layer. To simplify the analysis, we assume all hidden layers of the DBNN have the same number of hidden units. The table shows that the complexity of the VGP and EVGP models are governed by the matrix inversion ${{ {\boldsymbol{K}} }}_{mm}^{-1}$. Because we assume completely independent VGP and EVGP models for each output of the system, their complexity also depends on the number of outputs $o$. The complexity of the DBNN model is governed by the matrix-vector product between the weight matrices and the hidden activation vectors. All models have constant space complexity w.r.t. the training dataset size ${\left\lvert {{\cal D}}\right\rvert}$. Furthermore, all models have linear time complexity w.r.t. ${\left\lvert {{\cal D}}\right\rvert}$ if we assume that training requires to visit each sample in ${{\cal D}}$ at least once.
**Metrics:** for comparison, we used two metrics: 1) prediction error, 2) containing ratios (CR). Prediction error is computed as the difference between sampled values ($y$) and the expected estimated output (${ \mathop{{\mathbb E}}_{} { \left[ \hat{y}^{(i)} \right] } }$): $$\begin{aligned}
\text{Error} = \dfrac{1}{{{{\left\lvert {{\cal D}}\right\rvert}}}} \sum_{i=1}^{{{\left\lvert {{\cal D}}\right\rvert}}}{\left\lVert y^{(i)} - { \mathop{{\mathbb E}}_{} { \left[ \hat{y}^{(i)} \right] } } \right\rVert}\end{aligned}$$ where ${{{\left\lvert {{\cal D}}\right\rvert}}}$ is the number of samples in the respective dataset. The expected output (${ \mathop{{\mathbb E}}_{} { \left[ \hat{y}^{(i)} \right] } }$) for the EVGP model is equal to $\mu_{\hat{{{ {\boldsymbol{g}} }}} | \hat{{{ {\boldsymbol{x}} }}}}$ in Eq. [(\[eq:prediction\])]{}. For the DBNN model, the expectation is estimated using Monte-Carlo.
The containing ratios (CR) are the percentage of values covered by the estimated distribution $\hat{y}$. We consider containing ratios for one, two and three standard deviations (CR-1, CR-2, and CR-3 respectively).
**Results:** Table \[table:results\] shows the prediction error and CR scores obtained in the testing dataset. EVGP-IF and EVGP-IFG refers to the use of an IF or IFG prior, respectively. We can observe a considerable improvement on the testing error and CR-3 scores when using EVGP models. We also see a progressive improvement on the testing error when using more detailed priors. Using IFG prior results in lower prediction errors when compared with the IF prior. Also, the residual models have a lower error than the standard VGP model. The EVGP-IFG model provided the estimations with the lowest prediction error and CR-3 scores close to 100%.
\
Table \[table:results\] also shows that the EVGP model required the lowest number of parameters. This translates into lower training and inference times with lower memory cost.
Figure \[fig:mean\_norm\] shows a comparison of the prediction error on the test dataset as we increase the number of training samples. For this experiment, we kept the testing dataset fixed while samples were progressively aggregated into the training dataset. The figure shows the mean, max and min values obtained for four independent runs. Figure \[fig:mean\_norm\_all\] shows a comparison that includes all models. As expected, the prediction error is reduced as we increase the size of our training dataset. The figure shows that the EVGP provides the most accurate predictions while requiring less number of samples.
Figure \[fig:mean\_norm\_all\] also shows how the performance of VGP and RES-VGP plateaus, struggling to take advantage of larger datasets. Although the RES-DBNN performs poorly with small training datasets, the high capacity of the RES-DBNN model allows it to take advantage or large datasets and improve accuracy, reducing the performance gap w.r.t. the EVGP models as more data is available. Thanks to the lower computational time-cost of the RES-DBNN (see Table \[table:complexity\]), this model can use a larger set of parameters without incurring in excessive training times.
[r]{}[0.4]{}\
Figure \[fig:mean\_norm\_evgp\] shows a scaled version that only considers the EVGP model with different priors. This figure shows that the IFG prior provides more accurate predictions when compared to the IF prior. In the case of the pendulum, the IFG prior provides a highly accurate model of the system, requiring only a small number of training samples. Figure \[fig:mean\_norm\_evgp\] also shows how as training data is aggregated, the accuracy gap between IF and IFG priors is reduced.
The priors that we use are extremely simple, they are ignoring friction and coriolis/centrifugal effects. Nonetheless, we observe a considerable performance improvement after providing our data-driven model basic information with the IF and IFG priors.
**Understanding the learned model:** one of the advantages of incorporating domain knowledge is that the learned model is easy to interpret by the domain expert. For example, in the case of the Acrobot, the value of the parameter ${{ {\boldsymbol{\beta}} }}$ can be visualized to understand and debug what the model has learned. Figure \[fig:beta\_acrobot\] shows the value of ${{ {\boldsymbol{b}} }}$ learned with the IF (Fig. \[fig:if\_beta\]) and IFG priors (Fig. \[fig:ifg\_beta\]).
We observe in Figure \[fig:beta\_acrobot\] that the learned parameters follow a similar structure given by the prior (see Eq. \[eq:acrobot\_priors\]). In our experiments, we did not enforce the sparse structure from the priors, i.e. zero parameters in the prior are allowed to have non-zero values in the posterior.
Figure \[fig:if\_beta\] shows that when using the IF prior, the EVGP model compensates for the missing gravity information by assigning negative values to $(\dot{q_1}, q_1)$ and $(\dot{q_2}, q_2)$. The reason for this behavior is that $q_1 \approx \sin{q_1}$ for small $q_1$, however this approximation is no longer valid for large $q_1$. When using IFG priors (Fig. \[fig:ifg\_beta\]), we observe that the model no longer assigns negative values to $(\dot{q_1}, q_1)$ and $(\dot{q_2}, q_2)$. The reason is that IFG provides the values of $sin(q_1)$ and $sin(q_1 + q_2)$ which help to model the effect of gravity more accurately.
Related work
============
Incorporating prior scientific knowledge in machine-learning algorithms is an ongoing effort that has recently gained increased attention. Convolutional neural networks (CNNs) have been used in modeling and simulation applications such as forecasting of sea surface temperature [@de2017deep] and efficient simulation of the Navier-Stokes equations [@tompson2017accelerating]. Gaussian Processes (GP) provide a general purpose non-parametric model that has been used for system identification and control under uncertainty [@deisenroth2011pilco] [@bijl2017system]. Previous work has explored using GPs to include partial model information [@hall2012modelling]. In [@gray2018hybrid] a GP model is used to correct a highly simplified physics model and improve accuracy. Our work is based on the GP model with explicit features presented in [@rasmussen2004gaussian]. Variations of this model are commonly used in calibration of computer models [@kennedy2001bayesian] [@li2016integrating].
Despite the advantages of GP models for modeling complex non-linear relations with uncertainty, GPs are computationally expensive. A large bulk of research has focused on improving the computational requirements of GPs. Sparse GP approximation methods are some of the most common approaches for reducing GP computation cost [@quinonero2005unifying]. Bayesian approximation techniques such as Variational Inference provide a rich toolset for dealing with large quantities of data and highly complex models [@kingma2013auto] [@titsias2009variational]. Variational approximations of a sparse GP have been explored in [@titsias2009variational] [@hensman2013gaussian]. In [@frigola2014variational] a variational GP model is presented for nonlinear state-space models. In [@gal2016improving] [@marino2019modeling], Deep Bayesian Neural Networks (DBNNs) are proposed as an alternative to GPs in order to improve scalability in reinforcement learning problems. Given the popularity of GP models and Variational Inference, there is an increased interest on developing automated variational techniques for these type of models [@nguyen2014automated] [@kucukelbir2017automatic].
Conclusion {#section:conclusion}
==========
In this paper, we presented the EVGP model, a variational approximation of a Gaussian Process which uses domain knowledge to define the mean function of the prior. We compared the performance of the EVGP model against purely data-driven approaches and demonstrated improved accuracy and interpretability after incorporating simple priors derived from Neutonian mechanics. The EVGP also provided higher accuracy with smaller training datasets. The priors provided a rough but simple approximation of the mechanics, informing the EVGP of important structure of the real system.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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Variational Inference {#section:variational_inference}
=====================
In a Bayesian learning approach, we model the parameters ${\omega}$ of our model $p({{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{x}} }}, {\omega})$ using probability distributions. The parameters are given a prior distribution ${ { p { \left( {\omega}\right) } } }$ that represents our prior knowledge about the model before looking at the data. Given a dataset ${{\cal D}}$, we would like to obtain the posterior distribution of our parameters following the Bayes rule: $$\begin{aligned}
p({\omega}| {{\cal D}}) = \frac{{ { p { \left( {{\cal D}}| {\omega}\right) } } } { { p { \left( {\omega}\right) } } }}
{{ { p { \left( {{\cal D}}\right) } } }}\end{aligned}$$ Variational Inference (VI) provides a tool for approximating the posterior $p({\omega}| {{\cal D}})$ using a variational distribution ${ { p_\phi { \left( {\omega}\right) } } }$ parameterized by $\phi$. In other words, with VI we find the value of $\phi$ such that ${ { p_\phi { \left( {\omega}\right) } } } \approx p({\omega}| {{\cal D}})$. The parameters $\phi$ of the distribution ${ { p_\phi { \left( {\omega}\right) } } }$ are found by maximizing the Evidence Lower Bound (ELBO) between the approximate and real distributions: $$\begin{aligned}
\phi \leftarrow \operatorname*{arg\,max}_{\phi}
{ \mathop{{\mathbb E}}_{{ { p_\phi { \left( {\omega}\right) } } }
} { \left[ \log { { p { \left( {{\cal D}}\mid {\omega}\right) } } } \right] } }
- D_{KL}{ \left( { { p_\phi { \left( {\omega}\right) } } } \mid \mid { { p { \left( {\omega}\right) } } } \right) }\end{aligned}$$ Maximizing the ELBO is equivalent to minimizing the KL divergence between the variational distribution and the real distribution.
Having obtained the variational approximation ${ { p_\phi { \left( {\omega}\right) } } }$, we can approximate the predictive distribution $p({{ {\boldsymbol{y}} }}| {{ {\boldsymbol{x}} }}, {{\cal D}})$ as follows: $$\begin{aligned}
p({{ {\boldsymbol{y}} }}| {{ {\boldsymbol{x}} }}, {{\cal D}}) = & { \mathop{{\mathbb E}}_{p({\omega}\mid {{\cal D}})} { \left[ p({{ {\boldsymbol{y}} }}| {{ {\boldsymbol{x}} }}, {\omega}) \right] } } \\
\approx & { \mathop{{\mathbb E}}_{{ { p_\phi { \left( {\omega}\right) } } }} { \left[ p({{ {\boldsymbol{y}} }}| {{ {\boldsymbol{x}} }}, {\omega}) \right] } }\end{aligned}$$ which is the approximated variational predictive distribution ${ { p_\phi { \left( {\hat{{{ {\boldsymbol{y}} }}}}\mid {\hat{{{ {\boldsymbol{X}} }}}}\right) } } }$ (see section \[section:evgp-prediction\]): $$\begin{aligned}
{ { p_\phi { \left( {\hat{{{ {\boldsymbol{y}} }}}}\mid {\hat{{{ {\boldsymbol{X}} }}}}\right) } } } =
{ \mathop{{\mathbb E}}_{p_\phi({\omega})} { \left[ { { p { \left( {\hat{{{ {\boldsymbol{y}} }}}}\mid {\hat{{{ {\boldsymbol{X}} }}}}, {\omega}\right) } } } \right] } }\end{aligned}$$
ELBO {#appendix:elbo}
====
Given the training dataset ${{\cal D}}= ({{ {\boldsymbol{y}} }}, {{ {\boldsymbol{X}} }})$, the parameters $\phi$ of ${ { p_\phi { \left( {\omega}\right) } } }$ are learned by minimizing the negative Evidence Lower Bound (ELBO). For the EVGP, the negative ELBO takes the following form: $$\begin{aligned}
{ { {\mathcal{L}}_1 { \left( \phi \right) } } } = -{ \mathop{{\mathbb E}}_{{ { p_\phi { \left( {\omega}\right) } } }} { \left[
\ln { \mathop{{\mathbb E}}_{g \mid {\omega}} { \left[ { { p { \left( {{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{g}} }}\right) } } } \right] } } \right] } }
+ {\mathcal{L}}_{KL}
\label{eq:elbo_1}\end{aligned}$$ where ${\mathcal{L}}_{KL}$ denotes the KL divergence between the variational posterior and the prior ${\mathcal{L}}_{KL} = D_{KL}{ \left( { { p_\phi { \left( {\omega}\right) } } } \mid \mid { { p { \left( {\omega}\right) } } } \right) }$ Note that the inner expectation in Eq. [(\[eq:elbo\_1\])]{} is taken w.r.t. ${g \mid {\omega}}$, presented in Eq. [(\[eq:g\_given\_w\])]{}. Following a similar approach than [@hensman2013gaussian], we apply Jensen’s inequality in the inner expectation of Eq. [(\[eq:elbo\_1\])]{}: $$\begin{aligned}
&\log { \mathop{{\mathbb E}}_{g \mid {\omega}} { \left[ { { p { \left( {{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{g}} }}\right) } } } \right] } }
\geq { \mathop{{\mathbb E}}_{g \mid {\omega}} { \left[ \log { { p { \left( {{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{g}} }}\right) } } } \right] } } \\
& { \mathop{{\mathbb E}}_{g \mid {\omega}} { \left[ \log { { P { \left( {{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{g}} }}\right) } } } \right] } } =
\log { {\cal N} \left( {{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{\mu}} }}_{{{ {\boldsymbol{g}} }}\mid {\omega}}, {{ {\boldsymbol{\Sigma}} }}_y \right) }
-\dfrac{1}{2} {\text{Tr}{ \left( {{ {\boldsymbol{\Sigma}} }}_y^{-1} {{ {\boldsymbol{\Sigma}} }}_{{{ {\boldsymbol{g}} }}\mid {\omega}} \right) }}\end{aligned}$$ where ${{ {\boldsymbol{\mu}} }}_{{{ {\boldsymbol{g}} }}\mid {\omega}} = {{ {\boldsymbol{H}} }}_x {{{ {\boldsymbol{\beta}} }}}+ {{ {\boldsymbol{\mu}} }}_{{{ {\boldsymbol{f}} }}\mid {\omega}}$, ${{ {\boldsymbol{\Sigma}} }}_{{{ {\boldsymbol{g}} }}\mid {\omega}} = {{ {\boldsymbol{\Sigma}} }}_{{{ {\boldsymbol{f}} }}\mid {\omega}}$. This allows us to express the ELBO in a way that simplifies the computation of the expectations w.r.t. the parameters ${\omega}$. Now, the variational loss in Eq. [(\[eq:elbo\])]{} can be obtained by simply computing the expectation w.r.t. the model parameters: $$\begin{aligned}
{ { {\mathcal{L}}{ \left( \phi \right) } } }
= & { \mathop{{\mathbb E}}_{{ { p_\phi { \left( {\omega}\right) } } }} { \left[
{ \mathop{{\mathbb E}}_{g \mid {\omega}} { \left[ - \log { { p { \left( {{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{g}} }}\right) } } } \right] } } \right] } } + {\mathcal{L}}_{KL}
\nonumber \\
= & - { \mathop{{\mathbb E}}_{{ { p_\phi { \left( {\omega}\right) } } }} { \left[
\log { {\cal N} \left( {{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{\mu}} }}_{{{ {\boldsymbol{g}} }}\mid {\omega}}, {{ {\boldsymbol{\Sigma}} }}_y \right) }
-\dfrac{1}{2} {\text{Tr}{ \left( {{ {\boldsymbol{\Sigma}} }}_y^{-1} {{ {\boldsymbol{\Sigma}} }}_{{{ {\boldsymbol{g}} }}\mid {\omega}} \right) }}
\right] } }
+ {\mathcal{L}}_{KL} \\
= & - \log { {\cal N} \left( {{ {\boldsymbol{y}} }}\mid {{ {\boldsymbol{H}} }}_x {{ {\boldsymbol{b}} }}+ {{ {\boldsymbol{K}} }}_{xm} {{ {\boldsymbol{K}} }}_{mm}^{-1} {{ {\boldsymbol{a}} }}, {{ {\boldsymbol{\Sigma}} }}_y \right) }
\nonumber \\
& + \dfrac{1}{2} { \left[
{\text{Tr}{ \left( {{ {\boldsymbol{M}} }}_1 {{ {\boldsymbol{A}} }}\right) }}
+ {\text{Tr}{ \left( {{ {\boldsymbol{M}} }}_2 {{ {\boldsymbol{B}} }}\right) }}
+ {\text{Tr}{ \left( {{ {\boldsymbol{\Sigma}} }}_y^{-1} {{ {\boldsymbol{\Sigma}} }}_{{{ {\boldsymbol{f}} }}\mid {\omega}} \right) }}
\right] } \nonumber \\
& + {\mathcal{L}}_{KL}\end{aligned}$$ where ${{ {\boldsymbol{M}} }}_1 = { \left( {{ {\boldsymbol{K}} }}_{mm}^{-1} {{ {\boldsymbol{K}} }}_{mx} \right) } {{ {\boldsymbol{\Sigma}} }}_y^{-1} { \left( {{ {\boldsymbol{K}} }}_{xm} {{ {\boldsymbol{K}} }}_{mm}^{-1} \right) }$, and ${{ {\boldsymbol{M}} }}_2 = {{ {\boldsymbol{H}} }}_x^T {{ {\boldsymbol{\Sigma}} }}_y^{-1} {{ {\boldsymbol{H}} }}_x$. The value of the KL-divergence is simply the sum of the divergence for both parameters: $$\begin{aligned}
{\mathcal{L}}_{KL} = &
D_{KL}{ \left( { {\cal N} \left( {{ {\boldsymbol{a}} }}, {{ {\boldsymbol{A}} }}\right) } \mid \mid { {\cal N} \left( 0, {{ {\boldsymbol{K}} }}_{mm} \right) } \right) }
+ D_{KL}{ \left( { {\cal N} \left( {{ {\boldsymbol{b}} }}, {{ {\boldsymbol{B}} }}\right) } \mid\mid { {\cal N} \left( {{ {\boldsymbol{\mu}} }}_{\beta}, {{ {\boldsymbol{\Sigma}} }}_{\beta} \right) } \right) }
\end{aligned}$$
Mini-batch optimization {#appendix:minibatches}
=======================
In order to make the model scalable to very large datasets, the ELBO can be optimized using mini-batches. Following [@gal2016dropout], assuming the samples are i.i.d., the loss for a mini-batch $({{ {\boldsymbol{y}} }}, {{ {\boldsymbol{X}} }})$ composed of ${\left\lvert {{ {\boldsymbol{X}} }}\right\rvert}$ number of samples can be expressed as follows: $$\begin{aligned}
{ { {\mathcal{L}}{ \left( \phi \right) } } } = - \dfrac{1}{{\left\lvert {{ {\boldsymbol{X}} }}\right\rvert}}
{ \mathop{{\mathbb E}}_{p_\phi(w)} { \left[ { { \ln { \left( p({{ {\boldsymbol{Y}} }}|{{ {\boldsymbol{X}} }},w) \right) } } } \right] } }
+ \dfrac{1}{{\left\lvert {{\cal D}}\right\rvert}} {\mathcal{L}}_{KL}
\end{aligned}$$ where ${\left\lvert {{\cal D}}\right\rvert}$ is the total number of samples in the training dataset.
[^1]: Obrained from http://www.gatsby.ucl.ac.uk/\~snelson/SPGP\_dist.tgz
|
---
author:
- 'M.-A. Miville-Deschênes'
- 'Q. Salomé'
- 'P. G. Martin'
- 'G. Joncas'
- 'K. Blagrave'
- 'K. Dassas'
- 'A. Abergel'
- 'A. Beelen'
- 'F. Boulanger'
- 'G. Lagache'
- 'F. J. Lockman'
- 'D. J. Marshall'
bibliography:
- 'Article\_v11.bib'
title: |
Structure formation in a colliding flow:\
The Herschel[^1] view of the Draco nebula
---
[The objective of this study is to better understand the process of structure formation in a colliding flow and to describe the effects of matter entering the disk on the interstellar medium.]{} [We conducted -SPIRE observations of the Draco nebula. The [*clumpfind*]{} algorithm was used to identify and characterize the small-scale structures of the cloud.]{} [The high-resolution SPIRE map reveals the fragmented structure of the interface between the infalling cloud and the Galactic layer. This front is characterized by a Rayleigh-Taylor (RT) instability structure. From the determination of the typical length of the periodic structure ($2.2$pc) we estimated the gas kinematic viscosity. This allowed us to estimate the dissipation scale of the warm neutral medium ($0.1$pc), which was found to be compatible with that expected if ambipolar diffusion were the main mechanism of turbulent energy dissipation. The statistical properties of the small-scale structures identified with [*clumpfind*]{} are found to be typical of that seen in molecular clouds and hydrodynamical turbulence in general. The density of the gas has a log-normal distribution with an average value of $10^3$cm$^{-3}$. The typical size of the structures is $0.1$-$0.2$pc, but this estimate is limited by the resolution of the observations. The mass of these structures ranges from $0.2$ to $20$M$_{\odot}$ and the distribution of the more massive structures follows a power-law $dN/d\log(M) \sim M^{-1.4}$. We identify a mass-size relation with the same exponent as that found in molecular clouds ($M\sim L^{2.3}$). On the other hand, we found that only 15% of the mass of the cloud is in gravitationally bound structures. ]{} [We conclude that the collision of diffuse gas from the Galactic halo with the diffuse interstellar medium of the outer layer of the disk is an efficient mechanism for producing dense structures. The increase of pressure induced by the collision is strong enough to trigger the formation of cold neutral medium out of the warm gas. It is likely that ambipolar diffusion is the mechanism dominating the turbulent energy dissipation. In that case the cold structures are a few times larger than the energy dissipation scale. The dense structures of Draco are the result of the interplay between magnetohydrodynamical turbulence and thermal instability as self-gravity is not dominating the dynamics. Interestingly they have properties typical of those found in more classical molecular clouds.]{}
{width="0.49\linewidth"} {width="0.49\linewidth"}
Introduction
============
One of the key questions regarding the formation of stars in galaxies is related to the way gas condenses and how it cycles from the hot and diffuse phase to cold and dense structures where stars form. This process is related to the formation of the cold neutral medium (CNM) and the atomic-molecular (or -H$_2$) transition. One of the frameworks in which this general process is understood is related to colliding flows where a region of the warm neutral medium (WNM) undergoes an increase of pressure that facilitates the rapid cooling of the gas to the stable CNM phase. This process has been studied in several numerical simulations [@hennebelle1999; @audit2005; @vazquez-semadeni2006a; @hennebelle2007a; @inoue2009; @saury2014]. One way of witnessing the formation of cold structures in a colliding flow is by looking at the effects of clouds falling onto the Milky Way disk.
Our Galaxy, as all galaxies, is an open and dynamical system. Matter is constantly arriving on the Galactic disk. Part of this gas is the result of the Galactic fountain: hot gas rises into the halo from stellar winds and supernovae then returns to the disk due to gravity [@shapiro1976; @bregman1980; @putman2012]. Matter also arrives from intergalactic space in the form of either gas stripped from satellite galaxies or gas from the intergalactic medium (IGM). The origin of the material falling into the disk can partly be determined by its metallicity.
The high-velocity clouds (HVC) in our Galaxy are considered possible direct evidence for extragalactic infalling gas. Measurements of the metallicity of HVC gas range from very low (0.1) up to solar [@collins2006]. The intermediate-velocity clouds (IVC) are closer to the Galactic layer. Most of these clouds are part of the Galactic fountain, but some should be matter of extragalactic origin that interact with the disk. At this point it is unclear if the amount of new material entering the disk is sufficient to maintain the star formation rate of galaxies [@peek2008; @sancisi2008].
This influx of matter from the halo also adds kinetic energy to the interstellar medium (ISM). It is a way to input the supernovae energy into interstellar turbulence at some distance from where it was produced. One important question relates to the physical properties of the infalling gas and the impact it has on the star formation cycle in galaxies [@heitsch2009a; @joung2012]. Given the significant velocity (10s to 100s of kms$^{-1}$) with which matter encounters the Galactic disk, it is expected to be shocked and undergo dynamical instabilities. The thermal distribution of the gas, its density structure, and the amount of molecular gas produced in the end is not well constrained observationally, partly because of the lack of observations that show this situation clearly.
The Draco nebula (hereafter Draco), an icon among the IVCs, probably provides the best opportunity to understand these processes and resulting physical conditions. By chance there is little structured local ISM gas in the direction of Draco, allowing a very clear view of the matter entering the disk even in integrated (dust or gas) emission.
In this paper we complement what is already known about Draco by presenting -SPIRE [@griffin2010a] maps of the nebula, revealing the fine details of its structure, and especially of its Rayleigh-Taylor front. We used these observations to quantify the typical length of the Rayleigh-Taylor instability, putting some constraints on the gas viscosity and properties of turbulence in Draco. The high resolution of the -SPIRE data also allows us to characterize the statistical properties of the small-scale structures formed in the postshock region, which enables us to quantify the outcome of the cloud collision in terms of structure formation.
The paper is organized as follows. In Sect. \[sec:draco\] we summarize what is known about Draco. Section \[sec:Obs\] presents the data used for this study, how we computed the map of the column density, and a description of the structure of the Rayleigh-Taylor instability front. The small-scale structure is analyzed in Sect. \[sec:structures\]. Our results are discussed in Sect. \[sec:discussion\] and summarized in Sect. \[sec:conclusion\].
The Draco nebula {#sec:draco}
================
![\[fig:NHI\_GBT\] column density maps in the direction of Draco for the LVC (bottom), IVC (middle), and HVC (top). Data were obtained at the Green Bank Telescope; they are part of the GHIGLS survey [@martin2015]. The column density maps are the same as those used in @planck_collaboration2011a.](Pictures/draco_Nhi_HVC.pdf "fig:"){width="0.98\linewidth"} ![\[fig:NHI\_GBT\] column density maps in the direction of Draco for the LVC (bottom), IVC (middle), and HVC (top). Data were obtained at the Green Bank Telescope; they are part of the GHIGLS survey [@martin2015]. The column density maps are the same as those used in @planck_collaboration2011a.](Pictures/draco_Nhi_IVC.pdf "fig:"){width="0.98\linewidth"} ![\[fig:NHI\_GBT\] column density maps in the direction of Draco for the LVC (bottom), IVC (middle), and HVC (top). Data were obtained at the Green Bank Telescope; they are part of the GHIGLS survey [@martin2015]. The column density maps are the same as those used in @planck_collaboration2011a.](Pictures/draco_Nhi_LVC.pdf "fig:"){width="0.98\linewidth"}
A molecular intermediate velocity cloud
---------------------------------------
Draco is the most studied diffuse IVC at high Galactic latitude ($l\approx91$[$^{\circ}$]{}, $b\approx38$[$^{\circ}$]{}). It was first observed at 21cm by [@goerigk1983] at a velocity $v~\sim-25$kms$^{-1}$, following calibration observations obtained with the Effelsberg 100m telescope [@kalberla1982]. More recently, a $5^\circ\times5^\circ$ region centered on Draco was observed at 21cm with the Green Bank Telescope (GBT) as part of the GHIGLS survey [@martin2015]. A smaller area was also covered at higher angular resolution with the interferometer of the Dominion Radio Astrophysical Observatory as part of the DHIGLS survey [@blagrave2017]. In this region of the sky, three specific velocity components are seen: IVC, HVC, and a low-velocity cloud, LVC, which corresponds to the gas in the solar neighborhood. These three velocity components show up in the median and standard deviation 21cm spectra (Fig. \[fig:spectrumGBT\]). Figure \[fig:NHI\_GBT\] presents their column density map based on the GHIGLS data. To identify the three components, we used the standard deviation spectrum of the 21cm emission (Fig. \[fig:spectrumGBT\] - right) following [@planck_collaboration2011a]. These maps indicate that the column density in this direction of the sky is dominated by Draco, both in absolute value and variations.
[@goerigk1983] noticed that the cloud coincides with a faint optical nebula seen in the Palomar Observatory Sky Survey (POSS). These authors estimated that the ratio of dust extinction to column density, $N_{\rm HI}$, is unusually high, a factor 10 higher than typical values for diffuse clouds, which led them to conclude that most of the gas is in molecular form. From the analysis of their 21cm data and its comparison with X-ray measurements, @goerigk1983 also suggested that Draco is the result of the interaction of Galactic halo gas entering the disk and that the details of its front-like structure is the result of the Rayleigh-Taylor (RT) instability.
The presence of molecular gas was attested by @mebold1985 and @rohlfs1989 who carried CO observations of the brightest part of the nebula and found clumps of strong molecular emission at the boundary of the nebula. CO emission has also been detected by [@planck_collaboration2014c]. In addition, @stark1997 reported the detection of CH emission at 3.3GHz and @park2009 presented far-UV (FUV) observations showing the molecular hydrogen fluorescence over the whole nebula.
Draco is part of the @magnani1985 catalog of high Galactic latitude molecular clouds (MBM 41 to 44). According to @magnani2010 there are only five other regions at high Galactic latitude where intermediate velocity molecular gas (CO) has been detected; Draco is the region at the lowest Galactic latitude and the largest on the sky (about $4^\circ\times 4^\circ$).
Molecular gas was also revealed indirectly by X-ray data; @moritz1998 estimated that up to 70% of the hydrogen is molecular in the brightest parts of the nebula. Draco also appears as a shadow in the soft X-ray ROSAT data, enabling to measure the column density of hydrogen independently [@burrows1991; @snowden1991]. These measurements also imply that most of the gas is in molecular form. Similar conclusions are reached by comparing dust and 21cm data [@herbstmeier1993; @planck_collaboration2011a].
One specific feature of Draco is that it shows unusually strong CO emission for a diffuse cloud. It has significant CO emission on lines of sight where the total column density (derived either from dust emission or X-rays) is much lower than the usual threshold of $0.5-1\times 10^{21}\,$cm$^{-2}$ where molecular gas is usually seen in the ISM. In addition, large values of the ratio $W({\rm ^{12}CO})/N({\rm H_2})$ were deduced by @herbstmeier1993 [@moritz1998]. Also, @herbstmeier1994 observed higher CO transitions (up to J=3-2) to look for unusual excitation conditions but they found line ratios compatible with the average values found in cirrus clouds. This is indicative of a high CO abundance that is potentially caused by an increase of the density due to the collision of a halo cloud entering the Milky Way disk.
{width="\linewidth"}
Hot gas
-------
There have been a few studies looking for hot gas in the area of Draco. @hirth1985 and @kerp1999 showed an excess of X-ray emission that seems to be spatially correlated with Complex C, but no clear relationship with Draco itself could be established. On the other hand, based on FUV observations, @park2009 reported the detection of several ionic lines in the direction of Draco, especially CIV, SiII, and OIII\]. The CIV and SiII lines are seen outside Draco, possibly coming from the hot ionized Galactic layer. In addition, there is an excess of CIV, correlated with the dust emission of Draco. This excess is also seen in H$\alpha$ and OIII\], but not in SiII. Because of the absence of SiII, the authors concluded that the CIV, H$\alpha,$ and OIII\] emission in Draco cannot be the result of photoionization but it could be from the radiative cooling of warm, shocked gas.
{width="\linewidth"}
Distance
--------
An important clue informing any self-consistent description is the location of Draco in the Galaxy. Many attempts to estimate its distance have been made using different techniques. Star counts and color excess methods were used by @mebold1985 [@goerigk1986; @penprase2000] giving distance estimates of $800 < d < 1300\,$pc. On the other hand, as pointed out by @lilienthal1991, both these methods have potential biases especially at low $A_V$. The only direct measure of the distance to Draco came from the detection of NaID absorption lines at the systemic velocity of Draco in the spectrum of one star with known parallax distance [@gladders1998]. Combined with the nondetection for other foreground stars, @gladders1998 estimated that $463^{+192}_{-136}\le d\le 618^{+243}_{-174}\,$pc.
In the following, we adopt a distance of 600 pc. At the latitude of Draco, this distance corresponds to a height above the Galactic plane of $z= 370\,$pc. As the half width at half maximum of the WNM in the Galactic disk is $\sim 265$pc [@dickey1990], Draco is located in the upper part of the diffuse Galactic disk (or lower Galactic halo). What is clear is that Draco is out of the Local Bubble and so it must shadow X-rays coming from the Galactic halo.
The fact that Draco is located in the outer part of the Galactic disk is compatible with the fact that its brightness in the optical is comparable to clouds with dust properties typical of the disk [@mebold1985]. It suggests that Draco is illuminated by a fairly standard interstellar radiation field and so is not likely to be located at kpc distances. A similar argument can be made from its infrared brightness.
-SPIRE map of Draco {#sec:Obs}
===================
Draco was observed with PACS (110 and 170$\mu$m) and SPIRE (250, 350 and 500$\mu$m) as part of the open-time program [*“First steps toward star formation: unveiling the atomic to molecular transition in the diffuse interstellar medium”*]{} (P.I. M-A Miville-Deschênes). A field of $3.85^\circ \times 3.85^\circ$ was observed in parallel mode. Unfortunately, an error occurred during the acquisition of the PACS data making them unusable. Therefore, the results presented here are solely based on SPIRE data, especially the 250$\mu$m map that has the highest angular resolution.
The SPIRE data were reduced using a standard procedure with HIPE v13. We used the product available on the HErSchel IdOc Database (HESIOD[^2]). The zero level of each map was set by correlation with data (see Appendix \[sec:convert\_to\_NH\]). The SPIRE 250$\mu$m map of Draco is shown in Fig. \[fig:map250\]. In what follows, we use the 250$\mu$m map converted to total hydrogen column density, $N_{\rm H}$. The details of this conversion are given in Appendix \[sec:convert\_to\_NH\].
{width="0.9\linewidth"}
The data reveals for the first time the structure of matter in Draco at physical scales down to 0.05pc (resolution of $17.6''$ at 250$\mu$m). The wispy and filamentary structure already seen in previous data (21cm, far-infrared) is striking here. At the front structure at low declination where RT type structure were already identified, the SPIRE data reveals a wealth of clumpy structures organized in finger type structures.
One of the most striking features of the observations of Draco is the structure of the front that shows periodic half shells that are similar to structures produced by the Rayleigh-Taylor instability. A close-up view of the shock front is given in Fig. \[fig:RTI\] revealing the typical arches. The size of the arches is variable and the identification of the fingers is not always obvious. Nevertheless, we found a variation of about a factor of two with an average angular size of $12'.5$. Assuming a distance of 600pc and that the structures are orientated perpendicular to the line of sight, the typical length of the instability structure in Draco is $\lambda_{max}\approx 2.2$pc. In Sect. \[sec:viscosity\] we discuss on how this typical length might provide some information on the viscosity of the gas and on some properties of interstellar turbulence.
The 250$\mu$m dust emission map reveals a higher dynamic range of the column density than 21cm data. The column density (Fig. \[fig:NHI\_GBT\]) ranges from 1 to $3\times10^{20}$cm$^{-2}$, while the dust emission indicates a range from 3 to $50\times10^{20}$cm$^{-2}$. It even reaches $N_{\rm H} = 1\times 10^{22}$cm$^{-2}$ for a few pixels in the brightest parts in the southern region of the nebula. Part of this difference is due to different angular resolution of the GBT (9’) and SPIRE (17.6”) data. Nevertheless, even when brought to the same angular resolution, a significant difference remains between the column density estimated with dust emission and 21cm data. This confirms previous studies based on such a comparison and that showed that a significant portion of the bright region of Draco is composed of molecular hydrogen [@herbstmeier1993; @planck_collaboration2011a].
Figure \[fig:SPIRE\_and\_WISE\] presents another view that combines SPIRE 250$\mu$m and WISE 12 $\mu$m data [@meisner2014]. This image highlights strong variations of the relative abundance of smaller dust grains with respect to bigger dust grains. In particular, in the front-like structure, where strong CO emission is observed, there is almost no 12$\mu$m emission, indicating a relative lack of smaller dust grains. One possibility is that smaller dust grains have disappeared through coagulation on bigger grains in a denser environment, similar to what is seen in molecular clouds. If that is the case, the color variations in Fig. \[fig:SPIRE\_and\_WISE\] could somehow reflect the variations of the gas density. Other possibilities might relate back to the pre-existing grain populations in the WNM and their response to the collision. The spectacular variations of the dust color ratio from dense to diffuse parts of the nebula encompass rich information on the evolution of interstellar dust in compressed environments. The image shown in Fig. \[fig:SPIRE\_and\_WISE\] clearly deserves a dedicated and detailed analysis that we leave for a future study.
![\[fig:PDF\_phys\_size\] Histogram of the size of each structure computed using the eigenvalues of the inertia matrix.](Pictures/PDF_phys_size.pdf)
Small-scale structures {#sec:structures}
======================
The -SPIRE data provide a striking view of the small-scale structure of Draco (Figures \[fig:map250\], \[fig:RTI\], and \[fig:SPIRE\_and\_WISE\]). In this section we characterize this structure and compare it with that seen in molecular clouds.
Structure identification
------------------------
To characterize the general morphology of the nebula we broke it into individual structures using the 2D version of the [*clumpfind*]{} algorithm [@williams1994] that has been used extensively to identify structures in molecular clouds, the Milky Way, and external galaxies. Because it identifies islands above some brightness thresholds, [*clumpfind*]{} does not make any assumption concerning the shape or size of the structures. On the other hand, as it identifies structures using contours, [*clumpfind*]{} is rather sensitive to noise because it breaks contours at low brightness level.
To identify structures on the original 250$\mu$m map, we had to set the lowest threshold value of [*clumpfind*]{} to a value higher than the noise level. The noise level at the pixel size is relatively high compared to the brightness of some of the diffuse emission seen at larger scales. The result was that only the brightest structures were identified by [*clumpfind*]{}, leaving out more diffuse parts of the nebula that are visible by eye. To minimize the effect of noise, we applied [*clumpfind*]{} on a version of the map convolved by a Gaussian kernel. Empirically we found that a kernel of FWHM=30.5”, corresponding to a decrease of the angular resolution by a factor of two, from 17.6” to 35.2”, eliminates the effect of noise. The smoothed map was then projected on a coarser grid with a pixel size that is twice the original size (from 6” to 12”).
We therefore applied [*clumpfind*]{} on the smoothed -SPIRE 250$\mu$m map, converted to $N_{\rm H}$, with 40 threshold values, equally spaced in log, from $1.25 \times 10^{21}$ to $2.0 \times 10^{22}$cm$^{-2}$ (corresponding to 5 to 80MJysr$^{-1}$). The lowest threshold value is rather conservative but it rejects any contamination by galaxies while providing little impact on the properties of the identified structures. We stress that the results presented here do not depend significantly on the details of these choices (size of the smoothing kernel and threshold levels of [*clumpfind*]{}). A total of 5028 structures were identified. Their properties are described next.
Physical properties
-------------------
For each structure we defined a typical size, $L$, its mass, $M$, and its density, $n$. The histograms of these quantities are shown in Figs. \[fig:PDF\_phys\_size\] to \[fig:PDF\_density\].
### Size of structures
For each structure identified we estimated a typical size, taking the brightness distribution of the structure into account; in the definition of the size, brighter pixels have a larger weight. This definition is less sensitive to noise and to the sensitivity of an observation. The method used to estimate $L$ is described in Appendix \[sec:size\].
![\[fig:PDF\_dN\_dlogM\] $dN/d \log(M)$ diagram for the identified structures. The red line is a power-law fit for range where $M \geq 2$M$_\odot$.](Pictures/m1_nh_conv_proj_250_doubleres_PDF_dN_dlogM.pdf)
At a distance of 600pc, the angular resolution of the 250$\mu$m map translates into a physical distance of 0.05pc. This is the FWHM of the smallest structure that can be identified in the map. The way the size is estimated (see Appendix \[sec:size\]), the smallest size that can be found corresponds to $L = 0.75\,$FWHM, so the minimum value $L$ can have is in fact 0.04pc. By degrading the resolution of the original image by a factor of two, the minimum $L$ is doubled to 0.08pc. This corresponds exactly to the lowest value of $L$ found here (see Fig. \[fig:PDF\_phys\_size\]). The distribution of $L$ has quite a narrow distribution; 50% of the structures are lower in size than twice the resolution ($L\leq0.16$pc) and 90% of them are lower in size than three times the resolution ($L\leq 0.24$pc). The largest value of $L$ is 0.47pc.
The fact that we are finding physical sizes that are similar to the resolution implies that the emission is varying strongly at small scales. This is compatible with the visual impression given by Figures \[fig:map250\], \[fig:SPIRE\_and\_WISE\], and \[fig:RTI\]. The range of sizes found here is likely to be a combination of the angular resolution of the -SPIRE data and of how close small-scale structures are. The size of a structure found by [*clumpfind*]{} is influenced by the distance to its neighbors. The fact that we find that 90% of the structures are lower in size than $0.25$pc is indicative that matter is structured significantly at scales smaller than that value.
![\[fig:mass\_size\] Two-dimensional histogram of $M$ vs. $L$. The grayscale is proportional to the density of points. The red line is not a fit. It is indicative of the trend seen at low values of $L$ and $M$.](Pictures/Mass_vs_L.pdf)
### Mass {#sec:ClMF}
The mass of each structure is defined as $$M = N_{\rm Htot} \, D^2 \, \tan^2(\delta)\, \mu m_{\rm H}
,$$ where $N_{\rm Htot}$ is the total column density of a structure, summed over all pixels, $D$ the distance, $\delta$ the pixel angular size, and $\mu$ the molecular weight. Here we assumed $\mu=1.4$ to take elements heavier than hydrogen into account.
The total amount of mass in the structures identified is $\sim 5.2\times 10^3\: M_\odot$. The mass distribution of the structures is shown in Fig. \[fig:PDF\_dN\_dlogM\]. The mass ranges from 0.1 to 20$M_\odot$ with a median value of 0.53M$_\odot$. We fitted the high mass part of the distribution using a power-law $dN/d\log(M) \propto M^{-\alpha}$ assuming a $1/\sqrt{N}$ uncertainty for each data point. The exact value of the power-law exponent depends on the range over which the fit is performed. For $M\geq 2$M$_\odot$ we obtain $\alpha=1.4$. We noticed that the low end of the mass distribution has a shape similar to a log-normal distribution.
The value of $\alpha$ found for Draco is significantly different of that found for giant molecular clouds in general [$\alpha \sim 0.8$ for $M>10^4$M$_{\odot}$, @solomon1987; @kramer1998; @heyer2001; @marshall2009]. On the other hand, a mass distribution with a similar shape (log-normal plus power-law tail) and a similar range in mass was found in a study of the core mass function (CMF) in Aquila by @konyves2010. These authors found that the high mass part of their $dN/d\log(M)$ is compatible with a power law with a slope of $\alpha=1.5$. We also note that @peretto2010 found similar $\alpha$ values for small scale fragments of molecular clouds. These similarities are intriguing given the large difference in physical conditions between the cores of Aquila rift, the fragments of molecular clouds, and the clumps of Draco, and considering that different methods were used to identify structures.
We note a general trend between the mass and size of the structures that is similar to what is observed in more massive molecular clouds. Figure \[fig:mass\_size\] shows a log-log density plot of the mass of the structures versus their size. The low end of the diagram is well modeled by a power-law $M = 45\, L^{2.3}$, where $L$ and $M$ have units of pc and $M_\odot$, respectively. On the other hand, the structures with sizes larger than 0.2pc or masses larger than 1$M_\odot$ seem to depart from this relation; these structures are systematically more massive than what the relation predicts.
Several studies have highlighted such a relationship with a very similar exponent. @roman-duval2010 found $M\propto r^{2.26}$ for molecular clouds having sizes in the range 0.7-30pc. @heithausen1998 found $M\propto r^{2.31}$ for scales ranging from 0.01 to 1pc. A similar mass-size relation is seen in numerical simulations where there is no gravity, and with or without heating and cooling processes included [@kritsuk2007; @federrath2009; @audit2010]. Because of the fact that this relation is seen in different physical conditions, including isothermal gas, it is often attributed to turbulence.
![\[fig:PDF\_density\] $dN/d\log(n)$ diagram. The blue line is a log-normal fit of the distribution of the density $n$.](Pictures/PDF_density.pdf)
### Gas density
The gas density of each structure is defined by $$n = \frac{3 M}{4\pi L^3} \frac{1}{(\mu+f_{\rm H2}) m_{\rm H}}
,$$ where $f_{\rm H_2} = M_{\rm H_2}/( M_{\rm HI} + M_{\rm H_2})$ is the portion of the hydrogen that is in molecular form. It is expected that $f_{\rm H2}$ varies across the Draco field; @herbstmeier1993 estimated that the molecular portion varies from 0.04 to 0.7. The details of the -H$_2$ transition in Draco is beyond the scope of the present paper. Here we assume a constant value of $f_{\rm H_2}=0.5$.
The histogram of $n$ (Fig. \[fig:PDF\_density\]) is very well represented by a log-normal distribution (i.e., $\log(n)$ is Gaussian distributed) with a median value of $1.0\times10^3$cm$^{-3}$ and a standard deviation of 0.14 in $\log_{10}(n)$. The density values given here assume that half of the hydrogen atoms are in H$_2$. The majority of the structures have a density higher than the $^{12}$CO $J=1-0$ critical density ($n_{\rm H2} \sim 750$cm$^{-3}$).
In general, it is observed that smaller molecular clouds have higher densities. This is related to the very clumpy and open structure of molecular clouds, which implies a volume filling factor lower than unity. In other words, the mass does not scale with $L^3$ (see Fig. \[fig:mass\_size\]), therefore the density estimated over large physical scales is an underestimate of the density at small scales, explaining why the average density in GMCs is found to be smaller than the critical density of CO. For instance, from the sample of molecular clouds of @roman-duval2010, one obtains that ${\left< n_{\rm H2} \right>} = 800 L^{-0.64}$. For the typical size of a structure in Draco ($L= 0.15$pc, see Fig. \[fig:PDF\_phys\_size\]) that relationship would predict ${\left< n_{\rm H2} \right>} = 2.7\times10^3$cm$^{-3}$, only a factor of 3 more than what we estimated. Given the large dispersion of the $n_{\rm H_2} - L$ relation observed in the ISM, the values obtained here seem typical for molecular clouds.
### Jeans mass
To evaluate the gravitational stability of the structures in Draco we compared their mass with the Jeans mass that gives the maximal mass of a stable isolated and spherical clouds [@lequeux2005], $$M_j = \left(\frac{1}{\mu m_{\rm H}}\right)^2 \, \left( \frac{5}{2} \frac{k T}{G} \right)^{3/2} \,
\left(\frac{4}{3} \pi n\right)^{-1/2}
,$$ where $\mu$ is the molecular weight (equals to 1.4 or 2.4 for fully atomic or molecular hydrogen, respectively).
Only 1% of the structures have a mass larger than the Jeans mass (Fig. \[fig:PDF\_Jeans\_Mass\]). These structures are the more massive. They encompass 15% of the total mass of the structure identified.
![\[fig:PDF\_Jeans\_Mass\] $dN / d\log(M / M_{\rm Jeans})$ diagram.](Pictures/PDF_M_over_Mjeans.pdf)
Discussion {#sec:discussion}
==========
The -SPIRE observations of Draco reveal two important aspects of the gas structure resulting from the collisions of two flows. The observations reveal a dense front with regularly spaced fingers of dense gas, a structure typical of the RT instability, and numerous and contrasted small-scale structures (clumps).
Formation scenario
------------------
According to what was proposed very early on [@kalberla1984; @mebold1989; @rohlfs1989], Draco seems to be the result of the compression of warm gas in the outer part of the Galactic WNM layer ($z\sim 370$pc) because of the collision with a cloud falling from the Galactic halo onto the disk. Whether the cloud at the origin of the collision was part of the Galactic fountain or from the intergalactic medium is difficult to establish at this point.
@benjamin1997 suggested that IVCs and HVCs have trajectories in the Galactic halo that are compatible with clouds falling on the Galactic disk, attracted by the Milky Way gravitational potential, and slowed down by their interaction with the Galactic ISM. In this scenario, IVCs and HVCs would have reached their terminal velocity.
According to these authors the terminal velocity depends on the Galactic height, $z$, of the cloud, its column density, $N_{\rm H}$, the gravitational acceleration of the galaxy, $g(z)$, the density distribution of the halo gas, $n(z)$, and on a parameter that quantify the efficiency of the friction, $C_{\rm d}$, $$\label{eq:terminal_velocity}
v_{\rm t}(z) = \sqrt{\frac{2\, N_{\rm H}\, g(z)}{C_{\rm d}\, n(z)}}.$$ The total gravitational acceleration (gas and stars) is approximated by $$g(z)= 9.5 \times 10^{-9}\, {\rm tanh} (z/{\rm 400 pc}).$$ The gas distribution is the sum of the layer [@dickey1990], the WIM layer [@reynolds1993], and of a hot ionized layer [@wolfire1995a], shown in Fig. \[fig:vterminal\] [for details, see @benjamin1997]. Like in @benjamin1997, we assumed $C_{\rm d}=1$.
@benjamin1997 stressed that this picture is compatible with the observations as clouds closer to the plane have lower absolute velocity than clouds far away. Draco fits that picture. Assuming a column density of $N_{\rm H} = 1.3 \times 10^{20}$cm$^{-2}$ and a height above the plane of $z = 370$pc, its observed velocity ($v= v_{\rm LSR}/\sin b = 40$kms$^{-1}$) is comparable to its expected terminal velocity [$v_{\rm t}\approx 50$kms$^{-1}$, see Fig. \[fig:vterminal\] and @benjamin1999a].
![\[fig:vterminal\] Terminal velocity (top) and deceleration (middle) as a function of Galactic height for a cloud of $N_{\rm H}=1.3 \times 10^{20}$cm$^{-2}$. The bottom panel is the gas density vs. $z$, including the , the WIM, and the HIM. The dotted lines indicate the range in distance of @gladders1998 and the dashed lines show the corresponding values on each curve.](Pictures/vterminal_vs_z.pdf "fig:"){width="0.98\linewidth"} ![\[fig:vterminal\] Terminal velocity (top) and deceleration (middle) as a function of Galactic height for a cloud of $N_{\rm H}=1.3 \times 10^{20}$cm$^{-2}$. The bottom panel is the gas density vs. $z$, including the , the WIM, and the HIM. The dotted lines indicate the range in distance of @gladders1998 and the dashed lines show the corresponding values on each curve.](Pictures/a_vterminal_vs_z.pdf "fig:"){width="0.98\linewidth"} ![\[fig:vterminal\] Terminal velocity (top) and deceleration (middle) as a function of Galactic height for a cloud of $N_{\rm H}=1.3 \times 10^{20}$cm$^{-2}$. The bottom panel is the gas density vs. $z$, including the , the WIM, and the HIM. The dotted lines indicate the range in distance of @gladders1998 and the dashed lines show the corresponding values on each curve.](Pictures/n_vs_z_draco.pdf "fig:"){width="0.98\linewidth"}
If Draco has indeed reached its terminal velocity, its average velocity does not inform us on its origin (Galactic fountain or extragalactic gas). Draco could have once been an HVC but it would have slowed down as it interacted with the ISM of the disk. One fact that could favor the Galactic fountain origin is the bright dust and CO emission, showing that it contains a significant amount of heavy elements. On the other hand, the matter that composes Draco today could be Galactic gas swept up by an originally low metallicity cloud that has been slowed down by the Galactic ISM. At $z=370$pc above the Galactic plane, the total column density (+H$^+$) encountered by a cloud coming from infinity is about $N_{\rm H} = 1\times10^{20}$cm$^{-2}$, estimated by integrating $n(z)$ (Fig. \[fig:vterminal\]) for $z>370$pc. This is similar to the column density of Draco itself. Therefore, a large portion of the mass of Draco could be composed of Galactic gas with near solar metallicity even though the original cloud had a low metallicity. A detailed study of the metallicity of the gas in Draco could provide some answers to that question.
There has been some speculation about the potential role of the HVC component seen at $v_{\rm LSR} \sim -150$kms$^{-1}$ in the formation of Draco [@hirth1985]. The morphology of the three components (Fig. \[fig:NHI\_GBT\]) could suggest that the IVC (Draco) is the result of the dynamical interaction between the HVC and LVC. Along those lines, @pietz1996 proposed that faint 21cm emission at velocities between the HVC and IVC could be due to an inelastic collision of HVC gas with the Galactic thick disk. This emission is extremely faint (see Fig. \[fig:spectrumGBT\]). If Draco is the result of gas being decelerated from -100 to -20kms$^{-1}$, one would expect to see significant 21cm emission at all velocities. One possibility could be that the original HVC was composed of two components: one component that has already hit the disk and produced Draco and a second component that is on its way and that we observe today at $v<-100$kms$^{-1}$.\
Whatever the origin and metallicity of the infalling cloud, it seems likely that it has accelerated and pressurized WNM gas at rest in the Galactic layer, producing a shocked front of denser gas that is progressing toward us in the diffuse ISM. The structure of Draco itself, with its long cometary plumes, is indeed reminiscent of a dense cloud moving in a more diffuse medium [@odenwald1987]. The structure of the front, with its sharp increase in column density, is also indicative of a shock. Assuming $T=8000$K and taking elements heavier than hydrogen into account, Draco moves with respect to the Galactic ISM with a Mach number of $M=5.8$. This scenario is compatible with the FUV observations of @park2009 that show the presence of hot gas due to shock heating. In addition, these conditions are favorable for the development of a RT instability at the interface and for the formation of CNM out of compressed WNM through the thermal instability [@saury2014].
Turbulence
----------
In this section we use the RT typical length, combined with velocity information from the 21cm data, to estimate parameters of interstellar turbulence, such as the rate of transferred energy, $\epsilon$, the Reynolds number, $Re$, and the turbulence dissipation scale, $l_d$. The values of these parameters estimated for Draco are summarized in Table \[tab:turbulence\].
### Rayleigh-Taylor instability and gas viscosity {#sec:viscosity}
The RT instability occurs when two fluids are accelerated toward each other. For the incompressible case, the fluids cannot interpenetrate freely; bubbles of the light fluid rise into the heavy fluid forming finger structures. It appears that such structures are also observed in compressible fluids, like supernovae remnants [@ellinger2012]. In our case, the fingers are pointing toward the Galactic plane, indicating that Draco is denser than the medium in which it is moving toward, i.e., the Galactic WNM layer.
For two incompressible fluids of constant viscosity separated by a horizontal boundary, [@chandrasekhar1961] showed that the RT instability typical length is given by $$\label{eq:Chandrasekhar}
\lambda_{\rm max}=4\pi {\left( \frac{\nu_{\rm kin}^2A}{a} \right)}^{1/3}
,$$ where $\nu_{\rm kin}$ is the kinematic viscosity, $a$ is the acceleration, and $$A=\frac{\rho_1-\rho_2}{\rho_1+\rho_2}$$ is the Atwood number that takes values between 0 and 1. Here $\rho_1$ and $\rho_2$ are the density of the heavier and lighter fluids, respectively.
The Chandrasekhar study of hydrodynamic stability was made for incompressible fluids, which is not the case in the interstellar medium. However, [@ribeyre2004] studied the compressible case for supernovae and concluded that compressibility slows down the growth of the RT instability, but it has no important impact on $\lambda_{\rm max}$. The typical length of the RT instability in supernovae is usually estimated with equation \[eq:Chandrasekhar\] [@ellinger2012]. The same assumption is made here. Interestingly, Eq. \[eq:Chandrasekhar\] offers us the opportunity to estimate the gas viscosity knowing the instability typical length, the gas acceleration, and $A$.
What acceleration should be considered here ? In the classical picture of two fluids, one on top of the other, $a$ is the gravitational acceleration. Here the gas is slowed down by the friction with the Galactic ISM. In fact the cloud is decelerating as it gets closer to the disk, slowly reaching $v=0$.
In this context, a cloud entering the disk is slowed down by the friction with the ISM. The deceleration can be estimated by $$a_{\rm t} = v_{\rm t} \, \frac{d v_{\rm t}}{dz}.$$ Assuming that Draco has reached its terminal velocity, and using Eq. \[eq:terminal\_velocity\] for $v_{\rm t}$, we estimated that its deceleration is $a=2.3\times 10^{-8}$cms$^{-2}$.
The parameter $A$ depends on the density of the two fluids. The density of the gas at a height above the plane of $z\approx370$pc, the assumed location of Draco, is $n=0.07$cm$^{-3}$ (see Fig. \[fig:vterminal\]). For Draco, the density can be estimated using the dust emission and 21cm data. The gas column density, averaged over the whole nebula, is about $1.3 \times 10^{20}$cm$^{-2}$. Draco spans about 3$^\circ$ on the sky, which translates into 30pc at a distance of 600pc. Assuming a depth of 30pc for the infalling cloud, its average density would be on the order of 1.4cm$^{-3}$. This is an averaged value that represents the density of the WNM in Draco. The density is certainly larger at the shock front where strong CO emission [as well as CH, @stark1997] is observed (the CO critical density is on the order of $10^3$cm$^{-3}$). Assuming $n_1=1.4$cm$^{-3}$ and $n_2=0.07$cm$^{-3}$, the Atwood number is $A=0.90$.
The RT length, $\lambda_{\rm max}$, is the one estimated from the periodic structure observed at the dense front (Fig. \[fig:RTI\]). Even though describing the front as RT-like structure is appealing, the determination of the typical scale is difficult and somewhat uncertain. The arch-like structures have sizes ranging from 1.5 to 3.2pc, with an average value of $\lambda_{\rm max}=2.2$. Using this average value, the kinematic viscosity computed with Eq. \[eq:Chandrasekhar\] is $\nu_{\rm kin} = 6.3 \times 10^{22}$cm$^2$s$^{-1}$.
It is interesting to compare $\nu_{\rm kin}$ to the more classical molecular viscosity $$\nu_{\rm mol}=\frac{1}{3}\frac{1}{\sigma n}\sqrt{\frac{3}{2}\frac{kT}{\mu m_{\rm H}}}.$$ For typical values of temperature and density for the CNM ($T=100$K, $n=30$cm$^{-3}$) and WNM in Draco ($T=8000$K, $n=1.4$cm$^{-3}$) and assuming $\sigma=1\times 10^{-15}$cm$^{-2}$ for the hydrogren cross-section [@lequeux2005], the molecular viscosity is $\nu_{\rm mol}=1.0 \times 10^{18}$cm$^2$s$^{-1}$ and $\nu_{\rm mol} = 2.0\times 10^{20}$cm$^2$s$^{-1}$ for the CNM and WNM, respectively. The value of the WNM (the largest of the two) is more than 300 times smaller than what is estimated using the RT typical length and Eq. \[eq:Chandrasekhar\]. That results depends on the distance to the power $3/2$. Even if we assumed a distance to Draco of 400pc, the kinematic viscosity would still be more than 200 times larger than the molecular viscosity. [This is also much larger than the uncertainty of less than a factor of 2 on $\lambda_{\rm max}$.]{}
### Energy transfer rate
An important parameter that characterizes the turbulent cascade of energy is the energy transfer rate by unit of mass, $$\epsilon = \frac{1}{2} \frac{\bar{v_l}^3}{l}.$$ Here $l$ and $\bar{v_l}$ represent the typical scale and typical average velocity of the flow. We assumed $l \sim 30$pc for the largest scale of the flow. The average velocity of the gas at that scale is best estimated by looking at the width of the 21cm line, averaged over the whole region. Using the 21cm GBT data (Fig. \[fig:NHI\_GBT\]), we estimated the velocity dispersion of the IVC component to be $\sigma_v=10.7$kms$^{-1}$, which is likely to be dominated by the warm phase. Removing the thermal contribution to the line width (assuming $T=8000$K), this reduces to $\sigma_v=7.0$kms$^{-1}$. Assuming a Gaussian distribution of velocity, this corresponds to an average velocity of $\bar{v_l} = \sqrt{8/\pi}\,\sigma_v=11.2$kms$^{-1}$.
The energy transfer rate of the WNM in Draco is $\epsilon = 3.9\times 10^{-3}$$L_\odot$$M_\odot^{-1}$. This value is slightly higher than that estimated for the in the solar neighborhood [$\epsilon \sim 10^{-3}$$L_\odot$$M_\odot^{-1}$; @hennebelle2012a].
### Reynolds number
By combining velocity information from the 21cm data with the measurement of $\nu_{\rm kin}$, one can estimate the Reynolds number of the flow $$Re = \frac{l\, \bar{v_l}}{\nu_{\rm kin}}.$$ For Draco, we find $Re=1600$, a value significantly smaller than the more traditional estimates for the ISM [$Re \sim 10^5$; @chandrasekhar1949]. This difference is explained by the fact that $\nu_{\rm kin} >> \nu_{\rm mol}$ as $Re$ is usually computed assuming that the dissipation of turbulent energy is dominated by molecular viscosity.
Quantity Symbol Value Units
------------------------ --------------------- -------------------- -------------------------
Largest scale $L$ 30 pc
Average velocity $\bar{v_l}$ 11.2 kms$^{-1}$
Rayleigh-Taylor length $\lambda_{\rm max}$ 2.2 pc
Viscosity $\nu_{\rm kin}$ $6.3\times10^{22}$ cm$^2$s$^{-1}$
Reynolds number Re $1600$
Energy transfer rate $\epsilon$ $3.9\times10^{-3}$ $L_\odot$$M_\odot^{-1}$
Dissipation scale $l_d$ $0.1$ pc
: \[tab:turbulence\] Properties of the turbulence in Draco.
### Dissipation scale
Turbulent energy in the ISM is dissipated via either viscous (hydrodynamical) or resistive (MHD) processes, depending on which process happens at the largest scale [@benjamin1999a; @hennebelle2013c]. The scale at which turbulent energy is dissipated is $$l_{\rm d} \approx \left( \frac{\nu^3}{\epsilon} \right)^{1/4}.$$ Like for $Re$, the scale at which turbulent energy is dissipated depends on what process is more efficient at transforming motion into heat. Assuming that $\nu=\nu_{\rm kin}$, the dissipation scale is $l_d=0.1$pc. This is close to two to three orders of magnitude larger than what is generally estimated for the WNM ($l_d \sim 0.003$pc) and the CNM ($l_d \sim 2$AU), assuming that the dissipative process is molecular viscosity [@lequeux2005].
The other potential dissipative process is the ion-neutral friction. Following @lequeux2005, the ambipolar diffusion typical scale is $$\label{eq:ldiss}
l_{\rm AD} = \sqrt{\frac{\pi}{\mu m_{\rm H}}} \frac{B}{2X\,\langle\sigma v\rangle \, n_n^{3/2}}
,$$ where $X$ is the ionization ratio, $n_n$ the density of neutrals, and $\langle \sigma v \rangle$ the collision rate between ions and neutral (assumed to be the Langevin rate $2 \times 10^{-9}$cm$^3$s$^{-1}$). For typical conditions in the ISM, from warm gas to molecular clouds, $l_{\rm AD}$ is often larger than the dissipation scale for molecular viscosity. This is the case in Draco; assuming $n_n=1.4$cm$^{-3}$ (the density of the compressed WNM) and $B=6$$\mu$G, which is a typical value for the WNM in the solar neighborhood [@beck2001], the ambipolar diffusion scale is also $l_{\rm AD}=0.1$pc.
The kinematic viscosity estimated from the RT-like structures indicates that the dissipation of energy happens at a scale of $\sim 0.1$pc; this is much larger than the dissipation scale of molecular viscosity but is in agreement with what is expected if ambipolar diffusion is the dominant process of energy dissipation.
### Wardle instability
Given that the magnetic field seems to play an important role in dynamics, one could consider that the structure of the front is the result of the Wardle instability [@wardle1990]. This instability occurs because of the velocity difference between neutrals and ions, when the Alfven Mach number, $M_A$, is greater than 5. The Alfven speed in a WNM with $n=0.1$cm$^{-3}$ and $B=6$$\mu$G is $V_A = 35.0$kms$^{-1}$. Considering that the velocity of the shock is 40kms$^{-1}$, the Alfven Mach number is $M_A=1.1$. This suggests that the Wardle instability is not dominant in Draco.
Dense structures
----------------
Using the two-dimensional version of [*clumpfind*]{} on the -SPIRE 250$\mu$m, converted to $N_{\rm H}$, structures of sizes from 0.08 to 0.5pc were revealed. Interestingly, these small structures share many properties with sub-structures found in more massive molecular clouds as well as with prestellar cores. First, the mass distribution ranging from 0.1 to 20$M_\odot$ with a median value of 0.53$M_\odot$ is very similar to a prestellar core mass distribution [@konyves2010]. In addition, the mass-size relation ($M\propto L^{2.3}$) is typical of that seen in more massive molecular clouds. Figure \[fig:SPIRE\_and\_WISE\] also shows that the 12$\mu$m emission from smaller dust grains is extremely weak compared to the big grain emisson at 250$\mu$m, as is observed in dense parts of molecular clouds [@stepnik2003].
On the other hand, unlike prestellar cores [@konyves2010] and unlike molecular clouds [@roman-duval2010], 85% of the structures found in Draco are not gravitationally bound. This is compatible with the fact that the density histogram is found to be log-normal, something generally associated with compressive supersonic flows without gravity [@kritsuk2007]. Also, even though the structures in Draco are dense enough ($n\sim 1000$cm$^{-3}$) to explain the strong CO emission observed, the density of these structures is 100 times less than that seen in prestellar cores.
We suggest that the formation of dense structures seen in Draco is the result of the transition from the WNM to cold and dense gas through the thermal instability of . This transition is favored by the compression of the warm gas at the interface between the Galactic layer and the infalling matter. As shown in several numerical simulations, the increase of density of the WNM sends the gas in a thermally unstable state that favors the transition to the CNM [@hennebelle1999; @audit2005; @vazquez-semadeni2006a; @inoue2009; @saury2014]. If the increase in density is sufficient, the formation of molecules is expected; using hydrodynamic simulations of a strong shock wave propagating into WNM, [@koyama2000] showed that it can produce a thin and dense H$_2$ layer via thermal instability. They also predicted fragmentation of the thermally collapsed layer into small molecular structures, similar to that seen in Draco. An increase of density would also favor faster evolution of interstellar dust through grain-grain collision. The low abundance of small dust grains in the densest parts of the front favors a coagulation process [sticking of smaller grains onto bigger grains, @kohler2012].
The fact that the CMF and fragments of molecular clouds mass spectrum have a similar power-law index has been used as an argument in favor of turbulence being responsible for the fragmentation and structuration of matter in dense environments. On the other hand, because of the fact that most of these objects are gravitationally bound, it is difficult to exclude the role of gravity. In this context, Draco is interesting as it reveals a clear example of dense structures that are formed through hydrodynamical processes combined with the thermal instability of with the exact same statistical properties as gravitationally bound systems.\
The small-scale structures of Draco have statistical properties of a thermally bistable turbulent flow. Numerical simulations [e.g., @saury2014] have shown that the formation of CNM structures happens when the WNM is thermally unstable (i.e., compressed) and if the cooling time is shorter than then dynamical time. These two conditions must be satisfied in order for the transition to occur. The cloud collision provides the necessary increase in WNM density but at the same time, the input of kinetic energy increases the amplitude of the turbulent motions, lowering the dynamical time. The formation of CNM can then only occur if part of the turbulent energy is dissipated such that dynamical time becomes larger than the cooling time. The small-scale structures of Draco have sizes slightly larger (0.1 to 0.5pc) than the WNM dissipation scale estimated here ($l_{\rm d} = 0.1$pc). This correspondance could be a coincidence; the minimum size of the structures found here is limited by the angular resolution and noise level of the data. Nevertheless, in the absence of higher resolution data, the observations presented here are compatible with the idea that the dissipation of turbulent energy of the WNM via the ambipolar diffusion provides favorable conditions for the formation of dense structures at scales close to $l_{\rm d}$.
Conclusion {#sec:conclusion}
==========
We presented -SPIRE observations of the Draco nebula, a diffuse high Galactic latitude interstellar cloud located about $370\: pc$ above the Galactic plane. Draco is likely to be the result of the compression of diffuse gas of the outer WNM layer by the collision of a cloud falling from the Galactic halo. It offers a unique opportunity to study the formation of dense structures in a colliding flow.
The data reveal the fine details of the structure of this cloud, especially of a front that shows a typical Rayleigh-Taylor instability structure. We were able to estimate the gas kinematic viscosity from the typical length of this structure. Combined with 21cm GBT data, we estimated the Reynolds number ($Re = 1600$), the energy transfer rate ($\epsilon=3.9 \times 10^{-3}$$L_\odot$$M_\odot^{-1}$), and the dissipation scale of the turbulent cascade ($l_d = 0.1$pc). The viscosity and dissipation scale are typical of values that would be found in the WNM in the case in which the turbulent energy dissipation is dominated by ambipolar diffusion.
The SPIRE 250$\mu$m map reveals high contrast structures with a wealth of small-scale clumps. Using the [*clumpfind*]{} algorithm, a total of 5028 structures were identified, with physical sizes ranging from 0.08 to 0.5pc, where the lower limit is set by the angular resolution of the data. The mass spectrum is very close to that of prestellar cores (in terms of shape and range), even though the densities in Draco are 100 times lower. The high mass part of the mass spectrum is well represented by a power-law $dN/d\log(M) \propto M^{-1.4}$, such as that seen for prestellar cores and molecular clouds fragments. On the other hand, the majority of the structures (85% in mass) are not gravitationally bound. This is corroborated by the fact that the density spectrum clearly follows a log-normal distribution.
The results presented here reveal dense structures formed in a colliding flow where gravity is not a dominant process. We showed that these structures are likely to be the result of the transition from the WNM to CNM through the thermal instability. We propose that the formation of dense structures is favored by the dissipation of the turbulent energy in the WNM through ambipolar diffusion, at a scale of $l_{\rm AD}=0.1$pc. The energy dissipation favors the formation of small CNM structures at scales slightly larger than $l_{\rm AD}$. The fact that these structures share many statistical properties with denser environments is an indication of the importance of interstellar turbulence and thermal instability in shaping the structure of the ISM.
Herschel-SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).
Estimating gas column density from dust emission
================================================
{width="0.48\linewidth"} {width="0.48\linewidth"} {width="0.48\linewidth"} {width="0.48\linewidth"}
\[sec:convert\_to\_NH\]
We are interested in using the large dust grain emission as a tracer of column density. To do so, it is customary to fit the spectral energy distribution of dust using a model to separate the effects of dust properties, dust temperature, and column density. The dust emission in the SPIRE wavelength range is dominated by big grains at thermal equilibrium with the ambient radiation field. It is often modeled as a modified blackbody spectrum, i.e.,$$\label{eq:modif-BB}
I_\nu= \tau_{\nu_0}\, B_\nu(T_{\rm obs}) \, {\left( \frac{\nu}{\nu_0} \right)}^{\beta_{\rm obs}}
,$$ where $\tau_{\nu 0}$ is the optical depth at a reference frequency $\nu_0$, $T_{\rm obs}$ represents the dust equilibrium temperature, and $\beta_{\rm obs}$ is the dust spectral index. For a constant dust emissivity and a constant dust-to-gas ratio, $\tau_{\nu_0}$ is proportional to gas column density. It can thus be used as a tracer of the structure of matter projected on the sky. To obtain $\tau_{\nu_0}$ it is common practice to fit the dust emission pixel by pixel, by combining PACS and SPIRE data for instance. This is unfortunately impossible in our case because of the failure of the PACS observations.
A similar fit has been performed over the whole sky by @planck_collaboration2014h using 100$\mu$m data and 350, 550, and 850$\mu$m data. The maps of $\tau_{\rm 353\,GHz}$, $T_{\rm obs}$, and $\beta_{\rm obs}$ for the Draco field are shown in Fig. \[fig:tau\_T\_beta\_planck\]. These maps have an angular resolution of 5’, 5’, and 30’, respectively. As a comparison, the SPIRE 250$\mu$m map convolved at 5’ is also shown in Fig. \[fig:tau\_T\_beta\_planck\]. The morphological resemblance of $I_{250\, \mu m}$ and $\tau_{\rm 353\, GHz}$ is striking, indicating that variations of dust properties and dust temperature do not dominate the observed emission fluctuations in the SPIRE bands. This is confirmed by looking at the maps of $T_{\rm obs}$ and $\beta_{\rm obs}$ computed by @planck_collaboration2014h. The $1 \sigma$ variations of $T_{\rm obs}$ and $\beta_{\rm obs}$ are only 3% and the structure of Draco is barely visible in these parameter maps. This is in accordance with the fact that there are no local sources of heating photons and a relatively low extinction; locally the radiation field appears to be very uniform.
In addition we note the presence of small-scale structures in $T_{\rm obs}$, apparently unrelated to variations of the dust equilibrium temperature. These fluctuations were observed and quantified by @planck_collaboration2014h; they are caused by the cosmic infrared background anisotropies (CIBA). In such a diffuse high Galactic latitude field, the CIBA appear very strongly in the map of $T_{\rm obs}$ (see @planck_collaboration2014h for details).
Even though the dust products are at significantly lower resolution than the SPIRE data, one could envisage estimating the dust column density by correcting for $T_{\rm obs}$ and $\beta_{\rm obs}$ obtained with . We conclude that the variations in $T_{\rm obs}$ that would be of interstellar origin are not strong enough to overcome the drawback of affecting the data with the CIBA. In addition the correlation of $I_{250 \mu m}$, convolved at 5’, with $\tau_{\rm 353 GHz}$ is extremely tight (see Fig. \[fig:tau\_vs\_I250\]), with residuals compatible with CIBA. For those reasons we decided to simply multiply $I_{250\, \mu m}$ by a single factor to translate it to hydrogen column density, $N_H$. To do so, we use the $I_{250\, \mu m}$-$\tau_{\rm 353\, GHz}$ correlation shown in Fig. \[fig:tau\_vs\_I250\] to set the zero level of the SPIRE map. Then, combined with the relationship $\tau_{\rm 353\, GHz} = 6.3 \times 10^{-27} \, N_{HI}$ that @planck_collaboration2014h found by comparing and 21cm data at high Galactic latitude, we converted the 250$\mu$m emission to hydrogen column density using the following relation: $$N_{\rm H} = 2.49\times 10^{20} \, I_{\rm 250\, \mu m}
,$$ where $I_{\rm 250\, \mu m}$ is the zero level corrected map, expressed in MJysr$^{-1}$, and $N_{\rm H}$ is the total hydrogen column density ( and H$_2$) expressed in cm$^{-2}$. Because we assumed constant values for $T_{\rm obs}$ and $\beta_{\rm obs}$, no color correction is needed in the correspondence between $I_{\rm 250\, \mu m}$, $\tau_{\rm 353\, GHz}$, and $N_{H}$.
![\[fig:tau\_vs\_I250\] -SPIRE 250$\mu$m intensity, smoothed to 5’, compared to optical depth at 353GHz. The red line is the linear relation $I_{250\,\mu m} = 6.37 \times 10^5 \, \tau_{\rm 353 \, GHz} - 1.46$. The intercept of this relation was removed from the $I_{\rm 250\, \mu m}$ map to set its zero level.](Pictures/I250_vs_tau353.pdf){width="\linewidth"}
Size of structures {#sec:size}
==================
Each structure is composed of a number of pixels in the map,where each pixel $i$ is defined by a position \[$X_i$, $Y_i$\] and a column density $N_{{\rm H}, i}$. The structures are not assumed to be round; we estimate the length of their short and long axis using the inertia matrix [@hennebelle2007a; @saury2014] $$M = \begin{bmatrix}
\sigma^2(X) & \sigma^2(XY) \\
\sigma^2(XY) & \sigma^2(Y) \\
\end{bmatrix}
,$$ where $$\sigma^2(X) = \frac{1}{N_{\rm H tot}} \, \sum_i N_{{\rm H}, i} \, ( X_i - \langle X \rangle )^2$$ $$\sigma^2(Y) = \frac{1}{N_{\rm H tot}} \, \sum_i N_{{\rm H}, i} \, (Y_i - \langle Y \rangle)^2$$ $$\sigma^2(XY) = \frac{1}{N_{\rm H tot}} \, \sum_i N_{{\rm H}, i} \, (X_i - \langle X \rangle) \, (Y_i - \langle Y \rangle)$$ with $$N_{\rm H tot} = \sum_i N_{{\rm H}, i}$$ $$\langle X \rangle = \frac{1}{N_{\rm H tot}} \, \sum_i N_{{\rm H}, i} \, X_i$$ $$\langle Y \rangle = \frac{1}{N_{\rm H tot}} \, \sum_i N_{{\rm H}, i} \, Y_i.$$
The two eigenvalues of $M$ correspond to the smallest and largest semi-axes of the structure, denoted $L_{\rm min}$ and $L_{\rm max}$, respectively. From these, we define a typical size for the structure, i.e., $$L = \left( L_{\rm max} \, L_{\rm min}^2 \right)^{1/3}.$$ Here we make the hypothesis that an elongated structure projected on the sky is more likely to have a depth along the line of sight that corresponds to the smallest projected size on the sky.
We check that for a uniformly weighted circle of radius $R$, this method gives $L=R$. More generally, $L_{\rm min}$ and $L_{\rm max}$ correspond to the half length of an ellipse, whatever its orientation with respect to the $X-Y$ plane. For a point source with a Gaussian distribution of brightness of width $\sigma$, the typical size is $L = 1.77 \, \sigma$ or $L = 0.75\, FWHM$.
We also consider another way of defining the size of a cloud, often used in the literature, where $R=\sqrt{A/\pi}$, with $A$ being the surface of the structure (in pc$^2$) proportional to the number of pixels of the structure, $N_{\rm pix}$. These two methods give very concordant results indicating that structures are not very elongated.
[^1]: is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
[^2]: http://idoc-herschel.ias.u-psud.fr
|
---
abstract: 'We provide insights into the influence of surface termination on the oxygen vacancy incorporation for the perovskite model material SrTiO$_3$ during annealing in reducing gas environments. We present a novel approach to control to tailor the oxygen vacancy formation by controlling the termination. We prove that a SrO-termination can inhibit the incorporation of oxygen vacancies across the (100)-surface and apply this to control their incorporation during thin film growth. Utilizing the conducting interface between LaAlO$_3$ and SrTiO$_3$, we could tailor the oxygen-vacancy based conductivity contribution by the level of SrO termination at the interface.'
author:
-
bibliography:
- 'library.bib'
title: 'SrTiO$_3$ termination control: A method to tailor the oxygen exchange kinetics'
---
surface kinetics; 2DEGs; defect chemistry; substrate termination; oxide interfaces
Impact Statement {#impact-statement .unnumbered}
================
For the first time the termination dependent oxygen exchange kinetics are reported for the perovskite model material SrTiO$_3$ and applied to the 2D electron gas model system LaAlO$_3$/SrTiO$_3$.
Introduction
============
Transition metal oxides have become a central topic in research over the past decade due to their manifold interesting properties. [@C.N.R.RaoSolid1989; @Christen2008] One of the most commonly used perovskites is SrTiO$_3$ (STO), which offers the advantage of a well known defect chemistry. Typical applications for STO as a functional material are resistive switching devices,[@Waser2009] catalysis[@Kawasaki2016] or thermoelectrics.[@Brooks2015]
Regardless of the application, the oxidation state of STO is a key parameter. In the field of thermoelectrics, oxygen vacancies are used to tailor the thermal conductivity of STO.[@Brooks2015; @Breckenfeld2012a] When using STO for (photo-)catalysis, specific doping with oxygen vacancies is utilized to increase the activity.[@Tan2014; @Mueller2015] Resistive switching of STO is based on the generation and redistribution of oxygen vacancies. Thus oxygen vacancies generated during growth or dedicated annealing steps define its switching properties.[@Janousch2007; @Szot2006]
The central role of oxygen vacancies for all STO applications has resulted in intense research efforts to understand and control their formation.[@Hensling2017; @Hensling2018; @Lee2016b; @Sambri2012; @Scullin2010; @Chen2002b; @Schneider2010; @Gunkel2010; @Moos1995b; @Moos1997a; @DeSouza2012; @Xu2016a] Apart from the classic influence factors temperature and pressure[@Moos1995b; @Moos1997a; @DeSouza2012] it was found that e.g. UV radiation[@Hensling2018; @Merkle2001; @Merkle2002; @Merkle2008; @Leonhardt2002; @Walch2017] plays a crucial role. STO substrates are easily reduced during thin film growth at low oxygen pressure. This is a drawback as it can mask the functional properties of the deposited thin film. Although a strong termination dependence of the oxygen exchange kinetics has been demonstrated for other perovskites,[@Tascon1981; @Huang2014; @Maiti2016] it has only been scarcely considered for STO(100),[@Alexandrov2009; @DeSouza2012] which can exhibit a TiO$_2$-termination, a SrO-termination or a mixed termination.[@Kawasaki1993; @Koster2000] The termination of STO especially plays a key role for applications, which rely on interfaces. Examples are the properties of magnetic heterojunctions,[@Zheng2010] interface dependent superconductivity,[@Rijnders2004a] and the formation of a two dimensional electron gas (2DEG), as observed at the interface of the model system LaAlO$_3$(LAO)/STO.[@Chen2011; @Ohtomo2004; @Breckenfeld2013; @Herranz2007; @Kalabukhov2007; @Cancellieri2010; @Huijben2009a; @Nishimura2004; @Nakagawa2006] The latter one is an outstanding candidate for an all-oxide field effect transistor. [@Woltmann2015; @Goswami2015; @Liu2012; @Hosoda2013; @Eerkes2013; @Liu2015; @Hurand2015]
Both properties, termination and oxidation state, thus play a central role in the field of STO applications. Yet their interplay has not been investigated so far, which is especially surprising as an influence of the termination layer on the oxygen exchange kinetics is well known for other perovskite systems. In this work we will thoroughly investigate the role of the STO termination on the oxygen exchange kinetics. We find that the TiO$_2$-termination is more favorable to form oxygen vacancies, while the SrO-termination can completely suppress their formation. Hence, we present a new method to tailor the oxygen vacancy formation in SrTiO$_3$, namely by a precise control of the termination. In the course of this we will present evidence for an instability of the SrO termination under ambient conditions. Finally we will apply this knowledge to LAO/STO and show that we can tailor the resistivity of both, the crystalline system grown under reducing conditions and the amorphous system. As the resistivity for both system depends on oxygen vacancies, we thus prove termination control is a new method for controlling the formation of oxygen vacancies.
Experimental
============
To systematically control different degrees of SrO-termination, SrO was deposited on TiO$_2$-terminated STO (treatment with buffered HF)[@Kawasaki1993; @Koster2000] from a SrO$_2$ target with a laser fluence of 0.81 $\frac{\textnormal{J}}{\textnormal{cm}^{2}}$ at an oxygen pressure of 10$^{-7}$ mbar and $800~^\circ\text{C}$ substrate temperature with a target substrate distance of 44 mm. The SrO coverage was controlled using reflective high energy electron diffraction. The deposition of both, crystalline and amorphous LAO, was performed at the same target substrate distance, at 10$^{-4}$ mbar and with a laser fluence of 1.3 $\frac{\textnormal{J}}{\textnormal{cm}^{2}}$ from a single crystalline LAO target. Amorphous LAO was grown at room temperature and crystalline LAO at $800~^\circ\text{C}$ with subsequent quenching (cool down time to below 400 $^{\circ}$C was about 40 s). The crystalline thin films were $8$ unit cells thick, the amorphous about 12 unit cells.
For the *in situ* annealing process samples were annealed under oxidizing conditions after termination to minimize adsorbates. Subsequently the annealing experiment was performed for 1 h at an oxygen pressure of 10$^{-6}$ mbar at $800~^\circ\text{C}$ with subsequent quenching (cool down time to below 400 $^{\circ}$C was about 40 s).
The *ex situ* samples were stored at room temperature under ambient conditions for 60 h before being exposed to the same annealing conditions.
Electrical characterization at room temperature was performed using a *Lakeshore 8400 Series* Hall measurement setup. Low temperature electrical characterization was performed with a physical property measurement setup. The XPS is a *PHI 5000 Versa Probe* and the AFM a *Omicron VT AFM XA*. The photoemission angle was $45^\circ$ and the spectra were fitted using *Casa XPS* with a Shirley background and a convolution of Gaussian and Lorentzian line shape.
Results
=======
Interplay of termination and oxygen vacancy incorporation
---------------------------------------------------------
In order to investigate the termination dependent oxygen vacancy incorporation, STO single crystals were SrO- and TiO$_2$-terminated selectively. After termination, we applied the *in situ* annealing process at 10$^{-6}$ mbar and $800~^\circ\text{C}$. The subsequent quenching of the samples preserves the defect equilibrium achieved at high temperature.
![a) Carrier concentration of TiO$_2$ terminated (left) and SrO terminated (right) STO after 1 h of annealing at 10$^{-6}$ mbar oxygen pressure and $800~^\circ\text{C}$ for an *in situ* sample (red) and a sample stored 60 h in air (blue). b) the topography of the SrO terminated sample after 60 h of air storage and c) the topography of a SrO terminated sample measured *in situ* before storage in air. The topography in b) was measured for the sample presented in c), after that sample was exposed to air for 60 h.[]{data-label="fig:STO"}](STO_SO){width="\linewidth"}
Figure \[fig:STO\] a) shows the resulting carrier concentration $n$, which is a measure for the oxygen vacancy concentration, for TiO$_2$ and SrO terminated STO (red square). The resulting carrier concentration for the TiO$_2$ terminated sample is $5\times10^{18}$ cm$^{-3}$, a typical value observed for the applied conditions.[@Hensling2017; @Frederikse1964a] The carrier concentration of the SrO terminated sample after the same treatment is, however, below the measurement limit ($<10^{10}$ cm$^{-3}$). This is a first hint to a termination dependency of the oxygen vacancy formation.
The observed behavior, however, changes drastically after SrO terminated STO was exposed to ambient conditions for 60 h. The annealing of *ex situ* samples results in high carrier densities independent of the termination (blue triangles). Ambient storage thus affects the reduction of SrO-terminated STO, while the carrier concentration of TiO$_2$-terminated STO remains unchanged.
Concomitant to the reduction behavior of SrO-terminated STO, also the topography changes, when exposed to ambient conditions . Figure \[fig:STO\] c) shows the topography of an *in situ* annealed sample and figure \[fig:STO\] b) shows the topography of the same sample, but stored 60 h in ambient. While the vicinal surface is atomically flat in the beginning, we can observe features of about 0.2 nm height decorating the unit cell step terraces after ambient storage. The topography of TiO$_2$-terminated STO showed no change after ambient storage.
Stability of the SrO termination
--------------------------------
We investigated the changes of the surface configuration during exposure to ambient conditions further using *in situ* and *ex situ* XPS in order to investigate the surface chemistry before and after ambient exposure. The C 1s (Figure \[fig:AFMXPS\] top), O 1s (Figure \[fig:AFMXPS\] center) and Sr 3d spectra (Figure \[fig:AFMXPS\] bottom) were recorded. The C 1s spectrum is of interest as exposure to ambient is expected to give significant rise to adventitious carbon.[@Barr1995; @Swift1982] The O 1s and Sr 3d spectra are of interest to probe chemical changes in the SrO termination layer. The center and right column show the comparison of the spectra before (*in situ*) and after (*ex situ*) exposure to the ambient for a SrO and a TiO$_2$-terminated sample, respectively.
{width="\linewidth"}
The fits in the left column of Figure \[fig:AFMXPS\] are representatively depicted for the SrO-terminated sample and were obtained in the same manner for the TiO$_2$-terminated sample. The chemical information is gained from these fits. The comparisons in the center and right column elucidate the differences occurring for both terminations before and after exposure to the ambient. The C 1s spectra (Figure \[fig:AFMXPS\] a)) are fitted with a carbonate component for the highest binding energy ($E_B$) and a hydrocarbon component ($E_B\approx286~$eV). The O 1s spectra are fitted using 4 components (Figure \[fig:AFMXPS\] b)). The lowest $E_B$ component represents metal oxide bonds in the STO bulk. The highest $E_B$ component represents carbonates.[@Shchukarev2004; @Lam2017; @Crumlin2012] Both of the intermediate $E_B$ can be ascribed to hydroxides and other non-carbonate contaminations, and are referred to as contaminations peaks. The Sr 3d spectra are composed of a doublet from the STO bulk and a second, surface-related doublet at higher binding energies, as is typically observed for STO (Figure \[fig:AFMXPS\] c)).[@Szot2000]
In order to estimate changes of the surface stoichiometry, we compared the the peak areas of the different core-levels. In case of the C 1s spectra we use the area ratio of the C 1s and Sr 3p$_{3/2}$ peaks (Figure \[fig:AFMXPS\] d) and g)) to obtain a C/Sr ratio. We further use the area of the carbonate and the hydrocarbon peak to obtain a relative carbonate contribution. For the O 1s spectra, we use the areas of the carbonate and the contamination peaks in relation to the metal oxide peak to obtain a relative carbonate contribution and a non-carbonate contamination concentration, respectively. The relative contribution of the Sr related surface component to the Sr 3d signal is obtained from the ratio of the doublet at high binding energy and the doublet from the bulk STO.
\[h\]
-------------------------------------- --------- ------- --------- -------
termination TiO$_2$ SrO TiO$_2$ SrO
C/Sr ratio (C 1s) 2.1 % 8.6 % 40 % 40 %
Carbonate contribution (C 1s) - - 3.0 % 6.5 %
Contamination concentration (O 1s) 24 % 21 % 28 % 30 %
Carbonate contribution (O 1s) - - 7.5 % 9.5 %
Sr related surface component (Sr 3d) 18 % 18 % 18 % 21 %
-------------------------------------- --------- ------- --------- -------
These ratios can be found in Table \[tab:XPS\] for SrO and TiO$_2$-terminated samples, measured *in situ* and after 60 h of ambient exposure (*ex situ*). After air exposure, the C/Sr ratio increases markedly, to 40%, for both, TiO$_2$- and SrO-termination. For both samples there is no discernible C 1s carbonate component, when measuring *in situ*. Ambient exposure gives rise to this component, 3.0% for the TiO$_2$-termination and 6.5% for the SrO-termination. The carbonate component is thus markedly the highest for an ambient exposed SrO-termination. Similarly, the O 1s spectra do not exhibit a carbonate component when measuring *in situ*, but after exposure to ambient, this component is 7.5% for the TiO$_2$-termination and 9.5% for the SrO termination. Again the carbonate component is thus the most pronounced for an ambient exposed SrO-termination. The 3d surface component of the Sr 3d spectra is 18% for the TiO$_2$-termination *in* and *ex situ* and for the *in situ* SrO-termination. Ambient exposure of the SrO-termination increases this spectral weight (21%). This means that the 3d surface component of the Sr 3d spectra only changes after ambient exposing the SrO-terminated STO.
The strong increase of the C/Sr ratio for both terminations is typical for XPS measurements after ambient exposure due to adventitious carbon.[@Barr1995; @Swift1982; @Piao2002; @Miller2002] The more pronounced carbonate component of the C 1s and O 1s spectra for the SrO-termination points towards the formation of SrCO$_3$, naturally occurring when SrO reacts with CO$_2$ of the atmosphere.[@Ropp2013] This is also substantiated by the increase of surface component obtained from the Sr 3d spectrum after ambient exposure exclusively for the SrO-termination, as the component could be SrCO$_3$ related. It thus seems that the SrO-termination is instable under ambient conditions due to the formation of SrCO$_3$, eliminating the oxygen vacancy formation inhibiting effect.
Application to LAO/STO heterostructures
---------------------------------------
As we have developed a method to tailor the oxygen vacancy incorporation in STO by control of the the surface termination under UHV conditions, we next transferred this new method to a thin film system, namely LAO/STO. Since its discovery by Ohtomo *et al.*[@Ohtomo2004] LAO/STO is by far the most researched oxide 2DEG system.
The formation of a 2DEG results from the evasion of the polar catastrophe by the transfer of half an electron to the TiO$_2$-terminated $n$-type interface, ultimately resulting in a 2DEG.[@Nakagawa2006] In the same way, one would expect half a hole to be transfered into the p-type interface for a SrO-termination. The potential build-up is, however, in this case compensated by positively charged oxygen vacancies rather than holes.[@Lee2018] This results in an insulating interface for the SrO-termination.[@Nakagawa2006] The $n$-type conducting interface was shown to prevail for interfaces with up to 83% SrO-termination, which can be explained by the formation of the 2DEG in the TiO$_2$-terminated areas and percolation paths in between those areas.[@Huijben2009a; @Nishimura2004]
If LAO/STO structures are grown at low oxygen pressures the conduction mechanism changes. We have previously shown that the growth of crystalline LAO at oxygen pressures $\leq10^{-3}$ mbar results in a shift from 2DEG conductivity to bulk conductivity, when quenching the sample immediately after growth.[@Hensling2017; @Xu2016a] The appearance of bulk conductivity in crystalline LAO/STO can be explained by the incorporation of oxygen vacancies in the STO bulk, which contribute electrons to the conduction band. During its low pressure growth, LAO sucks oxygen from the underlying STO substrate resulting in the formation of oxygen vacancies in the STO.[@Schneider2010] The shift towards bulk conductivity can easily be identified as it is accompanied by a shift towards lower resistivity. [@Amoruso2011; @Amoruso2012a; @Breckenfeld2013; @Herranz2007; @Hwang2004; @Sambri2012; @Ohtomo2004; @Xu2016a; @Cancellieri2010; @Hensling2017; @Kalabukhov2007; @Huijben2009a]
Similar to the bulk conductivity of crystalline LAO/STO grown at low pressures, the 2DEG conductivity of the interface between amorphous LAO and STO single crystals relies on the incorporation of oxygen vacancies into the STO lattice. The main difference is their confinement to the interface in case of amorphous LAO/STO.[@Sambri2012; @Trier2013; @Chen2011] Amorphous LAO/STO therefore can be expected to be sensitive to the oxygen vacancy incorporation at the interface. We thus expect to see differences in the conductivity depending on the termination for both, crystalline LAO/STO grown in the bulk-conducting regime (i.e. grown at low pressures) and for amorphous LAO/STO.
Figure \[fig:PPMS\] a) shows the sheet resistance of crystalline LAO/STO in dependence of temperature obtained for different STO terminations. The sheet resistance of LAO/STO with 0% SrO-termination is about $100~\Omega$ at room temperature and about $10^{-2}~\Omega$ below 10 K. This corresponds to a dominant metallic bulk conduction of STO, as expected for these growth conditions.[@Xu2016a] An increase of the SrO-termination to 50% results in an increase of the sheet resistance, to about $10^{4}~\Omega$ at room temperature and about $10^{3}~\Omega$ below 50 K. This is the typical temperature dependency of the sheet resistance for crystalline LAO/STO dominated by 2DEG conductivity.[@Amoruso2011; @Amoruso2012a; @Breckenfeld2013; @Herranz2007; @Hwang2004; @Sambri2012; @Ohtomo2004; @Xu2016a; @Cancellieri2010; @Hensling2017; @Kalabukhov2007; @Huijben2009a] Increasing the SrO-termination further to 100 % did, in agreement with observations reported in literature,[@Huijben2009a; @Nishimura2004] result in insulating samples, whose sheet resistance is above the measurement limit (10$^8~\Omega$).
![Sheet resistance in dependence of temperature of a) crystalline LAO/STO with different STO terminations and b) amorphous LAO/STO with different STO terminations. 100% SrO-termination results in insulating behavior for both heterostructures. 50% SrO-termination results in 2DEG conductivity for both, as does 0% SrO termination in the amorphous case. The 0% SrO-terminated crystalline LAO/STO shows metallic conductivity.[]{data-label="fig:PPMS"}](PPMS){width="\linewidth"}
Figure \[fig:PPMS\] b) shows the sheet resistance of amorphous LAO/STO in dependence of temperature and STO termination. Amorphous LAO/STO with 0% SrO-terminated STO has a sheet resistance of about $10^{4}~\Omega$ at room temperature and about $10^{3}~\Omega$ below 50 K. This is in good agreement with the temperature dependent resistivity of the 2DEG formed by amorphous LAO/STO, now solely relying on the incorporation of interface oxygen vacancies.[@Sambri2012; @Trier2013; @Chen2011] Increasing the SrO-termination of the STO substrate to 50% results in a slight increase of the sheet resistance by a factor of two. A further increase to 100% results, as for the crystalline case, in an insulating sample. Hence, only a negligible amount of oxygen vacancies has formed at the SrO-terminated interface.
Figure \[fig:PPMS\] shows that we can utilize the termination control of the STO substrate to tailor the sheet resistance in LAO/STO heterostructures. As the sheet resistance of both, crystalline LAO/STO grown in reducing conditions and amorphous LAO/STO, is defined by the oxygen vacancies[@Amoruso2011; @Amoruso2012a; @Breckenfeld2013; @Herranz2007; @Hwang2004; @Sambri2012; @Ohtomo2004; @Xu2016a; @Cancellieri2010; @Hensling2017; @Kalabukhov2007; @Huijben2009a; @Trier2013; @Chen2011], we have successfully tailored their incorporation. This also explains the shift from bulk dominated to interface dominated conductivity for crystalline LAO, when increasing the SrO-termination to 50% (Figure \[fig:PPMS\] a)). The oxygen vacancy incorporation is limited to the remaining TiO$_2$-terminated areas, as shown in the top sketch in Figure \[fig:PPMS\] a). The resulting confined conductivity is comparable to the conductivity of amorphous LAO/STO (Figure \[fig:PPMS\] b)), which is also dominated by oxygen vacancies confined to the interface, as shown in the according sketches. We can thus in the same way explain the increased resistivity for amorphous LAO/STO.
Discussion
==========
Considering all the results described above we present a more complete picture of the role of the termination of STO for its oxygen exchange kinetics. Applying conditions known to be reducing for STO single crystals,[@Hensling2017; @Frederikse1964a] we are only able to efficiently incorporate oxygen vacancies into the TiO$_2$-terminated single crystal. The SrO-terminated sample remains insulating. We conclude that the incorporation of oxygen vacancies is inhibited, as other kinetic and thermodynamic factors, e.g. the diffusion coefficient in the bulk, are not affected by termination.
The surface reaction of the oxygen vacancy incorporation is known to be a multi step process, which is influenced by several parameters.[@Merkle2002; @Merkle2008] Considering that Alexandrov *et al.*[@Alexandrov2009] have shown by *ab initio* calculations that the formation energy of oxygen vacancies is lower for TiO$_2$-terminated STO as compared to SrO-terminated STO, it is conceivable that the termination has an influence on one or more steps of the surface reaction. $$\textnormal{O}^x_\textnormal{O,surface} \rightleftharpoons \frac{1}{2}\textnormal{O}_2 + \textnormal{V}^{\bullet\bullet}_\textnormal{O} + 2\textnormal{e}^-$$ In particular a high formation energy of oxygen vacancies in SrO-terminated STO could fully depress the oxygen vacancy incorporation at SrO-terminated surfaces. The schematic of this is shown in figure \[fig:dis\].
Another possible mechanism behind the inhibited oxygen vacancy incorporation is SrO acting as a diffusion barrier for oxygen.[@Baeumer2015] However an increased diffusion barrier was only shown for SrO in a rock salt structure. For SrO in the perovskite structure of STO, the effect was not observed.[@Hensling2018a] We thus rule out the explanation of a SrO diffusion barrier, leaving the increased formation energy as decisive parameter.
![Schematic of the oxygen vacancy incorporation at the STO surface for different terminations. Due to the higher formation energy of oxygen vacancies for a SrO-termination, the oxygen vacancy incorporation is suppressed.[]{data-label="fig:dis"}](STO_idea_2){width="\linewidth"}
The observed behavior, however, changed drastically after SrO-terminated STO was exposed to ambient conditions. Similar vacuum annealing of *ex situ* samples now resulted in high carrier densities independent of termination, indicating that the blocking effect of the SrO termination layer was eliminated by air exposure. This was accompanied by the formation of morphological features, indicating a clustering of SrO related particles.[@Hensling2018a] Utilizing XPS measurements these particles were identified as SrCO$_3$.
If a SrO-termination inhibits the formation of oxygen vacancies, the formation of SrCO$_3$ can explain the elimination of the effect under ambient conditions. The formation of islands after ambient exposure results in pores in the oxygen blocking SrO-termination layer, revealing the TiO$_2$-terminated STO, which then allows pathways for the incorporation of oxygen vacancies. Moreover, the resulting SrCO$_3$ clusters do not necessarily have a high formation energy for oxygen vacancies.
To profit from our new method to tailor the oxygen vacancy incorporation properties we applied it to the model application of the 2DEG at the LAO/STO heterointerface. Crystalline LAO/STO dominated by bulk conductivity and amorphous LAO/STO both rely on oxygen vacancy incorporation. In case of crystalline LAO/STO fabricated under reducing conditions we observe a transition from bulk conducting STO (0 %) to interface conductivity (50 %) and finally to insulating (100 %) with increasing amount of SrO-termination . For amorphous LAO/STO we do as well observe a transition to insulating behavior, when increasing the SrO-termination to 100%. This confirms the inhibition of the incorporation of oxygen vacancies we observed for STO single crystals.
Considering crystalline LAO/STO this is especially interesting, as it was found that the conductivity in the initial report of 2DEG conductivity by Ohtomo *et al.*[@Ohtomo2004] was in fact dominated by bulk conductivity, hence by oxygen vacancies.[@Breckenfeld2013; @Herranz2007] Nevertheless, conductivity was only observed for TiO$_2$-terminated STO.[@Ohtomo2004] This effect could not be explained within the polar catastrophe scenario. With our findings we are able to explain this phenomenon *via* the inhibition of the oxygen vacancy incorporation for the SrO-terminated sample.
Conclusions
===========
With this work we have provided a novel way to tailor the oxygen vacancy incorporation in STO by controlling its termination. We have demonstrated that a SrO-termination of STO completely inhibits the incorporation of oxygen vacancies for otherwise reducing conditions. By systematically controlling the termination of STO it is thus possible to tailor the areas of oxygen vacancy incorporation. Due to the widespread use of STO as a substrate this result is highly interesting. A specific application for which we employed this new method are LAO/STO 2DEG heterostructures. By doing so, we were not only able to directly influence the conductivity of these heterostructures, but did also improve their understanding. Further applications include, but are not limited to: i) oxides that require low oxygen pressure growth due to thermodynamic reasons (e.g. LaVO$_3$[@Hotta2007; @He2012; @Vrejoiu2016a], EuTiO$_3$[@Shkabko2013a]), for which film properties would otherwise be masked by oxygen vacancies induced in STO; ii) metals, which induce the incorporation of oxygen vacancies at the interface.[@Santander-Syro2011]
Disclosure statement {#disclosure-statement .unnumbered}
====================
There are no conflicts to declare.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge funding from the W2/W3 program of the Helmholtz association. The research has furthermore been supported by the Deutsche Forschungsgemeinschaft (SFB 917 ‘Nanoswitches’). F.G. and M.R. thank the DFG GU/1604. CB has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 796142. We further thank R.A. de Souza and M. Müller for the helpful discussions.
|
---
author:
- Asja Jelić
- 'Cécile Appert-Rolland'
- Ludger Santen
title: A bottleneck model for bidirectional transport controlled by fluctuations
---
The understanding of the macroscopic behavior of complex systems out of equilibrium is one of the main challenges in modern statistical mechanics. A common feature of many non-equilibrium systems is the presence of a current in their stationary state, in contrast to equilibrium. A general framework describing these systems is still lacking, though the importance of current large deviations and their link with fluctuations theorems has been emphasized [@ft].
The absence of such general framework motivated the study of many oversimplified microscopical models, among which exclusion processes have been reference systems because they allow for extremely precise numerical results and exact analytical solutions in some cases. Exclusion processes are simple models defined on a discrete –usually one-dimensional– lattice, on which particles hop from site to site. Their role as a reference system is of particular importance for exact calculations of large deviations, the non-equilibrium counterpart of the free energy [@derrida0711; @chou_m_z11]. At the same time, exclusion processes are a flexible tool to model various physical systems. Indeed, they capture the correct collective behavior of systems with very different length scales from social individuals, such as pedestrians [@chowdhury_s_n05], vehicles [@review_traffic], ants [@prl_ants; @chowdhury_s_n05], to molecular systems as molecular motors [@chowdhury_s_n05; @chou_m_z11], and microscopical ones as quantum dots [@quantum_dot]. Furthermore exclusion processes are closely related [@asepvskpz] to growth phenomena described by the celebrated Kardar- Parisi-Zhang equation [@kpz], and the universality of these phenomena has been confirmed in recent experiments in liquid crystals [@exper].
For the asymmetric simple exclusion process (ASEP) the full current probability distribution has been calculated for different boundary conditions [@derrida93c; @current]. However, while the ASEP captures the proper behavior of particles moving on a single lane, it is obviously not appropriate to describe situations in which the particles (such as individuals or cars) move in few intersecting lanes. To describe such systems several one-dimensional exclusion processes must be coupled and the task of determining the current distributions becomes harder. Models for interacting parallel lanes (to describe e.g. the traffic on highways) have been introduced and studied [@reichenbach_f_f06]. In particular, in the so-called bridge models, two lanes share a finite number of sites (the ‘bridge’) [@evans95a]. Such systems have attracted a large interest due to the symmetry breaking that occurs in most cases [@evans95a; @bridge_models].
In this letter, we introduce a new model which couples two totally asymmetric simple exclusion process (TASEP) lanes with oppositely directed flows sharing a common bottleneck (see Fig. \[fig:model\]). In contrast to the bridge models, exchange of oppositely moving particles is not possible inside the bottleneck, because we impose that only particles going in one direction can go through the bottleneck at a given time. Therefore, particles going in the opposite direction have to wait until the [*whole*]{} bottleneck is empty before being allowed to go through. Our model can be seen as a representation of opposite pedestrian flows crossing e.g. at a door. The model is also relevant for other systems such as multiphase flows, or bidirectional molecular traffic across nuclear pores [@kapon08].
![Schematic representation of the model.[]{data-label="fig:model"}](fig1.png){width="45.00000%"}
We study this model with a combination of analytical and numerical techniques. The property that makes this model qualitatively different from previous ones is that the condition for reversing the flow inside the bottleneck (i.e. for having an empty bottleneck) is a rare event as soon as the bottleneck exceeds a few sites. Indeed, the dynamics of this system is driven by rare fluctuations, and is non trivial. For example, there exists a regime in which stop and go waves invade the whole system. Also, a counterintuitive feature of the model is that it can sustain in the bottleneck a current [*higher*]{} than the maximal current of a single lane system of infinite size.
The model {#sec:model}
=========
We define now the model more precisely. Particles move on two parallel tracks, modeled as two TASEP lanes. Both lanes share the same bottleneck of length $N_b$. We call ‘+’ (resp. ‘-’) the particles moving from left to right (right to left), represented as red (blue) particles in Fig. \[fig:model\]. We consider explicitly only the lanes of incoming particles (of length $L$). Thus outgoing lanes are shadowed in Fig. \[fig:model\]. Random sequential update is considered. Particles enter the lane with rate $\alpha$, hop forward with rate $p$ if the next site is empty, and leave the bottleneck with rate $\beta$. A particle enters the bottleneck with rate $p_0$ only if (i) there is no particle of the other species inside the bottleneck and (ii) the first site of the bottleneck is empty. We assume $p=p_0=1$, leaving only $\alpha$ and $\beta$ as the model parameters, in addition to the bottleneck and lanes lengths $N_b$ and $L$.
Phase diagram {#sec:phase-diagram}
=============
By using the stationary properties of the TASEP, we infer the phase diagram (given in Fig. \[fig:phase-diagram-MF\]) of the system with bottleneck from simple phenomenological arguments. In the following, currents refer in general to the current of one given species of particles (and not the total current).
![Phase diagram of the bottleneck model, predicted by the phenomenological approach. []{data-label="fig:phase-diagram-MF"}](fig2.png){width="40.00000%"}
For low $\alpha$ values, we expect a free flow (FF) phase driven by the entrance rate. For each species, the corresponding particle current and density are $$J_{in}=\alpha(1-\alpha); \;\;\;\; \rho_{in} = \alpha.
\label{eq_jin}$$ For large $\alpha$ and small $\beta$, we expect an exit driven jammed (JJ) phase. When a species goes through the bottleneck for a long enough time, the system should become equivalent to a one-lane TASEP in stationary state, and the current should be $\beta (1-\beta)$. As the use of the bottleneck is shared between the two types of particles, each species can go through only half of the time. Neglecting the effects of the transients, on average the current is $$J_{out}=\frac{1}{2} \beta(1-\beta).
\label{eq_jout}$$ We shall see that the particle distribution in incoming lanes is not homogeneous, and that, in contrast to current, density cannot be obtained from this mean field like approach.
The FF/JJ boundary is given by the solution of $J_{in} = J_{out}$, satisfying $\alpha_c(\beta) \to 0$ when $\beta \to 0$, i.e. $$\alpha_c (\beta)=\frac{1}{2}-\frac{1}{2}\sqrt{1-2\beta(1-\beta)}.
\label{boundary1}$$ The FF phase corresponds to $\alpha < \alpha_c(\beta)$.
When $\alpha>1/2$ and $\beta>1/2$, the single lane TASEP is in the maximal current (MC) phase, with a current equal to $1/4$. Again, if we neglect transients in the bottleneck, the average current must be half of this value: $$J_{MC}=1/8.
\label{eq_jmc1}$$ The MC/FF boundary, obtained from $J_{in} = J_{MC}$, is given by $
\alpha_c = (1-\sqrt{1/2})/2 \simeq 0.146,
$ while for the MC/JJ boundary, $J_{out} = J_{MC}$ provides $
\beta_c = \frac{1}{2}.
$
![ Average current as a function of $\alpha$, for different values of $\beta$. Symbols indicate numerical results obtained from Monte Carlo simulations for $L = 50$ and $N_b = 4$. The solid line gives the theoretical predictions for $\beta > 1/2$, i.e. both for the free flow and the maximal current phase. Dashed lines give predictions for $\beta < 1/2$ for the jammed phase with $\alpha > \alpha_c (\beta)$. []{data-label="fig:current-betadiff"}](fig3.png){width="42.30000%"}
Monte Carlo simulations confirm these mean field predictions both for current and density in the FF phase \[Eq. (\[eq\_jin\])\], and for the current in the JJ phase \[Eq. (\[eq\_jout\])\] for values of $\beta$ not close to the MC boundary (see Fig. \[fig:current-betadiff\]). By contrast, the current in the MC phase is systematically underestimated by this phenomenological approach. However, as seen in Fig. \[fig:current-nbdiff\], the maximal current value (\[eq\_jmc1\]) is recovered for long bottlenecks, for which a stationary state can be established in the bottleneck.
![Average current against $\beta$ for different $N_b$, and $L=50$, $\alpha=0.6$. The dashed line gives the theoretical predictions for $\alpha=0.6$, both for the jammed and MC phase. For large bottlenecks the current saturates to $J_{MC}=1/8$. []{data-label="fig:current-nbdiff"}](fig4.png){width="40.00000%"}
For shorter bottlenecks, currents [*higher*]{} than the stationary maximal current $1/4$ can be obtained.
Interestingly, with our bi-directional model, we can even reach the capacity of a finite-size system of size $N_b$ [@derrida93c] which would be fed by two incoming lanes, each transporting the maximal current and injecting particles on the same end of the bottleneck (in this case, flow would be unidirectional in the bottleneck and no flux reversal would be needed).
Hence the counterintuitive result according to which it is at least as efficient to have particles coming from both ends of the bottleneck than having all these particles coming from the same end. This high capacity can be obtained in spite of the fact that bidirectional traffic requires to empty the bottleneck at each flux reversal, a limitation which is compensated by very efficient transient states. Indeed, at each reversal of the current in the bottleneck, particles enter an empty system. Their motion is not hindered by predecessors, and thus high fluxes can be achieved.
Spatio-temporal structures {#sect_st}
==========================
To characterize the dynamics of the system, we use a domain wall approach [@kolomeisky98; @santen_a02] describing the system at a mesoscopic scale. We consider only the FF and JJ phases, since the domain wall approach is not appropriate to describe the MC phase (which has long-range correlations). In this approach, we neglect transients inside the bottleneck, and assume that currents and densities are given by stationary expressions.
First we describe the FF phase, and take the point of view of ‘+’ particles. When the bottleneck is closed, ‘+’ particles accumulate in front of the bottleneck, forming a queue of density $1$. The upstream end of the queue can be seen as a discontinuity (or a wall) separating the queue from the bulk. A new wall is created when the bottleneck opens. The queuing particles feed the bottleneck with a high effective injection rate, and a high density domain is installed in the bottleneck. Ignoring transients, this jammed domain imposed by the exit has density $1-\beta$ and current $\beta(1-\beta)$. Due to mass conservation, the wall between the jammed domain and the queue moves backwards with velocity $1-\beta$ until the whole queue is dissolved. Then, at the separation between the bulk FF phase and the jammed domain localized at the exit, a new wall forms and moves forward with velocity $
V = \frac{\left[\beta(1-\beta) - \alpha(1-\alpha)\right]}{(1-\alpha-\beta)}.
$ In order to understand which phase invades the bulk, we have to determine whether the FF domain will reach the entrance of the bottleneck [*before*]{} it closes again. This is the case if $$\alpha (1-\alpha) \le \frac{1}{2} \beta(1-\beta),
\label{dw1}$$ and assuming that the bottleneck is open and closed for the same period of time $\tau$ (which is true on average). Then the queue and the jammed domain stay localized near the bottleneck, and the bulk FF phase can be sustained. If condition (\[dw1\]) is not fulfilled, the bulk FF phase is slowly invaded by the queue, and cannot survive over long times. Note that Eq. (\[dw1\]) is indeed identical to condition (\[boundary1\]) for the FF/JJ boundary.
![ Spatio-temporal plot in the jammed phase ($\alpha = 0.6$, $\beta = 0.2$, $L=100$, $N_b=2$). Both incoming TASEP lanes are shown, with the bottleneck in the middle. Particles (empty sites) are represented as black (white) spots. []{data-label="fig:spacetime"}](fig5.png){width="41.50000%"}
Now we apply the same coarse-grained approach to the JJ phase to show that a homogeneous bulk density is not possible. Indeed, $\rho_{bulk}$ should satisfy $\rho_{bulk}(1-\rho_{bulk}) = J_{out}$. However, when the bottleneck is closed, a queue of density $1$ is formed. The wall between this queue and the bulk density moves backwards with velocity $\rho_{bulk}$. Instead, when the bottleneck is open, the wall separating the exit driven jammed domain and the queue moves backwards only with velocity $1-\beta$. Thus, it cannot catch up with the previous wall, and the queue can never be entirely dissolved. When the bottleneck closes again, a new queue of density $1$ is formed, whose rear end also moves with velocity $1-\beta$ inside the exit driven phase. Then, regions of queuing particles with density $1$ alternate with regions of density $1-\beta$ and current $\beta(1-\beta)$, resulting in an overall striped jammed phase. This is confirmed by spatio-temporal plot in Fig. \[fig:spacetime\] obtained from simulations. Averaging over the stripes gives an actual bulk density $$\rho_{out} = 1-\frac{\beta}{2},
\label{eq_rhoout}$$ also confirmed by the Monte Carlo simulations.
Distribution of the oscillation periods
=======================================
Until now, we considered the average value of the oscillation period $\tau$, which is a fluctuating variable. To perform a quantitative analysis, we define $T$ as the time during which the bottleneck is occupied by at least one particle of the type under consideration. After $T$, bottleneck is empty and a particle of either the same or opposite type can enter. Thus $T$ is not identical to $\tau$ but strongly related to it. We now consider the probability distribution $P(T)$, shown in Fig. \[fig:distT\].
We focus on the jammed state, though a similar analysis could be done in the FF phase. If we assume as before that a stationary state is established in the bottleneck, then the probability for the bottleneck to be empty should be $P_\emptyset=\beta^{N_b}$ at each time step and, if we neglect correlations between successive time steps, the distribution $P(T)=(1-P_{\emptyset})^{T-1}
P_{\emptyset}$ follows. For $N_b=1$, the stationary state is established very rapidly. For larger $N_b$, non negligible corrections are present because there is a non vanishing relaxation time $\theta$ towards stationarity. The accuracy of our prediction for $P(T)$ depends on the ratio between the typical times $T$ and the relaxation time $\theta$. The typical time $T$ is expected to increase more rapidly with the bottleneck size $N_b$ than the relaxation time $\theta$, making our prediction more accurate for large $N_b$. Indeed, in Fig. \[fig:error\] we observe that the relative error between our prediction and simulation results decreases again for large bottlenecks.
![ Distribution $P(T)$ for $N_b=4$, $\alpha = 0.6$, $\beta = 0.2$ and $L=400$. The blue solid line is the whole distribution. The other curves are the contributions to $P(T)$ corresponding to the passage of at most $N_{max}$ particles through the bottleneck during the time interval $T$. []{data-label="fig:distT"}](fig6.png){width="40.00000%"}
For all $N_b$, the prediction works better for small $\beta$, and the difference between theory and numerics increases when the MC phase is approached. Indeed, when $\beta$ increases, $P_\emptyset$ increases, and the typical $T$ value decreases. It can then become smaller than the relaxation time $\theta$ and a description of transients becomes necessary. Besides, the queues may not have enough time to form and thus do not overfeed the bottleneck.
![ Relative error between the numerical result and theoretical prediction for the decay rate of the tail of $P(T)$. Error is plotted against $N_b$, for various $\beta$, $L=100$ or $400$, $\alpha=0.6$. Lines are guides to the eyes. []{data-label="fig:error"}](fig7.png){width="42.50000%"}
Another feature visible in Fig. \[fig:distT\] is that there is an important contribution to $P(T)$ from events during which only a single or very few particles pass through the bottleneck. When these events occur, the system explores only very special configurations, which are not representative of the whole stationary distribution. Thus the system is still in a transient state when the next reversal occurs.
As a conclusion the dynamics for the reversal of the flux actually involves two quite different mechanisms (one based on stationarity and the other involving transients), and exhibits some kind of intermittency, with long periods of one-directional flows alternating with rapid switches. In order to have a full description of the dynamics of the flux reversal, not only the transient nature of the incoming flows in the bottleneck should be taken into account, but also the correlations between successive switches.
Conclusions
===========
We have introduced a new model for bidirectional transport with a bottleneck. We have shown that theoretical considerations, based on the stationarity hypothesis for the flow inside the bottleneck, allow to predict with a good accuracy the phase diagram, and the values of the currents in the free flow phase, in the jammed phase, and in the MC phase for long bottlenecks. For short bottlenecks, transients dominate the behavior of the system, and as a consequence large values of the current can be observed, in spite of the cost of having to empty the bottleneck at each reversal of the flux. These transients could be studied by exploiting the exact results obtained for a single TASEP with a step initial condition [@asepvskpz; @derrida_g09; @tracy_w09].
The jammed phase turns out to have a striped structure. In spite of the similarity with [@appert_s01], here the striped phase can be sustained in the bulk without any modification of the standard TASEP bulk rules. It should be noted that this striped phase results from a self-regulated dynamics for the flux reversal in the bottleneck. Though an assumption of constant reversal periods succeeds in predicting the striped phase density, we find that actually the structure of the distribution of the bottleneck occupation times $P(T)$ is more complex and emerges from two different mechanism. The tail of the distribution can be explained through our phenomenological approach assuming a 1-lane stationary state in the bottleneck. However, this prediction is valid only for small $\beta$, and in any case cannot explain the shape of the distribution around its maximum. A large part of the distribution is due to non-typical events where only a few particles go through the bottleneck. A complete understanding of $P(T)$ should involve the study of non-stationary distributions and correlations between flux reversals.
To conclude, the main feature that puts this new model apart from others in the literature is that fluctuations localized in the bottleneck can have a macroscopic effect on the whole system. It provides a sensitive test for different theoretical approaches and can be easily tested numerically. While a phenomenological approach assuming a stationary state in the bottleneck gives surprisingly good predictions for the free flow and part of the jammed phase, some other observations trigger much more complex questions involving non-stationary and correlated behaviors. While we concentrated on identical lanes without directional bias, one can, of course, consider asymmetries either in particles species or capacities of the lanes which are of interest for different applications.
[ This work was supported by the French Research National Agency (ANR) in the frame of the contract .L.S. (resp. A.J.) acknowledges support from the RTRA Triangle de la physique (Project ) (resp. Project ). We thank B. Derrida for inspiring discussions. ]{}
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---
abstract: 'We present a systematic analysis of the spectral and temporal properties of 17 gamma-ray bursts (GRBs) co-detected by Gamma-Ray Monitor (GBM) and Large Area Telescope (LAT) on board the [*Fermi*]{} satellite by May 2010. We performed a time-resolved spectral analysis of all the bursts with the finest temporal resolution allowed by statistics, in order to [reduce]{} temporal smearing of different spectral components. We found that the time-resolved spectra of 14 out of 17 GRBs are best modeled with the classical “Band” function over the entire Fermi spectral range, which may suggest a common origin for emissions detected by LAT and GBM. GRB 090902B and GRB 090510 require the superposition of a MeV component and an extra power law component, with the former having a sharp cutoff above $E_p$. For GRB 090902B, this MeV component becomes progressively narrower as the time bin gets smaller, and can be fit with a Planck function as the time bin becomes small enough. In general, we speculate that phenomenologically there may be three elemental spectral components that shape the time-resolved GRB spectra: a Band-function component (e.g. in GRB 080916C) that extends in a wide energy range and does not narrow with decreasing time bins, which may be of non-thermal origin; a quasi-thermal component (e.g. in GRB 090902B) with the spectra progressively narrowing with reducing time bins; and another non-thermal power law component extending to high energies. The spectra of different bursts may be decomposed into one or more of these elemental components. We compare this sample with the BATSE sample and investigate some correlations among spectral parameters. We discuss the physical implications of the data analysis results for GRB prompt emission, including jet composition (matter-dominated vs. Poynting-flux-dominated outflow), emission sites (internal shock, external shock or photosphere), as well as radiation mechanisms (synchrotron, synchrotron self-Compton, or thermal Compton upscattering).'
author:
- 'Bin-Bin Zhang, Bing Zhang, En-Wei Liang, Yi-Zhong Fan, Xue-Feng Wu, Asaf Pe’er, Amanda Maxham, He Gao,Yun-Ming Dong'
title: 'A Comprehensive Analysis of [*Fermi*]{} Gamma-Ray Burst Data. I. Spectral Components and Their Possible Physical Origins of LAT/GBM GRBs'
---
Introduction
============
Although observationally accessed much earlier, GRB prompt emission is stil less understood than afterglow. The fundamental uncertainties lie in the following three poorly known important properties of GRBs (e.g. Zhang & Mészáros 2004 for a review): (1) Ejecta composition: Are the ejecta mostly composed of baryonic matter or a Poynting flux? (2) Energy dissipation site: Is the emission from internal shocks (Rees & Mészáros 1994; Kobayashi et al. 1997), the photosphere (Paczýnski 1986; Goodman 1986; Mészáros & Rees 2000; Mészáros et al. 2002; Pe’er 2008), some magnetic dissipation regions (Lyutikov & Blandford 2003; Zhang & Yan 2011), or the external shock (Rees & Mészáros 1992; Mészáros & Rees 1993; Dermer & Mitman 1999)? (3) Is the radiation mechanism synchrotron/jitter radiation (Mészáros et al. 1994; Medvedev 2000), synchrotron self-Compton (Panaitescu & Mészáros 2000; Kumar & McMahon 2008), or Comptonization of thermal photons (e.g. Thompson 1994; Pe’er et al. 2005, 2006; Beloborodov 2010; Lazzati & Begelman 2009)?
Before [*Fermi*]{}, understanding of GRB prompt emission has progressed slowly. Observations of early X-ray afterglows by [*Swift*]{} revealed a steep decay phase that is smoothly connected to prompt emission (Tagliaferri et al. 2005; Barthelmy et al. 2005), which suggests that the prompt emission region is detached from the afterglow emission region, and that the prompt emission site is “internal” (Zhang et al. 2006). Other than this, the properties of prompt emission were poorly constrained. The main factor that hampers progress has been the narrow energy band of the gamma-ray detectors of previous missions. Theoretical models usually predict rich features in the prompt spectra (e.g. Pe’er et al. 2006, see Zhang 2007; Fan & Piran 2008 for reviews on high energy emission processes). However, within the narrow observational spectral window, these features cannot be fully displayed. Instead, most previous spectral analyses revealed an empirical “Band”-function (Band et al. 1993), which is a smoothly-joint broken power law, whose physical origin is not identified. For the bright BATSE GRB sample, the typical low and high energy photon indices are distributed around $\alpha \sim -1$ and $\beta \sim -2.2$, respectively, while the spectral peak energy $E_p$ is distributed around 200-300 keV (Preece et al. 2000). Later observations suggested that the distribution of $E_p$ can be much wider, extending to a few keV in the soft regime for X-ray flashes (Sakamoto et al. 2005) and to greater than 100 MeV in the hard regime (e.g. $\gtrsim$ 170 MeV for GRB 930506, Kaneko et al. 2008). Some BATSE GRBs were also detected by EGRET in the GeV range (Kaneko et al. 2008). For example, It was found that the GeV emission can last much longer than the MeV emission (e.g. GRB 940217, Hurley et al. 1994), and that it can form a distinct spectral component (e.g. GRB 941017, González et al. 2003). In the softer regime, an X-ray excess component with respect to the Band function was discovered in some BATSE GRBs (Preece et al. 1996). However, the previous data were not adequate to place meaningful constraints on the three main questions discussed above.
The [*Fermi*]{} satellite ushered in a new era of studying GRB prompt emission. The two instruments on board [*Fermi*]{}, the Gamma-ray Burst Monitor (GBM; Meegan et al. 2009) and the Large Area Telescope (LAT; Atwood et al. 2009), provide an unprecedented spectral coverage for 7 orders of magnitude in energy (from $\sim$8 keV to $\sim$300 GeV). Since the beginning of GBM/LAT science operation in August 2008 and as of the writing of this paper (May 2010), there have been 17 GRBs co-detected by LAT and GBM, with a detection rate comparable to the expectation assuming that the LAT-band emission is the simple extrapolation of the Band spectrum to the GeV range (Abdo et al. 2008; Lü et al. 2010). As will be shown below, the Band-function fits apply to most LAT GRBs, although some outliers do exist. Broad band spectral analyses have been published by the [*Fermi*]{} team for several individual GRBs, e.g. GRB 080916C (Abdo et al. 2009a), GRB 090510 (Abdo et al. 2009b, Ackermann et al. 2010), GRB 090902B (Abdo et al. 2009c, Ryde et al. 2010), GRB 080825C (Abdo et al. 2009d), and GRB 081024B (Abdo et al. 2010a), which revealed several interesting features, such as the nearly featureless Band spectra covering 6 orders of magnitude in all epochs for GRB 080916C, the existence of an extra power law component extending to high energies in GRB 090510 and GRB 090902B, the existence of a quasi-thermal emission component in GRB 090902B, the delayed onset of the LAT-band emission with respect to the GBM-band emission, as well as an extended rapidly decaying GeV afterglow for most GRBs.
These discoveries have triggered a burst of theoretical investigations of GRB prompt emission. Zhang & Pe’er (2009) argued that the lack of a thermal component in the nearly featureless spectra of GRB 080916C suggests a Poynting flux dominated flow for this burst. The conclusion was strengthened by a follow up study of Fan (2010, see also Gao et al. 2009). On the other hand, the quasi-thermal component in GRB 090902B (Ryde et al. 2010) is well-consistent with the photosphere emission of a hot fireball (Pe’er et al. 2010, Mizuta et al. 2010), suggesting that the burst is not highly magnetized. The possibility that the entire Band function spectrum is photosphere emission was discussed by several authors (Fan 2009; Toma et al. 2010; Beloborodov 2010; Lazzati & Begelman 2010; Ioka 2010). These models have specific predictions that can be tested by the available data. In the high energy regime, Kumar & Barniol Duran (2009, 2010), Ghisellini et al. (2010) and Wang et al. (2010) suggested that the GeV afterglow is of external shock origin, which requires some unconventional parameters (Li 2010a; Piran & Nakar 2010). On the other hand, the fact that LAT emission is the natural spectral extension of GBM emission in some GRBs suggests that the GeV emission may be of an internal origin similar to MeV emission (Zhang & Pe’er 2009). Finally, the delayed onset of the GeV emission has been interpreted as emergence of the upscattered cocoon emission (Toma et al. 2009), synchrotron emission from shock accelerated protons (Razzaque et al. 2009), as well as delayed residual internal shock emission (Li 2010b). Again these models have specific predictions that may be tested by a detailed analysis of the data.
Our goal is to systematically analyze the GRB data collected by the [*Fermi*]{} mission, aiming at addressing some of the above mentioned problems in prompt GRB emission physics. Here we report the first paper in the series, which focuses on a comprehensive analysis of the GRBs that were co-detected by LAT and GBM. This sample has a much broader spectral coverage than the GBM-only GRBs, and therefore carries much more information about GRB prompt emission. The plan of the paper is the following. In Section 2 we describe the details of our sample selection and data analysis method. The data analysis results are presented in Section 3, with emphases on the unique features of some GRBs. We also present spectral parameter distributions and some possible correlations. In Section 4, we summarize the results and speculate on the existence of at least three elemental spectral components, and discuss their possible physical origins and possible combinations. In Section 5, we present the comparison between the emissions detected in the GBM band and that detected in the LAT band and discuss their physical connections. Our conclusions are summarized in Section 6 with some discussion.
Sample and Data Reduction
=========================
As of May 2010, 17 GRBs have been co-detected by [*Fermi*]{} LAT and GBM. Our sample includes all 17 GRBs (Table 1). We downloaded the GBM and LAT data for these GRBs from the public science support center at the official [*Fermi*]{} web site http://fermi.gsfc.nasa.gov/ssc/data/. An IDL code was developed to extract the energy-dependent lightcurves and time-dependent spectra for each GRB. This code was based on the [*Fermi*]{} RMFIT package (V3.3), the [*Fermi*]{} Science Tools (v9r15p2) and the HEASOFT tools, which allows a computer to extract lightcurves and spectra automatically. The human involvement is introduced later to refine the analysis when needed. The code automatically performs the following tasks.
1. Extract the background spectrum and lightcurve of the GBM data. [*Fermi*]{} records GBM data in several formats. For background reduction we use the CSPEC format data because it has a wider temporal coverage than the event data (time-tagged event, TTE, format). The background spectrum and lightcurve are extracted from some appropriate time intervals before and after the burst[^1], and the energy-dependent background lightcurves are modeled with a polynomial function $B(E_{\rm ch},t)$, where $E_{\rm ch}$ is a specified energy band.
2. Extract the source spectrum and lightcurve of the GBM data. This is done with the event (TTE) data. GBM has 12 NaI detectors (8 keV–1 MeV) and 2 BGO detectors (200 keV–40 MeV). The overall signal-to-noise ratio (SNR) and peak count rate are calculated for each detector. The brightest NaI and BGO detectors are usually used for the analyses. If several detectors have comparable brightnesses, all of them (usually 2-4 detectors) are taken for the analyses. By subtracting the background spectrum and lightcurve obtained in the previous step, the time-dependent spectra and energy-dependent lightcurves of the source in the GBM band are then obtained.
3. Estimate the LAT-band background. Since only a small number of photons are detected by LAT for most GRBs, the background estimation should be performed cautiously. It is not straightforward to estimate an accurate LAT background using off-source regions around the trigger time. In our analyses, the LAT background is extracted using on-source region data long after the GBM trigger when the photon counts merge into a Poisson noise.
4. Extract the LAT-band spectrum and lightcurve. Both “diffuse” and “transient” photons (level 0-3) are included. Since the LAT point spread function (PSF) strongly depends on the incident energy and the convention point of the tracker (Ohno et al. 2010), the photons are grouped into FRONT and BACK classes and their spectra are extracted separately based on different detector response files. The region of interest (ROI) that contains significant counts of LAT photons is further refined when necessary (Atwood et al. 2009; Abdo et al. 2009d).
5. Extract the background-subtracted GBM and LAT lightcurves for different energy bands. In our analysis, the lightcurves are extracted in the following energy bands: 8–150 keV, 150–300 keV, 300 keV–MeV, 1–30 MeV, and the LAT band (above 100 MeV).
6. Make dynamically time-dependent spectral fits. Initially, the burst duration is divided in an arbitrary number of slices. The code then automatically refines the number of slices and the time interval for each slice, so that the photon counts in each bin [(typically minimum 20 counts for GBM spectra)]{} give adequate statistics for spectral fitting (the reduced $\chi^2$ is typically in the range of 0.75 - 1.5, a special case is GRB 090510, see Sect.\[sec:090510\]). The time slices are defined to be be as small as possible as long as the extracted spectra satisfy these statistical criteria. The GBM spectra of the selected NaI and BGO detectors and the LAT “FRONT” and “BACK” type spectra are all extracted for each slice. These spectra, together with the corresponding response files (using the same one as the CSPEC data for LAT, or generated using gtrsp for GBM) are input into XSPEC (V 12.5.1) simultaneously to perform spectral fitting. The following spectral functions are considered (in order of increasing free parameters): single power law (PL), blackbody (BB, Planck function), power-law with exponential cutoff (CPL), and Band function. The models are tested based on the following principles: (1) If a one-component model can adequately describe the data (giving reasonable reduced $\chi^2$, say, between 0.75 and 1.5), two-component models are not considered; (2) for one-component models, if a function with less free parameters can describe the data adequately, it is favored over the models with more parameters. [(3) In addition, the Akaike’s Information Criterion[^2] (AIC, Akaike 1974) is calculated to evaluate each model by considering both the fitting goodness ($\chi^2$) and the complexity of the model. We confirmed that the model with minimal AIC is the preferred model we choose based on the first two criteria. ]{} Nonetheless, since most GRBs have a Band-function spectra (see below), we also apply the Band function to those time bins that do not demand it in order to compare the fitting results between the Band function and other functions with less parameters (e.g. power law, blackbody, or power law with exponential cutoff).
To assess the quality of a spectral fit, we use the traditional $\chi^2$ statistics. Due to the low count rate of LAT photons, we use the Gehrels (1986) weighting method in the high energy regime. We also employed the C-stat method (as used by the [*Fermi*]{} team), and found that the two methods usually give consistent results. We chose the $\chi^2$ method since it gives more reliable error estimates. All the model fitting parameters and $\chi^2$ statistics are presented in Table 2. For each burst, we present the time-dependent spectral parameters in the designated time bins defined by the statistics of spectral fitting, as well as the time-integrated spectral fit during the entire burst in the last row.
Data Analysis Results
=====================
The data analysis results are presented in Figs. \[080825C\]-\[100414A\]. Each figure corresponds to one burst, and contains 10-11 panels. In the left panels, the lightcurves in 5 energy bands (8–150 keV, 150–300 keV, 300 keV–1 MeV, 1–30 MeV, and $>$ 100 MeV) are presented in linear scale, together with the temporal evolution of the spectral parameters ($\alpha$, $\beta$, $E_p$ for Band function, $kT$ for blackbody function, and $\Gamma$ for single power law photon index). The top right panel is an example photon spectrum with model fitting, typically taken at the brightest time bin. The time-dependent model spectra are presented in the mid-right panel. The time-slices for the time-resolved spectral fitting are marked with vertical lines in the left panel lightcurves. In the bottom right panel, the GBM and LAT lightcurves are presented and compared in logarithmic scale.
In the following, we discuss the results of several individual bright GRBs (Sect.\[sec:080916C\]-\[sec:090926A\]), and then discuss other GRBs in general (Sect.\[sec:otherGRBs\]). We then present statistics of spectral parameters (Sect.\[sec:distributions\]) and some possible correlations (Sect.\[sec:correlations\]).
GRB 080916C {#sec:080916C}
-----------
As shown in Fig.\[080916C\], GRB 080916C is a long GRB with a duration $\sim 66$ s. The entire lightcurve can be divided into 6 segments. The smallest time bins during the brightest epochs (first two) are 3.7 s and 5.4 s, respectively. This corresponds to a rest-frame time interval $\leq 1$ s (given its redshift 4.35, Greiner et al. 2009). In all the time intervals, we found that the Band-function gives excellent fits to the data, consistent with Abdo et al. (2009a). Initially there is a spectral evolution where the spectra “widen” with time ($\alpha$ hardening and $\beta$ softening), but later the spectral parameters essentially do not evolve any more. We note that the steep $\beta$ in the first time bin is mostly because of the non-detection in the LAT band. The tight upper limit above 100 MeV constrains the range of $\beta$ not to be too hard. On the other hand, with GBM data alone, the data of in first time bin can be still fit as a Band function, with $\beta \sim -2.12^{+0.158}_{-0.107}$ similar to the values at later epochs. This suggests an alternative interpretation to the data: The high energy spectral index may be similar throughout the burst. The delayed onset of LAT-band emission may be because initially there is a spectral cutoff around 100 MeV, which later moves to much higher energies (e.g. above 13.2 GeV in the second time bin).
It is interesting to note that the time integrated spectrum of GRB 080916C throughout the burst is also well fit with a Band function, where the spectral indices do not vary with time resolution. As an example, we present in Fig.\[080916C-090902B\] the $\nu F_\nu$ spectra of GRB 080916C for three time bins with varying time resolution. Remarkably, the parameters do not vary significantly: $\alpha \sim -1.12$, $\beta \sim -2.25$ for 3.5-8 s; $\alpha \sim -1.0$, $\beta \sim -2.29$ for 2-10 s; $\alpha \sim -1.0$, $\beta \sim -2.27$ for 0-20 s. This is in stark contrast with GRB 090902B discussed below.
GRB 090510 {#sec:090510}
----------
The short GRB 090510 was triggered with a precursor 0.5 s prior to the main burst. Two LAT photons were detected before the main burst. During the first time slice (0.45-0.5 s), no LAT band emission is detected, and the GBM spectrum can be well fit with a PL with an exponential cutoff (CPL hereafter). In the subsequent time slices, an additional PL component shows up, and the time-resolved spectra are best fit by the CPL + PL model. If one uses a Band + PL model to fit the data, the high energy spectral index $\beta$ of the Band component cannot be constrained. If one fixes $\beta$ to a particular value, it must be steeper than -3.5 in order to be consistent with the data. The CPL invoked in these fits has a low energy photon index $\Gamma_{\rm CPL} \sim -(0.6-0.8)$, which is very different from the case of a BB (where $\Gamma_{\rm CPL} \sim +1$). On the other hand, the high-energy regime (exponential cutoff) is very similar to the behavior of a BB.
Since this is a short GRB, we do not have enough photons to perform very detailed time-resolved spectral analysis. However, in order to investigate spectral evolution and the interplay between the MeV component and the extra PL component, we nonetheless make 4 time bins (see also Ackermann et al. 2010). As a result, the reduced $\chi^2$ of each segment is outside the range of $0.75 \leq\chi^2/{\rm dof} \leq 1.5$ as is required for other GRBs. Our reduction results are generally consistent with those of the Fermi team (Abdo et al. 2009b; Ackermann et al. 2010).
GRB 090902B
-----------
The spectrum of GRB 090902B is peculiar. Abdo et al. (2009c) reported that both the time-integrated and time-resolved spectra of this GRB can be fit with the Band+PL model. Ryde et al. (2010) found that the time-resolved spectra can be fit with a PL plus a multi-color blackbody model. This raises the interesting possibility that a blackbody-like emission component is a fundamental emission unit shaping the observed GRB spectra.
In order to test this possibility, we carried out a series of time-resolved spectral analysis on the data (Fig.\[080916C-090902B\]). We first fit the time-integrated data within the time interval 0-20 s, and found that it can be fit with a model invoking a Band function and a power law, but with a poor $\chi^2/{\rm dof} \sim
3.52$. Compared with the Band component of other GRBs, this Band component is very narrow, with $\alpha \sim -0.58$, $\beta \sim -3.32$. A CPL + PL model can give comparable fit, with $\Gamma_{\rm CPL} \sim -0.59$. Next we zoom into the time interval 8.5 - 11.5 s, and perform spectral fits. The Band+PL and CPL+PL models can now both give acceptable fits, with parameters suggesting a narrower spectrum. For the Band+PL model, one has $\alpha \sim -0.07$, $\beta \sim -3.69$ with $\chi^2/{\rm dof}
=1.26$. For the CPL+PL model, one has $\Gamma_{\rm CPL} \sim -0.08$ with $\chi^2/{\rm dof} =1.30$. Finally we room in the smallest time bin (9.5 - 10 s) in which the photon counts are just enough to perform adequate spectral fits. We find that the Band + PL model can no longer constrain $\beta$. The spectrum becomes even narrower, with $\alpha \sim 0.07$ and $\beta < -5$. The CPL+PL model can fit the data with a range of allowed $\Gamma_{\rm CPL}$. In particular, if one fixes $\Gamma_{\rm CPL} \sim +1$ (the Rayleigh-Jeans slope of a blackbody), one gets a reasonable fit with $\chi^2/{\rm dof} =0.92$. This encourages us to suspect that a blackbody (BB) + PL model can also fit the data. We test it and indeed found that the model can fit the data with $\chi^2/{\rm dof} =1.11$. These different models require different $\Gamma_{\rm PL}$ for the extra PL component, but given the low photon count rate at high energies, all these models are statistically allowed. Since the BB + PL model has less parameters than the CPL + PL and Band + PL models, we take this model as the simplest model for this smallest time interval.
Next, we tried to divide the lightcurve of GRB 090902B into as many as time bins as possible so that the photon numbers in each time bin are large enough for statistically meaningful fits to be performed. Thanks to its high flux, we managed to divide the whole data set (0-30 s) into 62 time bins. We find that the data in each time bin can be well fit by a BB+PL model, and that the BB temperature evolves with time. The fitting results are presented in Table 2 and Fig.\[090902B\]. The time-integrated spectrum, however, cannot be fit with such a model ($\chi^2/{\rm dof}=14732/276$). A Band+PL model gives a much improved fit, although the fit is still not statistically acceptable ($\chi^2/{\rm dof}=2024/275$). The best fitting parameters are $\alpha=-0.83$, $\beta=-3.68$, $E_p=847$ keV, and $\Gamma=-1.85$. Notice that the high energy photon index of the time-integrated Band spectrum is much steeper/softer than that observed in typical GRBs (Fig.\[distributions\]).
In Fig.\[090902B-lightcurves\], we display the lightcurves of both the thermal and the power-law components. It is found that the two components in general track each other. This suggests that the physical origins of the two components are related to each other (see Section 5 for more discussion).
An important inference from the analysis of GRB 090902B is that a Band-like spectrum can be a result of temporal superposition of many blackbody-like components. This raises the interesting possibility of whether all “Band” function spectra are superposed thermal spectra. From the comparison between GRB 090902B and GRB 080916C (Fig.\[080916C-090902B\]), we find that such speculation is far-fetched. As discussed above, GRB 080916C shows no evidence of “narrowing” as the time bin becomes small ($\sim 1$ s in the rest frame). In the case of GRB 090902B, a clear “narrowing” feature is seen. For the time integrated spectrum, GRB 080916C has a wide Band function (with $\alpha \sim -1.0$, $\beta \sim -2.27$), while GRB 090902B (0-20 s) has a narrow Band function (with $\alpha=-0.58$, $\beta=-3.32$) with worse reduced $\chi^2$. Another difference between GRB 090902B and GRB 080916C is that the former has a PL component, which leverages the BB spectrum on both the low-energy and the high-energy ends to make a BB spectrum look more similar to a (narrow) Band function. GRB 080916C does not have such a component, and the Band component covers the entire [*Fermi*]{} energy range (GBM & LAT). We therefore conclude that GRB 090902B is a special case, whose spectrum may have a different origin from GRB 080916C (and probably most other LAT GRBs as well, see Sect.\[sec:otherGRBs\] below).
GRB 090926A {#sec:090926A}
-----------
This is another bright long GRB with a duration $\sim 20$ s. In our analysis, the lightcurve is divided into 9 segments. The Band function gives an acceptable fit to all the time bins (Fig.\[090926A\]). We however notice a flattening of $\beta$ after $\sim 11$ s after the trigger. Also the Band function fit gives a worse reduced $\chi^2$ (although still acceptable) after this epoch. Since our data analysis strategy is to go for the simplest models, we do not explore more complicated models that invoke Band + PL or Band + CPL (as is done by the Fermi team, Abdo et al. 2010b). In any case, our analysis does not disfavor the possibility that a new spectral component emerges after $\sim 11$ s since the trigger (Abdo et al. 2010b).
Other GRBs {#sec:otherGRBs}
----------
The time resolved spectra of other 13 GRBs are all adequately described by the Band function, similar to GRB 080916C. The Band-function spectral parameters are generally similar to GRB 080916C. It is likely that these GRBs join GRB 080916C forming a “Band-only” type GRBs. In the current sample of 17 GRBs, only GRB 090510, GRB 090902B and probably GRB 090926A do not belong to this category and have an extra PL component extending to high energies. One caveat is that some GRBs in the sample are not very bright, so that we only managed to divide the lightcurves into a small number of time bins (e.g. 3 bins for GRB 080825C, 1 bin for GRB 081024B, 3 bins for GRB 090328, 3 bins for GRB 091031, 2 bins for GRB 100225A, and 1 bin for GRB 100325A). So one cannot disfavor the possibility that the observed spectra are superposition of narrower components (similar to GRB 090902B). However, at comparable time resolution, GRB 090902B already shows features that are different from these GRBs: (1) the Band component is “narrower”, and (2) there is an extra PL component. These two features are not present in other GRBs. We therefore suggest that most LAT/GBM GRBs are similar to GRB 080916C.
Spectral parameter distributions {#sec:distributions}
--------------------------------
Since the time-resolved spectra of most GRBs in our sample can be adequately described as a Band function, we present the distributions of the Band function parameters in this section. Since their MeV component may be of a different origin, GRB 090510 and GRB 090902B are not included in the analysis.
The distributions of the spectral parameters $\alpha$, $\beta$, and $E_p$ are presented in Figure \[distributions\], with a comparison with those of the bright BATSE GRB sample (Preece et al. 2000). It is found that the distributions peak at $\alpha=-0.9$, $\beta=-2.6$, and $E_p\sim 781$ keV, respectively. The $\alpha$ and $\beta$ distributions are roughly consistent with those found in the bright BATSE GRB sample (Preece et al. 2000). The $E_p$ distribution of the current sample has a slightly higher peak than the bright BATSE sample (Preece et al. 2000). This is likely due to a selection effect, namely, a higher $E_p$ would favor GeV detections.
Spectral parameter correlations {#sec:correlations}
-------------------------------
For time-integrated spectra, it was found that $E_p$ is positively correlated with the isotropic gamma-ray energy and the isotropic peak gamma-ray luminosity (Amati et al. 2002; Wei & Gao 2003; Yonetoku et al. 2004). For time resolved spectra, $E_p$ was also found to be generally correlated with flux (and therefore luminosity, Liang et al. 2004), although in individual pulses, both a decreasing $E_p$ pattern and a $E_p$-tracking-flux pattern have been identified (Ford et al. 1995; Liang & Kargatis 1996; Kaneko et al. 2006; Lu et al. 2010).
In Fig.\[L-Ep\], we present the $E_p$-luminosity relations. Fig.\[L-Ep\]a is for the global $E_p-L_{\gamma,\rm iso}^p$ correlation. Seven GRBs in our sample that have redshift information (and hence, the peak luminosity) are plotted against previous GRBs (a sample presented in Zhang et al. 2009). Since the correlation has a large scatter, all the GBM/LAT GRBs follow the same correlation trend. In particular, GRB 090902B, whose $E_p$ is defined by the BB component, also follows a similar trend. This suggests that even if there may be two different physical mechanisms to define a GRB’s $E_p$, both mechanisms seem to lead to a broad $E_p-L_{\gamma,\rm iso}^p$ relation. It is interesting to note that the short GRB 090510 (the top yellow point), even located at the upper boundary of the correlation, is still not an outlier. This is consistent with the finding (Zhang et al. 2009; Ghirlanda et al. 2009) that long/short GRBs are not clearly distinguished in the $L_{\gamma,\rm iso}^p - E_p$ domain.
In Fig.\[L-Ep\]b, we present the internal $E_p-L_{\gamma,\rm iso}$ correlation. It is interesting to note that although with scatter, the general positive correlation between $E_p$ and $L_{\gamma,\rm iso}$ as discovered by Liang et al. (2004) clearly stands. More interestingly, the BB-defined $E_p$ (in GRB 090902B) follows a similar trend to the Band-defined $E_p$ (e.g. in GRB 080916C and GRB 090926A), although different bursts occupy a different space region in the $E_p-L_{\gamma,\rm iso}$ plane.
In Fig.\[correlations\], we present various pairs of spectral parameters in an effort to search for possible new correlations. The GRBs with redshift measurements are marked in colors, while those without redshifts are marked in gray with an assumed redshift $z=1$. In order to show the trend of evolution, points for same burst are connected, with the beginning of evolution marked as a circle.
No clear correlation pattern is seen in the $E_p -\alpha$ and $E_p -\beta$ plots. Interestingly, a preliminary trend of correlation is found in the following two domains.
- An $\alpha-\beta$ anti-correlation: Fig.\[correlations\]a shows a rough anti-correlation between $\alpha$ and $\beta$ in individual GRBs. This suggests that a harder $\alpha$ corresponds to a softer $\beta$, suggesting a narrower Band function. In the time domain, there is evidence in some GRBs (e.g. GRB 080916C, GRB 090926A, and GRB 100414A, see Figs.\[080916C\],\[090926A\],\[100414A\]) that the Band function “opens up” as time goes by, but the opposite trend is also seen in some GRBs (e.g. GRB 091031, Fig.\[091031\]). [ The linear Pearson correlation coefficients for individual bursts are insert in Fig.\[correlations\]a ]{}
- A flux-$\alpha$ correlation: Fig.\[correlations\]b shows a rough correlation between flux and $\alpha$. Within the same burst, there is rough trend that as the flux increases, $\alpha$ becomes harder. The linear Pearson correlation coefficients for individual bursts are presented in Fig.\[correlations\]b inset. One possible observational bias is that when flux is higher, one tends to get a smaller time slice based on the minimum spectral analysis criterion. If the time smearing effect can broaden the spectrum, then a smaller time slice tends to give a narrower spectrum, and hence, a harder $\alpha$. This would be relevant to bursts similar to GRB 090902B, but not bursts similar to GRB 080916C (which does not show spectral evolution as the time resolution becomes finer). More detailed analyses of bright GRBs can confirm whether such a correlation is intrinsic or due to the time resolution effect discussed above.
Several caveats should be noted for these preliminary correlations: First, some bursts do not obey these correlations, so the correlations, if any, are not universal; Second, the currently chosen time bins are based on the requirement for adequate spectral analyses, so the time resolution varies in different bursts. For some bright bursts, a burst pulse can be divided into several time bins, while in some faint others, a time bin corresponds to the entire pulse; Third, the current sample is still too small. A time-resolved spectral analysis for more bright GBM GRBs may confirm or dispute these correlations.
Elemental Spectral Components and their physical origins
========================================================
Three phenomenologically identified elemental spectral components
-----------------------------------------------------------------
The goal of our time-resolved spectral analysis is to look for “elemental” emission units that shape the observed GRB prompt gamma-ray emission. In the past it has been known that time-integrated GRB spectra are mostly fit by the Band function (Band et al. 1993). However, whether this function is an elemental unit in the time-resolved spectra is not known. One speculation is that this function is the superposition of many simpler emission units. If such a superposition relies on adding the emission from many time slices, then these more elemental units should show up as the time bins become small enough.
One interesting finding of our time-resolved spectral analyses is that the “Band”-like spectral component seen in GRB 090902B is different from that seen in GRB 080916C and some other Band-only GRBs. While the Band spectral indices of GRB 080916C essentially do not change as the time bins become progressively smaller, that of GRB 090902B indeed show the trend of “narrowing” as the time bin becomes progressively smaller. With the finest spectral resolution, GRB 090902B spectra can be fit by the superposition of a PL component and a CPL function, including a Planck function. Even for the time-integrated spectrum, the “Band”-like component in GRB 090902B appears “narrower” than that of GRB 080916C. All these suggest that the “Band”-like component of GRB 090902B is fundamentally different from that detected in GRB 080916C and probably also other Band-only GRBs[^3]. Similarly, the time-resolved spectra of the short GRB 090510 can be well fit by the superposition of a PL component and a CPL spectrum (although not a Planck function). The PL component extends to high energies with a positive slope in $\nu F_\nu$. The CPL component may be modeled as a multi-color blackbody spectrum. We therefore speculate that the MeV component of GRB 090510 is analogous to that of GRB 090902B.
Phenomenologically, the power law component detected in GRB 090902B and GRB 090510 is an extra component besides the Band-like component. Such a component may have been also detected in the BATSE-EGRET burst GRB 941017 (González et al. 2003), and may also exist in GRB 090926A at later epochs.
We therefore speculate that phenomenologically there might be three elemental spectral components that shape the prompt gamma-ray spectrum. These include: (I) a [*Band function component*]{} (“Band” in abbreviation) that covers a wide energy range (e.g. 6-7 orders of magnitude in GRB 080916C) and persists as the time bins become progressively smaller. It shows up in GRB 080916C and 13 other LAT GRBs; (II) a [*quasi-thermal component*]{} (“BB” in abbreviation[^4]) which becomes progressively narrower as the time bin becomes smaller, and eventually can be represented as a blackbody (or multi-color blackbody) component as seen in GRB 090902B; (III) a [*power law component*]{} (“PL” in abbreviation) that extends to high energy as seen in GRBs 090902B and 090510, which has a positive slope in the $\nu F_\nu$ spectrum and should have an extra peak energy ($E_p$) at an even higher energy that is not well constrained by the data.
Figure \[Cartoon\] is a cartoon picture of the $\nu F_\nu$ spectrum that includes all three phenomenologically identified elemental spectral components. The time resolved spectra of the current sample can be understood as being composed of one or more of these components. For example, GRB 080916C and other 13 GRBs have Component I (Band), GRB 090902B and probably GRB 090510 have Components II (BB) and III (PL), and GRB 0900926A has Component I initially, and may have components I and III at later times.
Possible physical origins of the three spectral components
----------------------------------------------------------
### Band Component
The fact that the this component extends through a wide energy range (e.g. 6-7 orders of magnitude for GRB 080916C) strongly suggests that a certain non-thermal emission mechanism is in operation. This demands the existence of a population of power-law-distributed relativistic electrons, possibly accelerated in internal shocks or in regions with significant electron heating, e.g. magnetic dissipation. In the past there have been three model candidates for prompt GRB emission: synchrotron emission, synchrotron self-Compton (SSC), and Compton upscattering of a thermal photon source. In all these models the high energy PL component corresponds to emission from a PL-distributed electron population. The spectral peak energy $E_p$ may be related to the minimum energy of the injected electron population, an electron energy distribution break, or the peak of the thermal target photons.
Most prompt emission modeling (Mészáros et al. 1994; Pilla & Loeb 1998; Pe’er & Waxman 2004a; Razzaque et al. 2004; Pe’er et al. 2006; Gupta & Zhang 2007) suggest that the overall spectrum is curved, including multiple spectral components. Usually a synchrotron component is accompanied by a synchrotron self-Compton (SSC) component. For matter-dominated fireball models, one would expect the superposition of emissions from the photosphere and from the internal shock dissipation regions. As a result, the fact that 14/17 ($\sim 80\%$) of GRBs in our sample have a Band-only spectrum is intriguing. The three theoretically expected spectral features, i.e. the quasi-thermal photosphere emission, the SSC component (if the MeV component is of synchrotron origin), and a pair-production cutoff at high energies, are all not observed. This led to the suggestion that the outflows of these GRBs are Poynting flux dominated (Zhang & Pe’er 2009). Within such a picture, the three missing features can be understood as the following: (1) Since most energy is carried in magnetic fields and not in photons, the photosphere emission (BB component) is greatly suppressed; (2) Since the magnetic energy density is higher than the photon energy density, the Compton $Y$ parameter is smaller than unity, so that the SSC component is naturally suppressed; (3) A Poynting flux dominated model usually has a larger emission radius than the internal shock model (Lyutikov & Blandford 2003 for current instability and Zhang & Yan 2011 for collision-induced magnetic reconnection/turbulence). This reduces the two-photon annihilation opacity and increases the pair cutoff energy. This allows the Band component extend to very high energy (e.g. 13.2 GeV for GRB 080916C).
Another possibility, advocated by Beloborodov (2010) and Lazzati & Begelman (2010) in view of the [*Fermi*]{} data [(see also discussion by Thompson 1994; Rees & Mészáros 2005; Pe’er et al. 2006; Giannios & Spruit 2007; Fan 2009; Toma et al. 2010; and Ioka 2010)]{}, is that the Band component is the emission from a dissipative photosphere. This model invokes relativistic electrons in the regions where Thomson optical depth is around unity, which upscatter photosphere thermal photons to high energies to produce a power law tail. This model can produce a Band-only spectrum, but has two specific limitations. First, the high energy power law component cannot extend to energies higher than GeV in the cosmological rest frame, since for effective upscattering, the emission region cannot be too far above the photosphere. The highest photon energy detected in GRB 080916C is 13.2 GeV (which has a rest-frame energy $\sim 70$ GeV for its redshift $z=4.35$). This disfavors the dissipative photosphere model. [This argument applies if the LAT-band photons are from the same emission region as the MeV photons, as suggested by the single Band function spectral fits. It has been suggested that the LAT emission during the prompt phase originates from a different emission region, e.g. the external shock (Kumar & Barniol Duran 2009; Ghisellini et al. 2010). This requires that the two distinct emission components conspire to form a nearly featureless Band spectrum in all temporal epochs, which is contrived. As will be shown in Sect.\[sec:tracking\] later, there is compelling evidence that the LAT emission during the prompt emission phase is of an internal origin. In particular, the peak of the GeV lightcurve of GRB 080916C coincides with the second (the brightest) peak of GBM emission, and the 13.2 GeV photon coincides with another GBM lightcurve peak. All these suggest an internal origin of the GeV emission during the prompt phase.]{}
The second limitation of the dissipative photosphere model is that the photon spectral index below $E_p$ is not easy to reproduce. The simplest blackbody model predicts a Rayleigh-Jeans spectrum $\alpha=+1$. By considering slow heating, this index can be modified as $\alpha=+0.4$ (Beloborodov 2010). Both are much harder from the observed $\alpha \sim -1$ value. In order to overcome this difficulty, one may appeal to the superposition effect, i.e. the observed Band spectrum is the superposition of many fundamental blackbody emission units (e.g. Blinnikov et al. 1999; Toma et al. 2010; Mizuta et al. 2010; Pe’er & Ryde 2010). However, no rigorous calculation has been performed to fully reproduce the $\alpha=-1$ spectrum. Pe’er & Ryde (2010) show that when the central engine energy injection is over and the observed emission is dominated by the high-latitude emission, an $\alpha=-1$ can be reproduced with the flux decaying rapidly with $\propto t^{-2}$. During the phase when the central engine is still active, the observed emission is always dominated by the contribution along the line of sight, which should carry the hard low energy spectral index of the blackbody function. Observationally, the Band component spectral indices are not found to vary when the time bins are reduced (in stark contrast to the narrow Band-like component identified in GRB 090902B). This suggests that at least the temporal superposition of many blackbody radiation units is not the right interpretation for this component.
### Quasi-Thermal (BB) Component
The MeV component in GRB 090902B narrows with reduced time resolution and eventually turns into being consistent with a blackbody (or multi-color blackbody) as the time bin becomes small enough. This suggests a thermal origin of this component. Within the GRB content, a natural source is the emission from the photosphere where the photons advected in the expanding relativistic outflow turn optically thin for Compton scattering. In fact, the original fireball model predicts a quasi-thermal spectrum (Paczýnski 1986; Goodman 1986). In the fireball shock model, such a quasi-blackbody component is expected to be associated with the non-thermal emission components (Mészáros & Rees 2000; Mészáros et al. 2002; Daigne & Mochkovitch 2002; Pe’er et al. 2006).
Some superposition effects may modify the thermal spectrum to be different from a pure Planck function. The first is the temporal smearing effect. If the time bin is large enough, one samples photosphere emission from many episodes, and hence, the observed spectrum should be a multi-color blackbody. This effect can be diminished by reducing the time bin for time-resolved spectral analyses. GRB 090902B is such an example. The second effect is inherited in emission physics of relativistic objects. At a certain epoch, the observer detects photons coming from different latitudes from the line of sight, with different Doppler boosting factors. The result is an intrinsic smearing of the Planck function spectrum. Pe’er & Ryde (2010) have shown that after the central engine activity ceases, the high-latitude emission effect would give an $\alpha \sim -1$ at late times, with a rapidly decaying flux $F_\nu \propto t^{-2}$. This second superposition effect is intrinsic, and cannot be removed by reducing the time bins.
The case of the thermal component is most evidenced in GRB 090902B, and probably also in GRB 090510. In both bursts, the MeV component can be well fit with a CPL + PL spectrum. The exponential cutoff at the high energy end is consistent with thermal emission with essentially no extra dissipation. For GRB 090902B, the low energy spectral index $\Gamma_{\rm CPL}$ is typically $\sim 0$, and can be adjusted to $+1$ (blackbody). For GRB 090510, $\Gamma_{\rm CPL}$ is softer ($\sim -0.7$). Since it is a short GRB, the high-latitude effect may be more important. The softer low energy spectral index may be a result of the intrinsic high-latitude superposition effect (Pe’er & Ryde 2010).
### Power-Law (PL) Component
This component is detected in GRB 090902B and GRB 090510. Several noticeable properties of this component are: (1) For our small sample, this component is always accompanied by a low energy MeV component (likely the BB component). Its origin may be related to this low energy component; (2) It is demanded in both the low energy end and the high energy end, and amazingly the same spectral index can accommodate the demanded excesses in both ends. This suggests that either this PL component extends for 6-7 orders of magnitude in energy, or that multiple emission components that contribute to the excesses in both the low and high energy regimes have to coincide to mimic a single PL; (3) The spectral slope is positive in the $\nu F_\nu$ space, so that the main energy power output [ of this component]{} is at even higher energies (possibly near or above the upper bound of the LAT band).
Since the non-thermal GRB spectra are expected to be curved (Mészáros et al. 1994; Pilla & Loeb 1998; Pe’er & Waxman 2004a; Razzaque et al. 2004; Pe’er et al. 2006; Gupta & Zhang 2007; Asano & Terasawa 2009), the existence of the PL component is not straightforwardly expected. It demands coincidences of various spectral components to mimic a single PL component in the low and high energy ends. Pe’er et al. (2010) have presented a theoretical model of GRB 090902B. According to this model, the apparent PL observed in this burst is the combination of the synchrotron emission component (dominant at low energies), the SSC and Comptonization of the thermal photons (both dominant at high energies). A similar model was analytically discussed by Gao et al. (2009) within the context of GRB 090510.
One interesting question is how Component III (PL) differs from Component I (Band). Since both components are non-thermal, they may not be fundamentally different. They can be two different manifestations of some non-thermal emission mechanisms (e.g. synchrotron and inverse Compton scattering) under different conditions. On the other hand, since Component III seems to be associated with Component II (BB) (e.g. in GRB 090902B and GRB 090510), its origin may be related to Component II. One possible scenario is that Component III [(at least the part above component II)]{} is the Compton-upscattered emission of Component II (e.g. Pe’er & Waxman 2004b for GRB 941017). The fact that the lightcurves of the BB component and the PL component of GRB 090902B roughly track each other (Fig.\[090902B-lightcurves\]) generally supports such a possibility. [ Within this interpretation, one must attribute the PL part below the thermal peak as due to a different origin (e.g. synchrotron, see Pe’er et al. 2010).]{} Alternatively, Component I and III may be related to non-thermal emission from two different emission sites (e.g. internal vs. external or two different internal locations). Indeed, if the late spectra of GRB 090926A are the superposition of the components I and III, then both components can coexist, which may correspond to two different non-thermal emission processes and/or two different emission sites.
Possible Spectral Combinations of GRB Prompt Emission
-----------------------------------------------------
Using the combined GBM and LAT data, we have phenomenologically identified three elemental spectral components during the prompt GRB phase (Fig.\[Cartoon\]). Physically they may have different origins (see above). One may speculate that all the GRB prompt emission spectra may be decomposed into one or more of these spectral components. It is therefore interesting to investigate how many combinations are in principle possible, how many have been discovered, how many should not exist and why, and how many should exist and remain to be discovered. We discuss the following possibilities in turn below (see Fig.\[spectral-combinations\] for illustrations).
1. Component I (Band) only:
This is the most common situation, which is observed in 14/17 GRBs in our sample exemplified by GRB 080916C. Either the BB and PL components do not exist, or they are too faint to be detected above the Band component. If the BB component is suppressed, these bursts may signify non-thermal emission from an Poynting flux dominated flow.
2. Component II (BB) only:
No such case exists in the current sample. GRB 090902B, and probably also GRB 090510, have a BB component, but it is accompanied by a PL component in both GRBs. It remains to be seen whether in the future a BB-only GRB will be detected, or whether a BB component is always accompanied by a PL component. Since the case of GRB 090902B is rare, we suspect that the BB-only GRBs are even rarer, if they exist at all.
3. Component III (PL) only:
Our PL component stands for the high energy spectral component seen in GRB 090902B and GRB 090510, which likely has a high $E_p$ near or above the boundary of the LAT band. Observationally, there is no solid evidence for such PL-only GRBs[^5]. In our current sample which covers the widest energy band, the PL component only exists in 2 out of 17 GRBs, and is found to be associated with the BB component. The luminosity of the PL component is found to roughly track that of the thermal component (Fig.\[090902B-lightcurves\]). If the PL component is the Comptonization of a low energy photon source (e.g. the BB component), then PL-only GRBs may not exist in nature.
4. I + II:
Such a case is not found in our sample. If the Band component is the emission from the internal shocks and the BB component is the emission from the photosphere, then such a combination should exist and be common for fireball scenarios. An identification of such a case would confirm the non-thermal nature of the Band component (since the thermal component is manifested as the BB component). Observationally, an X-ray excess has been observed in 12 out of 86 ($\sim 14\%$) bright BATSE GRBs (Preece et al. 1996). This could be due to the contamination of a BB component in the X-ray regime. With the excellent spectral coverage of [*Fermi*]{}, we expect that such a spectral combination may be identified in some GRBs, even if technically it may be difficult because there are too many spectral parameters to constrain.
5. I + III:
Such a combination has not been firmly identified in our sample. Nonetheless, the spectral hardening of GRB 090926A after 11 s may be understood as the emergence of the PL component on top of the Band component seen before 11 s. Physically it may be related to two non-thermal spectral components or non-thermal emission from two different regions.
6. II + III:
Such a case is definitely identified in GRB 090902B, and likely in GRB 090510 as well. From the current sample, it seems that such a combination is not as common as the Band-only type, but nonetheless forms a new type of spectrum that deserves serious theoretical investigations. Physically, the high-energy PL component is likely the Compton up-scattered emission of the BB component, although other non-thermal processes (e.g. synchrotron and SSC) could also contribute to the observed emission (Pe’er et al. 2010).
7. I + II + III:
The full combination of all three spectral components (e.g. Fig.\[Cartoon\]) is not seen from the current sample. In any case, in view of the above various combinations (including speculative ones), one may assume that the full combination of the three spectral components is in principle possible. Physically this may correspond to one photosphere emission component and two more non-thermal components (either two spectral components or non-thermal emission from two different regions). Nonetheless, technically there are too many parameters to constrain, so that identifying such a combination is difficult.
LAT-band emission vs. GBM-band emission
=======================================
Besides the joint GBM/LAT spectral fits, one may also use temporal information to investigate the relationship between the emission detected in the GBM-band and that detected in the LAT band. In this section we discuss three topics: delayed onset of LAT emission, rough tracking behavior between GBM and LAT emissions, and long-lasting LAT afterglow.
Delayed onset of LAT emission
-----------------------------
The [*Fermi*]{} team has reported the delayed onset of LAT emission in several GRBs (GRBs 080825C, 080916C, 090510, 090902B, Abdo et al. 2009a,b,c,d). Our analysis confirms all these results. In Table 1, we mark all the GRBs in our sample that show the onset delay feature.
There have been several interpretations to the delayed onset of GeV emission discussed in the literature. Toma et al. (2009) suggested that GeV emission is the upscattered cocoon emission by the internal shock electrons. Razzaque et al. (2009) interpreted the GeV emission as the synchrotron emission of protons. Since it takes a longer time for protons to be accelerated and be cooled to emit GeV photons, the high energy emission is delayed. Li (2010b) interpreted GeV emission as the upscattered prompt emission photons by the residual internal shocks.
Although it is difficult to test these models using the available data, our results give some observational constraints to these models. First, except GRBs 090510 and 090902B whose GeV emission is a distinct spectral component, other GRBs with onset delay still have a simple Band-function spectrum after the delayed onset. This suggests that for those models that invoke two different emission components to interpret the MeV and GeV components, one needs to interpret the coincidence that the GeV emission appears as the natural extension of the MeV emission to the high energy regime.
For such delayed onsets whose GeV and MeV emissions form the same Band component, one may speculate two simpler explanations. One is that there might be a change in the particle acceleration conditions (e.g. magnetic configuration in the particle acceleration region). As shown in Sect.\[sec:080916C\], the early spectrum during the first time bin (before onset of LAT emission) of GRB 080916C may be simply a consequence of changing the electron spectral index. One may speculate that early on the particle acceleration process may not be efficient, so that the electron energy spectral index is steep. After a while (the observed delay), the particle acceleration mechanism becomes more efficient, so that the particle spectral index reaches the regular value. The second possibility is that there might be a change in opacity. The GBM data alone during the first time bin gives a similar $\beta$ as later epochs. It is possible that there might be a spectral cutoff slightly above the GBM band early on. A speculated physical picture would be that the particle acceleration conditions are similar throughout the burst duration, but early on the pair production opacity may be large (probably due to a lower Lorentz factor or a smaller emission radius), so that the LAT band emission is attenuated. The opacity later drops (probably due to the increase of Lorentz factor or the emission radius), so that the LAT band emission can escape from the GRB. [Within such a scenario, one would expect to see a gradual increase of maximum photon energy as a function of time. Figure 25 shows the LAT photon arrival time distribution of GRB 080916C. Indeed one can see a rough trend of a gradual increase of the maximum energy with time.]{}
One last possibility is that the LAT band emission is dominated by the emission from the external shock, which is delayed with respect to the GBM-band prompt emission. This possibility is discussed in more detail below in Sec. \[sec:LATafterglow\].
Rough tracking behavior {#sec:tracking}
-----------------------
Inspecting the multi-band lightcurves (Figs.\[080825C\]-\[100414A\] left panels), for bright GRBs (e.g. 080916C, 090217, 090323, 090902B) the LAT emission peaks seem to roughly track some peaks of the GBM emission (aside from the delayed onset for some of them). For example, the peak of the LAT lightcurve of GRB 080916C coincides with the second GBM peak. This is consistent with the spectral analysis showing that most time-resolved joint spectra are consistent with being the same (Band-function) spectral component. Even for GRB 090902B whose LAT band emission is from a different emission component from the MeV BB component, the emissions in the two bands also roughly track each other (Fig.\[090902B-lightcurves\]). This suggests that the two physical mechanisms that power the two spectral components are related to each other.
The rough tracking behavior is evidence against the proposal that the entire GeV emission is from the external forward shock (see Sec. \[sec:LATafterglow\] for more discussion). Within the forward shock model, the fluctuation in energy output from the central engine should be greatly smeared, since the observed flux change amplitude is related to $\Delta E/E \ll 1$ (where $E$ is the total energy already in the balstwave, and $\Delta E$ is the newly injected energy from the central engine), rather than $\Delta E$ itself within the internal models.
Long Term Emission in the LAT Band {#sec:LATafterglow}
----------------------------------
In order to study the long-term lightcurve behavior, we extract the GBM-band and LAT-band lightcurves in logarithmic scale and present them in the bottom right panel of Figs.\[080825C\]-\[100414A\]. We unevenly bin the LAT lightcurves with bin sizes defined by the requirement that the signal-to-noise ratio must be $>5$. For a close comparison, we correspondingly re-bin the GBM lightcurves using the same bin sizes. Some GRBs (e.g. 080916C, 090510, 090902B, and 090926A) have enough photons to make a well sampled LAT lightcurve.
In several GRBs, LAT emission lasts longer than GBM emission and decays as a single power law (Ghisellini et al. 2010). The decay indices of LAT emission are marked in the last panel of Figs.\[080825C\]-\[100414A\], which can be also found in Table 3. [ Due to low photon numbers, it is impossible to carry out a time resolved spectral analysis. In any case, the LAT-band photon indices of long-term LAT emission are estimated and also presented in Table 3.]{} In Table 1 we mark those GRBs with detected LAT emission longer than GBM emission and those without. The most prominent ones with long lasting LAT afterglow are GRBs 080916C, 090510, 090902B, and 090926A. Spectral analyses suggest that the LAT emission in GRBs 090510 and 090902B is a different spectral component from the MeV emission. The GBM lightcurves of these GRBs indeed follow a different trend by turning off sharply as compared with the extended PL decay in the LAT band. GRB 090926A, on the other hand, shows a similar decay trend in both GBM and LAT bands. GRB 080916C is special. Although the spectral analysis shows a single Band function component, the GBM lightcurve turns over sharply around 70-80 seconds, while the LAT emission keeps decaying with a single PL.
One caveat of LAT long-term lightcurves is that they depend on the level of background and time-bin selection. Due to the low count rate at late times, the background uncertainty can enormously change the flux level, and a different way of binning the data may change the shape of the lightcurve considerably. In our analysis, the background model is extracted from the time interval prior to the GBM trigger in the same sky region that contains the GRB. The bin-size is chosen to meet the $5\sigma$ statistics to reduce the uncertainty caused by arbitrary binning.
Our data analysis suggests a controversial picture regarding the origin of this GeV afterglow. Spectroscopically, the LAT-band emission is usually an extension of the GBM-band emission and forms a single Band-function component, suggesting a common physical origin with the GBM-band emission. If one focuses on the prompt emission lightcurves, the LAT-band activities seem to track the GBM-band activities. Even for GRB 090902B which shows a clear second spectral component, the PL component variability tracks that of the BB component well (Fig.\[090902B-lightcurves\]), suggesting a physical connection between the two spectral components. These facts tentatively suggest that at least during the prompt emission phase, the LAT-band emission is likely connected to the GBM-band emission, and may be of an “internal” origin similar to the GBM-band emission.
It has been suggested that the entire GeV emission originates from the external shock (e.g. Kumar & Barniol Duran 2009a, 2009b; Ghisellini et al. 2010; Corsi et al. 2009). This idea is based on the power law temporal decay law that follows the prompt emission. Such a GeV afterglow scenario is not straightforwardly expected for the following reasons. First, before [*Fermi*]{}, afterglow modeling suggests that for typical afterglow parameters, the GeV afterglow is initially dominated by the synchrotron self-Compton component (Mészáros & Rees 1994; Dermer et al. 2000; Zhang & Mészáros 2001; Wei & Fan 2007; Gou & Mészáros 2007; Galli & Piro 2007; Yu et al. 2007; Fan et al. 2008), or by other IC processes invoking both forward and reverse shock electrons (Wang et al. 2001). For very energetic GRBs such as GRB 080319B, one may expect a synchrotron-dominated afterglow all the way to an energy $\sim$ 10 GeV (Zou et al. 2009; Fan et al. 2008). Second, the required parameters for the external shock are abnormal to interpret the data. For example, the magnetic field strength at the forward shock needs to be much smaller than equipartition, consistent with simply compressing the ISM magnetic field without shock amplification (Kumar & Barniol Duran 2010). This, in turn, causes a problem in accelerating electrons to a high enough energy to enable emission of GeV photons (Li 2010a; Piran & Nakar 2010). Moreover, the circumburst number density of these long GRBs are required to be much lower than that of a typical ISM (e.g., Kumar & Barniol Duran 2010), which challenges the collapsar model. Finally, observed GeV decay slope is typically steeper than the predictions invoking a standard adiabatic forward shock (e.g. Figs.\[080916C\],\[090510\],\[090902B\],\[090926A\],\[100414A\], see also Ghisellini et al. 2010). One needs to invoke a radiative blastwave (Ghisellini et al. 2010) or a Klein-Nishina cooling-dominated forward shock (Wang et al. 2010) to account for the steepness of the decay slope.
The external shock model to interpret the entire GeV emission is challenged by the following two arguments. First, the GeV lightcurve peak coincides the second peak of the GBM lightcurve for GRB 080916C. This requires a fine-tuned bulk Lorentz factor of the fireball to make the deceleration time coincide the epoch of the second central engine activity. This is highly contrived. Second, the external shock component should not have decayed steeply while the prompt emission is still on going. To examine this last point, we have applied the shell-blastwave code developed by Maxham & Zhang (2009) to model the blastwave evolution of GRB 080916C using the observed data by assuming that the outflow kinetic energy traces the observed gamma-ray lightcurve (assuming a constant radiation efficiency). The resulting LAT-band lightcurve always displays a shallow decay phase caused by refreshing the forward shock by materials ejected after the GeV lightcurve peak time even for a radiative blastwave, in stark contrast to the data. This casts doubts on the external shock origin of GeV emission during the prompt phase (Maxham et al. 2011). We note that detailed modeling of GRB 090510 (He et al. 2010) and GRB 090902B (Liu & Wang 2011) with the external shock model both suggests that the prompt GeV emission cannot be interpreted as the emission from the external forward shock.
Collecting the observational evidence and the theoretical arguments presented above, we suggest that at least during the prompt emission phase (when GBM-band emission is still on), the LAT-band emission is not of external forward shock origin.
After the GBM-band prompt emission is over, the LAT-band emission usually decays as a PL. We note that the long-term GeV lightcurve can be interpreted in more than one way. (1) If one accepts that the prompt GeV emission is of internal origin, one may argue that the external shock component sets in before the end of the prompt emission and thereafter dominates during the decay phase (Maxham et al. 2011). This requires arguing for coincidence of the same decaying index for the early internal and the late external shock emission. Considering a possible superposition effect (i.e. the observed flux during the transition epoch includes the contributions from both the internal and external shocks), this model is no more contrived than the model that interprets prompt GeV emission as from external shocks, which requires coincidence of internal emission spectrum and the external shock emission spectrum to mimic the same Band spectrum in all time bins (Kumar & Barniol Duran 2009). (2) An alternative possibility is to appeal to an internal origin of the entire GeV long-lasting afterglow, which reflects the gradual “die-off” of the central engine activity. The difficulty of such a suggestion is that it must account for the different decaying behaviors between the GBM-band emission and LAT-band emission in some (but not all) GRBs (e.g. GRB 080916C). To differentiate between these possibilities, one needs a bright GRB co-triggered by [*Fermi*]{} LAT/GBM and [*Swift*]{} BAT, so that an early [*Swift*]{} XRT lightcurve is available along with the early GeV lightcurve. The external-shock-origin GeV afterglow should be accompanied by a PL decaying early X-ray lightcurve (Liang et al. 2009) instead of the canonical steep-shallow-normal decaying pattern observed in most [*Swift*]{} GRBs (Zhang et al. 2006; Nousek et al. 2006; O’Brien et al. 2006). A violation of such a prediction would suggest an internal origin of the GeV afterglow.
Conclusions and discussion
==========================
We have presented a comprehensive joint analysis of 17 GRBs co-detected by [*Fermi*]{} GBM and LAT. We carried out a time-resolved spectral analysis of all the bursts with the finest temporal resolution allowed by statistics, in order to [ reduce]{} temporal smearing of different spectral components. Our data analysis results can be summarized as the following:
- We found that the time-resolved spectra of 14 out of 17 GRBs are best modeled with the classical “Band” function over the entire [*Fermi*]{} spectral range, which may suggest a common origin for emissions detected by LAT and GBM. GRB 090902B and GRB 090510 are found to be special in that the data require the superposition between a MeV component and an extra power law component, and that the MeV component has a sharp cutoff above $E_p$. More interestingly, the MeV component of GRB 090902B becomes progressively narrower as the time bin gets smaller, and can be fit with a Planck function as the time bin becomes small enough. This is in stark contrast to GRB 080916C, which shows no evidence of “narrowing” with the reducing time bin. This suggests that the Band-function component seen in GRB 080916C is physically different from the MeV component seen in GRB 090902B.
- We tentatively propose that phenomenologically there can be three elemental spectral components (Fig.\[Cartoon\]), namely, (I): a Band-function component (Band) that extends to a wide spectral regime without “narrowing” with reduced time bins, which is likely of non-thermal origin; (II): a quasi-thermal component (BB) that “narrows” with reducing time bins and that can be reduced to a blackbody (or multi-color blackbody) function; and (III): a power-law component (PL) that has a positive slope in $\nu F_\nu$ space and extends to very high energy beyond the LAT energy band.
- Component I (Band) is the most common spectral component, which appears in 15 of 17 GRBs. Except GRB 090926A (which may have Component III at late times), all these GRBs have a Band-only spectrum in the time-resolved spectral analysis.
- Component II (BB) shows up in the time-resolved spectral analysis of GRB 090902B and possibly also in GRB 090510. The MeV component of these two GRBs can be fit with a power law with exponential cutoff (CPL). Since data demand the superposition with an additional PL component (Component III), the uncertainty in the spectral index of the PL component makes it possible to have a range of low energy photon indices for the CPL component. In particular, the MeV component of GRB 090902B can be adjusted to be consistent with a blackbody (Plank function). This is not possible for GRB 090510, whose low energy photon index is softer. In any case, the MeV component of GRB 090510 may be consistent with a multi-color blackbody.
- Component III (PL) shows up in both GRB 090902B and the short GRB 090510, and probably in the late epochs of GRB 090926A as well. It has a positive slope in $\nu F_\nu$, which suggests that most energy in this component is released near or above the high energy end of the LAT energy band.
- With the above three elemental emission components, one may imagine 7 possible spectral combinations. Most ($\sim 80\%$) of GRBs in our sample have the Band-only spectra. GRB 090902B has the BB+PL spectra in the time resolved spectral analyses, and GRB 090510 has a CPL + PL spectra. Both can be considered as the superposition between Components II and III. GRB 090926A may have the superposition between I and III at late epochs. Other combinations are not identified yet with the current analysis, but some combinations (e.g. I+II, I+II+III) may in principle exist.
- LAT-band emission has a delayed onset with respect to GBM-band emission in some (but not all) GRBs and it usually lasts much longer. In most cases (all except GRBs 090902B and 090510), however, the LAT and GBM photons are consistent with belonging to the same spectral component, suggesting a possible common origin. For bright bursts, the LAT-band activities usually roughly track the GBM-band activities. In the long-term, the LAT and GBM lightcurves sometimes (not always) show different decaying behaviors. The LAT lightcurves continuously decay as a power-law up to hundreds of seconds.
- A statistical study of the spectral parameters in our sample generally confirms the previously found correlations between $E_p$ and luminosity, both globally in the whole sample and individually within each burst. We also discover preliminary rough correlations between $\alpha$ and $\beta$ (negative correlation) and between flux and $\alpha$ (positive correlation). Both correlations need confirmation from a larger sample.
From these results, we can draw the following physical implications regarding the nature of GRBs.
The Band-only spectra are inconsistent with the simplest fireball photosphere-internal-shock model. This is because if the Band component is non-thermal emission from the internal shock, the expected photosphere emission should be very bright. A natural solution is to invoke a Poynting-flux-dominated flow. An alternative possibility is to interpret the Band component as the photosphere emission itself. However, the following results seem to disfavor such a possibility. (1) In some cases (e.g. GRB 080916C), the Band-only spectrum extends to energies as high as 10s of GeV; (2) The low-energy photon indices in the time-resolved spectra are typically $-1$, much softer than that expected in the photosphere models; (3) There is no evidence that the Band component is the temporal superposition of thermal-like emission components in the Band-only sample. We therefore suggest that GRB 080916C and probably all Band-only GRBs may correspond to those GRBs whose jet composition is dominated by a Poynting flux rather than a baryonic flux (Zhang & Pe’er 2009; Zhang & Yan 2011).
The existence of a bright photosphere component in GRB 090902B (see also Ryde et al. 2010; Pe’er et al. 2010) suggests that the composition of this GRB is likely a hot fireball without strong magnetization. It is rare, but its existence nonetheless suggests that GRB outflow composition may be diverse. Its associated PL component is hard to interpret, but it may be from the contributions of multiple non-thermal spectral components (Pe’er et al. 2010). The case of GRB 090510 may be similar to GRB 090902B. The low-energy spectral index of the MeV component is too shallow to be consistent with a blackbody, but the high-latitude emission from an instantaneously ejected fireball (which is relevant to short GRBs) would result in a multi-color blackbody due to the angular superposition effect (Pe’er & Ryde 2010).
The delayed onset of GeV emission may be simply due to one of the following two reasons: (1) The particle acceleration condition may be different throughout the burst. Initially, the electron spectral index may be steep initially (so that GeV emission is too faint to be detected), but later it turns to a shallower value so that GeV emission emerges above the detector sensitivity; (2) Initially the ejecta may be more opaque so that there was a pair-production spectral cutoff below the LAT band. This cutoff energy later moves to higher energies to allow LAT photons to be detected. Within this picture, the electron spectral index is similar throughout the burst. There are other models discussed in the literature to attribute GeV emission to a different origin from the MeV component. This is reasonable for GRB 090510 and GRB 090902B, but for most other GRBs this model is contrived since the GeV emission appears as the natural extension of the MeV Band-function to high energies.
The GeV emission during the prompt phase is very likely not of external forward shock origin. This is due to the following facts: (1) In most GRBs the entire [*Fermi*]{}-band emission is well fit by a single Band component. The GeV emission is consistent with being the extension of MeV to high energies. (2) During the prompt phase and except for the delayed onset in some GRBs, the LAT-band activities in bright GRBs generally track GBM-band activities. The latter property is relevant even for GRB 090902B which shows clearly two components in the spectra. (3) The peak of GeV lightcurve coincides the second peak of GBM lightcurve for GRB 080916C. A more reasonable possibility is that the GeV emission during the prompt phase has an “internal” origin similar to its MeV counterpart.
The origin of the long lasting GeV afterglow after the prompt emission phase (end of the GBM-band emission) is unclear. If it is from the external forward shock, one needs to introduce abnormal shock parameters, and to argue for coincidence to connect with the internal-origin early GeV emission to form a simple PL decay lightcurve. Alternatively, the long lasting GeV emission can be also of the internal origin. Future joint [*Fermi*]{}/[*Swift*]{} observations of the early GeV/X-ray afterglows of some bright GRBs will help to differentiate between these possibilities. The two tentative correlations ($\alpha-\beta$ and $\alpha$-flux) proposed in this paper need to be confirmed with a larger data sample, and their physical implications will be discussed then.
We thank Rob Preece for important instructions on [*Fermi*]{} data analysis. This work is partially supported by NASA NNX09AT66G, NNX10AD48G, and NSF AST-0908362 at UNLV. EWL, YZF, and XFW acknowledge National Basic Research Program of China (973 Program 2009CB824800). This work is partially supported by the National Natural Science Foundation of China (grant 10873002 for EWL, and grants 10633040, 10921063 for XFW). EWL is also supported by Guangxi Ten-Hundred-Thousand project (Grant 2007201), Guangxi Science Foundation (2010GXNSFC013011), and the program for 100 Young and Middle-aged Disciplinary Leaders in Guangxi Higher Education Institutions. YZF is also supported by a special grant from Purple Mountain Observatory and by the National Nature Science Foundation of China (grant 11073057). XFW is also supported by the Special Foundation for the Authors of National Excellent Doctorial Dissertations of P. R. China by Chinese Academy of Sciences. AP is supported by the Riccardo Giacconi Fellowship award of the Space Telescope Science Institute.
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[llllllllllll]{} GRB & $z$ & dur. \[sec\] & $E_p$ \[keV\] & $E_{\gamma,\rm iso}$ \[erg\] & Fluence ($1-10^4$ keV) & Spectral Type & Onset Delay& $E_{\rm max}$\
080825C & - & $22$ & $192\pm15$ & - & $4.84_{-0.57}^{+0.59}\times 10^{-5}$ & BAND& Y & $\sim 600$ MeV\
080916C & 4.35 & $66$ & $1443_{-303}^{+433}$ & $5.7_{-0.41}^{+0.54}\times 10^{54} $ & $1.55_{-0.11}^{+0.15}\times 10^{-4}$ & BAND & Y & $\sim 13.2$ GeV\
081024B & - & $0.8$ & $1258_{-522}^{+2405}$ & -& $(1,61\pm 3.8) \times 10^{-6}$ & BAND&Y & $\sim 3$ GeV\
081215A & - & $7.7$ & $1014_{-123}^{+140}$ & - & $8.74_{-0.99}^{+1.21}\times
10^{-5}$ & BAND& - & -\
090217 & - & $32.8$ & $552_{-71}^{+85}$ & - & $4.48_{-0.56}^{+0.69}\times 10^{-5}$ & BAND& N & $\sim 1$ GeV\
090323 & 3.57 & $150$ & $812_{-143}^{+181}$ & $>2.89_{-0.69}^{+6.56}\times 10^{54}$ & $>1.07_{-0.26}^{+0.24}\times 10^{-5}$ & BAND& N & $\sim 1$ GeV\
090328 & 0.736 & $80$ & $756_{-72}^{+85}$ & $1.02_{-0.083}^{+0.087}\times 10^{53}$ & $7.14_{-0.58}^{+0.61}\times 10^{-5}$ & BAND& ? & $>100$ MeV\
090510 & 0.903 & $0.3$ & $6010_{-1690}^{+2524}$ & $4.47_{-3.77}^{+4.06}\times
10^{52}$ & $2.06_{-1.74}^{+1.88} \times 10^{-5}$ & CPL+PL& Y& $\sim$ 31 GeV\
090626 & - & $70$ & $362_{-41}^{+47}$ & - & $7.81_{-0.38}^{+0.44}\times 10^{-5}$ & BAND& ? & $\sim 30$ GeV\
090902B & 1.822 & $21$ & $ 207\pm6$ \[BB\] & $(1.77\pm 0.01)\times
10^{52}$ & $(2.10\pm 0.02) \pm 10^{-4}$ & BB+PL& Y & $33.4_{-3.5}^{+2.7}$ GeV\
090926A & 2.1062 & $\sim 20$ & $ 412\pm 20$& $2.10_{-0.08}^{+0.09}\times 10^{54}$ & $1.93_{-0.07}^{+0.08}\times 10^{-4}$ & BAND& Y & $\sim $20 GeV\
091003 & 0.8969 & $21.1$ & $ 409_{-31}^{+34}$ & $7.85_{-0.57}^{+0.73}\times
10^{52}$ & $3.68_{-0.27}^{+0.34}\times 10^{-5}$& BAND& N & $>150$ MeV\
091031 & - & $\sim 40$ & $ 567_{-135}^{+197}$ & - & $3.17_{-0.51}^{+0.64}\times
10^{-5}$ & BAND&N & $1.2$ GeV\
100116A & - & $\sim 110$ & $ 1463_{-122}^{+163}$ & - & $7.34_{-1.26}^{+1.42}\times
10^{-5}$ & BAND& N& $\sim 2.2 $ GeV\
100225A & - & $13\pm 3 $ & $ 540_{-204}^{+381}$ & - & $1.21_{-0.57}^{+1.07}\times
10^{-5}$ & BAND&Y& $\sim 300$ MeV\
100325A & - & $8.3\pm 1.9 $ & $ 198_{-37}^{+44}$ & -& $6.15_{-1.81}^{+2.85}\times
10^{-6}$ & BAND&N& $\sim 800$ MeV\
100414A & 1.368 & $26.4\pm 1.6 $ & $ 520_{-39}^{+42}$ & $5.88_{-0.65}^{+0.69}\times 10^{53}$ & $1.20_{-0.10}^{+0.12}\times
10^{-5}$ & BAND&N & $\sim 2.6$ GeV\
[llllllllll]{}
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&0.00-6.75& $ -0.57_{-0.04}^{+0.05}$ & $ -2.29\pm0.04$ & $ 135_{-9}^{+10}$ & $ 0.114_{-0.007}^{+0.008}$ & $147.1$& $154$\
$2$&6.75-18.1& $ -0.75\pm0.06$ & $ -2.35_{-0.07}^{+0.09}$ & $ 141_{-14}^{+16}$ & $ 0.051_{-0.004}^{+0.005}$ & $132.7$& $154$\
$3$&18.1-25.0& $ -0.95_{-0.15}^{+0.17}$ & $ -2.17_{-0.08}^{+0.17}$ & $ 131_{-35}^{+56}$ & $ 0.027_{-0.006}^{+0.009}$ & $120.1$& $154$\
Total&0.00-25.0& $ -0.73\pm0.03$ & $ -2.33_{-0.03}^{+0.04}$ & $ 148\pm9$ & $ 0.058_{-0.003}^{+0.003}$ & $265.6$& $154$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&0.00-3.70& $ -0.69_{-0.04}^{+0.05}$ & $ -2.49_{-0.08}^{+0.13}$ & $ 342_{-37}^{+43}$ & $ 0.047_{-0.002}^{+0.003}$ & $99.5$& $124$\
$2$&3.70-9.10& $ -1.14\pm0.03$ & $ -2.32_{-0.05}^{+0.06}$ & $ 1680_{-348}^{+500}$ & $ 0.027\pm0.001$ & $153.0$& $124$\
$3$&9.10-17.0& $ -1.15_{-0.04}^{+0.05}$ & $ -2.29_{-0.05}^{+0.07}$ & $ 975_{-235}^{+361}$ & $ 0.016\pm0.001$ & $125.9$& $124$\
$4$&17.0-25.0& $ -0.99\pm0.04$ & $ -2.27_{-0.04}^{+0.06}$ & $ 447_{-60}^{+75}$ & $ 0.024\pm0.001$ & $114.3$& $124$\
$5$&25.0-41.0& $ -1.08\pm0.03$ & $ -2.49_{-0.07}^{+0.10}$ & $ 666_{-87}^{+111}$ & $ 0.017\pm0.001$ & $124.2$& $124$\
$6$&41.0-66.0& $ -1.09\pm0.04$ & $ -2.36_{-0.05}^{+0.06}$ & $ 696_{-128}^{+186}$ & $ 0.010\pm0.001$ & $162.8$& $124$\
Total&0.00-66.0& $ -1.05\pm0.02$ & $ -2.30\pm0.02$ & $ 664_{-46}^{+51}$ & $ 0.018\pm0.001$ & $427.5$& $124$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&-0.300-0.800& $ -1.15_{-0.16}^{+0.14}$ & $ -2.20(fixed)$ & $ 1478_{-551}^{+2810}$ & $ 0.007\pm0.001$ & $353.9$& $208$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&0.00-1.50& $ -0.65\pm0.05$ & $ -2.27_{-0.11}^{+0.14}$ & $ 753_{-88}^{+101}$ & $ 0.059\pm0.002$ & $80.0$ & $71$\
$2$&1.50-2.28& $ -0.52_{-0.07}^{+0.08}$ & $ -2.16_{-0.08}^{+0.10}$ & $ 280_{-39}^{+43}$ & $ 0.223_{-0.017}^{+0.020}$ & $63.6$ & $61$\
$3$&2.28-4.93& $ -0.60\pm0.06$ & $ -2.34_{-0.08}^{+0.09}$ & $ 178_{-17}^{+20}$ & $ 0.156_{-0.012}^{+0.013}$ & $66.1$ & $77$\
$4$&4.93-5.59& $ -0.49_{-0.08}^{+0.09}$ & $ -2.29_{-0.11}^{+0.15}$ & $ 214_{-31}^{+36}$ & $ 0.266_{-0.026}^{+0.032}$ & $45.0$ & $54$\
$5$&5.59-8.00& $ -0.72_{-0.14}^{+0.16}$ & $ -2.19_{-0.10}^{+0.13}$ & $ 102_{-22}^{+28}$ & $ 0.093_{-0.019}^{+0.029}$ & $47.5$ & $82$\
Total&0.00-8.00& $ -0.71\pm0.03$ & $ -2.16_{-0.03}^{+0.04}$ & $ 289_{-21}^{+22}$ & $ 0.110_{-0.004}^{+0.005}$ & $179.9$& $86$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&0.00-7.50& $ -0.59\pm0.04$ & $ -2.56_{-0.07}^{+0.10}$ & $ 365_{-30}^{+33}$ & $ 0.027\pm0.001$ & $165.1$& $156$\
$2$&7.50-13.1& $ -0.83\pm0.05$ & $ -2.66_{-0.14}^{+0.37}$ & $ 470_{-58}^{+70}$ & $ 0.021\pm0.001$ & $135.5$& $156$\
$3$&13.1-19.7& $ -0.96\pm0.09$ & $ -2.38_{-0.10}^{+0.22}$ & $ 257_{-51}^{+73}$ & $ 0.015\pm0.002$ & $131.1$& $156$\
$4$&19.7-30.0& $ -0.52_{-0.25}^{+0.43}$ & $ -2.22_{-0.09}^{+0.17}$ & $ 118_{-52}^{+65}$ & $ 0.008_{-0.003}^{+0.009}$ & $175.4$& $156$\
Total&0.00-30.0& $ -0.81\pm0.03$ & $ -2.54_{-0.04}^{+0.06}$ & $ 418_{-30}^{+33}$ & $ 0.015\pm0.001$ & $371.6$& $156$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&5.00-14.0& $ -0.97_{-0.04}^{+0.05}$ & $ -2.58_{-0.13}^{+0.25}$ & $ 792_{-136}^{+172}$ & $ 0.016\pm0.001$ & $98.4$ & $125$\
$2$&14.0-25.0& $ -1.11\pm0.04$ & $ -2.54_{-0.10}^{+0.18}$ & $ 826_{-141}^{+198}$ & $ 0.017\pm0.001$ & $127.2$& $125$\
$3$&35.0-50.0& $ -1.08\pm0.03$ & $ -2.64_{-0.15}^{+0.39}$ & $ 557_{-69}^{+84}$ & $ 0.018\pm0.001$ & $151.5$& $125$\
$4$&50.0-60.0& $ -0.88\pm0.04$ & $ -2.81_{-0.24}^{+1.13}$ & $ 449_{-44}^{+52}$ & $ 0.026\pm0.001$ & $115.2$& $125$\
$5$&60.0-135.& $ -1.31_{-0.01}^{+0.02}$ & $ -2.62_{-0.07}^{+0.11}$ & $ 987_{-116}^{+694}$ & $ 0.010\pm0.001$ & $496.7$& $125$\
$6$&135.-145.& $ -1.30\pm0.06$ & $ -2.34_{-0.12}^{+0.32}$ & $ 294_{-57}^{+74}$ & $ 0.017_{-0.001}^{+0.002}$ & $208.3$& $125$\
Total&0.00-150.& $ -1.22\pm0.01$ & $ -2.68_{-0.04}^{+0.06}$ & $ 880_{-50}^{+64}$ & $ 0.012\pm0.001$ & $857.3$& $125$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&3.00-8.00& $ -0.92_{-0.03}^{+0.04}$ & $ -2.38_{-0.10}^{+0.16}$ & $ 662_{-86}^{+99}$ & $ 0.024\pm0.001$ & $188.0$& $217$\
$2$&12.0-20.0& $ -0.96\pm0.02$ & $ -2.38_{-0.06}^{+0.09}$ & $ 727_{-67}^{+80}$ & $ 0.024\pm0.001$ & $199.3$& $217$\
$3$&20.0-30.0& $ -1.15\pm0.03$ & $ -2.30_{-0.07}^{+0.09}$ & $ 616_{-69}^{+81}$ & $ 0.020\pm0.001$ & $250.7$& $217$\
Total&0.00-30.0& $ -1.05\pm0.01$ & $ -2.44_{-0.04}^{+0.05}$ & $ 791_{-50}^{+58}$ & $ 0.018\pm0.001$ & $472.5$& $217$\
\
\
Seq & Time & $\Gamma_{\rm CPL}$ & $E_0$ & $K_{\rm CPL}$ & $\Gamma_{\rm PL}$ & $K_{\rm PL}$ & $\chi^2$ & $dof$\
& s & & keV & $\frac{photons}{keV cm^{2}s}@1 keV$ & & $\frac{photons}{keV cm^{2}s}@1 keV$ & &\
$1$&0.450-0.600& $ -0.76\pm0.08$ & $ 2688_{-765}^{+1360}$ & $ 1.85_{-0.63}^{+0.85}$ & $ ---$ & $ ---$ & $83.7$& $230$\
$2$&0.600-0.800& $ -0.60_{-0.13}^{+0.14}$ & $ 4286_{-1130}^{+1760}$ & $ 0.47_{-0.26}^{+0.53}$ & $ -1.73_{-0.07}^{+0.06}$ & $ 23.2_{-12.3}^{+13.0}$ & $154.9$& $251$\
$3$&0.800-0.900& $ -0.75_{-0.31}^{+0.67}$ & $ 777_{-464}^{+1900}$ & $ 0.97_{-0.93}^{+3.41}$ & $ -1.60_{-0.07}^{+0.11}$ & $ 14.3_{-11.6}^{+17.9}$ & $52.0$& $178$\
$4$&0.900-1.00& $ ---$ & $ ---$ & $ ---$ & $ -1.62\pm0.06$ & $ 11.5_{-5.8}^{+7.4}$ & $38.0$& $134$\
Total&0.450-1.00& $ -0.76_{-0.07}^{+0.08}$ & $ 3624_{-612}^{+759}$ & $ 1.06_{-0.39}^{+0.54}$ & $ -1.66_{-0.03}^{+0.05}$ & $ 11.9_{-5.6}^{+6.2}$ & $215.0$& $272$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&0.00-9.00& $ -0.99_{-0.02}^{+0.03}$ & $ -2.47_{-0.03}^{+0.04}$ & $ 193_{-11}^{+12}$ & $ 0.079\pm0.003$ & $340.3$& $186$\
$2$&15.0-20.0& $ -1.42\pm0.03$ & $ -2.47_{-0.08}^{+0.13}$ & $ 391_{-50}^{+60}$ & $ 0.040\pm0.002$ & $155.6$& $186$\
$3$&20.0-27.0& $ -1.28_{-0.02}^{+0.03}$ & $ -2.58_{-0.08}^{+0.13}$ & $ 504_{-54}^{+63}$ & $ 0.034\pm0.001$ & $136.5$& $186$\
$4$&30.0-40.0& $ -1.30\pm0.03$ & $ -2.49_{-0.06}^{+0.10}$ & $ 444_{-50}^{+63}$ & $ 0.025\pm0.001$ & $211.7$& $186$\
Total&0.00-60.0& $ -1.40\pm0.01$ & $ -2.62_{-0.03}^{+0.04}$ & $ 482_{-25}^{+27}$ & $ 0.025\pm0.001$ & $743.3$& $186$\
\
\
Seq & Time & $kT$ (keV) & $K_{\rm BB}$ & $\Gamma_{\rm PL}$ & $K_{\rm PL}$ & $\chi^2$ & $dof$\
& s & keV & $\frac{L_{39}}{D_{10}^2}$& &$\frac{photons}{keV cm^{2}s}@1 keV$\
$1$&0.00-1.50 & $ 75.60_{-1.79}^{+1.86}$ & $ 38.84_{-1.03}^{+1.02}$ & $ -1.88\pm0.02$ & $ 43.0_{-3.8}^{+3.9}$ & $330.6$& $264$\
$2$&1.50-2.25 & $ 98.74_{-3.41}^{+3.57}$ & $ 57.13_{-2.19}^{+2.25}$ & $ -1.84_{-0.04}^{+0.03}$ & $ 31.1_{-4.3}^{+5.3}$ & $226.3$& $237$\
$3$&2.25-2.81 & $ 121.20_{-4.79}^{+5.00}$ & $ 84.54_{-3.72}^{+3.79}$ & $ -1.81_{-0.04}^{+0.03}$ & $ 27.5_{-4.3}^{+4.6}$ & $217.5$& $238$\
$4$&2.81-3.23 & $ 82.52_{-3.97}^{+4.32}$ & $ 58.00_{-2.88}^{+3.05}$ & $ -1.80_{-0.04}^{+0.03}$ & $ 33.6_{-5.3}^{+6.4}$ & $199.0$& $217$\
$5$&3.23-3.83 & $ 100.90_{-3.57}^{+3.76}$ & $ 69.22_{-2.71}^{+2.81}$ & $ -1.83_{-0.04}^{+0.03}$ & $ 34.7_{-4.8}^{+6.0}$ & $190.7$& $240$\
$6$&3.83-4.46 & $ 86.81_{-2.79}^{+2.92}$ & $ 60.01_{-2.14}^{+2.20}$ & $ -1.83_{-0.04}^{+0.03}$ & $ 33.4_{-4.7}^{+5.7}$ & $218.3$& $236$\
$7$&4.46-4.99 & $ 90.79_{-4.43}^{+4.78}$ & $ 47.82_{-2.52}^{+2.65}$ & $ -1.83_{-0.04}^{+0.03}$ & $ 38.6_{-5.2}^{+6.4}$ & $207.4$& $225$\
$8$&4.99-5.45 & $ 109.50_{-4.11}^{+4.32}$ & $ 88.50_{-3.68}^{+3.82}$ & $ -1.82_{-0.05}^{+0.04}$ & $ 31.5_{-5.2}^{+6.6}$ & $185.3$& $228$\
$9$&5.45-5.86 & $ 116.20_{-4.94}^{+5.20}$ & $ 85.70_{-4.13}^{+4.22}$ & $ -1.82_{-0.05}^{+0.04}$ & $ 34.6_{-5.9}^{+7.2}$ & $180.5$& $227$\
$10$&5.86-6.28& $ 132.60_{-4.21}^{+4.36}$ & $ 141.20_{-5.14}^{+5.27}$ & $ -1.81_{-0.05}^{+0.04}$ & $ 32.5_{-5.3}^{+6.5}$ & $186.5$& $233$\
$11$&6.28-6.61& $ 157.40_{-6.50}^{+6.74}$ & $ 155.60_{-7.36}^{+7.77}$ & $ -1.81_{-0.06}^{+0.04}$ & $ 38.0_{-6.0}^{+8.6}$ & $186.2$& $228$\
$12$&6.61-7.19& $ 171.10_{-4.85}^{+5.01}$ & $ 174.10_{-5.80}^{+5.97}$ & $ -1.86_{-0.03}^{+0.02}$ & $ 87.2_{-7.3}^{+8.6}$ & $229.0$& $248$\
$13$&7.19-7.65& $ 174.20_{-5.35}^{+5.55}$ & $ 207.90_{-7.37}^{+7.57}$ & $ -1.87_{-0.03}^{+0.02}$ & $ 124.3_{-10.3}^{+12.1}$ & $231.3$& $244$\
$14$&7.65-8.00& $ 217.80_{-7.29}^{+7.47}$ & $ 307.00_{-12.20}^{+12.50}$& $ -1.87\pm0.02$ & $ 203.5_{-13.2}^{+15.0}$ & $223.0$& $243$\
$15$&8.00-8.50& $ 204.80_{-5.48}^{+5.62}$ & $ 288.60_{-9.01}^{+9.22}$ & $ -1.91\pm0.01$ & $ 344.6_{-15.7}^{+17.3}$ & $319.9$& $248$\
$16$&8.50-9.00& $ 206.60_{-5.83}^{+5.97}$ & $ 281.00_{-9.16}^{+9.35}$ & $ -1.93_{-0.02}^{+0.01}$ & $ 375.7_{-19.3}^{+21.5}$ & $260.2$& $249$\
$17$&9.00-9.50& $ 206.20_{-5.83}^{+5.99}$ & $ 270.50_{-8.91}^{+9.11}$ & $ -1.92\pm0.01$ & $ 445.6_{-18.9}^{+20.5}$ & $325.6$& $248$\
$18$&9.50-10.0& $ 135.90_{-3.18}^{+3.26}$ & $ 209.90_{-5.45}^{+5.53}$ & $ -1.96_{-0.02}^{+0.01}$ & $ 553.2_{-26.0}^{+28.6}$ & $271.2$& $244$\
$19$&10.0-10.5& $ 168.80_{-4.47}^{+4.58}$ & $ 236.40_{-7.04}^{+7.18}$ & $ -1.94\pm0.02$ & $ 378.4_{-20.9}^{+23.8}$ & $258.3$& $244$\
$20$&10.5-11.0& $ 195.70_{-5.89}^{+6.03}$ & $ 246.60_{-8.50}^{+8.70}$ & $ -1.90\pm0.01$ & $ 352.5_{-16.0}^{+17.7}$ & $348.6$& $247$\
$21$&11.0-11.5& $ 145.20_{-4.34}^{+4.50}$ & $ 179.10_{-5.81}^{+5.98}$ & $ -1.93\pm0.02$ & $ 332.2_{-18.3}^{+20.8}$ & $278.5$& $242$\
$22$&11.5-12.0& $ 153.10_{-4.32}^{+4.43}$ & $ 169.30_{-5.56}^{+5.68}$ & $ -1.92\pm0.02$ & $ 253.5_{-16.2}^{+18.8}$ & $241.9$& $241$\
$23$&12.0-12.4& $ 61.07_{-2.90}^{+3.09}$ & $ 44.61_{-2.24}^{+2.31}$ & $ -1.90\pm0.02$ & $ 242.6_{-15.9}^{+18.4}$ & $194.7$& $214$\
$24$&12.4-13.2& $ 35.36_{-0.88}^{+0.92}$ & $ 31.80_{-0.90}^{+0.91}$ & $ -1.92\pm0.01$ & $ 271.2_{-11.9}^{+12.8}$ & $324.6$& $231$\
$25$&13.2-13.3& $ 42.30_{-1.59}^{+1.68}$ & $ 87.55_{-3.83}^{+3.92}$ & $ -1.84\pm0.03$ & $ 213.7_{-22.7}^{+27.0}$ & $141.4$& $180$\
$26$&13.3-13.6& $ 45.32_{-1.97}^{+2.10}$ & $ 57.60_{-2.72}^{+2.79}$ & $ -1.87\pm0.02$ & $ 276.6_{-20.6}^{+23.4}$ & $175.3$& $192$\
$27$&13.6-13.8& $ 53.27_{-1.94}^{+2.02}$ & $ 69.62_{-2.85}^{+2.90}$ & $ -1.87_{-0.03}^{+0.02}$ & $ 203.7_{-17.3}^{+20.6}$ & $169.2$& $199$\
$28$&13.8-14.1& $ 66.19_{-2.72}^{+2.92}$ & $ 89.79_{-3.80}^{+3.93}$ & $ -1.84\pm0.02$ & $ 187.8_{-13.8}^{+15.3}$ & $275.3$& $206$\
$29$&14.1-14.2& $ 105.70_{-4.91}^{+5.22}$ & $ 201.80_{-9.99}^{+10.2}$ & $ -1.82\pm0.03$ & $ 169.6_{-18.2}^{+20.2}$ & $177.9$& $204$\
$30$&14.2-14.4& $ 120.40_{-5.70}^{+5.93}$ & $ 199.60_{-10.00}^{+10.40}$& $ -1.83_{-0.03}^{+0.02}$ & $ 159.9_{-15.2}^{+18.7}$ & $180.7$& $211$\
$31$&14.4-14.6& $ 51.74_{-2.30}^{+2.45}$ & $ 57.16_{-2.79}^{+2.86}$ & $ -1.86_{-0.03}^{+0.02}$ & $ 186.8_{-16.2}^{+18.8}$ & $164.6$& $194$\
$32$&14.6-14.8& $ 99.11_{-4.00}^{+4.23}$ & $ 155.80_{-6.57}^{+6.88}$ & $ -1.85\pm0.03$ & $ 160.5_{-15.4}^{+19.3}$ & $173.6$& $211$\
$33$&14.8-15.0& $ 71.48_{-3.09}^{+3.30}$ & $ 115.90_{-5.38}^{+5.55}$ & $ -1.82\pm0.03$ & $ 149.0_{-15.9}^{+19.0}$ & $165.7$& $196$\
$34$&15.0-15.1& $ 102.20_{-5.26}^{+5.60}$ & $ 220.80_{-11.7}^{+12.2}$ & $ -1.81\pm0.03$ & $ 159.0_{-18.3}^{+21.9}$ & $184.4$& $202$\
$35$&15.1-15.2& $ 102.10_{-4.22}^{+4.40}$ & $ 233.10_{-10.1}^{+10.5}$ & $ -1.81\pm0.03$ & $ 144.6_{-15.4}^{+18.9}$ & $212.1$& $199$\
$36$&15.2-15.5& $ 127.0_{-3.73}^{+3.85}$ & $ 223.0_{-7.18}^{+7.36}$ & $ -1.85_{-0.0234}^{+0.0201}$ & $ 160.7_{-12.5}^{+14.3}$ & $216.60$& $215$\
$37$&15.5-15.7& $ 150.70_{-5.99}^{+6.16}$ & $ 254.80_{-11.30}^{+11.80}$& $ -1.83\pm0.03$ & $ 120.5_{-12.4}^{+15.9}$ & $168.4$& $221$\
$38$&15.7-16.2& $ 59.42_{-1.74}^{+1.81}$ & $ 63.99_{-2.12}^{+2.15}$ & $ -1.88\pm0.02$ & $ 169.4_{-12.4}^{+14.3}$ & $197.2$& $221$\
$39$&16.2-16.3& $ 84.53_{-3.69}^{+3.95}$ & $ 132.10_{-6.08}^{+6.36}$ & $ -1.84\pm0.03$ & $ 168.9_{-16.5}^{+20.3}$ & $190.3$& $203$\
$40$&16.3-16.5& $ 90.82_{-3.47}^{+3.67}$ & $ 160.90_{-6.63}^{+6.85}$ & $ -1.83\pm0.03$ & $ 158.1_{-15.4}^{+18.3}$ & $177.3$& $206$\
$41$&16.5-16.7& $ 94.44_{-4.25}^{+4.55}$ & $ 143.00_{-6.81}^{+7.11}$ & $ -1.84\pm0.03$ & $ 160.6_{-15.8}^{+19.1}$ & $169.6$& $210$\
$42$&16.7-16.9& $ 78.69_{-4.10}^{+4.46}$ & $ 96.94_{-5.29}^{+5.55}$ & $ -1.83_{-0.04}^{+0.03}$ & $ 137.2_{-15.1}^{+18.4}$ & $155.4$& $198$\
$43$&16.9-17.1& $ 47.97_{-2.47}^{+2.65}$ & $ 40.30_{-2.26}^{+2.33}$ & $ -1.84_{-0.03}^{+0.02}$ & $ 138.7_{-13.1}^{+15.3}$ & $144.2$& $191$\
$44$&17.1-17.5& $ 63.52_{-2.19}^{+2.29}$ & $ 75.35_{-2.87}^{+2.93}$ & $ -1.86_{-0.03}^{+0.02}$ & $ 148.8_{-13.1}^{+15.6}$ & $171.4$& $206$\
$45$&17.5-17.8& $ 68.97_{-3.26}^{+3.46}$ & $ 54.62_{-2.76}^{+2.85}$ & $ -1.85_{-0.03}^{+0.02}$ & $ 113.7_{-10.6}^{+12.6}$ & $191.9$& $209$\
$46$&17.8-18.3& $ 46.21_{-1.50}^{+1.56}$ & $ 38.75_{-1.36}^{+1.39}$ & $ -1.87\pm0.02$ & $ 142.8_{-9.5}^{+10.4}$ & $248.0$& $228$\
$47$&18.3-18.9& $ 57.27_{-1.85}^{+1.95}$ & $ 52.36_{-1.75}^{+1.80}$ & $ -1.88\pm0.02$ & $ 166.4_{-9.8}^{+10.6}$ & $334.0$& $233$\
$48$&18.9-19.4& $ 57.29_{-1.87}^{+1.97}$ & $ 49.10_{-1.71}^{+1.75}$ & $ -1.88\pm0.02$ & $ 156.0_{-9.7}^{+10.7}$ & $302.1$& $220$\
$49$&19.4-19.6& $ 49.44_{-1.86}^{+1.96}$ & $ 81.63_{-3.39}^{+3.50}$ & $ -1.83\pm0.03$ & $ 147.1_{-15.7}^{+18.8}$ & $167.7$& $189$\
$50$&19.6-19.7& $ 54.68_{-2.14}^{+2.24}$ & $ 88.95_{-3.81}^{+3.88}$ & $ -1.83_{-0.03}^{+0.02}$ & $ 164.9_{-16.3}^{+18.9}$ & $171.8$& $192$\
$51$&19.7-19.9& $ 57.57_{-2.43}^{+2.54}$ & $ 94.89_{-4.21}^{+4.29}$ & $ -1.83_{-0.03}^{+0.02}$ & $ 178.0_{-16.0}^{+18.0}$ & $202.2$& $194$\
$52$&19.9-20.1& $ 72.81_{-3.90}^{+4.16}$ & $ 91.88_{-5.08}^{+5.28}$ & $ -1.85_{-0.03}^{+0.02}$ & $ 197.8_{-17.5}^{+20.5}$ & $170.6$& $196$\
$53$&20.1-20.3& $ 43.33_{-3.07}^{+3.37}$ & $ 42.35_{-2.88}^{+2.99}$ & $ -1.82_{-0.03}^{+0.02}$ & $ 136.6_{-14.9}^{+16.6}$ & $165.1$& $189$\
$54$&20.3-20.6& $ 50.94_{-2.41}^{+2.52}$ & $ 53.85_{-2.59}^{+2.64}$ & $ -1.86\pm0.02$ & $ 193.9_{-15.3}^{+17.2}$ & $221.4$& $205$\
$55$&20.6-20.9& $ 46.04_{-1.71}^{+1.79}$ & $ 51.23_{-2.12}^{+2.16}$ & $ -1.87\pm0.02$ & $ 192.5_{-14.8}^{+16.7}$ & $192.6$& $196$\
$56$&20.9-21.0& $ 42.49_{-2.04}^{+2.20}$ & $ 55.46_{-2.79}^{+2.90}$ & $ -1.84\pm0.03$ & $ 148.9_{-16.0}^{+18.6}$ & $171.3$& $183$\
$57$&21.0-21.3& $ 36.47_{-2.20}^{+2.44}$ & $ 23.88_{-1.53}^{+1.59}$ & $ -1.87_{-0.03}^{+0.02}$ & $ 152.9_{-14.5}^{+17.0}$ & $143.5$& $189$\
$58$&21.3-21.7& $ 42.84_{-1.19}^{+1.23}$ & $ 50.72_{-1.63}^{+1.67}$ & $ -1.88_{-0.03}^{+0.02}$ & $ 155.2_{-12.7}^{+14.8}$ & $186.5$& $212$\
$59$&21.7-21.9& $ 47.05_{-2.70}^{+2.89}$ & $ 46.19_{-2.80}^{+2.89}$ & $ -1.84_{-0.03}^{+0.02}$ & $ 161.9_{-15.5}^{+17.8}$ & $152.6$& $195$\
$60$&21.9-22.2& $ 49.53_{-3.13}^{+3.39}$ & $ 42.03_{-2.83}^{+2.94}$ & $ -1.84_{-0.03}^{+0.02}$ & $ 153.6_{-15.1}^{+17.5}$ & $147.1$& $188$\
$61$&22.2-23.0& $ 31.13_{-3.30}^{+4.08}$ & $ 5.72_{-0.60}^{+0.62}$ & $ -1.90\pm0.02$ & $ 126.0_{-9.4}^{+10.2}$ & $187.3$& $233$\
Total&0.00-30.0& $ 96.71_{-0.484}^{+0.461}$& $ 71.65_{-0.36}^{+0.34}$ & $ -1.93\pm0.01$ & $ 175.1_{-1.3}^{+1.2}$ & $14732.0$& $276$\
& Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\Gamma_{\rm PL}$ & $K_{\rm PL}$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& & $\frac{photons}{keV cm^{2}s}@1 keV$\
Total&0.00-23.0& $ -0.83\pm0.01$ & $ -3.68_{-0.20}^{+0.12}$ & $ 724_{-12}^{+13}$ & $ 0.099\pm0.001$ & $ -1.85_{-1.85}^{+1.85}$ & $ 43.4\pm1.5$ & $2024.3$& $275$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&0.00-2.81& $ -0.53_{-0.03}^{+0.04}$ & $ -2.43_{-0.05}^{+0.06}$ & $ 235_{-15}^{+16}$ & $ 0.106\pm0.004$ & $189.0$& $210$\
$2$&2.81-3.75& $ -0.48\pm0.03$ & $ -2.75_{-0.13}^{+0.21}$ & $ 255_{-14}^{+15}$ & $ 0.303_{-0.010}^{+0.011}$ & $168.6$& $196$\
$3$&3.75-5.62& $ -0.57\pm0.02$ & $ -2.35\pm0.02$ & $ 208\pm8$ & $ 0.344\pm0.009$ & $269.1$& $213$\
$4$&5.62-7.50& $ -0.73\pm0.02$ & $ -2.50_{-0.08}^{+0.13}$ & $ 326\pm15$ & $ 0.191\pm0.004$ & $229.7$& $210$\
$5$&7.50-9.38& $ -0.63\pm0.03$ & $ -2.81_{-0.13}^{+0.17}$ & $ 183_{-8}^{+9}$ & $ 0.255_{-0.008}^{+0.009}$ & $169.6$& $209$\
$6$&9.38-11.2& $ -0.75\pm0.02$ & $ -2.52_{-0.08}^{+0.10}$ & $ 193_{-8}^{+9}$ & $ 0.327_{-0.009}^{+0.010}$ & $228.1$& $213$\
$7$&11.2-13.1& $ -0.80\pm0.03$ & $ -2.29_{-0.05}^{+0.06}$ & $ 154_{-10}^{+11}$ & $ 0.242_{-0.012}^{+0.014}$ & $186.1$& $212$\
$8$&13.1-15.9& $ -0.99\pm0.05$ & $ -2.36_{-0.11}^{+0.22}$ & $ 161_{-19}^{+22}$ & $ 0.081_{-0.007}^{+0.008}$ & $164.7$& $213$\
$9$&15.9-20.0& $ -1.26\pm0.08$ & $ -2.07_{-0.04}^{+0.07}$ & $ 216_{-48}^{+68}$ & $ 0.025_{-0.003}^{+0.004}$ & $170.9$& $214$\
Total&0.00-20.0& $ -0.74\pm0.01$ & $ -2.34\pm0.01$ & $ 226\pm4$ & $ 0.165\pm0.002$ & $777.1$& $216$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&7.00-15.0& $ -1.33\pm0.05$ & $ -2.41_{-0.10}^{+0.20}$ & $ 426_{-77}^{+101}$ & $ 0.012\pm0.001$ & $234.5$& $246$\
$2$&15.0-18.0& $ -1.01\pm0.04$ & $ -2.52_{-0.10}^{+0.19}$ & $ 337_{-38}^{+43}$ & $ 0.040\pm0.002$ & $152.4$& $243$\
$3$&18.0-20.0& $ -0.85\pm0.03$ & $ -2.55_{-0.07}^{+0.10}$ & $ 357_{-26}^{+28}$ & $ 0.094\pm0.003$ & $218.9$& $242$\
$4$&20.0-26.0& $ -1.36_{-0.05}^{+0.06}$ & $ -2.35_{-0.08}^{+0.15}$ & $ 429_{-97}^{+143}$ & $ 0.014\pm0.001$ & $189.2$& $246$\
Total&0.00-26.0& $ -1.09_{-0.01}^{+0.02}$ & $ -2.58_{-0.04}^{+0.05}$ & $ 474_{-25}^{+27}$ & $ 0.024\pm0.001$ & $446.2$& $246$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&0.00-8.00& $ -0.89\pm0.06$ & $ -2.44_{-0.07}^{+0.09}$ & $ 496_{-84}^{+111}$ & $ 0.013\pm0.001$ & $177.1$& $186$\
$2$&8.00-15.0& $ -0.86_{-0.05}^{+0.06}$ & $ -2.50_{-0.08}^{+0.13}$ & $ 357_{-47}^{+55}$ & $ 0.020\pm0.001$ & $173.3$& $186$\
$3$&15.0-25.0& $ -0.78_{-0.10}^{+0.11}$ & $ -2.55_{-0.12}^{+0.26}$ & $ 467_{-104}^{+157}$ & $ 0.006\pm0.001$ & $187.1$& $186$\
Total&0.00-25.0& $ -0.87_{-0.03}^{+0.04}$ & $ -2.55_{-0.05}^{+0.06}$ & $ 458_{-33}^{+51}$ & $ 0.012\pm0.001$ & $347.2$& $186$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&-2.00-5.00& $ -1.03_{-0.11}^{+0.13}$ & $ -2.54_{-0.24}^{+2.54}$ & $ 384_{-124}^{+201}$ & $ 0.006\pm0.001$ & $104.8$& $155$\
$2$&80.0-90.0 & $ -1.03_{-0.04}^{+0.05}$ & $ -2.80_{-0.21}^{+0.97}$ & $ 791_{-142}^{+192}$ & $ 0.010\pm0.001$ & $127.8$& $155$\
$3$&90.0-95.0 & $ -1.00\pm0.01$ & $ -3.22_{-0.25}^{+1.51}$ & $ 1459_{-121}^{+161}$ & $ 0.033\pm0.001$ & $156.9$& $155$\
$4$&95.0-110. & $ -1.03\pm0.05$ & $ -2.63_{-0.11}^{+0.23}$ & $ 677_{-120}^{+169}$ & $ 0.009\pm0.001$ & $127.0$& $155$\
Total&0.00-110. & $ -1.11_{-0.02}^{+0.01}$ & $ -3.13_{-0.09}^{+0.11}$ & $ 2867_{-283}^{+430}$ & $ 0.004\pm0.001$ & $415.6$& $155$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&0.00-6.00& $ -0.53_{-0.19}^{+0.22}$ & $ -2.43_{-0.19}^{+0.87}$ & $ 263_{-74}^{+120}$ & $ 0.010\pm0.002$ & $51.8$& $94$\
$2$&6.00-12.0& $ -0.93_{-0.13}^{+0.15}$ & $ -2.30_{-0.12}^{+0.26}$ & $ 507_{-181}^{+351}$ & $ 0.009_{-0.001}^{+0.002}$ & $40.3$& $93$\
Total&0.00-12.0& $ -0.77_{-0.11}^{+0.12}$ & $ -2.37_{-0.10}^{+0.18}$ & $ 375_{-86}^{+129}$ & $ 0.010\pm0.001$ & $64.5$& $94$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&-3.00-10.0& $ -0.72_{-0.10}^{+0.11}$ & $ -2.60_{-0.21}^{+1.89}$ & $ 155_{-26}^{+32}$ & $ 0.014\pm0.002$ & $151.6$& $125$\
\
\
Seq & Time & $\alpha$ & $\beta$ & $E_{0}$ & $K$ & $\chi^2$ & $dof$\
& s & & & keV & $\frac{photons}{keV cm^{2}s}@100 keV$& &\
$1$&1.00-7.25& $ -0.19_{-0.05}^{+0.06}$ & $ -2.54_{-0.10}^{+0.16}$ & $ 256_{-20}^{+22}$ & $ 0.036\pm0.002$ & $124.3$& $156$\
$2$&7.25-14.3& $ -0.25_{-0.04}^{+0.05}$ & $ -2.89_{-0.24}^{+0.51}$ & $ 281_{-20}^{+19}$ & $ 0.040_{-0.001}^{+0.002}$ & $124.5$& $156$\
$3$&14.3-19.6& $ -0.56_{-0.03}^{+0.04}$ & $ -2.53_{-0.10}^{+0.16}$ & $ 361_{-26}^{+28}$ & $ 0.047\pm0.002$ & $135.1$& $156$\
$4$&19.6-25.5& $ -0.76\pm0.03$ & $ -2.45_{-0.07}^{+0.11}$ & $ 386_{-28}^{+30}$ & $ 0.052\pm0.002$ & $131.9$& $156$\
Total&1.00-26.0& $ -0.52\pm0.02$ & $ -2.62_{-0.05}^{+0.07}$ & $ 344_{-12}^{+12}$ & $ 0.042\pm0.001$ & $281.7$& $156$\
[lll]{} Name & $\alpha_{\rm LAT}$ & ${\bar\Gamma_{\rm LAT}}$\
080825C & $-0.47 \pm 0.74$ &$ -1.71$\
080916C & $-1.33 \pm 0.08$& $ -1.77$\
081024B & $-1.37 \pm 0.41$ & $ -1.98$\
081215A & - & -\
090217 & $-0.81 \pm 0.23$&$ -1.97$\
090323 & $-0.52 \pm 0.67$ & $ -1.75$\
090328 & $-0.96 \pm 0.44$& $ -1.82$\
090510 & $-1.70 \pm 0.08$& $ -1.94 $\
090626 &- & $ -1.53$\
090902B & $-1.40 \pm 0.06$ & $ -1.76$\
090926A & $-2.05 \pm 0.14$ &$ -2.03$\
091003 & $< -0.93 $ & $ -1.74$\
091031 & $-0.57 \pm 0.28$ & $ -1.73$\
100116A & - &$ -1.68$\
100225A & - & $ -1.77$\
100325A & $<-1.04 $ & $ -1.53$\
100414A & $-1.64 \pm 0.89$& $ -1.85 $\
-- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Joint temporal and spectral analysis of GBM and LAT data for GRB 080825C. [*Left panels:*]{} the background-subtracted GBM and LAT lightcurves (from top: 8-150 keV, 150-300 keV, 300 keV - 1 MeV, 1-30 MeV, $>$100 MeV), and evolution of spectra parameters ($\alpha$, $\beta$, $E_p$). [*Right panels:*]{} an example (the brightest episode) of the observed photon spectrum as compared with the spectral model ([*top*]{}), the best fit $\nu F_\nu$ spectra of all time bins ([*middle*]{}), and the comparison between the GBM (green) and LAT (blue) count rate lightcurves in log-scale ([*bottom*]{}).[]{data-label="080825C"}](f1b.ps "fig:")
![Joint temporal and spectral analysis of GBM and LAT data for GRB 080825C. [*Left panels:*]{} the background-subtracted GBM and LAT lightcurves (from top: 8-150 keV, 150-300 keV, 300 keV - 1 MeV, 1-30 MeV, $>$100 MeV), and evolution of spectra parameters ($\alpha$, $\beta$, $E_p$). [*Right panels:*]{} an example (the brightest episode) of the observed photon spectrum as compared with the spectral model ([*top*]{}), the best fit $\nu F_\nu$ spectra of all time bins ([*middle*]{}), and the comparison between the GBM (green) and LAT (blue) count rate lightcurves in log-scale ([*bottom*]{}).[]{data-label="080825C"}](f1c.ps "fig:")
![Joint temporal and spectral analysis of GBM and LAT data for GRB 080825C. [*Left panels:*]{} the background-subtracted GBM and LAT lightcurves (from top: 8-150 keV, 150-300 keV, 300 keV - 1 MeV, 1-30 MeV, $>$100 MeV), and evolution of spectra parameters ($\alpha$, $\beta$, $E_p$). [*Right panels:*]{} an example (the brightest episode) of the observed photon spectrum as compared with the spectral model ([*top*]{}), the best fit $\nu F_\nu$ spectra of all time bins ([*middle*]{}), and the comparison between the GBM (green) and LAT (blue) count rate lightcurves in log-scale ([*bottom*]{}).[]{data-label="080825C"}](f1d.ps "fig:")
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-- ----------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 080916C.[]{data-label="080916C"}](f2b.ps "fig:")
![Same as Figure 1, but for GRB 080916C.[]{data-label="080916C"}](f2c.ps "fig:")
![Same as Figure 1, but for GRB 080916C.[]{data-label="080916C"}](f2d.ps "fig:")
-- ----------------------------------------------------------------------------------
-- ----------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 081024B.[]{data-label="081024B"}](f3b.ps "fig:")
![Same as Figure 1, but for GRB 081024B.[]{data-label="081024B"}](f3c.ps "fig:")
![Same as Figure 1, but for GRB 081024B.[]{data-label="081024B"}](f3d.ps "fig:")
-- ----------------------------------------------------------------------------------
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 081215A. This burst was at an angle of 86 degrees to the LAT boresight. The data cannot be obtained with the standard analysis procedures. Using a non-standard data selection, over 100 counts above background were detected within a 0.5 s interval in coincidence with the main GBM peak (McEnery et al. 2008). We thus add this GRB in our sample, but do not add its LAT data in our analysis.[]{data-label="081215A"}](f4b.ps "fig:")
![Same as Figure 1, but for GRB 081215A. This burst was at an angle of 86 degrees to the LAT boresight. The data cannot be obtained with the standard analysis procedures. Using a non-standard data selection, over 100 counts above background were detected within a 0.5 s interval in coincidence with the main GBM peak (McEnery et al. 2008). We thus add this GRB in our sample, but do not add its LAT data in our analysis.[]{data-label="081215A"}](f4c.ps "fig:")
![Same as Figure 1, but for GRB 081215A. This burst was at an angle of 86 degrees to the LAT boresight. The data cannot be obtained with the standard analysis procedures. Using a non-standard data selection, over 100 counts above background were detected within a 0.5 s interval in coincidence with the main GBM peak (McEnery et al. 2008). We thus add this GRB in our sample, but do not add its LAT data in our analysis.[]{data-label="081215A"}](f4d.ps "fig:")
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-- --------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 090217.[]{data-label="090217"}](f5b.ps "fig:")
![Same as Figure 1, but for GRB 090217.[]{data-label="090217"}](f5c.ps "fig:")
![Same as Figure 1, but for GRB 090217.[]{data-label="090217"}](f5d.ps "fig:")
-- --------------------------------------------------------------------------------
-- --------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 090323.[]{data-label="090323"}](f6b.ps "fig:")
![Same as Figure 1, but for GRB 090323.[]{data-label="090323"}](f6c.ps "fig:")
![Same as Figure 1, but for GRB 090323.[]{data-label="090323"}](f6d.ps "fig:")
-- --------------------------------------------------------------------------------
-- --------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 090328.[]{data-label="090328"}](f7b.ps "fig:")
![Same as Figure 1, but for GRB 090328.[]{data-label="090328"}](f7c.ps "fig:")
![Same as Figure 1, but for GRB 090328.[]{data-label="090328"}](f7d.ps "fig:")
-- --------------------------------------------------------------------------------
-- --------------------------------------------------------------------------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 090510. The applied model is cut-off power-law plus power-law (CPL + PL).[]{data-label="090510"}](f8b.ps "fig:")
![Same as Figure 1, but for GRB 090510. The applied model is cut-off power-law plus power-law (CPL + PL).[]{data-label="090510"}](f8c.ps "fig:")
![Same as Figure 1, but for GRB 090510. The applied model is cut-off power-law plus power-law (CPL + PL).[]{data-label="090510"}](f8d.ps "fig:")
-- --------------------------------------------------------------------------------------------------------------------------------------------------
-- --------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 090626.[]{data-label="090626"}](f9b.ps "fig:")
![Same as Figure 1, but for GRB 090626.[]{data-label="090626"}](f9c.ps "fig:")
![Same as Figure 1, but for GRB 090626.[]{data-label="090626"}](f9d.ps "fig:")
-- --------------------------------------------------------------------------------
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Same as Fig. 1, but for GRB 090902B. The applied model is blackbody plus power law (BB + PL). []{data-label="090902B"}](f10b.ps "fig:")
![Same as Fig. 1, but for GRB 090902B. The applied model is blackbody plus power law (BB + PL). []{data-label="090902B"}](f10c.ps "fig:"){width="2.05in" height="2in"}
![Same as Fig. 1, but for GRB 090902B. The applied model is blackbody plus power law (BB + PL). []{data-label="090902B"}](f10d.ps "fig:")
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-- -----------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 090926A.[]{data-label="090926A"}](f11b.ps "fig:")
![Same as Figure 1, but for GRB 090926A.[]{data-label="090926A"}](f11c.ps "fig:")
![Same as Figure 1, but for GRB 090926A.[]{data-label="090926A"}](f11d.ps "fig:")
-- -----------------------------------------------------------------------------------
-- ---------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 091003.[]{data-label="091003"}](f12b.ps "fig:")
![Same as Figure 1, but for GRB 091003.[]{data-label="091003"}](f12c.ps "fig:")
![Same as Figure 1, but for GRB 091003.[]{data-label="091003"}](f12d.ps "fig:")
-- ---------------------------------------------------------------------------------
-- ---------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 091031.[]{data-label="091031"}](f13b.ps "fig:")
![Same as Figure 1, but for GRB 091031.[]{data-label="091031"}](f13c.ps "fig:")
![Same as Figure 1, but for GRB 091031.[]{data-label="091031"}](f13d.ps "fig:")
-- ---------------------------------------------------------------------------------
-- -----------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 100116A.[]{data-label="100116A"}](f14b.ps "fig:")
![Same as Figure 1, but for GRB 100116A.[]{data-label="100116A"}](f14c.ps "fig:")
![Same as Figure 1, but for GRB 100116A.[]{data-label="100116A"}](f14d.ps "fig:")
-- -----------------------------------------------------------------------------------
-- -----------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 100225A.[]{data-label="100225A"}](f15b.ps "fig:")
![Same as Figure 1, but for GRB 100225A.[]{data-label="100225A"}](f15c.ps "fig:")
![Same as Figure 1, but for GRB 100225A.[]{data-label="100225A"}](f15d.ps "fig:")
-- -----------------------------------------------------------------------------------
-- -----------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 100325A.[]{data-label="100325A"}](f16b.ps "fig:")
![Same as Figure 1, but for GRB 100325A.[]{data-label="100325A"}](f16c.ps "fig:")
![Same as Figure 1, but for GRB 100325A.[]{data-label="100325A"}](f16d.ps "fig:")
-- -----------------------------------------------------------------------------------
-- -----------------------------------------------------------------------------------
![Same as Figure 1, but for GRB 100414A.[]{data-label="100414A"}](f17b.ps "fig:")
![Same as Figure 1, but for GRB 100414A.[]{data-label="100414A"}](f17c.ps "fig:")
![Same as Figure 1, but for GRB 100414A.[]{data-label="100414A"}](f17d.ps "fig:")
-- -----------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![A comparison between GRB 080916C and GRB 090902B. [*Upper panel*]{}: The case of GRB 080916C. The Band parameters are $(\alpha, \beta)= (-1.0, -2.27) (-1.0, -2.29), (-1.12, -2.25)$ for 0-20 s, 2-10 s, and 3.5-8 s, respectively. Little spectral parameter variation is seen with reducing time bins. [*Lower panel*]{}: The case of GRB 090902B. (1) For 0-20 s, the Band+PL model ($\alpha=-0.58$, $\beta=-3.32$, $\Gamma_{\rm PL} =-2.0$ with $\chi^2/{\rm dof}=3.52$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.59$, $\Gamma_{\rm PL}=-2.0$ with $\chi^2{\rm dof}=3.7$) give marginally acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ (Rayleigh-Jeans) and the BB+PL model give unacceptable fits. (2) For 8.5-11.5 s, the Band+PL model ($\alpha=-0.07$, $\beta=-3.69$, $\Gamma_{\rm PL} =-1.97$ with $\chi^2/{\rm dof}=1.26$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.08$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=1.3$) give acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=3.7$) and the BB+PL model ($\chi^2/{\rm dof}=4.9$) give marginally acceptable fits. (3) 9.5-10 s, the Band+PL model ($\alpha=0.07$, $\beta<-5$, $\Gamma_{\rm PL} =-2.05$ with $\chi^2/{\rm dof}=0.69$) can only give an upper limit on $\beta$. The CPL+PL model ($\Gamma_{\rm CPL}=-0.0004$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=0.63$) give marginally acceptable fit to the data. On the other hand, the CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=0.92$) and the BB+PL model ($\chi^2/{\rm dof}=1.11$) give acceptable fits. Clear narrowing trend is seen when the time bins get smaller.[]{data-label="080916C-090902B"}](f18a.ps "fig:") ![A comparison between GRB 080916C and GRB 090902B. [*Upper panel*]{}: The case of GRB 080916C. The Band parameters are $(\alpha, \beta)= (-1.0, -2.27) (-1.0, -2.29), (-1.12, -2.25)$ for 0-20 s, 2-10 s, and 3.5-8 s, respectively. Little spectral parameter variation is seen with reducing time bins. [*Lower panel*]{}: The case of GRB 090902B. (1) For 0-20 s, the Band+PL model ($\alpha=-0.58$, $\beta=-3.32$, $\Gamma_{\rm PL} =-2.0$ with $\chi^2/{\rm dof}=3.52$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.59$, $\Gamma_{\rm PL}=-2.0$ with $\chi^2{\rm dof}=3.7$) give marginally acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ (Rayleigh-Jeans) and the BB+PL model give unacceptable fits. (2) For 8.5-11.5 s, the Band+PL model ($\alpha=-0.07$, $\beta=-3.69$, $\Gamma_{\rm PL} =-1.97$ with $\chi^2/{\rm dof}=1.26$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.08$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=1.3$) give acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=3.7$) and the BB+PL model ($\chi^2/{\rm dof}=4.9$) give marginally acceptable fits. (3) 9.5-10 s, the Band+PL model ($\alpha=0.07$, $\beta<-5$, $\Gamma_{\rm PL} =-2.05$ with $\chi^2/{\rm dof}=0.69$) can only give an upper limit on $\beta$. The CPL+PL model ($\Gamma_{\rm CPL}=-0.0004$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=0.63$) give marginally acceptable fit to the data. On the other hand, the CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=0.92$) and the BB+PL model ($\chi^2/{\rm dof}=1.11$) give acceptable fits. Clear narrowing trend is seen when the time bins get smaller.[]{data-label="080916C-090902B"}](f18b.ps "fig:") ![A comparison between GRB 080916C and GRB 090902B. [*Upper panel*]{}: The case of GRB 080916C. The Band parameters are $(\alpha, \beta)= (-1.0, -2.27) (-1.0, -2.29), (-1.12, -2.25)$ for 0-20 s, 2-10 s, and 3.5-8 s, respectively. Little spectral parameter variation is seen with reducing time bins. [*Lower panel*]{}: The case of GRB 090902B. (1) For 0-20 s, the Band+PL model ($\alpha=-0.58$, $\beta=-3.32$, $\Gamma_{\rm PL} =-2.0$ with $\chi^2/{\rm dof}=3.52$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.59$, $\Gamma_{\rm PL}=-2.0$ with $\chi^2{\rm dof}=3.7$) give marginally acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ (Rayleigh-Jeans) and the BB+PL model give unacceptable fits. (2) For 8.5-11.5 s, the Band+PL model ($\alpha=-0.07$, $\beta=-3.69$, $\Gamma_{\rm PL} =-1.97$ with $\chi^2/{\rm dof}=1.26$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.08$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=1.3$) give acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=3.7$) and the BB+PL model ($\chi^2/{\rm dof}=4.9$) give marginally acceptable fits. (3) 9.5-10 s, the Band+PL model ($\alpha=0.07$, $\beta<-5$, $\Gamma_{\rm PL} =-2.05$ with $\chi^2/{\rm dof}=0.69$) can only give an upper limit on $\beta$. The CPL+PL model ($\Gamma_{\rm CPL}=-0.0004$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=0.63$) give marginally acceptable fit to the data. On the other hand, the CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=0.92$) and the BB+PL model ($\chi^2/{\rm dof}=1.11$) give acceptable fits. Clear narrowing trend is seen when the time bins get smaller.[]{data-label="080916C-090902B"}](f18c.ps "fig:")
![A comparison between GRB 080916C and GRB 090902B. [*Upper panel*]{}: The case of GRB 080916C. The Band parameters are $(\alpha, \beta)= (-1.0, -2.27) (-1.0, -2.29), (-1.12, -2.25)$ for 0-20 s, 2-10 s, and 3.5-8 s, respectively. Little spectral parameter variation is seen with reducing time bins. [*Lower panel*]{}: The case of GRB 090902B. (1) For 0-20 s, the Band+PL model ($\alpha=-0.58$, $\beta=-3.32$, $\Gamma_{\rm PL} =-2.0$ with $\chi^2/{\rm dof}=3.52$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.59$, $\Gamma_{\rm PL}=-2.0$ with $\chi^2{\rm dof}=3.7$) give marginally acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ (Rayleigh-Jeans) and the BB+PL model give unacceptable fits. (2) For 8.5-11.5 s, the Band+PL model ($\alpha=-0.07$, $\beta=-3.69$, $\Gamma_{\rm PL} =-1.97$ with $\chi^2/{\rm dof}=1.26$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.08$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=1.3$) give acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=3.7$) and the BB+PL model ($\chi^2/{\rm dof}=4.9$) give marginally acceptable fits. (3) 9.5-10 s, the Band+PL model ($\alpha=0.07$, $\beta<-5$, $\Gamma_{\rm PL} =-2.05$ with $\chi^2/{\rm dof}=0.69$) can only give an upper limit on $\beta$. The CPL+PL model ($\Gamma_{\rm CPL}=-0.0004$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=0.63$) give marginally acceptable fit to the data. On the other hand, the CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=0.92$) and the BB+PL model ($\chi^2/{\rm dof}=1.11$) give acceptable fits. Clear narrowing trend is seen when the time bins get smaller.[]{data-label="080916C-090902B"}](f18d.ps "fig:") ![A comparison between GRB 080916C and GRB 090902B. [*Upper panel*]{}: The case of GRB 080916C. The Band parameters are $(\alpha, \beta)= (-1.0, -2.27) (-1.0, -2.29), (-1.12, -2.25)$ for 0-20 s, 2-10 s, and 3.5-8 s, respectively. Little spectral parameter variation is seen with reducing time bins. [*Lower panel*]{}: The case of GRB 090902B. (1) For 0-20 s, the Band+PL model ($\alpha=-0.58$, $\beta=-3.32$, $\Gamma_{\rm PL} =-2.0$ with $\chi^2/{\rm dof}=3.52$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.59$, $\Gamma_{\rm PL}=-2.0$ with $\chi^2{\rm dof}=3.7$) give marginally acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ (Rayleigh-Jeans) and the BB+PL model give unacceptable fits. (2) For 8.5-11.5 s, the Band+PL model ($\alpha=-0.07$, $\beta=-3.69$, $\Gamma_{\rm PL} =-1.97$ with $\chi^2/{\rm dof}=1.26$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.08$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=1.3$) give acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=3.7$) and the BB+PL model ($\chi^2/{\rm dof}=4.9$) give marginally acceptable fits. (3) 9.5-10 s, the Band+PL model ($\alpha=0.07$, $\beta<-5$, $\Gamma_{\rm PL} =-2.05$ with $\chi^2/{\rm dof}=0.69$) can only give an upper limit on $\beta$. The CPL+PL model ($\Gamma_{\rm CPL}=-0.0004$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=0.63$) give marginally acceptable fit to the data. On the other hand, the CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=0.92$) and the BB+PL model ($\chi^2/{\rm dof}=1.11$) give acceptable fits. Clear narrowing trend is seen when the time bins get smaller.[]{data-label="080916C-090902B"}](f18e.ps "fig:") ![A comparison between GRB 080916C and GRB 090902B. [*Upper panel*]{}: The case of GRB 080916C. The Band parameters are $(\alpha, \beta)= (-1.0, -2.27) (-1.0, -2.29), (-1.12, -2.25)$ for 0-20 s, 2-10 s, and 3.5-8 s, respectively. Little spectral parameter variation is seen with reducing time bins. [*Lower panel*]{}: The case of GRB 090902B. (1) For 0-20 s, the Band+PL model ($\alpha=-0.58$, $\beta=-3.32$, $\Gamma_{\rm PL} =-2.0$ with $\chi^2/{\rm dof}=3.52$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.59$, $\Gamma_{\rm PL}=-2.0$ with $\chi^2{\rm dof}=3.7$) give marginally acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ (Rayleigh-Jeans) and the BB+PL model give unacceptable fits. (2) For 8.5-11.5 s, the Band+PL model ($\alpha=-0.07$, $\beta=-3.69$, $\Gamma_{\rm PL} =-1.97$ with $\chi^2/{\rm dof}=1.26$) and the CPL+PL model ($\Gamma_{\rm CPL}=-0.08$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=1.3$) give acceptable fits to the data. The CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=3.7$) and the BB+PL model ($\chi^2/{\rm dof}=4.9$) give marginally acceptable fits. (3) 9.5-10 s, the Band+PL model ($\alpha=0.07$, $\beta<-5$, $\Gamma_{\rm PL} =-2.05$ with $\chi^2/{\rm dof}=0.69$) can only give an upper limit on $\beta$. The CPL+PL model ($\Gamma_{\rm CPL}=-0.0004$, $\Gamma_{\rm PL}=-2.1$ with $\chi^2{\rm dof}=0.63$) give marginally acceptable fit to the data. On the other hand, the CPL+PL model with $\Gamma_{\rm CPL}=1$ ($\chi^2/{\rm dof}=0.92$) and the BB+PL model ($\chi^2/{\rm dof}=1.11$) give acceptable fits. Clear narrowing trend is seen when the time bins get smaller.[]{data-label="080916C-090902B"}](f18f.ps "fig:")
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![Distributions of the Band-function parameters $\alpha$, $\beta$, and $E_p$ in our sample (red) in comparison with the BATSE bright sources sample (green). The BATSE sample is adopted from Preece et al. (2000).[]{data-label="distributions"}](f19a.ps "fig:") ![Distributions of the Band-function parameters $\alpha$, $\beta$, and $E_p$ in our sample (red) in comparison with the BATSE bright sources sample (green). The BATSE sample is adopted from Preece et al. (2000).[]{data-label="distributions"}](f19b.ps "fig:") ![Distributions of the Band-function parameters $\alpha$, $\beta$, and $E_p$ in our sample (red) in comparison with the BATSE bright sources sample (green). The BATSE sample is adopted from Preece et al. (2000).[]{data-label="distributions"}](f19c.ps "fig:")
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![A comparison between the lightcurves of the blackbody component (red) and the power-law component (green) in GRB 090902B. The total lightcurve (the sum of the two components, [*dotted*]{} line) is also shown for comparison.[]{data-label="090902B-lightcurves"}](f20.ps)
![The global $L_{\gamma,\rm iso}^{p}$ vs. $E_p(1+z)$ correlation (panel a) and internal $L_{\gamma,\rm iso}$ vs. $E_p(1+z)$ correlation (panel b) for the 8 [*Fermi*]{}/LAT GRBs with known redshifts. The grey dots in (a) are previous bursts taken from Zhang et al. (2009).[]{data-label="Lp_Ep"}](f21a.ps "fig:") ![The global $L_{\gamma,\rm iso}^{p}$ vs. $E_p(1+z)$ correlation (panel a) and internal $L_{\gamma,\rm iso}$ vs. $E_p(1+z)$ correlation (panel b) for the 8 [*Fermi*]{}/LAT GRBs with known redshifts. The grey dots in (a) are previous bursts taken from Zhang et al. (2009).[]{data-label="Lp_Ep"}](f21b.ps "fig:")
\[L-Ep\]
![The two dimension plots of various pairs of spectral parameters. (a) $\alpha-\beta$, with linear Pearson correlation coeffcients for individual bursts marked in the inset; (b) $\alpha-$flux, with linear Pearson correlation coeffcients for individual bursts marked in the inset; (c) $E_p-\alpha$; (d) $E_p-\beta$. For those burst without redshift, $z=2.0$ is assumed (grey symbols and lines).[]{data-label="correlations"}](f22a.ps "fig:") ![The two dimension plots of various pairs of spectral parameters. (a) $\alpha-\beta$, with linear Pearson correlation coeffcients for individual bursts marked in the inset; (b) $\alpha-$flux, with linear Pearson correlation coeffcients for individual bursts marked in the inset; (c) $E_p-\alpha$; (d) $E_p-\beta$. For those burst without redshift, $z=2.0$ is assumed (grey symbols and lines).[]{data-label="correlations"}](f22b.ps "fig:")\
![The two dimension plots of various pairs of spectral parameters. (a) $\alpha-\beta$, with linear Pearson correlation coeffcients for individual bursts marked in the inset; (b) $\alpha-$flux, with linear Pearson correlation coeffcients for individual bursts marked in the inset; (c) $E_p-\alpha$; (d) $E_p-\beta$. For those burst without redshift, $z=2.0$ is assumed (grey symbols and lines).[]{data-label="correlations"}](f22c.ps "fig:") ![The two dimension plots of various pairs of spectral parameters. (a) $\alpha-\beta$, with linear Pearson correlation coeffcients for individual bursts marked in the inset; (b) $\alpha-$flux, with linear Pearson correlation coeffcients for individual bursts marked in the inset; (c) $E_p-\alpha$; (d) $E_p-\beta$. For those burst without redshift, $z=2.0$ is assumed (grey symbols and lines).[]{data-label="correlations"}](f22d.ps "fig:")\
![A cartoon picture of three elemental spectral components that shape GRB prompt emission spectra: (I) a Band-function component that is likely of the non-thermal origin; (II) a quasi-thermal component; and (III) an extra power-law component that extends to high energy, which is expected to have a cut-off near or above the high energy end of the LAT energy band.[]{data-label="Cartoon"}](f23.ps)
![Five possible spectral combinations with the three spectral components.[]{data-label="spectral-combinations"}](f24a.eps "fig:") ![Five possible spectral combinations with the three spectral components.[]{data-label="spectral-combinations"}](f24b.eps "fig:") ![Five possible spectral combinations with the three spectral components.[]{data-label="spectral-combinations"}](f24c.eps "fig:") ![Five possible spectral combinations with the three spectral components.[]{data-label="spectral-combinations"}](f24d.eps "fig:") ![Five possible spectral combinations with the three spectral components.[]{data-label="spectral-combinations"}](f24e.eps "fig:")
[^1]: [ An appropriate background time interval is typically when the lightcurve is “flat” with Poisson noise photons. For each burst, we select background time intervals as \[-$t_{b,1}$,-$t_{b,1}$\] before the burst and \[$t_{b,3}$,$t_{b,4}$\] after the burst, where $t_{b}$’s are typically in the order of tens to hundreds of seconds. The exact values vary for different bursts due to their different brightnesses and the corresponding orbit slewing phases.]{}
[^2]: AIC is defined by $ AIC = n \ln{ \left( \frac{\chi^{2}}{n} \right) } + 2 k$, where $n$ is the number of data points, $k$ is the number of free parameters of a particular model, and $\chi^{2}$ is the residual sum of squares from the estimated model (e.g. Shirasaki et al. 2008).
[^3]: Our finest time interval is around 1s in the rest frame of the burst. Theoretically, how time-integrated spectra broaden with increasing time bins is subject further study. Our statement is therefore relevant for time resolution longer than 1s.
[^4]: Notice that the abbreviation “BB” here not only denotes blackbody, but also includes various modifications to the blackbody spectrum such as multi-color blackbody.
[^5]: Most of Swift GRBs can be fit with a PL (Sakamoto et al. 2008). However, this is due to the narrowness of the energy band of the gamma-ray detector BAT on board Swift. The $E_p$ of many Swift GRBs are expected to be located outside the instrument band. In fact, using a Band function model and considering the variation of $E_p$ within and outside the BAT band, one can reproduce the apparent hardness of Swift GRBs, and obtain an effective correlation between the BAT-band photon index and $E_p$ (e.g. Zhang et al. 2007; Sakamoto et al. 2009). If a GRB is observed in a wider energy band, the spectrum should be invariably curved.
|
---
abstract: 'The conditions for validity of the Causal Entropy Bound (CEB) are verified in the context of non-singular cosmologies with classical sources. It is shown that they are the same conditions that were previously found to guarantee validity of the CEB: the energy density of each dynamical component of the cosmic fluid needs to be sub-Planckian and not too negative, and its equation of state needs to be causal. In the examples we consider, the CEB is able to discriminate cosmologies which suffer from potential physical problems more reliably than the energy conditions appearing in singularity theorems.'
address: |
(1) Department of Physics, Ben-Gurion University,\
Beer-Sheva 84105, Israel\
(2) The Racah Institute of Physics,Hebrew University of Jerusalem,\
Givat Ram, Jerusalem 91904, Israel\
[E-mail:]{} [ ramyb@bgumail.bgu.ac.il, foffa@bgumail.bgu.ac.il, mayo@cc.huji.ac.il]{}
author:
- 'Ram Brustein${}^{(1)}$ Stefano Foffa${}^{(1)}$ and Avraham E. Mayo${}^{(2)}$'
title: 'Causal Entropy Bound for Non-Singular Cosmologies'
---
2 cm
introduction
============
The validity of entropy bounds in Bekenstein’s non-singular cosmological model [@bmodel] has been recently challenged [@mayo]. In the course of the analysis some of the energy conditions that sources in the Einstein equations are assumed to obey [@Bousso; @Wald] were questioned. Here we determine the conditions that guarantee the validity of the CEB [@CEB] for non-singular cosmologies with classical sources, and discuss their relation to the energy conditions of the classic singularity theorems[@singth].
CEB is a covariant entropy bound which is applicable, in principle, to any space-like region [@CEB] in an arbitrary space-time dimension $D$ [@BFV]. It is an improvement of the Hubble Entropy Bound (HEB) [@GV1] (see also [@EL; @BR; @KL]), which was motivated by the following reasonable assumptions $(i)$ entropy is maximized by the largest stable black hole that can fit in a given region of space. $(ii)$ the largest stable black hole in a cosmological background is typically of size comparable to that of the Hubble horizon (this assumption is qualitatively supported by previous calculations [@Carr]). In cosmological backgrounds, the CEB refines HEB by defining the “horizon" concept through the identification of a critical (“Jeans"-like) causal connection scale $R_{\rm CC}$, above which perturbations are causally disconnected, so that black holes of larger size are unlikely to form.
In homogeneous and isotropic $D$ dimensional cosmological backgrounds $R_{\rm CC}$ depends on the Hubble parameter $H(t)$, its time-derivative $\dot{H}(t)$, and the scale factor $a(t)$ [@BFV], $$\begin{aligned}
\label{rcch} R_{\rm CC}^{-2}&=&\frac{D-2}{2}{\rm Max}\left[
\dot{H} + \frac{D}{2}H^2 + \frac{D-2}{2}\frac{k}{a^2}, -\dot{H} +
\frac{D-4}{2}H^2 + \frac{D-2}{2}\frac{k}{a^2}\right] \nonumber \\
&=&\frac{4\pi G_{\rm N}}{D-1} {\rm Max} \biggl[
\rho - (D-1)p\, , (2D-5)\rho + (D-1)p
\biggr],\end{aligned}$$ where $k=0,\pm1$ determines the spatial curvature. To derive the second equality we have used Einstein’s equations, $G_{\mu\nu} = 8 \pi
G_N T_{\mu\nu}$ and a perfect-fluid form for the energy-momentum tensor. Notice that $R_{\rm CC}$ is well defined if $\rho$ is positive because the maximum in Eq.(\[rcch\]) is larger than the average of the two entries in the brackets, and the average is equal to $2(D-2)\rho$.
Previously [@CEB] three cases which were believed to exhaust all possible types of cosmologies were considered [^1]:
1. $|\dot H|\sim H^2 \sim |k|/a^2$, or $|\dot H|\sim H^2 \gg |k|/a^2$. In this case effective energy density and pressure are of the same order, $\rho\sim p$, and all length scales that may be considered in entropy bounds, such as particle horizon, apparent horizon, $R_{\rm CC}$, and the Hubble length, are parametrically equal. This case includes non-inflationary FRW universes with matter and radiation.
2. $H^2 \gg |k|/a^2,|\dot H|$. In this case $|\rho+ p| \ll \rho$, and the universe is inflationary. Here the naive holographic bound fails miserably, but HEB, CEB and Bousso’s modification of the holographic entropy bound [@Bousso] do well.
3. $|\dot H|\gg H^2,\ |k|/a^2$. In this case $|\rho| \ll p$. Since $\rho$ and $p$ are the effective energy density and pressure, there are no problems with causality. This case occurs, for instance, near the turning point of an expanding universe which recollapses, or near a bounce of a contracting universe which reexpands.
4. $k/a^2 \gg |\dot H|,\ H^2$ so that spatial curvature determines the causal connection scale. This occurs, for example, when both $H$ and $\dot H$ vanish as in a closed Einstein Universe, or in the static version of Bekenstein’s non-singular Universe [@bmodel].
Here we discuss this last case and show that the same conditions that guarantee validity of CEB in the first three cases suffice to guarantee its validity in the fourth case.
CEB states that the maximal entropy $S_{\rm CEB}$ that can be contained in a space-like region of proper volume $V$ is given by (our units are such that $\hbar=c=1$ and $G_{\rm N}=M_{\rm
P}^{-(D-2)}= \ell_{\rm
P}^{D-2})$, $$\begin{aligned}
\label{scebD} S_{\rm CEB}=\beta n_{\rm H} S^{\rm BH}=\beta
\frac{V}{V(R_{\rm CC})} \frac{{\cal A}(R_{\rm CC})}{4 \ell_{\rm
P}^{D-2}},\end{aligned}$$ where $n_{\rm H} \equiv \frac{V}{V(R_{\rm CC})}$ is the number of causally connected regions in the volume considered, $V(x)$ denotes the volume of a region of size $x$, ${\cal A}(x)$ denotes the area of this region, and $\beta$ is a fudge factor reflecting current uncertainty on the actual limiting size for black-hole stability. For a spherical volume in flat space we have $V(x)=\Omega_{D-2}
x^{D-1}/(D-1)$, and ${\cal A}(x)=\Omega_{D-2} x^{D-2}$, with $\Omega_{D-2}=2\pi^{(D-1)/2}/ \Gamma\left(\frac{D-1}{2}\right)$, but in general the result is different and depends on the spatial-curvature radius. Since $\frac{{\cal A}(R_{\rm CC})}{V(R_{\rm
CC})}\sim \frac{D-1}{R_{\rm CC}}$, $$\label{scebE}
S_{\rm CEB}=\alpha (D-1) \frac{V}{G_N R_{\rm CC}},$$ where $\alpha$ is a numerical parameter of order one.
Conditions for validity of CEB were determined in [@CEB; @BFV]. Loosely speaking, energy densities are required to be sub-Planckian, and the total energy density of the cosmic fluid is required to be positive. In particular, for a universe with a large number of fields $\N$, in thermal equilibrium at temperature $T$, the CEB was found to be valid for temperatures not exceeding a value of order $M_{\rm
P}/\N^{{1\over D-2}}$ (see also [@Bek3; @GSL]).
CEB in non-singular cosmologies
===============================
Einstein Universe with radiation
--------------------------------
The simplest example of a non-singular cosmology is a static Einstein model in $D$ dimensions. This model requires positive curvature, and two types of sources: cosmological constant and dust; we denote by $\rho_{\Lambda}$ and $\rho_{\rm m}$ the energy densities associated with each of the two components. To provide entropy we need an additional source, which we choose to be radiation consisting of $\N$ species in thermal equilibrium at temperature $T$. The energy density of the radiation is given by $\rho_{\rm r}=\N T^D$, and the entropy density of the radiation is given by $s_{\rm r}=\N T^{D-1}$ (we ignore here numerical factors since we will be interested in scaling of quantities). The total entropy of the system is given entirely by the entropy of the radiation $S_{\rm r}=s_{\rm r} V$.
In term of these sources, Einstein’s equations can be written in the following way: $$\begin{aligned}
\label{ein1}
H^2 + \frac{1}{a^2}&=& \frac{16 \pi G_{\rm N}}{(D-2)(D-1)}\rho_{\rm tot}=
\frac{16 \pi G_{\rm N}}{(D-2)(D-1)}\left(\rho_{\Lambda}+\rho_{\rm m}+
\rho_{\rm r}\right)\\
\label{ein2}
\dot{H} - \frac{1}{a^2}&=&
-\frac{8\pi G_{\rm N}}{(D-2)}\left(\rho_{\rm tot} + p_{\rm tot}\right)=
-\frac{8\pi G_{\rm N}}{(D-2)(D-1)}
\left[D\rho_{\rm r}+(D-1)\rho_{\rm m}\right]\, ,\end{aligned}$$ where we have used in Eq. (\[ein2\]) the equations of state relating pressure to energy density: $p_{\Lambda}=-\rho_{\Lambda}$, $p_{\rm m}=0$, and $(D-1)p_{\rm
r}=\rho_{\rm r}$.
For given $\rho_{\rm m}$ and $\rho_{\rm r}$, one can choose $\rho_{\Lambda}$ and the scale factor $a$ such that $H$ and $\dot{H}$ vanish in Eqs. (\[ein1\]) and (\[ein2\]), and thus obtains a static solution. In particular, the condition given by Eq. (\[ein2\]) determines the scale factor in terms of $\rho_{\rm
m}$ and $\rho_{\rm r}$, $$\label{e2vanish}
a^2=\frac{(D-2)(D-1)}{8\pi G_{\rm N}}\frac{1}{D\rho_{\rm r} + (D-1)\rho_{\rm m}}\, .$$ Note that since both $H$ and $\dot H$ vanish identically, $R_{\rm CC}$ is determined solely by the scale factor $a$ given in Eq.(\[e2vanish\]), as discussed previously.
We now wish to determine under which conditions (if any) some violations of CEB may occur in this model. Recall that according to Eq.(\[scebE\]) the CEB bounds the total entropy of a region contained in a comoving volume $V$ by $S_{\rm CEB}=\alpha
(D-1)\frac{V}{G_{\rm N} R_{\rm CC}}$, and that in the static case under consideration $R_{\rm CC}=2 a/(D-2)$. The square of the ratio of $S_{\rm CEB}$ and the entropy of the system $S_{\rm r}$, is given by $$\begin{aligned}
\label{Soverceb}
\left(\frac{S_{\rm CEB}}{S_{\rm r}}\right)^2 &=& \left(
\frac{\alpha(D-1)} {s_{\rm r} R_{\rm CC}
G_{\rm N}}\right)^2 \nonumber \\
&=& \left[ 2\pi \alpha^2 (D-1)(D-2)\right] \left[{D +
(D-1){\rho_{\rm m}\over \rho_{\rm r}}}\right] \left[\frac{1}{\N}
\left(\frac{M_P}{T}\right)^{D-2}\right] \, .\end{aligned}$$ Since the second factor in expression (\[Soverceb\]) is larger than unity if $\rho_{\rm m}$ and $\rho_{\rm r}$ are positive, and neglecting the overall prefactor which is independent of the sources in the model, we conclude that the CEB is valid provided that $$\label{subP}
\N\left({T\over M_{\rm P}}\right)^{D-2}\laq 1.$$ This is the same condition discussed in [@BFV], and should be interpreted as a requirement that temperatures are sub-Planckian, in the case of many number of species $\N$ (see also [@Bek3; @GSL]).
We therefore conclude that, as long as the temperature of radiation stays well below Planckian, CEB is upheld. The fact that the model is gravitationally unstable to matter perturbations does not seem to be particularly relevant to the issue of validity of the CEB.
Bekenstein’s Universe
---------------------
A non-singular cosmological model which can describe time-dependent cosmologies was found years ago by Bekenstein[@bmodel]. This is a 4D Friedman-Robertson-Walker universe which is conformal to the closed Einstein Universe. It contains dust, consisting of $N$ particles of mass $\mu$ ($N$ is constant and $\mu$ is positive), coupled to a classical conformal massless scalar field $\psi$, and $\N$ species of radiation in thermal equilibrium. The action for the dust-$\psi$ system is given by $$\begin{aligned}
{\cal S}=-\frac{1}{2}\int\sqrt{-g}\left[\left(\nabla\psi\right)^2 +
\frac{1}{6} \psi^2 R\right]d^4 x - \int\left(\mu +
f\psi\right)d\tau.\end{aligned}$$ It includes in addition to the usual action for free point particles of rest mass $\mu$, a dust-scalar field interaction whose strength is determined by the coupling $f$. Accordingly, we may define the effective mass of the dust particles: $\mu_{\rm
eff}=\mu+f \psi$.
The total energy density and pressure in Bekenstein’s Universe are given by $$\begin{aligned}
\label{rototmay}
\rho_{\rm tot}=\rho_{\rm r}+\rho_{\psi}+\rho_{\rm m},\quad
p_{\rm tot}&=&p_{\rm r}+p_\psi+p_{\rm m} ,\end{aligned}$$ where $\{\rho_{\rm r},p_{\rm r}\}$, $\{\rho_{\psi},p_\psi\}$, and $\{\rho_{\rm m},p_{\rm m}\}$ are the energy densities and pressures associated with the radiation, scalar field and dust respectively. They depend on the scale factor in the following way $$\begin{aligned}
\label{equations}
\rho_{\rm r}&=&{\cal C} \N a^{-4}=\N T^4, \nonumber \\
\rho_\psi &=& {1\over 2}f^2N^2 a^{-4}, \\
\rho_{\rm m} &=&N\mu_{\rm eff}
a^{-3}=N\mu a^{-3} -2\rho_\psi, \nonumber\end{aligned}$$ and their equations of state $\gamma_{\rm r}=p_{\rm r}/\rho_{\rm
r}$, $\gamma_{\psi}=p_\psi/\rho_{\psi}$, $\gamma_{\rm m}=p_{\rm
m}/\rho_{\rm m}$ are the following $$\begin{aligned}
\label{eoss}
\gamma_{\rm r}&=&1/3, \nonumber \\
\gamma_\psi &=& -1/3, \\
\gamma_{\rm m} &=&0. \nonumber\end{aligned}$$ The dependence of $\psi$ on $a$ $\psi=-f N a^{-1}$, yields $\mu_{\rm eff}=\mu-f^2 N a^{-1}$. ${\cal C}$ is an integration constant and the only source of entropy is the radiation whose entropy density is given by $s_{\rm r}=\N T^3$.
The solution for the scale factor $a$ is given in terms of the conformal time $\eta$ by $$a(\eta)=a_0(1+B \sin\eta).$$ We assume that $a_0$, the mean value of the scale factor, is macroscopic, so it is large in our Planck units. If $B=0$ the solution describes a static universe very similar to the closed Einstein Universe discussed previously. For $0<B<1$ the solution describes a “bouncing universe": the universe bounces off at $\eta=3 \pi/2$ when the scale factor is minimal $a=a_{\rm
min}=a_0(1-B)$, expands until it turns over at $\eta=5\pi/2$ when its scale factor is maximal $a=a_{\rm max}=a_0(1+B)$, and continues to oscillate without ever reaching a singularity. The equations of motion require that the energy densities of the sources obey the following equalities at all times [@bmodel]: $$2 {a\over a_0} \left({\rho_\psi-\rho_{\rm r}\over2\rho_\psi+
\rho_{\rm m}} \right)=1-B^2={a_{\rm min} a_{\rm max}\over
{a_0}^2}.
\label{equality}$$ Since $2\rho_\psi+\rho_{\rm m}=N\mu a^{-3}>0$, $\rho_{\rm r}>0$, and $B^2<1$, it follows that a necessary condition for a bounce is that $\rho_{\rm r}<\rho_\psi $. This implies that the total pressure $\frac{1}{3} (\rho_{\rm r}-\rho_{\psi})$ is always negative. Moreover, Eq.(\[equality\]) for $a=a_{\rm min}$ implies that $\rho_{\rm m}\leq-2\rho_{\rm r}<0$ there. But then, the conclusion must be that in order to avoid a singularity, $\mu_{\rm eff}<0$ at least at the bounce. It is possible, however, to find a range of initial conditions and parameters such that $\mu_{\rm eff}$ is positive near the turnover.
The result that $\rho_{\rm r}$ and $\rho_\psi$ are manifestly positive definite, but $\rho_{\rm m}$ can (and in fact must) be negative some of the time, suggest that it might be possible to parametrically decrease $\rho_{\rm tot}$ by lowering $\mu_{\rm
eff}$ (making it large and negative) by increasing the coupling strength $f$, so that the amounts of radiation and entropy are kept constant. As it turns out this is exactly the case in which the CEB can be potentially violated. Using Einstein’s equations to express $R_{\rm CC}$ in terms of the total energy density and pressure, we find the ratio $\left(S_{\rm CEB}/S_{\rm r}\right)^2$: $$\begin{aligned}
\label{rat1}
\left({S_{\rm CEB}\over S_{\rm r}}\right)^2 \sim {G_{\rm N}}^{-2}
\left(\frac{\rho_{\rm r}}{\cal N}\right)^{-3/2}
\frac{1}{{\cal N}^2} G_{\rm N}\,
{\rm Max} \left[
{\rho_{\rm tot}\over 3}-p_{\rm tot} ,\rho_{\rm tot}+p_{\rm tot}
\right].\end{aligned}$$ A system for which the ratio above is smaller than one would violate the CEB. Recalling that the maximum on the r.h.s. of (\[rat1\]) is always larger than the mean of the two entries and rearranging we find $$\left({S_{\rm CEB}\over S_{\rm r}}\right)^2 \gaq
\left[\frac{1}{\N}\frac{M_P^2}{T^2}\right]{\rho_{\rm tot}\over
\rho_{\rm r}}. \label{ratio}$$ Since we assume that the model is sub-Planckian, namely that the first factor is larger than one as in Eq.(\[subP\]), the only way in which CEB could be violated is if somehow the second factor was parametrically small. As discussed above, it does seem that the second term $\rho_{\rm tot}/\rho_{\rm r}$ can be made arbitrarily small by decreasing $\rho_{\rm tot}$ while keeping $\rho_{\rm r}$ constant. Consequently, it is apparently possible to make the ratio $S_{\rm
CEB}/ S_{\rm r}$ smaller than one and obtain a CEB violating cosmology. But this can be achieved only if the effective mass of the dust particles is negative (and large) as can be seen from Eq. (\[rototmay\]).
Violations of the CEB (and as a matter of fact, of any other entropy bound such as Bekenstein’s [@BEB], or Bousso’s [@Bousso]) go hand in hand with large negative energy densities in the dust sector. In the model under discussion, this manifests itself in the form of dust particles with highly negative effective masses. Occurrence of such negative energy density would most probably render the model unstable (see below). We argue that any analysis of entropy bounds should be performed for stable models. This is particularly relevant for the CEB, whose definition involves explicitly the largest scale at which stable black holes could be formed. Note, however, that instability does not necessarily lead to violations of the CEB as in the previous case.
To support this argument let us outline possible instabilities in the dust scalar field system when the dust particles mass is negative. To do this we need to be more specific about the model. Consider a possible field theoretic model for the dust as a fermionic field $\chi$. In this case the dust-scalar field action is given by $$\begin{aligned}
{\cal S}=-\frac{1}{2}\int d^4 x \sqrt{-g}\left[\left(\nabla\psi\right)^2
+ \frac{1}{6} \psi^2 R + i \bar\chi \dslash \chi + \mu \bar\chi
\chi + f \psi \bar\chi\chi\right].\end{aligned}$$ The equations of motion determine the constant non-vanishing values of $\psi$ (for simplicity consider the static case only) and $\bar\chi \chi$. We see that when the effective mass $\mu+ f
\psi$ becomes negative the model becomes unstable due to $\chi$ pair production, and will prefer a state with a $\bar\chi\chi$ condensate, which will feed back into $\psi$. Correspondingly, such rapid creation of pairs would be accompanied by strong fluctuations. It is not clear whether under these circumstances the condition for bounce $\rho_{\rm r}<\rho_\psi$ will continue to hold indefinitely, or whether a collapse to a singularity will ensue after a finite number of cycles of the universe. A complete discussion of the time-dependent situation is beyond the scope of this paper but it is clear that violations of CEB are related with a potential instability in the dust sector, and cannot be simply taken as a bona-fide example of CEB violation.
The fact that Bekenstein’s universe is non-singular indicates that the singularity theorems of Penrose and Hawking [@singth] are somehow eluded. And indeed the Strong Energy Condition (SEC) is violated in the model: $\rho_{\rm tot}+3p_{\rm
tot}=2\rho_{\rm r}+\rho_{\rm m}$ is negative at the bounce, positive at the turnover and changes continuously in between. As we show later, violation of some energy conditions does not necessarily mandate a violation of the CEB. We will argue that in this sense the CEB has a better discriminating power than energy conditions (see below).
Conditions for validity of CEB\
with general classical sources
===============================
We may summarize the lessons of the previous examples by imposing conditions on sources in a generic cosmological setting such that CEB is obeyed. This analysis is not restricted to a static universe, nor to a closed one, and contains the previous examples as particular cases.
We consider a cosmic fluid consisting of radiation, an optional cosmological constant, and additional unspecified classical dynamical sources which do not include any contributions from the cosmological constant or radiation. For simplicity we assume that the additional sources have negligible entropy. This is the most conservative assumption: if some of the additional sources have substantial entropy our conclusions can be strengthened. We use the previous notations for the total, cosmological, and radiation energy densities, $\rho_{\rm tot}$, $\rho_{\Lambda}$ and $\rho_{\rm r}$ respectively, and denote by $\rho^*$ the combined energy density of the additional sources. Thus $$\rho_{\rm tot}=\rho_{\rm r}+\rho_{\Lambda}+\rho^*.$$ We use the same notation for the relative pressures, and for the equation of state $\gamma^*\equiv\rho^*/p^*$, which may be time-dependent.
In term of these sources, the causal connection scale can be written as $$\begin{aligned}
\label{Rccrho}
&& R_{\rm CC}^{-2}=\frac{4\pi G_{\rm N}}{D-1}\times\nonumber\\
&&{\rm Max} \Biggl\{D\rho_{\Lambda} + \biggl[1 -
(D-1)\gamma^*\biggr]\rho^*\, , (D-4)\rho_{\Lambda} + \biggl[(2D-5) +
(D-1)\gamma^*\biggr]\rho^* + 2(D-2)\rho_{\rm r}\Biggr\}.\end{aligned}$$
We may now express the ratio of $(S_{\rm CEB}/S_{\rm r})^2$, neglecting as usual prefactors of order one $$\begin{aligned}
&& \left(\frac{S_{\rm CEB}}{S_{\rm r}}\right)^2 \sim
\frac{1}{\N}\left(\frac{M_{\rm P}}{T}\right)^{D-2} \times
\nonumber \\ && {\rm Max}\left\{ D\frac{\rho_{\Lambda}}{\rho_{\rm r}}+
\biggl[ 1 - (D-1)\gamma^*\biggr]\frac{\rho^*}{\rho_{\rm r}}\,,
(D-4)\frac{\rho_{\Lambda}}{\rho_{\rm r}}+\biggl[ (2D-5) +
(D-1)\gamma^*\biggr]\frac{\rho^*}{\rho_{\rm r}} +
2(D-2)\right\} .\label{ratiogen}\end{aligned}$$ As was already pointed out in the previous section, a condition for any CEB violations is that this ratio be parametrically smaller than one. Notice that the first factor is larger than one by our requirement that the radiation energy density be sub-Planckian. Thus the only remaining possibility for violating CEB is that the second factor be parametrically smaller than unity. As we show below, this can occur only if at least one of the additional sources has negative energy density.
The r.h.s. of (\[ratiogen\]) is larger than the average of the two entries, so that $$\label{newrat} \left(\frac{S_{\rm CEB}}{S_{\rm r}}\right)^2 \gaq
{1\over\N}\left({M_{\rm P}\over T}\right)^{D-2} (D-2) {\rho_{\rm
tot}\over\rho_{\rm r}}$$ Therefore, since $\rho_{\rm tot}>0$, a necessary condition for this expression to be smaller than unity is that $\rho_{\rm
tot}\ll\rho_{\rm r}$, which we may reexpress as $$\begin{aligned}
\label{condL}
\frac{\rho_{\Lambda}}{\rho_{\rm r}}\sim
-\left(1+\frac{\rho^*}{\rho_{\rm r}}\right).\end{aligned}$$ This is not a sufficient condition since the equations of motion could dictate, for example, that the first factor on the r.h.s. of eq.(\[newrat\]) could be parametrically larger than unity at the same time. By substituting condition (\[condL\]) into Eq. (\[ratiogen\]), we obtain $$\begin{aligned}
&& \left(\frac{S_{\rm CEB}}{S_{\rm r}}\right)^2 \sim
\frac{1}{\N}\left(\frac{M_{\rm P}}{T}\right)^{D-2} \times
\nonumber \\ && {\rm Max}\left\{-\biggl[(D-1)(1+\gamma^*)\frac{\rho^*}
{\rho_{\rm r}} + D\biggr]\,,(D-1)(1+\gamma^*)\frac{\rho^*}
{\rho_{\rm r}} + D\right\} .\label{ratnongen}\end{aligned}$$ Therefore, an additional necessary condition for $S_{CEB}/S_{\rm
r}$ to be smaller than one is that $$\begin{aligned}
\label{cond*}
(1 + \gamma^*)\rho^*\simeq-\frac{D}{(D-1)}\rho_{\rm r}\, .\end{aligned}$$ Condition (\[cond\*\]) can be satisfied in two ways:
\(i) $1+\gamma^*>0$ and $\rho^*<0$. This obviously requires that at least one of the sources has negative energy density. In this case (barring pathologies) the magnitude of $\rho^*$ is comparable to that of $\rho_{\rm r}$.
\(ii) $1+\gamma^*<0$ and $\rho^*>0$. However, for classical dynamical sources, this typically clashes with causality which requires that the pressure and energy density of each of the additional dynamical sources obey $|p_i|<|\rho_i|$; hence if all $\rho_i>0$ then necessarily .
Consequently, condition (\[cond\*\]) cannot be satisfied if all of the dynamical sources have positive energy densities and equations of state $|\gamma_i|\le 1$. Bekenstein’s Universe discussed in the previous section fits well within our framework: the total energy density is positive, but the overall contribution to $\rho_{\rm tot}$ of all the sources, excluding radiation (since the cosmological constant vanishes in this case), is negative and almost cancels the contribution of radiation, leaving a small positive $\rho_{\rm tot}$.
To summarize, if all dynamical sources (different from the cosmological constant) have positive energy densities $\rho_i >
0$ and have causal equations of state ($|\gamma_i|\le 1$), and if radiation temperatures are sub-Planckian, CEB is upheld.
The CEB (and entropy bounds in general) refines the classic singularity theorems in that it allows cosmologies for which the singularity theorems are not applicable because some of the energy conditions are violated, but do not seem to be problematic in any of their properties, or indicates possible problems already when the singularity theorems seem perfectly valid. For example, the scale factor for a closed deSitter Universe (i.e. a closed Universe containing a positive cosmological constant $\Lambda$) in $D=4$ is given by $a(t)=(\frac{\Lambda}{3})^{-1/2}\cosh{\sqrt{\frac{\Lambda}{3}}t}$, showing a bounce at $t=0$. This is not surprising since the sources of this model violate the SEC. The reliability of the SEC as a criterion of discriminating physical and unphysical solutions is therefore questionable (as is well known in the context of inflationary cosmology). Alternatively, in a contracting 4D radiation dominated universe, the singularity theorems imply the the solution will reach a future singularity, but the CEB indicates problems already when $T\sim M_{\rm P}/\N^{1/2}$.
In general, the total energy-momentum tensor of a closed “bouncing" universe violates the SEC, but it can obey the CEB. In order to see this explicitly let us consider the “bounce” condition, i.e. $H=0$, $\dot{H}>0$ for a closed Universe; by using the Einstein equations (\[ein1\]-\[ein2\]), we can express this condition in terms of the sources as follows: $$\begin{aligned}
\label{bounce}
\rho_{\rm tot}>0,\quad (D-3)\rho_{\rm tot}+(D-1)p_{\rm tot}<0.\end{aligned}$$ The second of these conditions is (in $D=4$) precisely the condition for violation of the SEC. In terms of $\rho_{\rm r}$, $\rho_{\Lambda}$ and $\rho^*$ this reads $$\label{condbounce}
2 \rho_{\Lambda} - (D-2)\rho_{\rm r} - \biggl[(D-3) +
(D-1)\gamma^*\biggr]\rho^*>0\, .$$ In comparison, a necessary condition that the CEB is violated can be obtained from Eqs.(\[condL\]) and (\[cond\*\]), $$\label{3.10}
2 \rho_{\Lambda} - (D-2)\rho_{\rm r} - \biggl[(D-3) +
(D-1)\gamma^*\biggr]\rho^*\sim 0\, ,$$ where the l.h.s of (\[3.10\]) can be either positive or negative. So we find that there is a range of parameters for which the CEB can be obeyed in some bouncing cosmologies but not in others.
In a spatially flat universe ($k=0$), the conditions for a bounce are slightly different: $\rho_{\rm tot}=0$ and $\rho_{\rm
tot}+p_{\rm tot}<0$. At the bounce these conditions imply violation of the Null Energy Condition (NEC). As discussed previously, classical sources are not expected to violate the NEC, but effective quantum sources, such as Hawking radiation, are known to violate the NEC (see [@BM1; @visser] for a more comprehensive discussion of this point). In terms of $\rho_{\rm
r}$, $\rho_{\Lambda}$ and $\rho^*$ the condition for a bounce reads $$\left(1+\frac{1}{D-1}\right) \rho_{\rm r} + (1+\gamma^*)\rho^*>0.$$ In comparison, a necessary condition that the CEB is violated can be obtained from Eq.(\[cond\*\]), $$\label{3.12}
\left(1+\frac{1}{D-1}\right) \rho_{\rm r} +
(1+\gamma^*)\rho^*\sim 0\, ,$$ where the l.h.s of (\[3.12\]) can be either positive or negative. So, again, we find that there is a range of parameters for which the CEB can be obeyed in some spatially flat bouncing cosmologies but not in others.
The CEB appears to be a more reliable criterion than energy conditions when trying to decide whether a certain cosmology is reasonable: taking again deSitter Universe as an example, we can add a small amount of radiation to it, and still have a bouncing model if $\rho_{\Lambda}$ is the dominant source, and SEC will not be obeyed (see Eq.(\[condbounce\])). Nevertheless, the general discussion in this section shows that in this case the CEB is not violated as long as radiation temperatures remain subPlanckian, despite the presence of a bounce. This happens, in part, because the CEB is able to discriminate better between dynamical and non-dynamical sources (such as the cosmological constant), and imposes constraints that involve the former ones only, such as Eq. (\[cond\*\]).
Conclusions
===========
We have reached the following conclusions by studying the validity of the CEB for non-singular cosmologies:
1. Violation of the CEB necessarily requires either high temperatures ${\cal N} \left(\frac{T}{M_{\rm P}}\right)^{D-2} \gaq 1$, or dynamical sources that have negative energy densities with a large magnitude, or sources with acausal equation of state. Of course, neither of the above is sufficient to guarantee violations of the CEB.
2. Classical sources of this type are suspect of being unphysical or unstable, but each source has to be checked on a case by case basis. In the examples we discussed in sect. II, the sources were indeed found to be unstable or are strongly suspected to be so.
3. Sources with large negative energy density could allow, in principle, to increase the entropy within a given volume, while keeping its boundary area and the total energy constant. This would lead to violation of all known entropy bounds, and of any entropy bound which depends in a continuous way on the total energy or on the linear size of the system.
4. The CEB is more discriminating than singularity theorems. In the examples we have considered it allows non-singular cosmologies for which singularity theorems cannot be applied, but does not allow them if they are associated with specific dynamical problems.
It is a pleasure to thank J. Bekenstein for many enlightening discussions. This research was supported by grants No. 174/00-2 (RB and SF) and No. 129/00-1 (AEM) of the Israel Science Foundation. SF wishes to acknowledge support from the Kreitman Foundation.
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[^1]: In [@CEB] space curvature was implicitly included in the total energy budget as a regular additional source.
|
---
abstract: 'We present a general-purpose data compression algorithm, Regularized L21 Semi-NonNegative Matrix Factorization (L21 SNF). L21 SNF provides robust, parts-based compression applicable to mixed-sign data for which high fidelity, individual data point reconstruction is paramount. We derive a rigorous proof of convergence of our algorithm. Through experiments, we show the use-case advantages presented by L21 SNF, including application to the compression of highly overdetermined systems encountered broadly across many general machine learning processes.'
author:
- |
Anthony D. Rhodes\
Portland State University\
`arhodes@pdx.edu` Bin Jiang\
Portland State University\
`bjiang@pdx.edu`
bibliography:
- 'references.bib'
title: 'Regularized L21-Based Semi-NonNegative Matrix Factorization '
---
Background
==========
Data reduction algorithms represent an essential component of machine learning systems. The use of such reductions are supported by topological properties of data, including the well-known Manifold Hypothesis [@NIPS2010_3958].
Our work presents a novel compression algorithm which renders a parts-based compression applicable to mixed-sign data for which high fidelity, individual data point reconstruction is paramount. We achieve this result by solving a constrained optimization problem for matrix reconstruction with respect to L2-1 loss. We present the details of our algorithm in Section 2. In Section 3 we demonstrate proof of convergence, and in Section 4 we provide experimental results.
As a precursor to our work, we consider the Non-negative Matrix Factorization (NMF) framework [@lee99; @NIPS2000_1861; @4359171; @NIPS2005_2757; @10.1016/j.csda.2008.01.011; @Kong]. NMF is defined as the problem of finding a matrix factorization of a given non-negative matrix $\mathbf{X}^{m \times n}$ so that $\mathbf{X}\approx \mathbf{WH}$ for non-negative factors $\mathbf{W}^{m \times r}$ and $\mathbf{H}^{r \times n}$; compression is achieved when $r < min(m,n)$.
A significant feature of this methodology is that it gives rise to a parts-based decomposition of $\mathbf{X}.$ In this way, each column of $\mathbf{W}$ represents a basis element in the reduced space; the columns of $\mathbf{H}$ can correspondingly be interpreted as embodying the coordinates for each basis element that render an approximation of the columns of $\mathbf{X}$. Since the number of basis vectors ($r$) is often relatively small, this set of vectors represents a useful latent structure in the data. Finally, because each component in the factorization is restricted to be non-negative, their interaction in approximating $\mathbf{X}$ is strictly *additive*, meaning that the columns of $\mathbf{W}$ yield a parts-based, compressed decomposition of $\mathbf{X}$.
Lee and Seung [@lee99; @NIPS2000_1861] provided a well-known solution to NMF in the form of two multiplicative algorithms based on the standard square Euclidean distance and “divergence”, respectively, defined: $$\|\mathbf{X}-\mathbf{WH}\|_{F}^{2}=\sum_{ij}(\mathbf{X}_{ij}-\mathbf{WH}_{ij})^{2}
\label{l2norm}$$
$$D(\mathbf{X}||\mathbf{WH}) = \sum_{ij} (\mathbf{X}_{ij}\text{log} \frac{\mathbf{X}_{ij}}{\mathbf{WH}_{ij}}-\mathbf{X}_{ij}+\mathbf{WH}_{ij})
\label{divnorm}$$
where both the distance and divergence are bounded below by zero, and vanish when $\mathbf{X=WH}$. Divergence furthermore reduces to KL-Divergence [@kullback1951] when $\sum_{ij}\mathbf{X}_{ij}=\sum_{ij}\mathbf{WH}_{ij}=1$. Subsequent to [@NIPS2000_1861], related research for NMF has used projected gradients [@10.1162/neco.2007.19.10.2756], non-negative least square [@10.1007/s10618-012-0265-y], and neural approaches [@Vu2016CombiningNM], among other methods.
L21 SNF
=======
When the data matrix $\mathbf{X}$ is not strictly non-negative, NMF is inapplicable. Nevertheless, in many common use cases, a parts-based decomposition is a desideratum for data compression with mixed-sign data. [@4685898] introduce a useful compromise toward this end, the Frobenius norm based Semi-Nonnegative Matrix Factorization, in which one (and only one) of the factor matrices is constrained to be non-negative. However, the Frobenius norm in is known for its instability with respect to noise and outliers [@Kong; @Liu].
In place of the loss functions given by and , we propose to instead employ a generally more robust measure that leverages together L2 and L1 loss, termed L2-1 loss [@NIPS2010_3988]. The definition of the L2-1 norm is as follows: $$\|\mathbf{X} \|_{2,1}=\sum_{i=1}^{n} \sqrt{ \sum_{j=1}^{m}\mathbf{X}_{ji}^{2} }
\label{l21norm}$$ We accordingly define L2-1 loss for matrix factorization by: $$\|\mathbf{X-WH} \|_{2,1}=\sum_{i=1}^{n} \sqrt{ \sum_{j=1}^{m}\mathbf{(X}_{ji}-\mathbf({\mathbf{WH}})_{ji})^{2} }
\label{l21normerror}$$ Note in particular that we define L2-1 loss as a sum of L2 vector magnitudes with respect to each column of $\mathbf{X}$. When applied, for example, to a set of convolutional filters [@denton] (consider each column of $\mathbf{X}$ as a “flattened” filter), L2-1 loss can viewed as a robust measure that weighs the distance per filter component using L2 cost, while summing over filters with L1 cost. [@Kong] use the L2-1 norm to solve NMF, but their method cannot be directly extended to accommodate mixed-sign data.
Let $\mathbf{X} \in \mathbb{R}^{m\times n},\mathbf{W} \in \mathbb{R}^{m\times k},\mathbf{H} \in \mathbb{R}_+^{k\times n}$; we define the optimization problem framing L2-1 semi non-negative matrix factorization:\
$$\operatorname*{arg\,min}_{\mathbf{W},\mathbf{H}} \|\mathbf{X}-\mathbf{WH}\|_{2,1}+\overline{\alpha}\|\mathbf{W} \|_{2}^{2}\text{ subject to } \mathbf{H}\geq 0
\label{optimization}$$\
where $\overline{\alpha}=\frac{\alpha}{2}$ (for simplicity of notation) and $\alpha\geq 0$ is a hyperparameter.
We present the following Regularized L21 SNF algorithm which provides an iterative solution to ; in the following section we prove convergence of our algorithm.
Initialize $\mathbf{H}(0)$ as non-negative matrix, initialize $\mathbf{W}(0)$ (e.g. use k-means)\
[ [ (1) $\mathbf{H}_{ij}(t+1)=\mathbf{H}_{ij}(t)\sqrt{\frac{(\Phi^{+}\mathbf{D}(t))_{ij}+(\Omega^{-}\mathbf{H}(t)\mathbf{D}(t))_{ij}}
{(\Phi^{-}\mathbf{D}(t))_{ij}+(\Omega^{+}\mathbf{H}(t)\mathbf{D}(t))_{ij}}}$ ]{}\
$(2)\mathbf{W}(t+1)=[\mathbf{XD}(t)\mathbf{H}(t)^{T}][\alpha\mathbf{I}+\mathbf{H}(t)\mathbf{D}(t)\mathbf{H}(t)^{T}]^{-1}$ ]{} where $\mathbf{W}^{T}(t)\mathbf{W}(t)=\Omega,$ $\Omega=\Omega^{+}-\Omega^{-}$, $\mathbf{W}^{T}(t)\mathbf{X}=\Phi,$ $\Phi=\Phi^{+}-\Phi^{-}$, and $\mathbf{D}(t)_{ii}=1/ \|\mathbf{x}^{(i)}-\mathbf{W}(t)\mathbf{h}(t)^{(i)} \|_{2}; \mathbf{x}^{(i)} \in \mathbb{R}^{m\times 1}$ denotes the *i*th column of $\mathbf{X}$, and $\mathbf{h}^{(i)} \in \mathbb{R}_+^{k\times 1}$ denotes the *i*th column of $ \mathbf{H}$. $\Omega^{+}$, $\Omega^{-}$ are positive and negative entries of $\Omega$.
Proof of Convergence
====================
We define the *proxy loss* function using matrix trace:\
$$\begin{gathered}
\begin{split}
\mathscr{L}(\mathbf{X},\mathbf{WH})=tr[(\mathbf{X}-\mathbf{WH})\mathbf{D}(\mathbf{X}-\mathbf{WH})^{T}]+\alpha tr[\mathbf{W}^{T}\mathbf{W}]
\\ \text{where } \mathbf{D} \in \mathbb{R}^{n\times n},
\mathbf{D}_{ii}= 1/\|\mathbf{x}^{(i)}-\mathbf{W}\mathbf{h}^{(i)} \|_{2}
\end{split}
\label{proxyloss}\end{gathered}$$
Next, we subsequently derive iterative update formulas based on the loss function given in and show that these updates incur a monotonic loss in .\
$$\mathscr{L}(\mathbf{X},\mathbf{WH})=
tr[\mathbf{XDX}^{T}]-2tr[\mathbf{W}^{T}\mathbf{XD}\mathbf{H}^{T}]+tr[\mathbf{WHDH}^{T}\mathbf{W}^{T}]+\alpha tr[\mathbf{W}^{T}\mathbf{W}]
\label{proxyloss2}$$\
Solving $\nabla_{\mathbf{W}}\mathscr{L}=0$, gives the solution: $$\mathbf{W}=[\mathbf{XDH}^{T}][\alpha\mathbf{I}+\mathbf{HD}\mathbf{H}^{T}]^{-1}
\label{wformula}$$
Now we prove that optimality of by demonstrating that is convex; we first consider $\frac{\partial\mathscr{L}}{\partial W_{ij}}$: $$\frac{\partial\mathscr{L}}{\partial W_{ij}}=2(\mathbf{WHD}\mathbf{H}^{T})_{ij}-2(\mathbf{XDH}^{T})_{ij}+2\alpha(\mathbf{W})_{ij}
\label{lwij}$$\
Expanding the first term on the RHS of renders the following simplification. $$\frac{\partial\mathscr{L}}{\partial W_{ij}}=2\sum_{l=1}^{k}\mathbf{W}_{il}(\mathbf{HD}\mathbf{H}^{T})_{lj}-2(\mathbf{XDH}^{T})_{ij}+2\alpha(\mathbf{W})_{ij}
\label{lwij2}$$\
The Hessian of $\mathscr{L}$ is consequently: $$\frac{\partial\mathscr{L}}{\partial W_{ij}\partial W_{pq}}
=2(\mathbf{HD}\mathbf{H}^{T}+\alpha\mathbf{I})_{qj}\delta_{ip} \quad \quad 1\leq i,p \leq m \quad 1\leq j,q \leq k
\label{lwijpq}$$
Therefore, the Hessian of $\mathscr{L}$ is a block diagonal matrix with each block being of the form $2\mathbf{HDH}^T+2\alpha \mathbf{I}$. It follows that $\mathscr{L}$ is convex, thus the formula given for $\mathbf{W}$ in minimizes $\mathscr{L}$ in .
The previous derivation of and the associated demonstration of optimality furnish a proof for the following Lemma. We now consider as an iterative update rule at step $t$, where we regard $\mathbf{H}(t)$ as fixed at the time of the $t$-th update for $\mathbf{W}$, denoted by $\mathbf{W}(t+1)$. Define $\mathbf{D}(t)_{ii}= 1/\|\mathbf{x}^{(i)}-\mathbf{W}(t)\mathbf{h}(t)^{(i)} \|_{2}$, which depends on $\mathbf{W}(t)$ and is also regarded as fixed at the time of $t$-th update for $\mathbf{W}(t+1)$. The iterative update for matrix $\mathbf{W}$ is given by: $$\mathbf{W}(t+1)=[\mathbf{XD}(t)\mathbf{H}(t)^{T}][\alpha \mathbf{I}+\mathbf{H}(t)\mathbf{D}(t)\mathbf{H}(t)^{T}]^{-1}
\label{wtplus1}$$
**Lemma 1.** Let $\mathbf{W}(t)$ and $\mathbf{W}(t+1)$ represent consecutive updates for $\mathbf{W}$ as prescribed by . Under this updating rule, the following inequality holds: $$\begin{gathered}
\begin{split}
tr[(\mathbf{X}-\mathbf{W}(t+1)\mathbf{H}(t))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t+1)\mathbf{H}(t))^{T}] +\alpha tr[\mathbf{W}^{T}(t+1)\mathbf{W}(t+1)]\\
\leq tr[(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))^{T})]+\alpha tr[\mathbf{W}^{T}(t)\mathbf{W}(t)]\\
\end{split}
\label{lemma1}\end{gathered}$$\
**Proof.** The proof of Lemma 1 follows directly from the optimality of the update formula in . $\square$\
**Lemma 2.** Under the update rule of , the following inequality holds where $\overline{\alpha}=\frac{\alpha}{2}$: $$\begin{gathered}
\begin{split}
\left(\|\mathbf{X}-\mathbf{W}(t+1)\mathbf{H}(t)\|_{2,1}-\|\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t)\|_{2,1}\right) \\
+\overline{\alpha} tr[\mathbf{W}(t+1)\mathbf{W}^T(t+1)]-\overline{\alpha} tr[\mathbf{W}(t)\mathbf{W}^T(t)]\\
\leq \frac{1}{2}\big\{ tr[(\mathbf{X}-\mathbf{W}(t+1)\mathbf{H}(t))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t+1)(\mathbf{H}(t))^T] \\ -tr[(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))^T] \big\} \\
+\overline{\alpha} tr[\mathbf{W}(t+1)\mathbf{W}^T(t+1)]-\overline{\alpha} tr[\mathbf{W}(t)\mathbf{W}^T(t)]
\end{split}
\label{lemma2}\end{gathered}$$ **Proof.** The proof is analogous to the proof of Lemma 3 in [@Kong], so we skip it here for brevity. $\square$\
**Theorem 1.** Updating $\mathbf{W}$ using formula while fixing $\mathbf{H}$ yields a monotonic decrease in the objective function defined by .\
\
**Proof.** By Lemma 1, the right hand side expression in Lemma 2 satisfies: $$\begin{gathered}
\begin{split}
\frac{1}{2}tr[(\mathbf{X}-\mathbf{W}(t+1)\mathbf{H}(t))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t+1)(\mathbf{H}(t))^T]+\overline{\alpha} tr[\mathbf{W}(t+1)\mathbf{W}^T(t+1)] \\
- \frac{1}{2}tr[(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))^T]
-\overline{\alpha} tr[\mathbf{W}(t)\mathbf{W}^T(t)] \leq 0.
\end{split}
\label{theo1rhs}\end{gathered}$$ So does the left hand side expression in Lemma 2: $$\begin{gathered}
\begin{split}
\|\mathbf{X}-\mathbf{W}(t+1)\mathbf{H}(t)\|_{2,1} + \overline{\alpha} tr[\mathbf{W}(t+1)\mathbf{W}^T(t+1)] \\
-\|\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t)\|_{2,1}
-\overline{\alpha} tr[\mathbf{W}(t)\mathbf{W}^T(t)] \leq 0.
\end{split}
\label{theo1lhs}\end{gathered}$$ Thus proving Theorem 1. $\square$\
Next we derive an iterative update formula for $\mathbf{H}$, with $\mathbf{H} \geq 0$; subsequently we prove convergence of this update rule by showing that the proxy loss $\mathscr{L}(\mathbf{X},\mathbf{WH})$ given in is monotonically decreasing for fixed $\mathbf{W}$. Since the second term of , $\alpha tr[\mathbf{W}^{T}\mathbf{W}]$, is fixed during the $\mathbf{H}$ update, we ignore it here. Define the corresponding *truncated proxy loss*: $F(\mathbf{H})=tr[(\mathbf{X}-\mathbf{WH})\mathbf{D}(\mathbf{X}-\mathbf{WH})^{T}]$.
Based on formula as well as $tr(\mathbf{AB})=tr(\mathbf{BA})$, $F(\mathbf{H})$ can be further simplified to $$F(\mathbf{H})=
tr[\mathbf{XDX}^{T}]-2tr[\mathbf{H}^{T}\mathbf{W}^{T}\mathbf{XD}]+tr[\mathbf{W}^{T}\mathbf{WHDH}^{T}]
\label{fhformula}$$
To prove convergence for $\mathbf{H}$, we utilize an auxiliary function, denoted $\mathscr{A}(\mathbf{H},\mathbf{H'})$ as in [@NIPS2000_1861; @10.1016/j.csda.2008.01.011].\
**Definition.** $\mathscr{A}$ is an auxiliary function for $F(\mathbf{H})$ if: $$\mathscr{A}(\mathbf{H},\mathbf{H'})\geq F(\mathbf{H}), \quad \mathscr{A}(\mathbf{H},\mathbf{H})= F(\mathbf{H})
\label{auxiliarydef}$$ **Lemma 3.** If $\mathscr{A}$ is an auxiliary function of $F(\mathbf{H})$, then $F(\mathbf{H})$ is non-increasing under the update: $$\mathbf{H^{t+1}}=\operatorname*{arg\,min}_{\mathbf{H}} \mathscr{A}(\mathbf{H}, \mathbf{H^{t}})
\label{lemma3}$$ **Proof.** $$F(\mathbf{H^{t+1}})\leq \mathscr{A}(\mathbf{H^{t+1}},\mathbf{H^{t}})\leq \mathscr{A}(\mathbf{H^{t}},\mathbf{H^{t}})=F(\mathbf{H^{t}}).
\label{auxiliaryproof}$$
We now consider an explicit solution for $\mathbf{H}$ in the form of an iterative update, for which we subsequently prove convergence. Since $\mathbf{H}$ is non-negative, it is helpful to decompose both the $k \times k$ matrix $\mathbf{W}^{T}\mathbf{W}=\Omega$ and the $k \times n$ matrix $\mathbf{W}^{T}\mathbf{X}=\Phi$ into their positive and negative entries as follows: $$\Omega^{+}_{ij}=\frac{1}{2}(|\Omega_{ij}|+\Omega_{ij}), \quad
\Omega^{-}_{ij}=\frac{1}{2}(|\Omega_{ij}|-\Omega_{ij}).
\label{omegapositive}$$ **Lemma 4.** Under the iterative update: $$\mathbf{H}_{ij}(t+1)=\mathbf{H}_{ij}(t)\sqrt{\frac{(\Phi^{+}\mathbf{D}(t))_{ij}+(\Omega^{-}\mathbf{H}(t)\mathbf{D}(t))_{ij}}
{(\Phi^{-}\mathbf{D}(t))_{ij}+(\Omega^{+}\mathbf{H}(t)\mathbf{D}(t))_{ij}}}
\label{lemma4}$$ where $\mathbf{W}^{T}(t)\mathbf{W}(t)=\Omega,$ $\Omega=\Omega^{+}-\Omega^{-}$, $\mathbf{W}^{T}(t)\mathbf{X}=\Phi$, $\Phi=\Phi^{+}-\Phi^{-}$, and $\mathbf{D}(t)_{ii}= 1/\|\mathbf{x}^{(i)}-\mathbf{W}(t)\mathbf{h}(t)^{(i)} \|_{2} $, the following relation holds for some auxiliary function $\mathscr{A}(\mathbf{H},\mathbf{H}')$: $$\mathbf{H}(t+1)=\operatorname*{arg\,min}_{\mathbf{H}} \mathscr{A}(\mathbf{H},\mathbf{H}(t))
\label{htplus1}$$ **Proof.** Using the notation introduced above, the truncated proxy loss $F(\mathbf{H})$ in can be rewritten in the following form: $$\begin{gathered}
\begin{split}
F(\mathbf{H})=tr[\mathbf{XD}\mathbf{X}^{T}]
-2tr[\mathbf{H}^{T}\mathbf{\Phi}^{+}\mathbf{D}]+2tr[\mathbf{H}^{T}\mathbf{\Phi}^{-}\mathbf{D}]\\
+tr[\Omega^{+}\mathbf{H}\mathbf{D}\mathbf{H}^{T}]-tr[\Omega^{-}\mathbf{H}\mathbf{D}\mathbf{H}^{T}]
\end{split}
\label{fhterms}\end{gathered}$$
In the subsequent steps we provide an auxiliary function $\mathscr{A}(\mathbf{H},\mathbf{H}')$ for $F(\mathbf{H})$. Following [@NIPS2010_3988], in order to construct an auxiliary function that furnishes an upper-bound for $F(\mathbf{H})$, we define $\mathscr{A}(\mathbf{H},\mathbf{H})$ as a sum comprised of terms that represent upper-bounds for each of the positive terms appearing in and lower-bounds for each of the negative terms, respectively.\
First, we derive a lower-bound for the second term of , using $a \geq 1+log\text{ }a, \text{ } \forall \text{ } a>0$: $$tr[\mathbf{H}^{T}\mathbf{\Phi}^{+}\mathbf{D}]=\sum_{ij}\mathbf{H}_{ij} (\mathbf{\Phi}^{+}\mathbf{D})_{ij} \geq\sum_{ij}(\mathbf{\Phi}^{+}\mathbf{D})_{ij}\mathbf{H}'_{ij}(1+log\frac{\mathbf{H}_{ij}}{\mathbf{H}'_{ij}})
\label{fh2ndterm}$$
Second, using the fact that $a\leq \frac{a^2+b^2}{2b} \text{ } \forall$ $a,b>0$, we derive an upper bound for the third term on the RHS of : $$tr[\mathbf{H}^{T}\mathbf{\Phi}^{-}\mathbf{D}]=\sum_{ij}\mathbf{H}_{ij} (\mathbf{\Phi}^{-} \mathbf{D})_{ij}\leq
\sum_{ij}(\mathbf{\Phi}^{-} \mathbf{D})_{ij}
\frac{(\mathbf{H}_{ij})^2+(\mathbf{H}'_{ij})^2}{2\mathbf{H'}_{ij}}
\label{fh3rdterm}$$
Third, we apply the following inequality [@4685898] to bound the fourth term on the RHS of .\
**Proposition 1.** For any matrices $\mathbf{A}\in \mathbb{R}^{n \times n}_{+}$, $\mathbf{B}\in \mathbb{R}^{k \times k}_{+}$, $\mathbf{S}\in \mathbb{R}^{n \times k}_{+}$, $\mathbf{S}'\in \mathbb{R}^{n \times k}_{+}$, with $\mathbf{A}$ and $\mathbf{B}$ symmetric: $$tr[\mathbf{S}^{T}\mathbf{ASB}] \leq \sum_{i=1}^{n} \sum_{p=1}^{k} \frac{{(\mathbf{AS'B})}_{ip}\mathbf{S}^{2}_{ip}}{\mathbf{S}'_{ip}}
\label{proposition1}$$
Considering the fourth term of the RHS of , we have: $$tr[\Omega^{+}\mathbf{HD}\mathbf{H}^{T}]=tr[\mathbf{H}^{T}\Omega^{+}\mathbf{HD}]
\leq \sum_{ij}\frac{(\Omega^{+}\mathbf{H}'\mathbf{D})_{ij}\mathbf{H}^2_{ij}}{\mathbf{H}'_{ij}}
\label{fh4thterm}$$ Finally, we consider the fifth term on the RHS of equation .
**Proposition 2.** $$tr[\Omega^{-}\mathbf{HD}\mathbf{H}^{T}]\geq \sum_{ijk}\Omega^{-}_{ik}\mathbf{H}'_{kj}\mathbf{D}_{jj}\mathbf{H}'_{ij}\Big(1+log\frac{\mathbf{H}_{kj}\mathbf{H}_{ij}}{\mathbf{H'}_{kj}\mathbf{H}'_{ij}} \Big)
\label{proposition2}$$ **Proof.** $$\begin{gathered}
\begin{split}
tr[\Omega^{-}\mathbf{HD}\mathbf{H}^{T}]=tr[\mathbf{H}^{T}\Omega^{-}\mathbf{HD}]=\sum_{ij}(\Omega^{-}\mathbf{HD})_{ij}\mathbf{H}_{ij} \\
= \sum_{ijk}\Omega^{-}_{ik}(\mathbf{HD})_{kj}\mathbf{H}_{ij} = \sum_{ijk}\Omega^{-}_{ik}\mathbf{H}_{kj}\mathbf{D}_{jj}\mathbf{H}_{ij}
\end{split}
\label{proposition2proof1}\end{gathered}$$ Once again we employ the inequality $a \geq 1+log\text{ }a$, whereupon: $$tr[\Omega^{-}\mathbf{HD}\mathbf{H}^{T}]\geq \sum_{ijk}\Omega^{-}_{ik}\mathbf{H}'_{kj}\mathbf{D}_{jj}\mathbf{H}'_{ij}\Big(1+log\frac{\mathbf{H}_{kj}\mathbf{H}_{ij}}{\mathbf{H'}_{kj}\mathbf{H}'_{ij}} \Big)
\label{proposition2proof2}$$ Thus proving Proposition 2. $\square$
Putting all the formulas , , and together, we define the auxiliary function $\mathscr{A}(\mathbf{H},\mathbf{H}')$: $$\begin{gathered}
\begin{split}
\mathscr{A}(\mathbf{H},\mathbf{H}')=tr[\mathbf{XD}\mathbf{X}^{T}]
+2\sum_{ij}(\mathbf{\Phi}^{-}\mathbf{D})_{ij}\frac{(\mathbf{H}_{ij})^2+(\mathbf{H}'_{ij})^2}{2\mathbf{H'}_{ij}}
+\sum_{ij}\frac{(\Omega^{+}\mathbf{H}'\mathbf{D})_{ij}\mathbf{H}^2_{ij}}{\mathbf{H}'_{ij}}\\
-2\sum_{ij}(\mathbf{\Phi}^{+}\mathbf{D})_{ij}\mathbf{H}'_{ij}(1+log\frac{\mathbf{H}_{ij}}{\mathbf{H}'_{ij}})
-\sum_{ijk}\Omega^{-}_{ik}\mathbf{H}'_{kj}\mathbf{D}_{jj}\mathbf{H}'_{ij}\Big(1+log\frac{\mathbf{H}_{kj}\mathbf{H}_{ij}}{\mathbf{H'}_{kj}\mathbf{H}'_{ij}} \Big)
\end{split}
\label{lhhprime}\end{gathered}$$ Observe that $\mathscr{A}(\mathbf{H},\mathbf{H'})\geq F(\mathbf{H})$ and $\mathscr{A}(\mathbf{H},\mathbf{H})= F(\mathbf{H})$, as required for an auxiliary function, where $F(\mathbf{H})$ denotes the truncated proxy loss as defined in equation . By the aforementioned Lemma, it follows that $F(\mathbf{H})$ is non-increasing under the update: $\mathbf{H}(t+1)=\operatorname*{arg\,min}_{\mathbf{H}} \mathscr{A}(\mathbf{H},\mathbf{H}(t))$.
We now demonstrate that the minimum of $\mathscr{A}(\mathbf{H},\mathbf{H}')$ coincides with the update rule in . Since $$\begin{gathered}
\begin{split}
\frac{\partial\mathscr{A}(\mathbf{H},\mathbf{H}')}{\partial\mathbf{H}_{ij}} =2(\mathbf{\Phi^{-}D})_{ij}\Big(\frac{\mathbf{H}_{ij}}{\mathbf{H}'_{ij}}\Big)
+2\frac{(\Omega^{+}\mathbf{H'D})_{ij}\mathbf{H}_{ij}}{\mathbf{H'}_{ij}} \\
-2(\mathbf{\Phi}^{+}\mathbf{D})_{ij}\Big(\frac{\mathbf{H'}_{ij}}{\mathbf{H}_{ij}}\Big)
-2\frac{(\Omega^{-}\mathbf{H'D})_{ij}\mathbf{H}'_{ij}}{\mathbf{H}_{ij}} =0\\
\end{split}
\label{lhhprimedev}\end{gathered}$$ We solve for $\mathbf{H}_{ij}$, arriving at the update formula given in . Thus corresponds with a critical point for $\mathscr{A}(\mathbf{H},\mathbf{H'})$.
Computing the corresponding Hessian of $\mathscr{A}(\mathbf{H},\mathbf{H'})$ yields: $$\frac{\partial\mathscr{A}(\mathbf{H},\mathbf{H'})}{\partial \mathbf{H}_{ij}\partial \mathbf{H}_{kl}}=
\begin{cases}
2\frac{(\Phi^{-}\mathbf{D})_{ij}}{\mathbf{H'}_{ij}}
+2\frac{(\Omega^{+}\mathbf{H'D})_{ij}}{\mathbf{H}'_{ij}}
+2\frac{(\Phi^{+}\mathbf{D})_{ij}\mathbf{H'}_{ij}}{\mathbf{H}_{ij}^2}
+2\frac{(\Omega^{-}\mathbf{H'D})_{ij}\mathbf{H'_{ij}}}{\mathbf{H}_{ij}^2} \quad \text{ if } (i,j)==(k,l) \\
0 \hspace{8.6cm} \text{otherwise}
\end{cases}
\label{lhhprimehessian}$$
Hence $\mathscr{A}(\mathbf{H},\mathbf{H'})$ is convex, as was to be shown.
Finally, to conclude the proof of Lemma 4, we show that the iterative update formula given by additionally enforces non-negativity for the matrix $\mathbf{H}$. To this end, we define a matrix $\mathbf{\Lambda} \in \mathbb{R}^{k \times n}$ of Lagrangian multipliers. This gives the following associated Lagrangian: $$\begin{gathered}
\begin{split}
F(\mathbf{H})_{\Lambda}=tr[\mathbf{XD}\mathbf{X}^{T}]
-2tr[\mathbf{H}^{T}\mathbf{\Phi}^{+}\mathbf{D}]+2tr[\mathbf{H}^{T}\mathbf{\Phi}^{-}\mathbf{D}]\\
+tr[\Omega^{+}\mathbf{H}\mathbf{D}\mathbf{H}^{T}]-tr[\Omega^{-}\mathbf{H}\mathbf{D}\mathbf{H}^{T}]- \Lambda \odot \mathbf{H}
\end{split}
\label{lagrangian}\end{gathered}$$ where $\odot$ denotes the Hadamard product. The gradient of the Lagrangian is therefore:\
$$\nabla_{H}F(\mathbf{H})_{\Lambda}=-2\Phi^{+}\mathbf{D}+2\Phi^{-}\mathbf{D}+2\Omega^{+}\mathbf{HD}-2\Omega^{-}\mathbf{HD}-\Lambda
\label{lagrangiangrad}$$
The Karush-Kuhn-Tucker (KKT) conditions [@10.5555/1355334] dictate that a necessary condition for optimality with the prescribed non-negative constraints is $\mathbf{H}^{*}\odot \mathbf{\Lambda}=0$, where $\mathbf{H}^{*}$ is optimal. This indicates that an optimal solution necessarily satisfies: $$-\Phi^{+}\mathbf{D}+\Phi^{-}\mathbf{D}+\Omega^{+}\mathbf{HD}-\Omega^{-}\mathbf{HD}-\frac{1}{2}\mathbf{\Lambda}=0
\label{kkt1}$$ which implies the following by the KKT slackness condition: $$\mathbf{H}_{ij}(-\Phi^{+}\mathbf{D}+\Phi^{-}\mathbf{D}+\Omega^{+}\mathbf{HD}-\Omega^{-}\mathbf{HD})_{ij}=0
\label{kkt2}$$ Equivalently, the optimal solution satisfies: $$\mathbf{H}_{ij}^2(-\Phi^{+}\mathbf{D}+\Phi^{-}\mathbf{D}+\Omega^{+}\mathbf{HD}-\Omega^{-}\mathbf{HD})_{ij}=0
\label{kktslackness}$$ Solving for $\mathbf{H}_{ij}$ renders formula . This concludes the proof of Lemma 4. $\square$\
**Lemma 5.** Let $H(t)$ and $H(t+1)$ represent consecutive updates for $\mathbf{H}$ as prescribed by . Under this updating rule, the following inequality holds: $$\begin{gathered}
\begin{split}
tr[(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t+1))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t+1))^{T}] \\
\leq tr[(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))^{T})]
\end{split}
\label{lemma5}\end{gathered}$$ **Proof.** The proof of Lemma 5 follows directly from Lemma 4 and Lemma 3. $\square$\
**Lemma 6.** Under the update rule of , the following inequality holds: $$\begin{gathered}
\begin{split}
\|\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t+1)\|_{2,1}-\|\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t)\|_{2,1} \\
\leq \frac{1}{2}\Big[ tr[(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t+1))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t)(\mathbf{H}(t+1))^T] - \\ tr[(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))\mathbf{D}(t)(\mathbf{X}-\mathbf{W}(t)\mathbf{H}(t))^T] \Big]
\end{split}
\label{lemma6}\end{gathered}$$
**Proof.** The proof of Lemma 6 follows analogously from the proof of Lemma 2.
**Theorem 2.** Updating $\mathbf{H}$ using formula while fixing $\mathbf{W}$ yields a monotonic decrease in the objective function defined by (5).\
\
**Proof.** The proof of Theorem 2 is similar to Theorem 1, via Lemma 5 and Lemma 6.
Experimental Results
====================
We perform two general experiments to compare the performance of L21 SNF Algorithm with SNF [@4685898]: (1) general data compression via matrix factorization, and (2) qualitative facial image data reconstruction via matrix factorization. To compare general data compression performance, we begin with randomized, mixed-sign data matrices $\mathbf{X}$ (in the range $[-20,20]$) of dimension $10,000 \times 128$. In each case, we perform different degrees of compression; through separate trials, we reduce $\mathbf{X}$ to dimension ${10,000 \times 64}$, ${10,000 \times 32}$, ${10,000 \times 16}$, and ${10,000 \times 8}$. In particular, these extreme matrix dimensions are inspired by the potential use-case applications of highly overdetermined systems (e.g., deep CNN compression, genomic data compression, etc.).\
Using cluster-based initialization schemes for $\mathbf{W}$ and $\mathbf{H}$, we compare reconstruction loss using (2) different metrics: (i) normalized Frobenius loss (NFL), i.e., $\frac{\|X-WH\|_F}{\|X\|_F}$ and (ii) normalized L21 loss (NL21) , i.e. $\frac{\|X-WH\|_{2,1}}{\|X\|_{2,1}}$. For initialization, we run K-means for five iterations; this yields initial cluster centroids and indicators. We initialize the basis matrix $\mathbf{W}$ to the rendered cluster centroids; we set the initial coordinates instantiated by $\mathbf{H}$ per the basis set to 1.2 when the datum belongs to the cluster, and 0.2 otherwise. Table 1 and Figure 1 summarizes these findings. Across our experiments we optimize the regularization hyperparameter $\alpha $ using random search [@journals/jmlr/BergstraB12] over the interval \[0,1\].\
Overall, the L21 SNF algorithm demonstrates a substantial improvement in comparison with SNF [@4685898] and PCA in reducing L21-based reconstruction loss across each of our experiments, while at the same time maintaining generally strong results for L2-based reconstruction loss (see Figure 1, Table 1). In particular, L21 SNF exhibits significant gains in the case of severely overdetermined systems. In experimental trials of reducing random, mixed-sign matrices of initial dimension $10,000 \times 128$, for instance, L21 SNF shows a relative improvement of 26% over SNF for the 50% compression task, while exhibiting only a 4% increase in L2 loss, comparatively; similarly, for the 75% compression task, L21 SNF demonstrates an 11% relative improvement over SNF with respect to L21 loss, and only a 1% increase in L2 loss compared with SNF.\
Lastly, we compare compression quality rendered by L21 SNF with SNF and PCA for the task of compression on a batch of images. For this experiment, we randomly sampled 200 images from the *Large-scale CelebFaces Attributes* (CelebA) dataset [@10.1109/ICCV.2015.425]. Each image is of dimension ${89 \times 108}$; we flattened and concatenated this batch of images, rendering a data matrix $\mathbf{X}$ of dimension ${9,612 \times 200}$. We then ran each of the L21 SNF, SNF, and PCA algorithms for 250 iterations, reducing the original matrix to size ${9,612 \times 100}$. The results of this experiment are shown in Figure 2.\
Figure 2 in particular provides a qualitative illustration of the stark contrast in performance among L21 SNF, SNF [@4685898] and PCA for compression applied to severely overdetermined datasets. While the reconstruction fidelity for L21 SNF is comparable with the original images from the CelebA dataset, both the SNF and PCA techniques performed poorly by comparison, as each introduces a significant amount of distortion and image artifacts in the reconstruction process.
Discussion
==========
We present a new, robust data compression algorithm which renders a parts-based compression of mixed-sign data. Theorems 1 and 2 furnish proofs of the convergence of the iterative updates given by our L21 SNF algorithm. Through experiments, we demonstrate the use-case advantages of our algorithm over the classic NMF and SNF algorithms, particularly in the case of highly overdetermined systems. In future work, we aim to generalize these results to more complex constraints, including sparsity. We anticipate that our algorithm can potentially be applied to a variety of relevant real-world applications in the future, including the compression of deep CNNs, problems in computational biology, and general clustering paradigms.
Compression NFL (ours) NL21 (ours) NFL (SNF) NL21 (SNF)
------------------- --------------- ------------------------------------ --------------- ---------------------------------- --
${10k \times 64}$ 0.704 (0.694) $\mathbf{0.498 \text{ } (0.498)}$ 0.674 (0.673) $\mathbf{0.672 \text{ }(0.620)}$
${10k \times 32}$ 0.865 (0.855) $\mathbf{0.749 \text{ } (0.749) }$ 0.845 (0.846) $\mathbf{0.845 \text{ }(0.844)}$
${10k \times 16}$ 0.935 (0.929) 0.874 (0.874) 0.925 (0.924) 0.924 (0.923)
${10k \times 8}$ 0.968 (0.964) 0.937 (0.937) 0.962 (0.962) 0.962 (0.962)
: Summary of loss measures for L21 SNF algorithm (ours) vs SNF run for 100 iterations, beginning with random, mixed-sign matrix of dimension ${10,000 \times 128}$.
{width="70.00000%"}
![Results for compression of batch of 200 face images sampled from the CelebA [@10.1109/ICCV.2015.425] dataset; we show a sample of seven randomly selected images after compression. The original image batch of dimension ${9,612 \times 200}$ was compressed to ${9,612 \times 100}$; each algorithm was run for 250 iterations. Top: ground-truth images; Second from Top: L21 SNF (ours) rendered result; Second from Bottom: SNF results; Bottom: PCA results. ](face_data_PCA_SNF.png){width="75.00000%"}
|
---
author:
- 'P. Bechtle'
- 'O. Brein'
- 'S. Heinemeyer'
- 'G. Weiglein'
- 'K. Williams'
title: New HiggsBounds from LEP and the Tevatron
---
[ address=[DESY, Notkestrasse 85, 22607 Hamburg, Germany]{} ]{}
[ address=[Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, 79106 Freiburg, Germany]{} ]{}
[ address=[Instituto de Física de Cantabria (CSIC-UC), Santander, Spain]{} ]{}
[ address=[Institute for Particle Physics Phenomenology, Durham University, Durham, DH1 3LE, UK]{} ]{}
[ address=[Bethe Center for Theoretical Physics, Physikalisches Institut der Universität Bonn, Nussallee 12, 53115 Bonn, Germany]{} ]{}
Introduction
============
The search for Higgs bosons is a major cornerstone of the physics programs of past, present and future high energy colliders. The LEP and Tevatron experiments, in particular, turned the non-observation of Higgs bosons into constraints on the Higgs sector, which can be very useful in reducing the available parameter space of particle physics models. Such constraints will continue to be important far into the LHC era as they will need to be taken into account in the interpretation of any new physics scenario.
The constraints are provided by experiments in the form of limits on cross sections of individual signal topologies (such as $e^+e^-\to h_iZ\to b\bar{b}Z$ or $p\bar p\to h_i Z\to b\bar b l^+l^-$) or in the form of combined limits for a specific model, such as the SM. The latter type of analyses include detailed knowledge of the overlap between the individual experimental searches, and therefore have a high sensitivity, whereas the former can be used to test a wide class of models.
Comparing the predictions of a particular model with the existing experimental bounds on the various search topologies can be quite a tedious task as it involves the implementation of experimental results that are distributed over many different publications and combining these results requires a procedure to ensure the correct statistical interpretation of the exclusion bounds obtained on the parameter space of the model.
The program [HiggsBounds]{} [@higgsbounds] is a tool designed to facilitate the above task so that wide classes of models can easily be checked against the state-of-the-art results from Higgs searches. This should be useful for applications in Higgs phenomenology and model building (see, e.g., [@diffH; @Master3]). [HiggsBounds]{} takes theoretical Higgs sector predictions, e.g. for a particular parameter scenario of a model beyond the SM, as input. It determines which Higgs search analysis has the highest exclusion power according to a list of expected exclusion limits from LEP and the Tevatron (an expected limit corresponds to the bound that one would obtain in the hypothetical case of an observed distribution that agrees precisely with the background expectation). In order to ensure the correct statistical interpretation of the obtained exclusion bound as a 95% CL, the comparison of the model with the experimental limits has to be restricted to the single channel that possesses the highest statistical sensitivity. For this channel, the program then compares the theoretical prediction for the Higgs production cross section times decay branching ratio with the actual experimental limit and determines whether or not the considered parameter point of the model is excluded at 95% CL.
The Code [HiggsBounds]{}
========================
The code roughly works as follows. The user provides the Higgs sector predictions of the model (point in the parameter space) under consideration. For each neutral Higgs boson $h_i \; (i=1,\ldots,n_\HIGGS)$ in the model, this will usually include the mass, total decay width, branching ratios and Higgs production cross sections: $$\label{basic input}
M_{h_i} \,,
\Gamma_{\TOT}(h_i)\,,
\BR_\MOD(h_i\to ...)\,,
\sigma_\MOD(P) / \sigma_{\REF}(P)\, .$$ Where it exists, $\sigma^\SM(P)$ is used as the reference cross section. Variations on this input format are offered. As an example, the branching ratios and production cross section can be replaced by the effective couplings of the Higgs boson(s) to SM particles. More details are given in the [HiggsBounds]{} manual [@higgsbounds].
A complete list of the experimental analyses included in the first release of [HiggsBounds]{} is given in [@higgsbounds]. The included results from LEP and the Tevatron consist of tables of expected (based on MC simulations with no signal) and observed 95% CL cross section limits, with a variety of normalizations. The set mainly consists of analyses for which model-independent limits were published. However, we also included some dedicated analyses carried out for the case of the SM. These analyses are only taken into account as a possible exclusion bound if the Higgs boson in question would appear sufficiently ‘SM-like’ to this analysis. Roughly speaking, this requires that the ratios of all involved couplings to the SM couplings are approximately equal.
For each Higgs process $X$ (here, we treat each combination of Higgs bosons in each experimental analysis as a separate $X$), [HiggsBounds]{} uses the input to calculate the quantity $Q_\MOD(X)$, which, up to a normalization factor, is the predicted cross section times branching ratio for $X$. In order to ensure the correct statistical interpretation of the results, it is crucial to only consider the experimentally observed limit for one particular $X$. Therefore, [HiggsBounds]{} must first determine $X_0$, which is defined as the process $X$ with the highest statistical sensitivity for the model point under consideration. In order to do this, the program uses the tables of expected experimental limits to obtain a quantity $Q_\EXPEC$ corresponding to each $X$. The process with the largest value of $Q_\MOD/Q_\EXPEC$ is chosen as $X_0$. [HiggsBounds]{} then determines a value for $Q_\OBS$ for this process $X_0$, using the appropriate table of experimentally observed limits. If $$Q_\MOD(X_0) / Q_\OBS(X_0) > 1 \,,
\label{eq:modvsobs}$$ [HiggsBounds]{} concludes that this particular parameter point is excluded at 95 % CL.
The [HiggsBounds]{} package (current version [1.2.0]{}) can be obtained from\
[http://www.ippp.dur.ac.uk/HiggsBounds]{}
\
The code has both a Fortran 77 and Fortran 90 version. It can be operated in a command line mode that can process input files in a variety of formats, as a subroutine suitable for inclusion in user applications, and as an online version, available at its home page. The package includes sample programs which demonstrate how [HiggsBounds]{} can be used in conjunction with the widely used MSSM Higgs sector programs [FeynHiggs]{} [@feynhiggs; @mhiggslong; @mhiggsAEC; @mhcMSSMlong] and [CPsuperH]{} [@cpsh].
Newly implemented and updated bounds
====================================
After the first release of [HiggsBounds]{} more search channels from LEP [@LEPHiggsSM; @LEPHiggsMSSM; @LEPflavindep; @LEPHgaga] and the Tevatron [@CDFHVbbMET; @D0HVbbMET; @CDFHWW; @D0hbbtautau; @D0hgaga] have been implemented on top of what is described in [@higgsbounds]. These newly implemented bounds are summarized in Tab. \[tab:new\].
---------------------------------------------------------------------------------- ----------------------------------------------------------------------------------
[$e^+e^-\to h_k \, Z\to X + Z$ ]{} [@LEPflavindep] [$e^+e^-\to h_k \, Z \to \gamma\gamma\; Z$]{} [@LEPHgaga]
[$p\bar p \to h_k V \to b\bar b$ $+$ missing $E_T$ [@CDFHVbbMET; @D0HVbbMET] ]{} [$p \bar p \to H + X \to WW + X$ [@CDFHWW; @D0HWW]]{}
[$p \bar p \to h_k b \to \tau^+\tau^- b$ [@D0hbbtautau]]{} [$p \bar p \to H + X$ (SM combined)]{} [@CDFD0HWW; @CDFhxx; @D0hxxSM; @CDFhxxSM]
---------------------------------------------------------------------------------- ----------------------------------------------------------------------------------
: Summary of newly implemented bounds into [HiggsBounds]{} after its first release [@higgsbounds]. $h_k$ denotes a generic neutral Higgs boson, while $H$ denotes a SM-like Higgs boson. []{data-label="tab:new"}
The code is constantly kept up-to-date by the inclusion of updated results published by the Tevatron experiments. This is illustrated in Tab. \[tab:update\], where we list the search channels that have been updated since the initial release.
---------------------------------------------------------------------------------- --------------------------------------------------------------------------------------
[$p \bar p \to h_k\,Z \to b \bar b \, ll$]{} [@D0hZbb; @CDFhZbb; @CDFHSMcomb] [$p \bar p \to h_k\,W \to b \bar b \, l \nu_l$]{} [@D0hWbb; @CDFhWbb; @D0HW; @CDFHW]
[$p \bar p \to h_k\,W \to WWW \to l \nu_l \, l \nu_l + X$]{} [@D0hWWW; @CDFhWWW] [$p \bar p \to H + X \to \gamma\gamma + X$]{} [@D0hgaga; @D0Hgaga]
[$p \bar p \to h_k \to \tau^+\tau^-$]{} [@CDFD0htautau; @CDFhtautau; @D0htautau]
---------------------------------------------------------------------------------- --------------------------------------------------------------------------------------
: Summary of Tevatron bounds that have been updated within [HiggsBounds]{} after its first release [@higgsbounds]. $h_k$ denotes a generic neutral Higgs boson, while $H$ denotes a SM-like Higgs boson. []{data-label="tab:update"}
Still missing are the bounds on charged Higgs bosons from both LEP and the Tevatron. They will be implemented into [HiggsBounds]{} in the near future.
We are grateful for the valuable assistance of A. Read, P. Igo-Kemenes, M. Owen, T. Junk, M. Herndon and S. Pagan Griso. This work has been supported in part by the European Community’s Marie-Curie Research Training Network under contract MRTN-CT-2006-035505 ‘Tools and Precision Calculations for Physics Discoveries at Colliders’ (HEPTOOLS). P.B. was partially supported by the Helmholtz Young Investogator Grant VH-NG-303 and the DFG Collaborative Research Center SFB 676.
[99]{}
|
---
abstract: 'We construct in this paper a large class of superoscillating sequences, more generally of $\mathscr F$-supershifts, where $\mathscr F$ is a family of smooth functions (resp. distributions, hyperfunctions) indexed by a real parameter $\lambda\in \R$. The key model we introduce in order to generate such families is the evolution through a Schrödinger equation $(i\partial/\partial t - \mathscr H(x))(\psi)=0$ with a suitable hamiltonian $\mathscr H$, in particular a suitable potential $V$ when $\mathscr H(x) = -(\partial^2/\partial x^2)/2 + V(x)$. The family $\mathscr F$ is in this case $\mathscr F= \{(t,x) \mapsto \varphi_\lambda(t,x)\,;\, \lambda \in \R\}$, where $\varphi_\lambda$ is evolved from the initial datum $x\mapsto e^{i\lambda x}$. Then $\mathscr F$-supershifts will be of the form $\{\sum_{j=0}^N C_j(N,a) \varphi_{1-2j/N}\}_{N\geq 1}$ for $a\in \R\setminus [-1,1]$, taking $C_j(N,a) =\binom{N}{j}(1+a)^{N-j}(1-a)^j/2^N$. We prove the locally uniform convergence of derivatives of the supershift towards corresponding derivatives of its limit. We analyse in particular the case of the quantum harmonic oscillator, which forces us, in order to take into account singularities of the evolved datum, to enlarge the notion of supershifts for families of functions to a similar notion for families of hyperfunctions, thus beyond the frame of distributions.'
author:
- 'F. Colombo[^1], I. Sabadini$^*$, D.C. Struppa[^2], A. Yger[^3]'
title: '**Superoscillating sequences and supershifts for families of generalized functions**'
---
[**MSC numbers**]{}: 42A16, 30D15, 46F15.
Introduction
============
The Aharonov-Berry superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in the study of Aharonov weak measurements, [@aav; @abook; @av; @b5]. The literature related to superoscillations is very large; without claiming completeness, we mention [@berry; @berry-noise-2013; @berry2; @b1; @b4; @kempf2; @leeferreira; @kempf1; @lindberg]. Quite recently, this class of functions has been investigated from the mathematical point of view, see [@acsst4; @acsst3; @acsst1; @acsst6; @JFAA; @AOKI; @QS2; @BCS; @harmonic; @CSY; @CSSY] and the monograph [@acsst5]. Their theory is now very well developed, even though there are still open problems associated with superoscillatory functions, in particular as it concerns their longevity, when evolved according to a wide class of partial differential equations. Let $a>1$ be a real number. The archetypical superoscillatory sequence is the sequence of complex valued functions $\{x\mapsto F_N(x,a)\}_{N\geq 1}$ defined on $\mathbb{R}$ by $$\label{sect1-eq1}
F_N(x,a)=\Big(\cos\Big(\frac{x}{N}\Big)+ia\sin \Big(\frac{x}{N}\Big) \Big)^N
=\sum_{j=0}^N C_j(N,a)e^{i(1-2j/N){x}}$$ where $$\label{sect1-eq2}
C_j(N,a)={N\choose j}\left(\frac{1+a}{2}\right)^{N-j}\left(\frac{1-a}{2}\right)^j,$$ and ${N\choose j}$ denotes the binomial coefficient. The first thing one notices is that if we fix $x \in \mathbb{R}$, and we let $N$ go to infinity, we immediately obtain that $$\lim_{N \to \infty} F_N(x,a)=e^{iax}.$$ This new representation, together with the computation of the limit of $x\mapsto F_N(x,a)$ when $N$ goes to infinity, explains why such a sequence $\{x\mapsto F_N(x,a)\}_{N\geq 1}$ is called [*superoscillatory*]{}. Even though, for every $N$, the frequencies $(1-2j/N)$ that appear in the Fourier representation of $F_N$ are bounded by one, the limit function is $x\mapsto e^{iax}$, where $a$ can be an arbitrarily large real number. If one considers the map $\lambda \in \R \mapsto \varphi_\lambda$, where $\varphi_\lambda~: x\in \R \mapsto e^{i\lambda x}$, one says that $\lambda \mapsto \{x \mapsto F_N(x,\lambda)\}_{N\geq 1}$ realizes a [*supershift*]{} for $\lambda \mapsto \varphi_\lambda$, or also that $\lambda \mapsto \varphi_\lambda$ admits $\lambda \mapsto
\{x\mapsto F_N(x,\lambda)\}_{N\geq 1}$ as a [*supershift*]{}. Such a notion will be made precise in Definition \[sect5-def1\]. Let $t\in \R$ or $t\in \R^+$ be a real parameter and $P= \gamma_0 + \gamma_1 X + \cdots + \gamma_d X^d \in \C[X]$ with $\gamma_d\not=0$. For any $\lambda \in \R$ and $N\in \N^*$, let $$\psi_{P,N}(t,x,\lambda) = \sum\limits_{j=0}^N C_j(N,\lambda)\, e^{i(1-2j/N)x} e^{it P(1-2j/N)},$$ so that $$\Big[i\, \frac{\partial}{\partial t} +
\Big(\sum\limits_{\kappa =0}^d \gamma_\kappa \Big(-i \frac{\partial}{\partial x}\Big)^\kappa\Big)\Big] (\psi_{P,N}(t,x,\lambda)) \equiv 0\quad {\rm on}\ \R^2_{t,x},$$ together with the initial condition $$\big[\psi_{P,N}(t,x,\lambda)\big]_{t=0} = F_N(x,\lambda).$$ In the particular case where $P$ is even with real coefficients, namely $P = \sum_{\kappa'=0}^{d'} \gamma_{2\kappa'} X^{2\kappa'}$ with $\gamma_{2\kappa'}\in \R$ for $\kappa'=0,...,d'$, $\gamma_{2d'}\neq 0$ and $\check P= \sum_{\kappa'=0}^{d'} (-1)^{\kappa'+1}\gamma_{2\kappa'} X^{2\kappa'}$, the function $(t,x)\in \R^2_{t,x}\longmapsto \psi_{P,N}(a,t,x)$ is the global solution of the Schrödinger type partial differential equation $\big(i\partial/\partial t - \check P(\partial/\partial x)\big)(\psi)\equiv 0$ in $\R^2_{t,x}$ evolved from the initial datum $x\mapsto F_N(x,\lambda)$ on the line $\{t=0\}$ in $\R^2_{t,x}$. Let $D_x := \partial/\partial x$ and $\odot$ denote the composition law between differential operators in the variable $x$ with coefficients depending on the parameter $t$. Observe then that one can rewrite formally for any $N\in \N^*$ $$\begin{gathered}
\psi_{P,N}(t,x,\lambda) = \sum\limits_{j=0}^N C_j(N,\lambda)\,
e^{ix(1-2j/N)}\, \prod_{\kappa =0}^d
\Big(\sum\limits_{\ell =0}^\infty
\frac{(i^{1-\kappa}\, t \gamma_\kappa)^\ell}{\ell!}
(i(1-2j/N))^{\kappa \ell}\Big) \\
= \sum\limits_{j=0}^N
C_j(N,\lambda) \, \Big(\bigodot_{\kappa = 0}^d\big(
\sum\limits_{\ell =0}^\infty
\frac{(i^{1-\kappa}\, t \gamma_\kappa)^\ell}{\ell!}\, D_x^{\kappa\,\ell}\big)\Big)\, (e^{ix(1-2j/N)}) \\
=
\Big(\bigodot_{\kappa = 0}^d
\big(\sum\limits_{\ell =0}^\infty
\frac{(i^{1-\kappa}\, t \gamma_\kappa)^\ell}{\ell!}\, D_x^{\kappa\,\ell}\big)\Big) \big(\sum\limits_{j=0}^N C_j(N,\lambda)\, e^{ix(1-2j/N)}\big).\end{gathered}$$ We will justify such a rewriting in section \[sect3\] and exploit it in order to prove (theorem \[sect3-thm3\]) that for any $(\mu,\nu)\in \N^2$ $$\frac{\partial^{\mu+\nu}}{\partial t^\mu
\partial x^\nu}
(\psi_{P,N}(t,x,\lambda)) \stackrel{N\longrightarrow \infty}{\longrightarrow}
\frac{\partial^{\mu+\nu}}{\partial t^\mu
\partial x^\nu} \, (e^{itP(\lambda)}\, e^{i\lambda x})$$ Let $a\in \R \setminus [-1,1]$. We deduce in this way from the original “superoscillating” sequence $\{x \mapsto F_N(x,a)\}_{N\geq 1}$ a large class of superoscillating sequences which are of the form $\{x\mapsto \psi_{P,N}(t,x,a)\}_{N\geq 1}$ where $t\in \R$ is interpreted as a real parameter, the superoscillating convergence being uniform with respect to the parameter $t$ ; moreover the superoscillating convergence property propagates through any differential operator in $t,x$, the convergence being uniform on any compact subset of $\R^2_{t,x}$. As we already mentioned, the class of superoscillating sequences $\{x\mapsto \psi_{P,N}(t,x,a)\}_{N\geq 1}$ introduced previously includes examples where $(t,x)\mapsto \psi_{P,N}(t,x,a)$ is realized through the evolution in $t$ (from the initial value $t=0$) of the solution of a Cauchy problem (with initial datum on $\{t=0\}$) attached to a Schrödinger operator $i\partial/\partial t - \check P(\partial/\partial x)$, where $P$ is a real even differential operator with constant coefficients (the case where $\check P(\partial/\partial x) = - \partial^2/\partial x^2$ corresponds for example to the classical case of Schrödinger equation for a free particle). Indeed if, for $\lambda\in \R$, one denotes the response (at the time $t$) to the input datum $x\mapsto e^{i\lambda x}$ (when $t=0$) with the function $(t,x)\longmapsto \varphi_\lambda(t,x)$, then for any $a\in \R$ the function $(t,x)\mapsto \psi_{P,N}(t,x,a)$ can be expressed as $$\psi_{P,N}(t,x,a) = \sum\limits_{j=0}^N C_j(N,a)\, \varphi_{1-2j/N}(t,x).$$ Since, when $\lambda>1$ is arbitrarily large and $N\in \N^*$, the function $(t,x)\mapsto \psi_{P,N}(t,x,\lambda)$ arises from [*shifted*]{} functions $(t,x)\mapsto \varphi_\lambda(t,x)$ with $|\lambda|\leq 1$, one can still say that $\lambda \in \R \mapsto \varphi_\lambda$ (considered from $\R$ into the space of functions of the variables $(t,x)$ in the phase domain, here $\R^2_{t,x}$) admits $\lambda \mapsto \big\{(t,x)\mapsto \psi_{P,N}(t,x,\lambda)\big\}_{N\geq 1}$ as a [*supershift*]{}. Given a Schrödinger operator $$i\frac{\partial}{\partial t} + \frac{1}{2} \frac{\partial^2}{\partial x^2} + V(x)$$ with a suitable real potential $V$ and Green function $G_V~: (t,x,x') \mapsto
G_V(t,x,0,x')$ such that $$\varphi_\lambda (t,x) = \int_\R G_V(t,x,0,x')\, e^{i\lambda x'}\, dx' \\\quad {\rm or}
\int_{\R^+} G_V(t,x,0,x')\, e^{i\lambda x'}\, dx'$$ for $\lambda \in \R$ and $(t,x)$ in the phase domain (as a regularized integral on $\R$ or $\R^+$ in a sense that will be precised in section \[sect4\]), we will settle sufficient conditions that ensure in particular that the integral operator $$\begin{gathered}
T(x'\mapsto f(x'))(t,x) = \int_{\R} G_V(t,x,0,x')\, (x'\mapsto f(x'))\, dx'\\{\rm or}\quad
\int_{\R^+} G_V(t,x,0,x')\, (x'\mapsto f(x'))\, dx'\end{gathered}$$ is such that for any $\lambda \in \R$, $$\begin{gathered}
T \Big(x' \mapsto \sum\limits_{j=0}^N C_j(N,\lambda)\, e^{i(1-2j/N) x'}\Big)(t,x)
= \sum\limits_{j=0}^N
C_j(N,\lambda)\, \varphi_{1-2j/N}(t,x)\\
\stackrel{N\longrightarrow \infty}{\longrightarrow} \, T(x'\mapsto e^{i\lambda x'})(t,x) =
\varphi_\lambda(t,x)\end{gathered}$$ locally uniformly in some open subset $\mathscr U$ of the phase space (in $\R^2_{t,x}$) on which $V$ is smooth and which is entirely determined by the explicit expression of the Green function $G_V$ (theorem \[sect5-thm1\]). In such a situation the map $\lambda \in \R \mapsto (\varphi_\lambda)_{|\mathscr U}$ admits then the sequence $$\big\{\lambda \mapsto \big(\sum\limits_{j=0}^N C_j(N,a)\, T(
x'\mapsto e^{i(1-2j/N)x'})\big)_{|\mathscr U}\big\}_{N\geq 1} =
\big\{\lambda \mapsto \big(\sum\limits_{j=0}^N C_j(N,a)\, \varphi_{1-2j/N}\big)_{|\mathscr U}\big\}_{N\geq 1}$$ (where $a\in \R\setminus [-1,1]$) as a supershift. Moreover, for any $\lambda \in \R$, $\varphi_\lambda \in \mathscr C^\infty(\mathscr U,\C)$ and for any $(\mu,\nu)\in \N^2$, $$\frac{\partial^{\mu+\nu}}{\partial t^\mu
\partial x^\nu} \Big(
\sum\limits_{j=0}^N C_j(N,a) \, \varphi_{1-2j/N}\Big)
\stackrel{N\longrightarrow \infty}{\longrightarrow}
\frac{\partial^{\mu+\nu}}{\partial t^\mu
\partial x^\nu} \, (\varphi_a)$$ locally uniformly in $\mathscr U$. Interesting situations occur when $\lambda\in \R \mapsto
\varphi_\lambda$ makes sense as a continuous map from $\R$ into $\mathscr D'(\mathscr U',\C)$ for some open subset $\mathscr U'
\supsetneq \mathscr U$ in the phase space. Such is the case in the example of the quantum harmonic oscillator, where $V(x)=x^2/2$, the phase space is $\R^{+*} \times \R$ and $$\mathscr U = (\R^+ \times \R) \setminus \{\big(k\, \pi/2,x\big)\,;\,
k\in \N,\ x\in \R\} \subset \mathscr U' = \R^{+*} \times \R.$$ In such case, given $k'\in \N$ and $x_0\in \R$, it is impossible to interpret $$\label{sect1-eq3}
\big\{\lambda \mapsto \big(\sum\limits_{j=0}^N C_j(N,a)\, \varphi_{1-2j/N}\big)_{{\rm about}\ ((2k'+1)\pi/2,x_0)}\big\}_{N\geq 1}$$ (when $a\in \R \setminus [-1,1]$) as a supershift for $\lambda \longmapsto (\varphi_\lambda)_{|{\rm about}\, ((2k'+1)\pi/2,x_0)}$ (all maps being considered here as distribution-valued about $((2k'+1)\pi/2,x_0)$), while it is possible to do so about a point $(k''\pi,x_0)$, where $k''\in \N^*$. In order to interpret as a supershift for $\lambda \mapsto (\varphi_\lambda)_{{\rm about}\ ((2k'+1)\pi/2,x_0)}$, one needs to consider $(\varphi_\lambda)_{|{\rm about}\ ((2k'+1)\pi/2,x_0)}$ as a hyperfunction (in $t$) times a distribution (in $x$) instead of distribution in $(t,x)$. We will discuss such questions in section \[sect6\].
The plan of the paper is the following: the paper contains five sections, besides this introduction. In section 2 we introduce the spaces $A_p(\mathbb C)$, $A_{p,0}(\mathbb C)$, and we define some infinite order differential operators with nonconstant coefficients which will play a crucial role to prove our main results. In section 3 we recall the definition of generalized Fourier sequence and (complex) superoscillating sequence in one and several variables together with some examples; we then study two Cauchy-Kowalevski problems (one of which of Schrödinger type) and we show that superoscillations persist in time. In section 4 we address the problem of explaining the process of regularization of formal Fresnel-type integrals which is a necessary step to obtain further results in the paper. Fresnel-type integrals are shown to be continuous on $A_1(\mathbb C)$ in section 5, in which we also treat a Cauchy problem for the Schrödinger equation with centrifugal potential and also for the quantum harmonic oscillator. Finally, in section 6, we investigate the evolution of superoscillating initial data with respect to the notion of supershift for the quantum harmonic oscillator, and we focus on singularities. It is interesting to note that in this case one needs to extend the concept of supershift in the case of hyperfunctions. [**Notations.**]{} We use the notations with capital letters $Z,d/dZ,W,d/dW,\check Z$ in the expressions of formal differential operators, besides the usual notation $z$ for the complex variable and $t$ (time) $x,x'$ (space) real variables.
On continuity of some convolution operators
===========================================
Let $f$ be a non-constant entire function of a complex variable $z$. We define $$M_f(r)=\max_{|z|=r}|f(z)|,\ \ \ \text{ for}\ \ r\geq 0.$$ The non-negative real number $\rho$ defined by $$\rho=\limsup_{r\to\infty}\frac{\ln\ln M_f(r)}{\ln r}$$ is called the [*order*]{} of $f$. If $\rho$ is finite then $f$ is said to be of [*finite order*]{} and if $\rho=\infty$ the function $f$ is said to be of [*infinite order*]{}. In the case $f$ is of finite order we introduce the non-negative real number $$\sigma=\limsup_{r\to\infty}\frac{\ln M_f(r)}{ r^\rho},$$ and call it the [*type*]{} of $f$. If $\sigma\in (0,\infty)$ we say $f$ is [*of normal type*]{}, while we say it is [*of minimal type*]{} if $\sigma=0$ and [*of maximal type*]{} if $\sigma=\infty$. Constant entire functions are considered of minimal type and order zero. In the sequel we will extensively make use of [*weighted spaces*]{} $A_p(\C)$ or $A_{p,0}(\C)$ of entire functions whose definition follows ; such spaces are classical, see e.g. [@BG_book; @Taylor].
\[sect2-def1\] Let $p$ be a strictly positive number. We define the space $A_p(\C)$ as the $\C$-algebra of entire functions such that there exists $B>0$ such that $$\sup\limits_{z\in \C} \big(|f(z)|\, \exp(-B|z|^p\big) <+\infty.$$ The space $A_{p,0}(\C)$ consists of those entire functions such that $$\forall\, \varepsilon>0,\ \sup\limits_{z\in \C}
\big(|f(z)|\, \exp(-\varepsilon|z|^p) < +\infty.$$
To define a topology in these spaces we follow [@BG_book Section 2.1]. For $p>0$, $B>0$ and for any entire function $f$, we set $$\|f\|_B:=\sup_{z\in{\mathbb C}}\{|f(z)|\exp(-B|z|^p)\}.$$ Let $A_p^B(\C)$ denote the $\C$-vector space of entire functions satisfying $\|f\|_B<\infty$. Then $\|\cdot\|_B$ defines a norm on $A_p^B(\C)$ so that $(A_p^B(\C),\|\ \|_B)$ is a Banach space and the natural inclusion mapping $A_p^B\hookrightarrow A_p^{B'}$ (when $0<B\leq B'$) is a compact operator from $(A_p^B(\C),\|\ \|_B)$ into $(A_p^B(\C),\|\ \|_{B'})$. For any sequence $\{B_n\}_{n\geq 1}$ of positive numbers, strictly increasing to infinity, we can introduce an LF-topology on $A_p(\C)$ given by the inductive limit $$A_p(\C):=\lim_{\longrightarrow} A_p^{B_n}(\C) .$$ Since this topology is stronger than the topology of the pointwise convergence, it is independent of the choice of the sequence $\{B_n\}_{n\geq 1}$. Thus, in this inductive limit topology, given $f$ and a sequence $\{f_N\}_{N\geq 1}$ in $A_p(\C)$, we say that $f_N\to f$ in $A_p(\C)$ if and only if there exists $n\in \N^*$ such that $f, f_N\in A_p^{B_n}(\C)$ for all $N\in \N^*$, and $\|f_N-f\|_{B_n}\to 0$ for $N\to\infty$. The topology on $A_{p,0}(\C)$ is given as the projective limit $$A_{p,0}(\C):=\lim_{\longleftarrow} A_p^{\varepsilon_n}(\C)$$ where $\{\varepsilon_n\}_{n\geq 1}$ is a strictly decreasing sequence of positive numbers converging to $0$. It can be proved, see [@BG_book Section 6.1], that $A_p(\C)$ and $A_{p,0}(\C)$ are respectively a DFS space and an FS space. When $p>1$, $A_{p,0}(\C)$ is the strong dual of $A_{p'}(\C)$ (where $1/p+1/p'=1$), the duality being realized as $$\mu \in (A_{p'}(\C))' \longmapsto
\Big[{\rm Fourier-Borel\ Transform\ of}\ \mu~: w\in \C \longmapsto \mu_z(e^{-zw})\Big] \in A_{p,0}(\C).$$ In the extreme case $p=1$, $A_1(\C)$ (also denoted as ${\rm Exp}(\C)$) is isomorphic to the space $\widehat{H(\C)}$ of analytic functionals, the duality being realized as $$T \in \widehat{H(\C)} \longmapsto \Big[
{\rm Fourier-Borel\ Transform\ of}\ T~: w\in \C \mapsto
T_z(e^{-zw})\Big] \in A_1(\C).$$ Here $H(\C)$ is equipped with its usual topology of uniform convergence on any compact subset. The following result is an immediate consequence of the definition of the topology in the spaces $A_p(\C)$ for $p>0$.
\[sect2-prop1\] Let $\bff = \{f_N\}_{N\geq 1}$ be a sequence of elements in $A_p(\C)$. The two following assertions are equivalent:
- the sequence $\bff$ converges towards $0$ in $A_p(\C)$ ;
- the sequence $\bff$ converges towards $0$ in $H(\C)$ and there exists $A_{\bff}\geq 0$ and $B_\bff\geq 0$ such that $$\label{sect2-eq1}
\forall\, N\in \N^*,\quad \forall\, z\in \C,\
|f_N(z)| \leq A_{\bff}\, e^{B_{\bff} |z|^p
}.$$
The first assertion means that there exists $B>0$ with $\lim_{N\rightarrow \infty} \|f_N\|_{B} =0$ (in particular $\|f_N\|_B \leq 1$ for $N\geq N_1$), which implies that the sequence $\bff$ converges to $0$ in $H(\C)$ and that $|f_N(z)|
\leq A e^{B |z|^p}$ with $B$ and $A=\sup(\tilde A_1,...,\tilde A_{N_1},1)$ independent of $N$ ($\tilde A_j=\sup_\C (|f_j(z)|\, e^{-B|z|^p})$ for $j=1,...,N_1$). Conversely, assume that the second assertion holds and take $B>B_\bff$, so that, given $\varepsilon>0$, there exists $R_\varepsilon >0$ such that $$\forall\, N\in \N^*,\
\sup\limits_{|z|\geq R_\varepsilon} |f_N(z)|\, e^{-B |z|^p}
\leq A_{\bff}\, e^{(B_\bff -B) R_\varepsilon^p} < \varepsilon.$$ On the other hand, since $\bff$ converges to $0$ uniformly on any compact subset of $\C$, in particular on $\overline{D(0,R_\varepsilon)}$, there exists $N_\varepsilon \in \N^*$ such that $$N\geq N_\varepsilon \Longrightarrow
\sup\limits_{|z|\leq R_\varepsilon}
|f_N(z)|\, e^{-B |z|^p} \leq \sup\limits_{|z|\leq R_\varepsilon}
|f_N(z)| < \varepsilon.$$ Therefore $\sup_{N\geq N_\varepsilon} \|f_N\|_B <\varepsilon$ and the sequence $\bff$ converges to $0$ in $A_p(\C)$.
To prove our main results we need an important lemma that characterizes entire functions in $A_p(\C)$ in terms of the behaviour of their Taylor development, see Lemma 2.2 in [@AOKI].
\[sect2-lem1\] The entire function $f: z\mapsto \sum_{j=0}^\infty f_j z^j$ belongs to $A_p(\C)$ if and only if there exists $C =C_f >0$ and $b= b_f >0$ such that $f\in A^p_{C,b}(\C)$, where $$\label{sect2-eq2}
A^p_{C,b}(\C) =
\Big\{\sum_{j=0}^\infty f_j z^j \in A_p(\C)\,; \forall\, j\in \N,\ |f_j| \leq C\, \frac{b^j}{\Gamma (j/p +1)}\Big\}.$$
The following lemmas are refinements of results previously stated in [@AOKI], except that we need here some extra dependency with respect to auxiliary parameters. They will be of crucial importance in order to prove the main results in the next sections.
\[sect2-lem2\] Let $\mathscr T$ be a set of parameters and $\tau \in \mathscr T \mapsto \D(\tau)$ be a differential operator-valued map $$\tau \in \mathscr T \longmapsto \D(\tau) = \sum\limits_{j=0}^\infty b_j(\tau) \Big(\frac{d}{dW}\Big)^j$$ (with $b_j~: \mathscr T \rightarrow \C$ for $j\in \N$) whose formal symbol $$\mathbb F~: (\tau,W) \in \mathscr T \times \C \longmapsto
\sum\limits_{j=0}^\infty b_j(\tau) W^j$$ realizes for each $\tau\in \mathscr T$ an entire function of $W$ such that $$\label{sect2-eq2bis}
\sup\limits_{\tau \in \mathscr T,W\in \C}
\big(|\mathbb F(\tau,W)|\, e^{-B\, |W|^p}\big) = A < +\infty$$ for some $p\geq 1$ and $B\geq 0$. Then $\D(\tau)$ acts as a continuous operator from $A_1(\C)$ into itself uniformly with respect to the parameter $\tau \in \mathscr T$.
It follows from Lemma \[sect2-lem1\] that the coefficient functions $\tau \longmapsto b_j(\tau)$ satisfy then uniform estimates $$\forall\, j\in \N,\ \forall\, \tau \in \mathscr T,\
|b_j(\tau)| \leq C \frac{b^j}{\Gamma(j/p+1)}$$ for some positive constants $C=C(\D)$ and $b=b(\D)$ depending only on the finite quantity $A$ in and $B$. Let $f: W \mapsto \sum_{\ell =0}^\infty a_\ell W^\ell \in A_1(\C)$. There are then (see again Lemma \[sect2-lem1\]) positive constants $\gamma$ and $\beta$ such that $|a_\ell|\leq (\gamma /\ell!)\, \beta^\ell$ for any $\ell\in \N$. Consider the action of $\D$ on such $f$. One has (for the moment formally) $$\begin{gathered}
\label{sect2-eq3}
\forall\, \tau \in \mathscr T,\
\D(\tau)(f) = \sum\limits_{j=0}^\infty
b_j(\tau) (d/dW)^j(f)= \sum\limits_{j=0}^\infty
b_j(\tau) \Big(\sum\limits_{\ell = 0}^\infty \frac{(j+\ell)!}{\ell!}\, a_{\ell+j}\, W^\ell\Big) \\
= \sum\limits_{\ell =0}^\infty \Big( \sum\limits_{j=0}^\infty
\frac{(j+\ell)!}{\ell!} b_j(\tau) a_{\ell +j}\Big)\, W^\ell\end{gathered}$$ with $$\label{sect2-eq4}
\sum\limits_{j=0}^\infty
\frac{(j+\ell)!}{\ell!} |b_j(\tau)| \, |a_{\ell +j}|
\leq \gamma\, C \frac{\beta^\ell}{\ell!}\, \sum\limits_{j=0}^\infty \frac{(b\, \beta)^j}{\Gamma(j/p+1)} = K(b, C, \beta,\gamma)\,
\frac{\beta^\ell}{\ell!}.$$ Therefore the formal identity is in fact a true one for any $W\in \C$, which shows that $\D(\tau)[f]\in A_1(\C)$ for any $\tau \in \mathscr T$, with $$\forall\, \tau \in \mathscr T,\ \forall\, W\in \C,\quad
|\D(\tau)(f)| \leq K(b,C,\beta,\gamma)\, e^{\beta |W|}.$$ Let $\EuFrak f = \{f_N\}_{N\geq 1}$ be a sequence converging towards $0$ in $A_1(\C)$ which is equivalent to say that $\sup (b_{f_N} + C_{f_N})<+\infty$ and that $\EuFrak f$ converges towards $0$ in $H(\C)$, see Proposition \[sect2-prop1\]. Then the sequence $\{\D(\tau)(f_N)\}_{N\geq 1}= \D(\tau)(\EuFrak f)$ is such that $$\forall\, N\in \N^*,\ \forall\, \tau \in \mathscr T,\ \forall\, W\in \C,\quad |\D(\tau)(f_N)(W)|\leq A_{\EuFrak f}\, e^{B_{\EuFrak f} |W|}$$ for some positive constants $A_{\EuFrak f}$ and $B_{\EuFrak f}$ depending only on $\D$ and $\EuFrak f$. Let $\EuFrak B > B_{\EuFrak f}$ and $\varepsilon >0$. Let $R=R_\varepsilon$ large enough such that $$\forall\, \tau \in \mathscr T,\ \forall\, N\in \N^*,\ \forall\, W \in
\C\ {\rm with}\ |W|> R,\ |\D(\tau)(f_N)(W)| e^{- \EuFrak B |W|} \leq \varepsilon.$$ Since $\D(\tau)(f_N)(W) = \sum_{\ell =0}^\infty a_{N,\ell}(\tau) W^\ell$ with $|a_{N,\ell}(\tau)| \leq (C_{\EuFrak f}/\ell!)\, b_{\EuFrak f}^\ell$ for some constants $C_{\EuFrak f}$ and $b_{\EuFrak f}$ independent on $\tau\in \mathscr T$ and on $N$ (see ) and the sequence $\EuFrak f$ converges to $0$ in $H(\C)$, one can find $N=N_\varepsilon$ such that $$\forall\, N\geq N_\varepsilon,
\ \forall\, \tau \in \mathscr T,\ \forall\,
W\in \C\ {\rm with}\ |W|\leq R,\ |\D(\tau)(f_N)(W)| \leq \varepsilon.$$ Hence the sequence $\D(\tau)(\EuFrak f)$ converges towards $0$ in $A_1(\C)$, uniformly with respect to the parameter $\tau$.
Since the next lemma involves as set of parameters $\mathscr T$ the set which is now given as split in the form $\mathscr T = \EuFrak T \times \C_Z$, where $\C_Z$ is already a copy of the complex plane, one needs to duplicate $\C_Z$ into an extra copy of $\C$ denoted as $\C_W$.
\[sect2-lem3\] Let $\EuFrak T$ be a set of parameters and $\EuFrak t \in \EuFrak T\mapsto \D(\EuFrak t,Z)$ be a differential operator-valued map $$\EuFrak t \in \EuFrak T \longmapsto \D(\EuFrak t,Z) = \sum\limits_{j=0}^\infty b_j(\EuFrak t,Z) \Big(\frac{d}{dZ}\Big)^j$$ (with $b_j~: \EuFrak T\times \C \rightarrow \C$, holomorphic in $Z$ for $j\in \N$) such that $$\label{sect2-eq4bis}
\forall\, \varepsilon >0,\quad
\sup\limits_{\EuFrak t\in \EuFrak T, (Z,W)\in \C^2}
\Big(\Big(\sum\limits_{j=0}^\infty |b_j(\EuFrak t,Z)|\, |W|^j\Big)\,
\exp (- \varepsilon\, |Z|^{\check p} - B\, |W|^{p})\Big) = A^{(\varepsilon)} < +\infty$$ for some $\check p>1$, $p \geq 1$ and $B\geq 0$. Then $\D(\EuFrak t,Z)$ acts as a continuous operator from $A_1(\C)$ into $A_{\check p,0}(\C)$ uniformly with respect to the parameter $\EuFrak t\in \EuFrak T$.
The function $$\label{sect2-eq5}
\F~: (\EuFrak t,Z,W) \longmapsto \sum\limits_{j=0}^\infty
\Big(\sum\limits_{\kappa =0}^\infty b_{j,\kappa}(\EuFrak t) Z^\kappa\Big)\, W^j =
\sum\limits_{\kappa =0}^\infty Z^\kappa\, \Big(
\sum\limits_{j=0}^\infty b_{j,\kappa}(\EuFrak t) \, W^j\Big)$$ is well defined and depends as an entire function of two variables of the variables $Z$ and $W$ (which also justifies in the application of Fubini theorem). Cauchy formulae in $\C \times \C$ show that for any $\EuFrak t\in \EuFrak T$, for any $j,\kappa\in \N$, $$\begin{gathered}
\label{sect2-eq6}
|b_{j,\kappa}(\EuFrak t)| = \frac{1}{4\pi^2}\Big|
\int_{|Z|=\check r,|W|=r} F(\EuFrak t,Z,W)
\frac{dZ}{Z^{\kappa+1}} \wedge \frac{dW}{W^{j+1}}\Big|
\leq A^{(\varepsilon)}\, \inf_{\check r>0}\frac{e^{\varepsilon \check r^{\check p}}}{r^\kappa}\times \inf_{r>0} \frac{e^{B r^p}}{r^j} \\
= A^{(\varepsilon)}
\Big(\frac{1}{\kappa}\Big)^{\kappa/\check p}
\times \Big(\frac{1}{j}\Big)^{j/p}\, ((\varepsilon \check p\, e)^{1/\check p})^\kappa\, ((Bpe)^{1/p})^j \\
\leq C_{\eta}\,
\frac{1}{\Gamma(\kappa/\check p+1) \Gamma(j/p+1)}\, (\eta\, \check b)^\kappa\, b^j\end{gathered}$$ for each $\eta>0$, with constants $C_{\eta}$, $\check b$ and $b$ independent on the parameter $\EuFrak t$. Let now $\EuFrak f = \{f_N\}_{N\geq 1}$ be a sequence of elements in $A_1(\C)$ which converges to $0$ in $A_1(\C)$. All differential operators $$\D_{\kappa}(\EuFrak t)~: = \sum\limits_{j=0}^\infty
b_{j,\kappa}(\EuFrak t) (d/dW)^j\quad (\kappa \in \N)$$ act continuously on $A_1(\C)$, as seen in Lemma \[sect2-lem2\]. Moreover, one has (plugging in the estimates ) that $$\begin{gathered}
\forall\, f\in A^1_{\gamma,\beta}(\C),\ \forall\, \EuFrak t\in \EuFrak T,\
\forall\, \kappa \in \N,\
\forall\, \ell\in \N,\\
(\D_{\kappa}(\EuFrak t)(f))_\ell
\leq \gamma\, \tilde C_{\eta}
\frac{(\eta\, \check b)^\kappa}{\Gamma(\kappa/\check p+1)}\, E_{1/p,1}
(\beta\, b)\, \frac{\beta^\ell}{\ell!}\end{gathered}$$ where $E_{1/p,1}~: \zeta \in \C \longmapsto \sum_0^\infty \zeta^k/\Gamma (k/p + 1)$ is the entire (with order $1/p$ and type $1$) Mittag-Leffler function. One has therefore for such $f\in A^1_{\gamma,\beta}(\C_W)$ that $$\label{sect2-eq7}
\forall\, \EuFrak t \in \EuFrak T,\ \forall\, \kappa \in \N,\
\forall\, W\in \C,\quad
|\D_{\kappa}(\EuFrak t)(f)(W)|
\leq \gamma\, C_{\eta}\, E_{1/p,1}(\beta\, b)\, \frac{e^{\beta |W|}}{\Gamma (\kappa/\check p +1)}$$ and (taking now $W=Z$) $$\label{sect2-eq8}
\forall\, \EuFrak t \in \EuFrak T,\
\forall\, Z \in \C,\quad
\sum\limits_{\kappa =0}^\infty
|Z|^\kappa\, |\D_{\kappa}(\EuFrak t)(f)(Z)|
\leq
\gamma\, C_{\eta}\, E_{1/p,1}(\beta\, b)\, e^{\beta |Z|}\,
\sum\limits_{\kappa =0}^\infty \frac{(\eta\,\check b\, |Z|)^\kappa}{\Gamma (k/\check p +1)}.$$ Since the Mittag-Leffler function $E_{1/\check p,1}$ has order $\check p>1$, the estimates (uniform in the parameter $\EuFrak t$ as well as on the function $f\in A^1_{\gamma,\beta}(\C)$) show that the differential operator acts continously from $A_1(\C)$ into $A_{\check p,0}(\C)$, uniformly with respect to the parameter $\EuFrak t\in \EuFrak T$. One just needs to repeat here the end of the proof of Lemma \[sect2-lem2\].
We conclude this section by proving a quantitative lemma which reveals to be essential in the sequel. It is a refinement of Lemma 1 in [@CSSY].
\[sect2-lem5\] Let $a\in \C$ with $\alpha := \max(1,|a|)$ and, for any $z\in \C$, $$F_N(z,a) :=
\Big(\cos \Big(\frac{z}{N}\Big) + i\, a\, \sin\Big(\frac{z}{N}\Big)\Big)^N$$ as in (with $z, a\in \C$ instead of $x, a\in \R$). For any $N\in \N^*$ and any $z\in \C$, one has $$\label{sect2-eq9}
\begin{split}
& |F_N(z,a)| \leq \exp\big(|a|\, |z| + |{\rm Im}(z)|\big)\leq
\exp\big((|a|+1)\, |z|\big)
\\
& |F_N(z,a) - e^{iaz}| \leq
\frac{2}{3}\,
\frac{|a^2-1|}{N}\, |z|^2\, \exp\big( (\alpha+1) |z|\big).
\end{split}$$
Let $${\rm sinc}: z \in \C \longmapsto \frac{\sin z}{z} =
\int_0^1 t\, \cos (tz)\, dt$$ be the sinus cardinal function; it satisfies $|{\rm sinc}(z)|
\leq e^{|{\rm Im}(z)|}$ for any $z\in \C$. One has then the upper uniform estimates $$\begin{gathered}
\label{sect2-eq10}
\forall\, N\in \N^*,\ \forall\, z\in \C,\ |F_N(z,a)| =
\Big|
\cos \Big( \frac{z}{N}\Big) + i a
\sin\Big(\frac{z}{N}\Big)\Big|^N =
\Big|
\cos \Big(\frac{z}{N}\Big) + i\, \frac{az}{N}\, {\rm sinc}
\Big(\frac{z}{N}\Big)\Big|^N \\
\leq e^{|{\rm Im}(z)|} \Big( 1 + \frac{|az|}{N}\Big)^N
\leq \exp (|a|\, |z| + |{\rm Im}(z)|) \leq \exp \big((|a|+1)|z|\big),\end{gathered}$$ which is the first chain of inequalities in . For any $N\in \N^*$, one has also $$\begin{gathered}
\label{sect2-eq11}
\Big|\cos \Big(\frac{z}{N}\Big) - \cos \Big(\frac{a z}{N}\Big)\Big|
=
2\,\Big|\sin \Big(
\frac{(a-1)z}{2N}\Big)\, \sin \Big(\frac{(a+1)z}{2N}\Big)\Big| \\
\leq \frac{|a^2-1|}{2N^2}\, |z|^2\, \exp
\Big(\frac{|a-1|+|a+1|}{2N}\, |z|\Big)
\leq \frac{|a^2-1|}{2N^2}\, |z|^2\, \exp \Big(\frac{\alpha+1}{N}\, |z|\Big)\end{gathered}$$ and $$\begin{gathered}
\label{sect2-eq12}
\Big|
a \sin \Big(
\frac{z}{N}\Big) -
\sin \Big(
\frac{a z}{N}\Big)\Big|
=
\Big|
\sum\limits_{k=0}^\infty
\frac{(-1)^{k}}{(2k+1)!} (a-a^{2k+1})\, \Big(
\frac{z}{N}\Big)^{2k+1}\Big| \\
= \frac{|a^2-1|}{N^2}\, |z|^2\,
\Big|
\sum\limits_{k=1}^\infty
\frac{(-1)^{k}}{(2k+1)!}\, \Big(\sum\limits_{\ell=0}^{k-1}
a^{2\ell+1}\Big)
\, \Big(
\frac{z}{N}\Big)^{2k-1}\Big| \\
\leq \frac{|a^2-1|}{2N^2}\, |z|^2
\sum\limits_{k=1}^\infty
\frac{\alpha^{2k-1}}{(2k-1)!(2k+1)}\, \Big(
\frac{|z|}{N}\Big)^{2k-1} \\
\leq \frac{|a^2-1|}{6N^2}\, |z|^2
\, \sum\limits_{k=1}^\infty
\frac{1}{(2k-1)!}\, \Big(
\frac{\alpha |z|}{N}\Big)^{2k-1} \leq \frac{|a^2-1|}{6N^2}\, |z|^2
\, \exp \Big(
\frac{\alpha}{N}\, |z|\Big).\end{gathered}$$ It follows from the identity $A^N-B^N = (A-B) \sum\limits_{k=0}^{N-1}
A^k B^{N-1-k}$, together with estimates , and , that for any $N\in \N^*$ and $z\in \C$, $$\begin{gathered}
|F_N(z,a) - e^{iaz}| =
\Big|\cos \Big(\frac{z}{N}\Big) - \cos \Big(\frac{a z}{N}\Big)
+ i \Big(a \sin \Big(
\frac{z}{N}\Big) -
\sin \Big(
\frac{a z}{N}\Big)\Big)\Big| \\
\times \sum\limits_{k=0}^{N-1}
|F_N(z,a)|^{k/N} \,
\Big|\exp\big( ia z \frac{N-1-k}{N}\big)\Big| \\
\leq \frac{2}{3}\, \frac{|a^2-1|}{N^2}\, |z|^2\,
\exp\Big(\frac{\alpha+1}{N}\, |z|\Big)\,
\sum\limits_{k=0}^{N-1}
\exp\Big(k\, \Big(\frac{|a|+1}{N}\Big)\, |z|
+ \frac{N-1-k}{N}\, |a|\, |z|
\Big)\\
\leq \frac{2}{3}\, \frac{|a^2-1|}{N}\, |z|^2\, \exp\big(
(\alpha+1)|z|\big).\end{gathered}$$ The second inequality in is thus proved.
One can now state as a consequence of Proposition \[sect2-prop1\] and Lemma \[sect2-lem5\] the following theorem.
\[sect2-thm1\] For any $a\in \C$, the sequence $\{z\mapsto F_N(z,a)\}_{N\geq 1}$ converges to $z\mapsto e^{iaz}$ in $A_1(\C)$.
It follows from estimates that the sequence $\bff = \{z\mapsto F_N(z,a)\}_{N\geq 1}$ satisfies the estimates with $p=1$, $B_\bff = |a|+1$ and $C_\bff = 1$. Lemma \[sect2-lem5\] implies on the other hand that the sequence $\bff$ converges towards $z\mapsto e^{iaz}$ in $H(\C)$. The result is then a consequence of Proposition \[sect2-prop1\].
Uniform convergence of superoscillating sequences {#sect3}
=================================================
Let $m\in \N^*$ and $(\mathscr F(\R^m,\C))^{\N^*}$ be the family of all sequences $\boldsymbol Y=\{x\in \R^m \mapsto Y_N(x)\}_{N\geq 1}$ of complex valued functions defined on $\R^m$. We first recall in this section the notions of (complex) [*generalized Fourier sequence*]{} (CGFS) and (complex) [*superoscillating sequence*]{} (CSOscS) in $(\mathscr F(\R^m,\C))^{\N^*}$. We start first with the case $m=1$.
\[sect3-def1\] A sequence $\boldsymbol Y \in (\mathscr F(\R,\C))^{\N^*}$ is called a [complex generalized Fourier sequence]{} if each entry $Y_N$ is, after re-indexation, of the form $$\label{sect3-eq1}
Y_N : x\in \R
\longmapsto \sum\limits_{j=0}^N C_{j}(N)\, \exp (i k_{j}(N)x),$$ where $C_j(N)\in \C$ and $k_j(N)\in \R$ for any $N\in \N^*$ and $j\in \N$.
\[sect3-expl1\]
1. If $f \in L^1\big(\T,\C\big)$, where $\T =\R/(2\pi\Z)$, is any subsequence of the Fourier (resp. Fourier-Fejér) sequences $\{x\mapsto S_N(x)\}_{N\geq 1}$ (resp. $\{x\mapsto F_N(x)\}_{N\geq 1}$), where $$\begin{aligned}
S_N(x) &=& \sum\limits_{j=0}^{2N}
\Big(
\int_{\T} f(\theta)
e^{-i\, (j-N)\, \theta}\,
\frac{d\theta}{(2\pi)}\Big)\, e^{i\, (j-N)\, x}
\\
F_N(x)
&=&
\sum\limits_{j=0}^{2N}
\Big(1 - \frac{|j-N|}{N}\Big)\, \Big(
\int_{\T} f(\theta)
e^{-i\, (j-N)\, \theta}\,
\frac{d\theta}{(2\pi)}\Big)\, e^{i\, (j-N)\, x},\end{aligned}$$ then it realizes, after re-indexation, an archetypical example of a complex generalized Fourier sequence in $(\mathscr F(\R,\C))^{\N^*}$ This fact justifies the terminology.
2. When $m=1$ and $a\in \R$, the sequence $\{x\mapsto F_N(x,a)\}_{N\geq 1}$ is also an example of a complex generalized Fourier sequence in $(\mathscr F(\R,\C))^{\N^*}$. In this case, note that $C_j(N) = C_j(N,a)\in \R$ for any $j\in \N$.
3. Let $P= \sum_{\kappa \in \Z^*} \gamma_\kappa X^\kappa \in \C[X,X^{-1}]$ be a Laurent polynomial and $L(P)$ the diameter of its support. Any sequence $\{x\mapsto Y_N(x)\}_{N\geq 1}$ such that $$Y_N(x)= \sum_{j=0}^N C_j(N) P(e^{ik_j(N) x}) =
\sum\limits_{j=0}^N
\sum\limits_{\kappa\in \Z^*}\lambda_\kappa C_j(N)
e^{i\kappa \kappa_j(N) x} = \sum\limits_{j=0}^{ L(P)\, N}
\widetilde C_j(N)\, e^{i\widetilde \kappa_j(N) x}$$ is after re-indexation a complex generalized Fourier sequence in $(\mathscr F(\R,\C))^{\N^*}$.
\[sect3-def2\] A complex generalized Fourier sequence $\{x\mapsto Y_N(x)\}_{N\geq 1}$ in $(\mathscr F(\R,\C))^{\N^*}$ is called a [complex superoscillating sequence]{} if
- each entry $Y_N$ is of the form with $|k_j(N)|\leq 1$ for any $j\in \N$ such that $0\leq j\leq N$;
- there exists an open subset $U^{\rm sosc}\subseteq \R$ which is called a [superoscillation domain]{} such that $\{x\mapsto Y_N(x)\}_{N\geq 1}$ converges uniformly on any compact subset of $U^{\rm sosc}$ to the restriction to $U^{\rm sosc}$ of a trigonometric polynomial function $$Y_\infty: x \longmapsto P_\infty(e^{ik(\infty) x})$$ where $P_\infty\in \C[X,X^{-1}]$ is a Laurent polynomial with no constant term and $k(\infty) \in \R\setminus [-1,1]$.
\[sect3-rem1\] [If $\boldsymbol Y$ is a superoscillating sequence in the sense of Definition \[sect3-def2\], it is $Y_\infty$-superoscillating in the sense of Definition 1.1 in [@CSY], with [*superoscillation set*]{} any segment $[a,b]$ such that $b-a>0$ is included in the superoscillation domain $U^{\rm sosc}$. ]{}
\[sect3-expl2\]
1. Any subsequence of the Fourier (resp. Fourier-Fejér) sequences $\{x\mapsto S_N(x)\}_{N\geq 1}$ (resp. $\{x\mapsto F_N(x)\}_{N\geq 1}$) introduced in Example \[sect3-expl1\] (1) fails to be superoscillating since the condition $|k_j(N)|\leq 1$ is not fulfilled.
2. If $a\in \R \setminus [-1,1]$, the sequence $\{x\mapsto F_N(x,a)\}_{N\geq 1}$ is a superoscillating sequence in $(\mathscr F(\R,\C))^{\N^*}$ with superoscillation domain equal to $\R$, with $Y_\infty: x\in \R \mapsto e^{iax}$. This follows from Lemma \[sect2-lem5\] (namely from the inequalities for $a\in \R$ and $x\in \R$). This is the model that inspired us originally and that we will generalize in this paper.
Inspired by physical considerations which we will discuss later on, we extend as follows Definition \[sect3-def1\] and Definition \[sect3-def2\] to the higher dimensional setting where $m>1$. The model we will use in order to extend Definition \[sect3-def1\] will be the one in Example \[sect3-expl1\] (3).
\[sect3-def3\] A sequence $\boldsymbol Y \in (\mathscr F(\R^m,\C))^{\N^*}$ is called a complex [generalized Fourier sequence]{} if, after re-indexation, each entry $Y_N$ is of the form $$\label{sect3-eq2}
Y_N : x=(x_1,...,x_m)\in \R^m
\longmapsto \sum\limits_{j=0}^N C_{j}(N)\,
P\big(e^{ix_1 k_{j,1}(N)},...,e^{ix_m k_{j,m}(N)}\big)$$ where $P\in \C[X_1,...,X_m]\in \C[X_1^{\pm 1},...,X_m^{\pm 1}]$ is a Laurent polynomial (independent of $N$), $C_j(N)\in \C$ and $N\mapsto k_j(N)$ is a map from $\N^*$ to $\R^m$ for any $N\in \N^*$ and $j=0,...,N$.
\[sect3-expl3\] [Let $t,x$ be two real variables, $C_j(N)\in \C$, $\kappa_j(N)\in \R$, $k_j(N)\in \R$ for any $N\in \N^*$ and $0\leq j\leq N$. Then $$\Big\{x \mapsto \sum\limits_{j=0}^N C_j(N) e^{i \kappa_j(N)\, t} e^{i k_j(N)\, x}\Big\}_{N\geq 1}$$ is a complex generalized Fourier sequence in the two real variables $t,x$, the polynomial $P\in \C[T,X]$ being here $P(T,X)=TX$. ]{}
Definition \[sect3-def2\] extends to the multivariate case as follows.
\[sect3-def4\] A complex generalized Fourier sequence $\{x\mapsto Y_N(x)\}_{N\geq 1}$ in $(\mathscr F(\R^m,\C))^{\N^*}$ is called a [complex superoscillating sequence]{} if
- each entry $Y_N$ is of the form with additionally $|k_{j,\ell}(N)|\leq 1$ for any $j\in \N$ such that $0\leq j\leq N$ and $\ell=1,...,m$;
- there exists an open subset $U^{\rm sosc}\subseteq \R^m$ which is called a [superoscillation domain]{} such that $\{x\mapsto Y_N(x)\}_{N\geq 1}$ converges uniformly on any compact subset of $U^{\rm sosc}$ to the restriction to $U^{\rm sosc}$ of a trigonometric polynomial function $$Y_\infty: x \longmapsto P_\infty(e^{ik_1(\infty) x_1},...,e^{i k_m(\infty) x_m})$$ where $P_\infty\in \C[X_1^{\pm 1},...,X_m^{\pm 1}]$ is a Laurent polynomial with no constant term and $k_j(\infty) \in (\R\setminus [-1,1])^m$.
In order to illustrate Definition \[sect3-def4\] with an example which is derived from Example \[sect3-expl2\] (2), consider, for $p\in \N$ and $a\in \R \setminus [-1,1]$, the complex generalized Fourier sequence in two real variables $t,x$ $$\label{sect3-eq3}
\Big\{\psi_{p,N}(\cdot,\cdot,a)~: (t,x) \in \R^2 \longmapsto
\sum\limits_{j=0}^N C_j(N,a)\, e^{i (1-2j/N)^p\, t}\, e^{i(1-2j/N)x}\Big\}_{N\geq 1}$$ (see Example \[sect3-expl3\]). An immediate computation shows that for any $(t,x)\in \R^2$, $$\begin{aligned}
\frac{\partial}{\partial t}
\big(\psi_{p,N}(t,x,a)\big) &=& i\, \sum\limits_{j=0}^N
C_j(N,a)\, (1-2j/N)^p\,e^{i (1-2j/N)^p\, t}\, e^{i(1-2j/N) x} \\
\frac{\partial^p}{\partial x^p}
\big(\psi_{p,N}(t,x,a)\big) &=& i^p\, \sum\limits_{j=0}^N
C_j(N,a)\, (1-2j/N)^p\,e^{i (1-2j/N)^p\, t}\, e^{i(1-2j/N) x},\end{aligned}$$ which shows that $(t,x)\in \R^2 \longmapsto \psi_{p,N}(t,x,a)$ is the (unique) global solution of the Cauchy-Kowalevski problem $$\label{sect3-eq4}
\Big(i^{p-1} \frac{\partial}{\partial t} -
\frac{\partial^p}{\partial x^p}\Big) (\psi) \equiv 0,\quad
[\psi(t,x)]_{|t=0} = F_N(x,a).$$ One can extend analytically $\psi_{p,N}(\cdot,\cdot,a)$ as a function from $\R \times \C$ to $\C$, such that one has formally $$\begin{aligned}
\label{sect3-eq5}
\psi_{p,N}(t,z,a) &=& \sum\limits_{j=0}^N C_j(N,a)\,
\Big( \sum\limits_{\ell=0}^\infty
\frac{i^{\ell(1-p)}\, t^\ell}{\ell!} \big(i(1-2j/N)\big)^{p\ell}\Big)\, e^{i(1-2j/N)z} \nonumber \\
&=& \Big(\sum\limits_{\ell=0}^\infty
\frac{i^{\ell(1-p)}\, t^\ell}{\ell!} D^{p\ell}\Big)(F_N(\cdot,a)) (z) = \D_p(t) (F_N(\cdot,a)) (z).\end{aligned}$$ One can prove here the following result.
\[sect3-thm1\] The operator $\D_p(t)$ acts continuously from $A_1(\C)$ into itself. The generalized Fourier sequence is superoscillating with $\R^2$ as superoscillation domain and limit function $$Y_\infty~: (t,x) \longmapsto e^{it a^p}\, e^{iax},$$ ($P_\infty(T,X) = T X$, $k_1(\infty) = a^p$, $k_2(\infty)=a$) uniformly on any compact in $\R^2$. For any $(\mu,\nu)\in \N^2$, the sequence of functions $$\begin{gathered}
\label{sect3-eq6}
\frac{\partial^{\mu +\nu}}{\partial t^\mu
\partial x^\nu} (\psi_{p,N}(t,x,a)) =
i^{-\mu(1-p)} \frac{\partial^{p\mu + \nu}}{\partial x^{p\mu+\nu}}
(\psi_{p,N}(t,x,a)) \\
= i^{-\mu(1-p)}
((d/dW)^{p\mu+\nu}\odot \D_p(t))(F_N(\cdot,a)(x) \quad (N\in \N^*)\end{gathered}$$ converges uniformly on any compact in $\R^2$ to the function $$(t,x)\in \R^2 \mapsto ((d/dW)^{p\mu+\nu}\odot \D_p(t))(e^{ia^p t}\, e^{ia (\cdot)})(x).$$
The first assertion follows from Lemma \[sect2-lem2\] with $\R_t= \mathscr T$ as set of parameters and $p\geq 1$ as order of the symbol of the differential operator $\D_p(t)$ as a differential operator in $W$. Since $\{z\mapsto F_N(z,a)\}_{N\geq 1}$ converges to $z\mapsto e^{iaz}$ in $A_1(\C)$ (see theorem \[sect2-thm1\]), the sequence $\{z\mapsto \D_p(t)(F_N(\cdot,a))(z)\}_{N\geq 1}$ converges towards $z\mapsto \D_p(t)(e^{ia(\cdot)})(z)$ locally uniformly with respect to $t\in \R$. One can check that $\D_p(t)(e^{ia(\cdot)})(z) = e^{ia^pt} e^{iaz}$ thanks to an immediate computation. Since $(1-2j/N)^p$ and $(1-2j/N)$ lie in $[-1,1]$ for any $j\in \{0,...,N\}$ and $a\in \R \setminus [-1,1]$, the generalized Fourier sequence is superoscillating with $P_\infty(T,X)=TX$, $k_1(\infty)=a^p$ and $k_2(\infty)=a$, the superoscillation domain being here $\R^2$. The expressions of the partial derivatives in $t$ in terms of the partial derivatives in $x$ in follow from the fact that $\psi_{p,N}(\cdot,\cdot,a)$ satisfies the partial differential equation in the Cauchy-Kowalevski problem . The last assertion in the theorem results from the continuity of the differentiation $d/dz$ as an operator from $A_1(\C)$ into itself.
Let now $P\in \R[X]$ be an even polynomial $P(X)=\gamma_0 + \gamma_1 X^2 +\cdots + \gamma_{2d'} X^{2d'}$ and $a\in \R\setminus [-1,1]$. Consider in this case the generalized Fourier sequence $$\label{sect3-eq7}
\Big\{\psi_{P,N}(\cdot,\cdot,a)~: (t,x) \in \R^2 \longmapsto
\sum\limits_{j=0}^N C_j(N,a)\, e^{i P(1-2j/N)\, t}\, e^{i(1-2j/N)x}\Big\}_{N\geq 1}.$$ As in the previous case, an easy computation shows that the function $\psi_{P,N}(\cdot,\cdot,a)$ is the unique global solution (on the whole space $\R^2$) of the Cauchy-Kowalevski problem $$\label{sect3-eq8}
\Big(i \frac{\partial}{\partial t} -
\check P\Big(\frac{\partial}{\partial x}\Big)\Big) (\psi) \equiv 0,\quad
[\psi(t,x)]_{|t=0} = F_N(x,a)$$ where $\check P=\sum_{\kappa'=0}^{d'} (-1)^{\kappa'+1} \gamma_{2\kappa'}\, X^{2\kappa'}$, and the partial differential operator is here of Schrödinger type. Let us introduce the differential operator $\D_P(t)$ defined as $$\D_P(t) = \bigodot_{\kappa' = 0}^{d'}
\big(\sum\limits_{\ell =0}^\infty
\frac{(i^{1-2\kappa'}\, t \gamma_{2\kappa'})^\ell}{\ell!}\, (d/dW)^{2\kappa'\ell}\big)$$ with symbol in $A_{2d'}(\C_W)$ (the set of parameters $\mathscr T$ being again $\mathscr T=\R_t$).
\[sect3-thm2\] Let $P\in \R[X]$ be an even polynomial with degree $2d'$. For any $\lambda \in \R$, the Cauchy-Kowalevski problem (of Schrödinger type) $$\label{sect3-eq9}
\Big(i \frac{\partial}{\partial t} -
\check P\Big(\frac{\partial}{\partial x}\Big)\Big) (\psi) \equiv 0,\quad
[(t,x)\mapsto \psi(t,x)]_{|t=0} = [x\mapsto e^{i\lambda x}]$$ admits as unique global solution in $\R^2$ the function $(t,x) \mapsto \varphi_{\lambda} (t,x) = e^{it P(\lambda)} e^{i\lambda x}$. One has $\psi_{P,N}(\cdot,\cdot,\lambda) =
\sum_{j=0}^N C_j(N,\lambda)\, \varphi_{1-2j/N}$ and the sequence $\{(t,x)\mapsto \psi_{P,N}(t,x,\lambda)\}_{N\geq 1}$ converges uniformly on any compact set in $\R^2$ to $(t,x) \mapsto e^{itP(\lambda)} e^{i\lambda x}$. For any $(\mu,\nu)\in \N^2$, the sequence of functions $$\begin{gathered}
\label{sect3-eq10}
\frac{\partial^{\mu +\nu}}{\partial t^\mu
\partial x^\nu} (\psi_{P,N}(t,x,\lambda)) =
(-i)^\mu \Big(\check P\Big(
\frac{\partial}{\partial x}\Big)\Big)^{\odot^\mu}
\odot \Big(\frac{\partial}{\partial x}\Big)^{\odot^\nu} (\psi_{p,N}(t,x,\lambda)\\
= (-i)^\mu
\Big(\big(\check P (d/dW)\big)^{\odot^\mu}
\odot (d/dW)^\nu
\odot \D_P(t)\Big)\big(F_N(\cdot,\lambda)\big)(x) \quad (N\in \N^*)\end{gathered}$$ converges uniformly on any compact in $\R^2$ to the function $$(t,x)\in \R^2 \mapsto
(-i)^\mu
\Big(\big(\check P (d/dW)\big)^{\odot^\mu}
\odot \big(d/dW\big)^{\odot^\nu}
\odot \D_P(t)\Big)
\big(e^{iP(\lambda) t}\, e^{i\lambda (\cdot)}\big)(x).$$
One has $$\Big(i \frac{\partial}{\partial t} -
\check P\Big(\frac{\partial}{\partial x}\Big)\Big) (\varphi_{\lambda}) = (- P(\lambda) + P(\lambda))\, e^{it P(\lambda)} e^{i\lambda x} \equiv 0$$ and $\varphi_{\lambda}(0,x)=e^{i\lambda x}$ for all $x\in \R$. It follows from Lemma \[sect2-lem2\] that the operator $\D_P(t)$ acts continuously on $A_1(\C)$, locally uniformly with respect to the parameter $t\in \R$. Since the sequence $\{z\in \C \mapsto F_N(\cdot,\lambda)\}_{N\geq 1}$ converges to $z\mapsto e^{i\lambda z}$ in $A_1(\C)$, the sequence $\{z\in \C \mapsto \D_P(t)(F_N(\cdot,\lambda))(z)\}_{N\geq 1}$ converges to $z\mapsto \D_P(t)(e^{i\lambda (\cdot)})(z)=e^{it P(\lambda)} e^{i\lambda z}$ in $A_1(\C)$ locally uniformly with respect to the parameter $t\in \R$. The first equality in follows from the fact that $\psi_{P,N}(\cdot,\cdot,\lambda)$ is solution of the Cauchy-Kowalevski problem . The final assertion follows from the continuity of $d/dz~: A_1(\C)\rightarrow A_1(\C)$.
One can even drop the hypothesis about $P$ and take $P=\sum_{\kappa=0}^d \gamma_\kappa X^\kappa$ as polynomial of degree $d$ in $\C[X]$ with associate polynomial $\check P = \sum_{\kappa=0}^d (-i)^{\kappa+1} \gamma_\kappa X^\kappa$. The Cauchy-Kowalevski problem is not anymore of the Schrödinger type (since $\check P \notin \R[X]$ in general), which makes the only difference with the case previously studied. Nevertheless, one can state exactly the same result, with this time $$\D_P(t) = \bigodot_{\kappa = 0}^{d}
\big(\sum\limits_{\ell =0}^\infty
\frac{(i^{1-\kappa}\, t \gamma_{\kappa})^\ell}{\ell!}\, (d/dW)^{\kappa\ell}\big).$$
\[sect3-thm3\]Let $P\in \C[X]$ be a polynomial of degree $d$. All the assertions in Theorem \[sect3-thm2\] are valid, except that is not anymore a Cauchy-Kowalevski problem of the Schrödinger type. When $a\in \R\setminus [-1,1]$, the generalized Fourier sequence $\{x\mapsto \psi_{B,N}(t,x,a)\}_{N\geq 1}$ is superoscillating for any $t\in \R$. Moreover, given such $a$ and $P\in \R[X]$ such that $\sup_{[-1,1]} |P| \leq 1 < |P(a)|$, the generalized Fourier sequence $$\Big\{(t,x) \mapsto \psi_{P,N}(t,x,a) = \sum\limits_{j=0}^N
C_j(N,a)\, \varphi_{1-2j/N}(t,x)\Big\}_{N\geq 1}$$ is superoscillating as a generalized Fourier sequence in two variables $(t,x)$, with $\R^2$ as domain of superoscillation.
The proof follows that one of Theorem \[sect3-thm2\]. The fact that the sequence $\{x\mapsto \psi_{B,N}(t,x,a)\}_{N\geq 1}$ is superoscillating for any $t\in \R$ follows from the fact that it converges on any compact of $\R_x$ (locally uniformly in $t$) to $x\mapsto e^{itP(a)}\, e^{iax}$. As for the last assertion, to define $Y_\infty$ one takes $P_\infty (T,X)=T X$, $\kappa(\infty) = P(a)$ and $k(\infty) =a$ in Definition \[sect3-def4\].
Let now $E(X) = \sum_{\kappa=0}^\infty \gamma_\kappa X^\kappa \in \C[[X]]$ be a power series with radius of convergence $\rho\in ]0,+\infty]$, together with the convolution operator $$\D_E(t) := \lim\limits_{d\rightarrow +\infty}
\bigodot\limits_{\kappa=0}^d \Big(\sum\limits_{\ell =0}^\infty
\frac{(i^{1-\kappa} t \gamma_\kappa)^\ell}{\ell!}
\, (d/dW)^{\kappa \ell}\Big)$$ with formal symbol $$\F_E(t)~: W \longmapsto \exp \Big( it \sum\limits_{\kappa=0}^\infty
i^{1-\kappa}\, \gamma_\kappa W^\kappa\Big).$$ Since $F$ and $\sum_{\kappa =0}^\infty i^{1-\kappa} \gamma_\kappa X^\kappa$ share the same radius of convergence $\rho>0$, $\F_E(t)$ realizes, for each $t\in \R$) an holomorphic function in $D(0,\rho) \subset \C_W$ (with Taylor series about $0$ depending on $t\in \R$). More precisely, one has $$\forall\, t,W \in \R \times D(0,\rho),\quad
\F_E(t)(W) = \sum\limits_{j=0}^\infty \Big(\sum\limits_{\kappa=0}^\infty b_{j,\kappa} t^\kappa\Big) W^j = \sum\limits_{j=0}^\infty b_j (t)\, W^j,$$ where, for $R>0$, the radius of convergence of the power series $\sum_{j\geq 0} \Big(\sum_{\kappa \geq 0} |\beta_{j,\kappa}| R^\kappa\Big) X^j$ is at least equal to $\rho$. For any $\lambda \in ]-\rho,\rho[$ and $z\in \C$, one has formally $$\begin{gathered}
\label{sect3-eq11}
e^{it E(\lambda)} e^{iz\lambda} =
\lim\limits_{d\rightarrow +\infty}
\prod\limits_{\kappa=0}^d \Big(\sum\limits_{\ell =0}^\infty
\frac{(i^{1-\kappa} t \gamma_\kappa)^\ell}{\ell!} (i\lambda)^{\kappa \ell}\Big)\, e^{i\lambda z} \\
= \lim\limits_{d\rightarrow +\infty}
\prod\limits_{\kappa=0}^d \Big(\sum\limits_{\ell =0}^\infty
\frac{(i^{1-\kappa} t \gamma_\kappa)^\ell}{\ell!} (d/dW)^{\kappa \ell}\Big)\Big) (e^{i \lambda (\cdot)})(z) = \D_E(t) (e^{i\lambda (\cdot )})(z).\end{gathered}$$ One requires the following lemma in order to justify the formal relations .
\[sect3-lem1\] When $\rho=+\infty$, the convolution operator $\D_E(t)$ acts continuously locally uniformly with respect to $t\in \R$ from $A_1(\C)$ into itself. When $\rho \in ]0,+\infty[$ it acts continuously locally uniformly with respect to $t\in \R$ from the space $$\{f\in A_1(\C)\,;\, \forall\, \varepsilon>0,\
\exists\, C_\varepsilon>0\ {\rm such\ that}\ |f(W)|\leq C_\varepsilon
e^{(\rho-\varepsilon)|W|}\Big\} =\lim_{\longleftarrow} A_1^{B_{\rho,n}}(\C)$$ (where $\{B_{\rho,n}\}_{n\geq 1}$ is a strictly increasing sequence converging to $\rho$) into itself.
Suppose first that $\rho=+\infty$. Let $R>0$ and $K\subset [-R,R]\subset \R_t$ be a compact set. One recalls here that the radius of convergence of the power series $\sum_{j\geq 0} \big(\sum_{\kappa\geq 0} |b_{j,\kappa}| R^\kappa\big) X^j$ equals $+\infty$. Let $\gamma>0$, $\beta>0$ and $f=\sum_{\ell \geq 0} f_\ell\, W^\ell \in A_1^{\gamma,\beta}(\C)$. One can check as in the proof of Lemma \[sect2-lem2\] (compare to ) that, for any $t\in K$ and $j\in \N$, $$\sum\limits_{j=0}^\infty
\frac{(j+\ell)!}{\ell!} |b_j(t)| \, |f_{\ell +j}|
\leq \gamma \frac{\beta^\ell}{\ell!}\, \sum\limits_{j=0}^\infty
\Big(\sum\limits_{\kappa=0}^\infty |b_{j,\kappa}| R^\kappa\Big)
\beta^j
= K_{\D_E}(\beta,\gamma)\,
\frac{\beta^\ell}{\ell!}.$$ This is indeed enough to conclude as in the proof of Lemma \[sect2-lem2\] that $\D_E(t)$ acts continuously locally uniformly in $t$ from $A_1(\C)$ into itself. Consider now the case where $\rho\in ]0,+\infty[$. For any $R>0$, the radius of convergence of the power series $\sum_{j\geq 0} \big(\sum_{\kappa\geq 0} |b_{j,\kappa}|\, R^\kappa\big) X^j$ is now at least equal to $\rho$. Repeating the preceeding argument (but taking now $\beta \leq \rho-\varepsilon$ for some $\varepsilon >0$ arbitrary small), one concludes that $\D_E(t)$ acts continuously locally uniformly in $t$ from $\displaystyle{\lim_{\longleftarrow} A_1^{B_{\rho,n}}(\C)}$ into itself.
We can now state the last result of this section.
\[sect3-thm4\] Let $E=\sum_{\kappa=0}^\infty \gamma_\kappa X^\kappa \in \C[[X]]$ be a power series with radius of convergence $\rho\in ]2,+\infty]$. Then $\check E := \sum_{\kappa =0}^\infty (-i)^{\kappa +1}\gamma_\kappa D^\kappa$ acts continuously from $\displaystyle{\lim_{\longleftarrow} A_1^{B_{\rho,n}}(\C)}$ into itself. For any $t\in \R$ and $a\in \R$ with $1<|a|<\rho-1$, the generalized Fourier sequence $$\Big\{x \in \R \longmapsto \psi_{E,N} (t,x,a) = \sum\limits_{j=0}^N C_j(N,a)\,
e^{i E(1-2j/N)t} e^{ix(1-2j/N)}\Big\}_{N\geq 1}$$ is superoscillating. Moreover, for any such $a$ and $(\mu,\nu)\in \N^2$, the sequence of functions $$\begin{gathered}
\label{sect3-eq12}
\frac{\partial^{\mu +\nu}}{\partial t^\mu
\partial x^\nu} (\psi_{E,N}(t,x,a)) =
(-i)^\mu \Big(\check E^{\odot^{\mu}}
\odot (d/dW)^\nu\Big)\big(\psi_{p,N}(t,\cdot,a)\big)(x)\\
= (-i)^\mu
\Big(\check E^{\odot^{\mu}}
\odot (d/dW)^\nu \odot \D_E(t)\Big)\big(F_N(\cdot,a)\big)(x)\end{gathered}$$ converges then uniformly on any compact in $\R^2_{t,x}$ to the fonction $$(t,x) \longmapsto (-i)^\mu
\Big(\check E^{\odot^{\mu}}
\odot (d/dW)^\nu \odot \D_E(t)\Big)\big(e^{it E(a)}
\, e^{ia(\cdot)}\big)(x).$$
The fact that $\check E$ acts continuously from $\lim_{\longleftarrow} A_1^{B_{\rho,n}}(\C_W)$ into itself follows from Lemma \[sect3-lem1\], considering just $\check E$ (independent of the parameter $t$) instead of $\D_E(t)$. For any $\lambda\in \R$ with $|\lambda|<\rho$, the operator $\check E$ then acts on $e^{i\lambda(\cdot)}$ and it is immediate to check that for any $t\in \R$ $$\label{sect3-eq13}
\forall\, (t,x)\in \R^2,\
\Big[\Big(i\, \frac{\partial}{\partial t} - \check E\Big)
\big(e^{it E(\lambda)}\, e^{i\lambda W}\big)\Big]_{W=x} = 0~;$$ moreover $\big[(t,x) \mapsto e^{it E(\lambda)}\, e^{i\lambda x}\big]_{t=0}$ is $x\mapsto e^{i\lambda x}$. Therefore, for any $a\in \R$ and $N\in \N^*$, one has by linearity (since $\rho>1$) $$\label{sect3-eq14}
\forall\, (t,x)\in \R^2,\
\Big[\Big(i\, \frac{\partial}{\partial t} - \check E\Big)
\Big(\sum\limits_{j=0}^N C_j(N,a) e^{i E(1-2j/N)t}
e^{i(1-2j/N) W}\Big)\Big]_{W=x} = 0.$$ Lemma (applied this time with $\D_E(t)$), combined with Theorem \[sect2-thm1\] and the estimates in the first line of in Lemma \[sect2-lem5\], imply that as soon as one has $|a|<\rho-1$ the sequence $\{z\in \C \mapsto \D_E(t)(F(\cdot,a))(z)\}_{N\geq 1}$ converges (locally uniformly with respect to the parameter $t$) to $z\mapsto e^{iaz}$ in $A_1(\C)$. The last assertion in the particular case $\mu=\nu=0$ follows. The first equality in comes from the identity , while the second one comes from (as justified by Lemma \[sect3-lem1\]). The last assertion of the theorem when $\mu,\nu$ are arbitrary is then a consequence of the continuity of $d/dz$ from $\underset{\lim}{\longleftarrow} A_1^{B_{\rho,n}}(\C)$ into itself. The superoscillating character of the sequence $\{\psi_{P,N}(t,\cdot,a)\}_{N\geq 1}$ follows from Definition \[sect3-def2\].
[When $E\in \R[[X]]$, $1<|a|<\rho-1$ and $\sup_{[-1,1]}(E)\leq 1 < |E(a)|$, the generalized Fourier sequence $$\Big\{(t,x) \in \R^2 \longmapsto \psi_{E,N} (t,x,a) = \sum\limits_{j=0}^N C_j(N,a)\,
e^{i E(1-2j/N)t} e^{ix(1-2j/N)}\Big\}_{N\geq 1}$$ is also superoscillating, this time according to Definition \[sect3-def4\] (with $P_\infty(T,X)=TX$, $\kappa(\infty) = E(a)$ and $k(\infty) = a$).]{}
Regularization of formal Fresnel-type integrals {#sect4}
===============================================
In order to settle from the mathematical point of view the approach to non-absolutely convergent integrals on the half-line $\R^{+*}$ or the whole real line $\R$ through the so-called principle of [*regularization*]{} that we will invoke in the remaining sections \[sect5\] and \[sect6\] (with respect to supershift considerations related to Schrödinger equations with specific potentials), we need to explain what regularization of formal Fresnel-type integrals on $\R^{+*}$ or $\R$ means. Suppose that $\EuFrak T$ is a set of parameters. Let $G~: (\EuFrak t,Z)\in \EuFrak t \times \C \longmapsto G(\EuFrak t,Z)$ be a function which is entire as a function of $Z$ for each $\EuFrak t \in \EuFrak T$ fixed. Let also $\phi$ be a non-vanishing real function on $\EuFrak T$ that will play the role of a [*phase function*]{}. Let finally $\chi$ be a real number such that $\chi >-1$. In order to give a meaning to the formal integral $$\label{sect4-eq1}
\int_0^\infty (x')^\chi\, e^{-i\phi(\EuFrak t) (x')^2}\, G(\EuFrak t,x')\, dx'\quad (\chi >-1)$$ we distinguish the cases where $\phi(\EuFrak t)>0$ and $\phi(\EuFrak t)<0$. In the first case ($\phi(\EuFrak t)>0$), we rewrite this (for the moment formal) expression as $$\begin{gathered}
\label{sect4-eq2}
\int_0^\infty (x')^\chi\, e^{-i\phi(\EuFrak t) (x')^2}\, G(\EuFrak t,x')\, dx' = e^{-i(\chi+1)\pi/4}\, \int_{\R^{+*}\, e^{i\pi/4}} Z^\chi\, e^{-\phi(\EuFrak t)\, Z^2}\,
G(\EuFrak t, e^{-i\pi/4} Z)\, dZ \\=
\int_{\R^{+*}\, e^{i\pi/4}} Z^\chi\, e^{-\phi(\EuFrak t)\, Z^2}
\, F_+(\EuFrak t,Z)\, dZ\end{gathered}$$ with $F_+(\EuFrak t,Z) := e^{-i(\chi+1)\pi/4} G(\EuFrak t,e^{-i\pi/4} Z)$ for any $\EuFrak t \in \EuFrak T$ and $Z\in \C$. In the second case ($\phi(\EuFrak t)<0$), we rewrite it as $$\begin{gathered}
\label{sect4-eq3}
\int_0^\infty (x')^\chi\, e^{-i\phi(\EuFrak t) (x')^2}\, G(\EuFrak t,x')\, dx' =
e^{i(\chi+1)\pi/4}\, \int_{\R^{+*}\, e^{-i\pi/4}} Z^\chi\, e^{\phi(\EuFrak t)\, Z^2}\, G(\EuFrak t, e^{i\pi/4} Z)\, dZ \\
= \int_{\R^{+*}\, e^{-i\pi/4}} Z^\chi
e^{\phi(t)\, Z^2}\, F_-(\EuFrak t,Z)\, dZ\end{gathered}$$ with $F_-(\EuFrak t,Z) := e^{i(\chi+1)\pi/4} G(\EuFrak t,e^{i\pi/4} Z)$ for any $\EuFrak t \in \EuFrak T$ and $Z\in \C$. The following elementary lemma will reveal to be essential.
\[sect4-lem1\] Let $\EuFrak T,\phi,\chi$ as above and $F~: \EuFrak T \times \C \longrightarrow \C$ be a function with is entire in the complex variable $Z$ and satisfies the growth estimates $$\label{sect4-eq4}
\forall\, \varepsilon >0,\
\sup\limits_{\EuFrak t\in \EuFrak T,Z\in \C}
\big(|F(\EuFrak t,Z)|\, \exp(-\varepsilon |Z|^{\check p})\big) <+\infty$$ for some $\check p\in ]1,2]$, that is $F(t,\cdot)\in A_{\check p,0}(\C)$ uniformly in $\EuFrak t$. Then, for any $u=e^{i\theta}$ with $\theta \in ]-\pi/4,\pi/4[$, the integral $$\label{sect4-eq5}
\int_{\R^{+*}\, u} Z^\chi\, e^{- |\phi(\EuFrak t)|\, Z^2}\, F(\EuFrak t,Z)\, dZ$$ is absolutely convergent and remains independent of $u$ ; it equals in particular its value for $u=1$.
The absolute convergence follows from the estimates , together with the fact that if $u=e^{i\theta}$, ${\rm Re}((t u)^2) =
t^2 \cos (2\theta)>0$ for $t>0$. The fact that the integrals do not depend of $u$ follows from residue theorem (applied on the oriented boundary of conic sectors with apex at the origin).
In view of this lemma, the regularization of an integral of the Fresnel-type such as consists in the successive two operations :
1. first transform the formal expression into one of the representations or according to ${\rm sign}(\phi(\EuFrak t))$ ;
2. then invoke Lemma \[sect4-lem1\] (provided the required hypothesis are satisfied) and consider then the regularization of as $\int_0^\infty Z^\chi e^{-\phi(\EuFrak t) Z^2}
F_+(\EuFrak t,Z)\, dZ$ when $\phi(\EuFrak t)>0$ or $\int_0^\infty Z^\chi e^{\phi(\EuFrak t) Z^2}
F_-(\EuFrak t,Z)\, dZ$ when $\phi(\EuFrak t)<0$.
\[sect4-rem1\] [In order to give a meaning (if possible of course) to the formal integral expression $$\label{sect4-eq6}
\int_\R |x'|^\chi\, e^{-i\phi(\EuFrak t) (x')^2}\, G(\EuFrak t,x')\, dx',$$ one splits it as $$\int_0^\infty (x')^\chi \, e^{-i\phi(\EuFrak t) (x')^2}\, G(\EuFrak t,x')\, dx' +
\int_0^\infty (x')^\chi\, e^{-i\phi(\EuFrak t) (x')^2}\, G(\EuFrak t,-x')\, dx'$$ and proceed as above for the two formal expressions involved into this formal decomposition. ]{}
It is immediate to compare this approach to regularization to the alternative following one.
\[sect4-prop1\] Let $G \in A_{2,0}(\C)$ and $\chi >-1$. Then, for all $\varpi \in \R^*$ $$\lim\limits_{\varepsilon \rightarrow 0} \int_0^\infty
(x')^\chi\, e^{i\, \varpi (x')^2}\, e^{-\varepsilon (x')^2}\, G(x')\, dx'$$ exists and coincides with the integral regularized under the approach described above.
It is enough to prove the result when $\varpi = \pm 1$ since one reduces to one of these two cases up to a homothety on the real half line. One has $$\begin{aligned}
\int_0^\infty (x')^\chi\, e^{-\varepsilon (x')^2} e^{-i\, (x')^2}
\, G(x')\, dx' &=&
\int_{e^{i\pi/4} \R^{+*}} Z^\gamma\, e^{- (1-i\, \varepsilon)\, Z^2}\, F_+(Z)\, dZ
\\
\int_0^\infty (x')^\chi\, e^{-\varepsilon (x')^2} e^{i\, (x')^2}
\, G(x')\, dx' &=&
\int_{e^{-i\pi/4} \R^{+*}} Z^\chi\, e^{- (1 + i\, \varepsilon)\, Z^2}\, F_-(Z)\, dZ,\end{aligned}$$ where $F_+(Z) = e^{-i(1+\chi)\pi/4} F(e^{-i\pi/4} Z)$ and $F_-(Z) = e^{i(1+\chi)\pi/4} F(e^{i\pi/4} Z)$. Let $\rho_\varepsilon = \sqrt{1+\varepsilon^2}$, and $\xi_{\varepsilon} = {\rm arg}_{[0,\pi/2[} \sqrt{1+ i\varepsilon}$. One has then $$\label{sect4-eq7}
\begin{split}
& \int_0^\infty (x')^\gamma\, e^{-\varepsilon (x')^2} e^{-i\, (x')^2}
\, G(x')\, dx' = \Big(\frac{e^{i\xi_\varepsilon}}{\sqrt{\rho_\varepsilon}}\Big)^{1+\chi}\,
\int_{e^{i(\pi/4-\xi_\varepsilon)}\, \R^{+*}}
Z^\chi\, e^{-Z^2} F^+(e^{i\xi_\varepsilon}\, Z/\sqrt{\rho_\varepsilon})\, dZ
\\
&
\int_0^\infty (x')^\chi\, e^{-\varepsilon (x')^2} e^{i\, (x')^2}\, G(x') \, dx'=
\Big(\frac{e^{- i\xi_\varepsilon}}{\sqrt{\rho_\varepsilon}}\Big)^{1+\chi}\,
\int_{e^{-i(\pi/4 -\xi_\varepsilon)}\, \R^{+*}}
Z^\chi\, e^{-Z^2}\, F_-(e^{-i\xi_\varepsilon}\, Z/\sqrt{\rho_\varepsilon})\, dZ.
\end{split}$$ In the two integrals on the right-hand side of the equalities , the integration contour can be replaced by the half-line $\R^{+*}$ as a consequence of Lemma \[sect4-lem1\]. It is then possible to take the limit when $\varepsilon$ tends to $0$. Lebesgue’s domination theorem then applies and since $\rho_\varepsilon$ tends to $1$ and $\xi_\varepsilon$ to $0$, one gets $$\begin{aligned}
\lim\limits_{\varepsilon \rightarrow 0} \int_0^\infty (x')^\chi\, e^{-\varepsilon (x')^2} e^{-i\, (x')^2}
\, G(y)\, dy &=&
\int_0^\infty Z^\chi\, e^{- Z^2}\, F_+(Z)\, dZ
\\
\lim\limits_{\varepsilon \rightarrow 0}
\int_0^\infty (x')^\chi\, e^{-\varepsilon (x')^2} e^{i\, (x')^2}
\, G(x')\, dx' &=&
\int_0^\infty Z^\chi\, e^{- Z^2}\, F_-(Z)\, dZ.\end{aligned}$$ This concludes the proof of the Proposition.
Fresnel-type integral operators {#sect5}
===============================
Continuity on $A_1(\C)$ of Fresnel-type integral operators {#sect5-1}
----------------------------------------------------------
Let $\EuFrak T$ be a set of parameters and $\EuFrak t \in \EuFrak T\mapsto \D(\EuFrak t,Z)$ (as in the statement of Lemma \[sect2-lem3\]) be a differential operator-valued map $$\EuFrak t \in \EuFrak T \longmapsto \D(\EuFrak t,Z) = \sum\limits_{j=0}^\infty b_j(\EuFrak t,Z) \Big(\frac{d}{dZ}\Big)^j$$ (with $b_j~: T\times \C \rightarrow \C$, holomorphic in $Z$ for $j\in \N$) such that $$\label{sect5-eq1}
\forall\, \varepsilon >0,\
\sup\limits_{\EuFrak t \in \EuFrak T, (Z,W)\in \C^2}
\Big(\Big(\sum\limits_{j=0}^\infty |b_j(\EuFrak t,Z)|\ |W|^j\Big)\, \exp (- \varepsilon\, |Z|^{\check p} - B\, |W|^{p})\Big) = A^{(\varepsilon)} < +\infty$$ for some $\check p\in ]1,2], p \geq 1$ and $B\geq 0$. Let also $\phi$ be a non-vanishing real function on $\EuFrak T$ and $\chi>-1$. It follows from the estimates , together with Lemma \[sect4-lem1\], that the regularization approach described in section \[sect4\] allows to define the operator $$\label{sect5-eq2}
\EuFrak t \longmapsto \int_{0}^\infty Z^\chi\, e^{-i \phi(\EuFrak t)\, Z^2}
\sum\limits_{j=0}^\infty b_j(\EuFrak t,Z) \Big(\frac{d}{dZ}\Big)^j (\cdot )\, dZ.$$ One needs to consider for the moment these operators as acting on entire functions of the complex variable $Z$. For $\alpha \in \C$, let also $H_\alpha$ be the dilation operator $H_\alpha~: f \mapsto f(\alpha (\cdot))$ acting on such functions. The symbol $\odot$ still stands for the composition of operators. The discussion is with respect to the sign of $\phi(\EuFrak t)$.
- When $\phi(\EuFrak t)>0$, $$\begin{gathered}
\label{sect5-eq3}
\int_{0}^\infty Z^\chi\, e^{-i \phi(\EuFrak t)\, Z^2} \Big(
\sum\limits_{j=0}^\infty b_j(\EuFrak t,Z) \Big(\frac{d}{dZ}\Big)^j (\cdot )\Big)\, dZ \\
= e^{-i(1+\chi) \pi/4}
\int_0^\infty
y^\chi\, e^{-\phi(\EuFrak t)\, y^2} \, \Big(
\sum\limits_{j=0}^\infty b_j(\EuFrak t, e^{-i\pi/4} Z)\,
\Big( e^{ij \pi/4} \, \Big(\frac{d}{dZ}\Big)^j \odot H_{e^{-i\pi/4}}\Big)\, (\cdot)\Big)(y)\, dy.\end{gathered}$$
- When $\phi(\EuFrak t)<0$, $$\begin{gathered}
\label{sect5-eq4}
\int_{0}^\infty Z^\chi\, e^{-i \phi(\EuFrak t)\, Z^2}
\Big(\sum\limits_{j=0}^\infty b_j(\EuFrak t,Z) \Big(\frac{d}{dZ}\Big)^j (\cdot )\Big)\, dZ \\
= e^{i(1+\chi)\pi/4}
\int_0^\infty y^\chi\,
e^{\phi(\EuFrak t)\, y^2}
\, \Big(\sum\limits_{j=0}^\infty b_j(\EuFrak t, e^{i\pi/4}\, Z)\,
\Big( e^{-ij \pi/4} \, \Big(\frac{d}{dZ}\Big)^j \odot H_{e^{i\pi/4}}\Big)\, (\cdot)\Big)(y)\, dy.\end{gathered}$$
\[sect5-thm1\] Suppose that the parameter space $\EuFrak T$ is a topological space and that $\phi$ is continuous. Consider functions $B_j~: \EuFrak T \times \C \times \C\rightarrow \C$ ($j\in \N$) which are entire in the two complex entries and such that $$\begin{gathered}
\label{sect5-eq5}
\forall\, \varepsilon >0,\ \exists\, A^{(\varepsilon)}, B^{(\varepsilon)} \geq 0\ {\it such\ that}\ \forall\, \EuFrak t\in \EuFrak T,\
\forall Z\in \C,\forall\, \check Z\in \C,\ \forall\, W\in \C,\\
\sum\limits_{j=0}^\infty |B_j(\EuFrak t,Z,\check Z)|\ |W|^j \leq
A^{(\varepsilon)} \, e^{\varepsilon\, |Z|^{\check p} + B^{(\varepsilon)} |\check Z|^{\check p} + B\, |W|^{p}}\end{gathered}$$ for some $p\geq 1$, $\check p\in ]1,2]$, and $B\geq 0$. Then the operator $$\int_{0}^\infty Z^\chi\, e^{-i \phi(\EuFrak t)\, Z^2} \Big(
\sum\limits_{j=0}^\infty B_j(\EuFrak t,Z,\check Z) \Big(\frac{d}{dZ}\Big)^j (\cdot )\Big)\, dZ$$ (understood through the process of regularization as described above) acts continuously locally uniformly in $\EuFrak t$ from $A_1(\C)$ into $A_{\check p}(\C)$.
It is enough to consider $\EuFrak T$ as a neighborhood of a point $\EuFrak t_0$ in which $\phi(\EuFrak t)\geq \varepsilon_0>0$ (since $\phi$ is continuous). Let $\EuFrak f = \{Z\mapsto f_N(Z)\}_{N\geq 1}$ be a sequence of elements in $A_1(\C)$ that converges towards $0$ in $A_1(\C)$, which means (see Proposition \[sect2-prop1\]) that all $f_N$ belong to some $A_1^{C,b}(\C)$ for some constants $C,b>0$ independent on $N$ (namely $f_N=\sum_\ell a_{N,\ell} Z^\ell$ with $|a_{N,\ell}|\leq C\, b^\ell/\ell!$ for any $\ell\in \N$). It is clear that the operator $$\sum\limits_{j=0}^\infty B_j(\EuFrak t, e^{-i\pi/4} Z,\check Z)\,
\Big( e^{ij \pi/4} \, \Big(\frac{d}{dZ}\Big)^j \odot H_{e^{-i\pi/4}}\Big)$$ involved in the integrand of is governed by estimates of the form . It follows then from Lemma \[sect2-lem3\], taking into account estimates , that for each $N\in \N^*$ the function $$\begin{gathered}
H(f_N)~: (\EuFrak t, Z,\check Z) \in \EuFrak T \times \C \times \C \\ \longmapsto \sum\limits_{j=0}^\infty B_j(\EuFrak t, e^{-i\pi/4} Z,\check Z)\,
\Big( e^{ij \pi/4} \, \Big(\frac{d}{dZ}\Big)^j \odot H_{e^{-i\pi/4}}\Big)(f_N)(Z)\end{gathered}$$ is such that for each $\varepsilon>0$, there exists $\widetilde A^{(\varepsilon)}\geq 0$ (depending on $\EuFrak T$, $A^{(\varepsilon)}$, the $B_j$, $b$ and $C$, but not on the $N$) such that $$\forall\, (\EuFrak t,Z,\check Z)\in \EuFrak T \times \C \times \C,\quad
|H(f_N)(\EuFrak t,Z,\check Z)|\leq
\widetilde A^{(\varepsilon)}\, e^{\epsilon\, |Z|^{\check p} + B^{(\varepsilon)} |\check Z|^{\check p}}.$$ Take in particular $\varepsilon < \varepsilon_0$. Then the function $$\check Z \in \C \longmapsto
\int_0^\infty y^\chi\,
e^{-\phi(t) y^2}
H(f_N) (\EuFrak t,y,\check Z)\, dy$$ is in $A_{\check p,0}(\C)$ since it is estimated as $$\Big|
\int_0^\infty y^\chi\,
e^{-\phi(t) y^2}
H(f_N) (\EuFrak t,y,\check Z)\, dy\Big|
\leq
\widetilde A_{(\varepsilon)}\, \Big(\int_0^\infty
y^\chi\, e^{-\varepsilon_0 y^2} e^{\varepsilon y^{\check p}}\, dy\Big)\, e^{B^{(\varepsilon)} |\check Z|^{\check p}}\quad \forall\, \check Z \in \C$$ (remember that $\check p \in ]1,2]$). It remains to show that the sequence $$\Big\{\check Z \longmapsto \int_0^\infty y^\chi\, e^{-\phi(t) y^2}
H(f_N) (\EuFrak t,y,\check Z)\, dy\Big\}_{N\geq 1}$$ converges to $0$ in $A_{\check p,0}(\C)$. It is enough (see Proposition \[sect2-prop1\]) to prove that it converges to $0$ uniformly on any closed disk $\overline{D(0,r)}$ in $\C$. Fix $\varepsilon < \varepsilon_0$ and $\eta>0$. Choose then $R_\eta >>1 $ such that $$\begin{gathered}
\forall\, N\in \N,\quad
\Big|\int_{R_\eta}^\infty
y^\chi\, e^{-\phi(t) y^2}
H(f_N) (\EuFrak t,y,\check Z)\, dy\Big| \\
\leq \widetilde A^{(\varepsilon)}\, \Big(\int_0^\infty y^\chi\,
e^{-\varepsilon_0 y^2} e^{\varepsilon y^{\check p}}\, dy\Big)\,
e^{B^{(\varepsilon)}\, |\check Z|^{\check p}} \leq \eta\, e^{-
B^{(\varepsilon)} r^{\check p}}\,
e^{B^{(\varepsilon)} |\check Z|^{\check p}} \leq \eta.\end{gathered}$$ On $[0,R_\eta]$, one uses the uniform convergence of $\EuFrak f$ towards $0$ on any compact set, hence of $H[\EuFrak f]$ on any compact set, to conclude that for $N\geq N_\eta >>1$, one has $$\Big|\int_{0}^{R_\eta}
y^\chi\, e^{-\phi(t) y^2}
H(f_N) (\EuFrak t,y,\check Z)\, dy\Big|
\leq \eta \quad \forall\, \check Z\in \overline{D(0,r)}.$$ Note that our estimates show that the convergence towards $0$ in $A_{\check p,0}(\C)$ thus obtained is uniform in $\EuFrak t\in \EuFrak T$.
Superoscillations and supershifts
---------------------------------
Consider the Schrödinger equation $$\label{sect5-eq6}
i\, \frac{\partial \psi}{\partial t} (t,x) = \big(\mathscr H(x) (\psi)\big)(t,x)$$ where $\mathscr H$ denotes the Hamiltonian operator attached to the physical system which is under consideration. Suppose that $\boldsymbol Y = \{x\mapsto Y_N(x)\}_{N\geq 1}$ is a superoscillating sequence. Since $$\Big(i\,
\frac{\partial}{\partial t} - \mathscr H(x)\Big)(\psi)(t,x) =0,\quad
\big[\psi(t,x)]_{t=0} = Y(x)$$ is a Cauchy-Kowalevski problem (assuming that $x$ lies in some open set $U\subset \R$ where the Hamiltonian operator is regular), any entry $x\in U\mapsto Y_N(x)$ evolves in a unique way from $t=0$ towards $t>0$ as $(t,x)\mapsto \psi_N(t,x)$. Assume in addition that $x$ lies in the maximal superoscillation domain $U_{\rm max}^{\rm suposc}$ ; the limit function $x\in U \cap U_{\rm max}^{\rm sosc} \mapsto
Y_\infty(x)$ then also evolves from $U\cap U_{\rm max}^{\rm sosc}$ into some function $(t,x) \mapsto \psi_\infty(t,x)$. A natural question then occurs. As long as the evolution persists (let say for $t\in [0,T]$), is it true that the sequence $\{x\in U \mapsto \psi_N(t,x)\}_{N\geq 1}$ is such that its restriction to $U\cap U_{\rm max}^{\rm sosc}$ converges (uniformly on any compact subset of $U\cap U_{\rm max}^{\rm sosc}$) to $x\mapsto \psi_\infty(t,x)$? If this is the case, one will say that the superoscillating character of the sequence $\boldsymbol Y$ persists in time through the Schrödinger evolution operator $\partial/\partial t - \mathscr H$ which is here considered. In order to formulate such question in a different way, let us now consider the $(t,x)$ domain $[0,T] \times (U\cap U_{\rm max}^{\rm sosc}) =
\EuFrak T \times (U\cap U_{\rm max}^{\rm sosc}) = \mathscr T$ as a parameter set and focus on the map $\lambda \in \R \longmapsto \varphi_\lambda$, where $\varphi_\lambda~: \mathscr T \rightarrow \R$ is evolved to $[0,T] \times U$ (through the Schrödinger operator) from the initial datum $x\in U \mapsto e^{i\lambda x}$, then restricted to the parameter set $\mathscr T$. Previous considerations lead to the following definition, which is inspired by Definition \[sect3-def2\].
\[sect5-def1\] Let $\mathscr T$ be a locally compact topological space and $\mathscr F = \{\varphi_\lambda\,;\, \lambda \in \R\}$ be a family of $\C$-valued functions on $\mathscr T$ indexed by $\R$. A sequence $\boldsymbol \psi = \{\tau \in \mathscr T \mapsto \psi_N(\tau)\}_{N\geq 1}$ of $\C$-valued functions on $\mathscr T$ is called a [supershift for the family $\mathscr F$]{} (or [$\mathscr F$ admits $\boldsymbol \psi$ as a supershift]{}) if
- any entry $\psi_N$ is of the form $\psi_N = \sum_{j=0}^N C_j(N) \, \varphi_{k_j(N)}$ with $|k_j(N)|\leq 1$ for any $N\in \N^*$ and $0\leq j\leq N$;
- there exists an open subset $\mathscr U^{\rm ssh}$ of $\mathscr T$ called a [$\mathscr F$-supershift domain]{} such that the sequence $\{\tau \in \mathscr U^{\rm ssh} \mapsto \psi_N(\tau)\}$ converges locally uniformly towards the restriction to $\mathscr U^{\rm ssh}$ of a function $\psi_\infty$ which is a $\C$-finite linear combination of elements in $\mathscr F$ of the form $\varphi_{\nu k(\infty)}$ with $\nu\in \Z^*$, where $k(\infty) \in \R \setminus [-1,1]$.
\[sect5-expl1\]
1. If $\mathscr T =\R$ and $\mathscr F$ denotes the family of characters $x\in \R \mapsto e^{i\lambda x}$ indexed by the dual copy $\R_\lambda^\star$ of $\R_x$, $\mathscr F$-supershifts are the complex superoscillating sequences (see Definition \[sect3-def2\]).
2. Let $a\in \R\setminus [-1,1]$, $\mathscr T=\R^2_{t,x}$ and $\mathscr F = \{\varphi_\lambda\,;\, \lambda \in \R^\star\}$ as defined in Theorem \[sect3-thm2\] or Theorem \[sect3-thm3\]. For any $a\in \R\setminus [-1,1]$, the sequence $\{(t,x) \mapsto \psi_{P,N}(t,x,a)\}_{N\geq 1}$ is a $\mathscr F$-supershift which admits $\R^2_{t,x} = \mathscr T$ as $\mathscr F$-supershift domain.
When $\mathscr T$ is of the form $[0,T[\times U$, where $U$ is an open subset in $\R^{m-1}_x$ ($m\geq 2$) and $T\in ]0,+\infty]$, one can consider as well families $\mathscr F = \{\varphi_\lambda\,;\, \lambda \in \R\}$ of [*$\C$-valued distributions in $\R \times U$ with support in $[0,T[\times U$*]{}. In order to define in this new context the notion of $\mathscr F$-supershift, one needs to keep the first clause in Definition \[sect5-def1\] as it is and modify the second clause as follows : “[*there exists an open subset $\mathscr U^{\rm ssh}= \mathscr V^{\rm ssh} \cap \mathscr T$ (where $\mathscr V$ is an open subset in $\R \times U$), called a [$\mathscr F$-supershift domain]{}, such that the sequence $\{(\psi_N(\tau))_{|\mathscr V^{\rm ssh}}\}_{N\geq 1}$ converges weakly [*in the sense of distributions*]{} in $\mathscr V$ to the restriction to $\mathscr V$ of a [*distribution*]{} $\psi_\infty\in \mathscr D'(\R \times U,\C)$ which is a $\C$-finite linear combination of elements in $\mathscr F$ of the form $\varphi_{\nu k(\infty)}$ with $\nu\in \Z^*$, where $k(\infty) \in \R \setminus [-1,1]$*]{}”. One will need in section \[sect6\] a further extension of this concept of $\mathscr F$-supershift to the case where $\mathscr T = [0,T[ \times U$, $U\subset \R^{m-1}$ with $m\geq 2$ as above, but elements $\varphi_\lambda\in \mathscr F$ are now [*hyperfunctions in $\R \times U$ with support in $\mathscr T$*]{}. The sequence $\{(\psi_N(\tau))_{|\mathscr V^{\rm ssh}}\}_{N\geq 1}$ needs in this case to converge still in the weak sense, but this time [*in the sense of hyperfunctions*]{} in $\mathscr V$, towards the restriction to $\mathscr V$ of the [*hyperfunction*]{} $\psi_\infty$. The notion of $\mathscr F$-supershift can thus be enlarged to families $\mathscr F =\{\varphi_\lambda\,;\, \lambda \in \R\}$ of [*hyperfunctions*]{} in $\R\times U$ with support in $\mathscr T$.
The Schrödinger Cauchy problem with centrifugal potential {#sect5-2}
---------------------------------------------------------
We will consider in this subsection the case where $U=\{x\in \R\,;\, x>0\}$ and the hamiltonian in is $x\in U \mapsto \mathscr H(x) = - (\partial^2/\partial x^2)/2 + u/(2x^2)$, where $u$ denotes a real strictly positive physical constant. The corresponding Cauchy-Kowalevski problem (with $[0,+\infty[ \times U$ as phase space) is the [*Schrödinger Cauchy problem with centrifugal potential*]{}, see [@centrifugal] for more references. For this Cauchy-Kowalevski problem, the analysis of the evolution $t\mapsto \psi(t,\cdot)$ of the solution $(t,x) \in [0,\infty[ \times U \mapsto \psi(t,x)$ from an initial datum $x\in U\mapsto \psi(0,x)$ can be carried through thanks to the explicit form of the Green function $(t,x,x') \mapsto G(t,x,t'=0,x')$. Let $\nu = \sqrt{1+4u}/2$ and the Bessel function $J_\nu$ defined in $\Omega := \C\setminus ]-\infty,0]$ as $$\begin{gathered}
\label{sect5-eq7}
J_\nu~: z \in \Omega \longmapsto
\Big(\frac{z}{2}\Big)^\nu \sum\limits_{k=0}^\infty
\frac{(-1)^k}{\Gamma (k+1)\, \Gamma(\nu+k+1)}\, \Big(\frac{z}{2}\Big)^{2k} \\
= \Big(\frac{|z|}{2}\Big)^\nu
e^{i\, \nu\, {\arg}_{]-\pi,\pi[}(z)}\,
\sum\limits_{k=0}^\infty
\frac{(-1)^k}{\Gamma (k+1)\, \Gamma(\nu+k+1)}\, \Big(\frac{z}{2}\Big)^{2k} \\
= \Big(\frac{|z|}{2}\Big)^\nu
e^{i\, \nu\, {\arg}_{]-\pi,\pi[}(z)}\, E_\nu(z).\end{gathered}$$ Then the Green function $(t,x,x')\mapsto G(t,x,0,x')$ can be explicited in this case as $$\label{sect5-eq8}
G(t,x,0,x') = (-i)^{\nu+1}\,
\frac{\sqrt {xx'}}{t}\, \exp \Big(i \frac{x^2 + (x')^2}{2t}\Big)\, J_\nu \Big(\frac{xx'}{t}\Big)\quad (t>0,x,x'\in U)$$ (see [@kempf1; @SchulmanBOOK; @Green]).
\[sect5-prop1\] Let $\mathscr T = ]0,+\infty[\times ]0,+\infty[$, $\mathscr H:
x\in ]0,+\infty[ \mapsto -(\partial^2/\partial x^2 - u/x^2)/2$ for some physical constant $u>0$. For any $\lambda \in \R$, the initial datum $x\in ]0,+\infty[ \mapsto e^{i\lambda x}$ evolves through the Cauchy-Kovalewski Schrödinger equation to a function $(t,x) \mapsto \varphi_\lambda (t,x)$ which is $C^\infty$ in $\mathscr T$. For any $a\in \R \setminus [-1,1]$, the family $\{\varphi_\lambda\,;\, \lambda \in \R\}$ admits as a $\mathscr F$-supershift (in the sense of Definition \[sect5-def1\]) the sequence $$\{(t,x)\in \mathscr T \mapsto \psi_N(t,x,a)\}_{N\geq 1} =
\Big\{\sum\limits_{j=0}^N C_j(N,a)\, \varphi_{1-2j/N}\Big\}$$ with $\mathscr F$-supershift domain equal to $\mathscr T$. Moreover, for any $(\mu,\nu)\in \N^2$, the sequence of functions $$\frac{\partial^{\mu+\nu}}{\partial t^\mu
\partial x^\nu} (\psi_N(t,x,a)) = \frac{1}{(2i)^{\mu}}
\Big(\Big(-\frac{\partial^2}{\partial x^2} + \frac{u}{x^2}\Big)^{\odot^\mu}
\odot \frac{\partial^{\nu}}{\partial x^\nu}\Big) (\psi_N(t,x,a))$$ converges uniformly on any compact $K\subset\subset \mathscr T$ to the function $$(t,x) \in \mathscr T \longmapsto \frac{1}{(2i)^{\mu}}
\Big(\Big(-\frac{\partial^2}{\partial x^2} + \frac{u}{x^2}\Big)^{\odot^\mu}
\odot \frac{\partial^{\nu}}{\partial x^\nu}\Big)(\varphi_a(t,x)).$$
Let $\lambda \in \R$. The evolution of the initial datum $x\in U \mapsto e^{i\lambda x}$ through the Schrödinger equation is explicited (for the moment formally) thanks to the expression of the Green function as $$\label{sect5-eq9}
(t,x) \in \mathscr T
\longmapsto
\frac{(-i)^{\nu +1}}{2^\nu}\, e^{ix^2/(2t)}\,
\frac{x^{\nu+1/2}}{t^{\nu+1}}\,
\int_{0}^\infty
(x')^{\nu+1/2}\, e^{i (x')^2/(2t)}\, E_\nu \Big(\frac{xx'}{t}\Big) e^{i\lambda x'}\, dx'.$$ For any $M\in \N$ such that $2M>\nu-1/2$ and any $y>0$, one has $$\begin{gathered}
E_\nu(y) = \frac{1}{\sqrt \pi}
\, \Big(\frac{2}{y}\Big)^{\nu+1/2}\,
\Big(\cos \big(y -\nu \pi/4 -\pi/2\big) \Big( \sum\limits_{\kappa =0}^{M-1}
(-1)^\kappa \frac{a_{2\kappa}(\nu)}{y^{2\kappa}} + R_{2M}(\nu,y)\Big)
\\
+ \sin \big(y -\nu \pi/4 - \pi/2\big) \Big( \sum\limits_{\kappa =0}^{M-1}
(-1)^\kappa \frac{a_{2\kappa+1}(\nu)}{y^{2\kappa+1}} + R_{2M+1}(\nu,y)\Big)\Big)\end{gathered}$$ with $$|R_{2M}(\nu,y)| < \frac{|a_{2M}(\nu)|}{y^{2M}},\quad
|R_{2M+1}(\nu,y)| < \frac{|a_{2M+1}(\nu)|}{y^{2M+1}},$$ where $$a_k(\nu) = (-1)^k \frac{\cos (\pi \nu)}{\pi}
\, \frac{\Gamma (k+1/2 +\nu) \Gamma (k+1/2-\nu)}{2^k \Gamma(k+1)}\quad
\forall\, k\in \N$$ (see [@Watson pp. 207-209]). It follows from such developments, together with Proposition \[sect4-prop1\], that the integral in exists for any $(t,x)\in \mathscr T$ as a semi-convergent integral (of the Fresnel-type), whose value coincides with the regularized integral described in section \[sect4\]. Set now $$\begin{aligned}
\EuFrak T = ]0,+\infty[, && \phi~: t\in \EuFrak T
\longmapsto -\frac{1}{2t} \in ]-\infty,0[\\
B_j~: (t,Z,\check Z) \in \EuFrak T \times \C^2
&& \longmapsto \begin{cases} E_\nu \Big(\displaystyle{\frac{Z \check Z}{t}}\Big)\
{\rm if}\ j=0 \\
0 \ {\rm if}\ j\in \N^*.
\end{cases}, \quad \chi := \nu + \frac{1}{2},\end{aligned}$$ in order to fit with the setting described in Theorem \[sect5-thm1\]. Since $E_\nu\in A_1(\C)$ and $$\frac{|Z\, \check Z|}{t} =
\frac{1}{t}\times \varepsilon |Z| \times
\frac{|\check Z|}{\varepsilon} \leq \frac{1}{2t} \Big( \varepsilon^2\, |Z|^2 +
\frac{|\check Z|^2}{\varepsilon^2}\Big)\, \forall t>0,\ \forall\,
(Z,\check Z)\in \C^2$$ the operator with order $0$ given as $t \mapsto B_0(t,Z,\check Z) \, (d/dZ)^0$ satisfies the hypothesis of this theorem with $p=1$ and $\check p = 2$. Then the operator $$\D(t) = \int_0^\infty Z^{\nu+1/2}\, e^{-i\, Z^2/(2t)}\, E_\nu\Big(
\frac{Z\cdot \check Z}{t}\Big)(\cdot )\, dZ$$ acts continuously locally uniformly in $t \in ]0,+\infty[$ from $A_1(\C)$ into $A_2(\C)$. For any $\lambda \in \R$ and $t >0$, the function $x\in ]0,+\infty[ \mapsto \varphi_\lambda (t,x)$ is $C^\infty$ because of its expression . Moreover, when $a\in \R \setminus [-1,1]$, it follows from Theorem \[sect2-thm1\] that the sequence $$\Big\{z\in \C \mapsto \D(t)\Big(\sum\limits_{j=0}^N C_j(N,a) e^{i(1-2j/N)(\cdot)}\Big)(z)\Big\}_{N\geq 1}$$ converges in $A_2(\C)$ (locally uniformly with respect to $t>0$) to $z\mapsto \D(t)(e^{ia(\cdot)})$. One concludes then to the second assertion in the statement of the theorem. As for the last assertion, it follows from the fact that the action of $i\partial/\partial t$ and $\mathscr H(x)$ coincide on solutions of , together with the continuity of the operator $d/dz$ from $A_2(\C)$ into itself.
The Schrödinger Cauchy problem for the quantum harmonic oscillator {#sect5-3}
------------------------------------------------------------------
Let now $U=\R$ and the hamiltonian in be $x\in \R \mapsto \mathscr H(x) = - (\partial^2/\partial x^2)/2 + x^2/2$. The corresponding Cauchy-Kowalevski problem (with $[0,+\infty[ \times \R$ as phase space) is the [*Schrödinger Cauchy problem for the quantum harmonic oscillator*]{}, see [@acsst5 §5.3, §. 6.4] or [@harmonic] for more references. In this case again, the Green function can be explicited and is therefore handable. It is the locally integrable function in $]0,+\infty[\times \R \times \R$ defined as $$\begin{gathered}
\label{sect5-eq10}
G(t,x,t'=0,x') =
\sqrt{\frac{1}{2i\pi \sin t}} \, e^{\displaystyle{
i \, \Big(\frac{(x^2+(x')^2)\cos t - 2 xx'}{2 \sin t}}\Big)} \\
= \Big(\sqrt{\frac{1}{2i\pi \sin t}}\,
e^{\displaystyle{i\, \frac{{\rm cotan}\, t}{2}\, x^2}}\Big)\,
e^{\displaystyle{i\, \frac{{\rm cotan}\, t}{2}\, (x')^2}}\, e^{ \displaystyle{- i \frac{xx'}{\sin t}}}\quad (t>0, x, x'\in \R).\end{gathered}$$
\[sect5-prop2\] Let $\mathscr T = ]0,+\infty[\times \R$ and $\mathscr H~: x\in \R \mapsto -(\partial^2/\partial x^2 - x^2)/2$. For any $\lambda \in \R$, the initial datum $x\in \R \mapsto e^{i\lambda x}$ evolves through the Cauchy-Kovalewski Schrödinger equation to a $\C$-valued distribution $\varphi_\lambda
\in \mathscr D'(\mathscr T,\C)$ with singular support $\pi (2\N+1)/2 \times \R$. Let $\mathscr U = \mathscr T \setminus (\pi (2\N+1)/2 \times \R)$. For any $a\in \R \setminus [-1,1]$, the family $\{(\varphi_\lambda)_{|\mathscr U}\,;\, \lambda \in \R\}$, considered as a family of functions, admits as a $\mathscr F$-supershift (in the sense of Definition \[sect5-def1\]) the sequence $$\{(t,x)\in \mathscr U \mapsto \psi_N(t,x,a)\}_{N\geq 1} =
\Big\{\sum\limits_{j=0}^N C_j(N,a)\, (\varphi_{1-2j/N})_{|\mathscr U}\Big\}_{N\geq 1}$$ with $\mathscr F$-supershift domain equal to $\mathscr U$. Moreover, for any $(\mu,\nu)\in \N^2$, the sequence of functions from $\mathscr U$ to $\C$ $$\frac{\partial^{\mu+\nu}}{\partial t^\mu
\partial x^\nu} (\psi_N(t,x,a)) = \frac{1}{(2i)^{\mu}}
\Big(\Big(-\frac{\partial^2}{\partial x^2} + x^2 \Big)^{\odot^\mu}
\odot \frac{\partial^{\nu}}{\partial x^\nu}\Big) (\psi_N(t,x,a))$$ converges uniformly on any compact $K\subset\subset \mathscr U$ to the function $$(t,x) \in \mathscr T \longmapsto \frac{1}{(2i)^{\mu}}
\Big(\Big(-\frac{\partial^2}{\partial x^2} + x^2 \Big)^{\odot^\mu}
\odot \frac{\partial^{\nu}}{\partial x^\nu}\Big)(\varphi_a(t,x)).$$
Consider the two (for the moment formal) operators $$\label{sect5-eq11}
t\in\, ]0,+\infty[\,\setminus\, \pi \N^*/2\, \longmapsto\, |\sin t|\, \int_0^\infty
e^{\displaystyle{i\, \frac{\sin 2t}{4}\, Z^2}}
e^{ \displaystyle{- i \varpi\, \,{\rm sign}\, (\sin (t))\, \check Z Z }}\circ H_{\varpi |\sin t|}\, (\cdot)\, dZ$$ ($\varpi=\pm 1$) which appear (after performing the change of variables $Z\leftrightarrow |\sin t|\, Z$ on $[0,+\infty[$) in the splitting of $$t\in\, ]0,+\infty[\,\setminus\, \pi \N^*/2 \, \longmapsto \int_\R e^{\displaystyle{i\, \frac{{\rm cotan}\, t}{2}\, Z^2}}\,
e^{ \displaystyle{- i\, \check Z Z/\sin t}}\, (\cdot)\, dZ$$ (see Remark \[sect4-rem1\]). Set now $$\begin{aligned}
\EuFrak T = ]0,+\infty[\, \setminus\, \pi \N^*/2, && \phi~: t\in \EuFrak T
\longmapsto - \frac{\sin (2t)}{4}\\
B_j~: (t,Z,\check Z) \in \EuFrak T \times \C^2
&& \longmapsto \begin{cases} \exp (-i\, \varpi\, {\rm sign}\, (\sin (t))\, Z \check Z)\odot
H_{\varpi |\sin t|}
\
{\rm if}\ j=0 \\
0 \ {\rm if}\ j\in \N^*.
\end{cases},\quad \chi=0\end{aligned}$$ ($\varpi =\pm 1$) in order to fit with the setting described in Theorem \[sect5-thm1\]. As in the proof of Proposition \[sect5-prop1\], this theorem applies here and the two operators act continuously from $A_1(\C)$ to $A_2(\C)$ (locally uniformly with respect to the parameter $t\in \EuFrak T$). Note again that the Fresnel-type integrals , where $Z \mapsto e^{i\lambda Z}$ ($\lambda \in \R)$ is taken inside the bracket and $\check Z \in \R$, are semi-convergent and their values as semi-convergent integrals coincide with the values that are obtained by regularization as in section \[sect4\]. In fact, in the case where $\check Z = x\in \R$ and $t\in \EuFrak T$, the value of $$\Big(\sqrt{\frac{1}{2i\pi \sin t}}\,
e^{\displaystyle{i\, \frac{{\rm cotan}\, t}{2}\, \check Z^2}}\Big)
\int_\R e^{\displaystyle{i\, \frac{{\rm cotan}\, t}{2}\, Z^2}}\,
e^{ \displaystyle{- i\, \check Z Z/\sin t}}\, (\cdot)\, dZ$$ (understood as a regularized integral, see section \[sect4\], in particular Remark \[sect4-rem1\]) equals $$(\cos t)^{-1/2} e^{- i \check Z^2\, \frac{{\rm tan}(t)}{2}}\,
e^{- i \lambda^2\, \frac{{\rm tan}(t)}{2}} \odot H_{1/\cos t}(e^{i\lambda (\cdot)})(\check Z)$$ (see [@acsst5 Proposition 5.3.1]). Since $(t,x) \in ]0,+\infty[\times \R
\longmapsto (\cos t)^{-1/2} e^{- i x^2\, {\rm tan}(t)}\,
e^{- i \lambda^2\, {\rm tan}(t)}$ is a locally integrable function, the initial datum $x\in \R \mapsto e^{i\lambda x}$ evolves through the Schrödinger equation as a distribution $\varphi_\lambda$ (in fact defined by a locally integrable function). Let $\D(t)$ the differential operator $$\D~: t\in ]0,\infty[\, \setminus\,
\pi\, \frac{2\N+1}{2} \longmapsto \sum\limits_{j=0}^\infty
\frac{1}{j!} \Big( i \frac{\sin 2t}{4}\Big)^j
(d/dW)^{2j}.$$ Since $$(\cos t)^{-1/2} e^{- i \check Z^2\, \frac{{\rm tan}(t)}{2}}\,
e^{- i \lambda^2\, \frac{{\rm tan}(t)}{2}} \odot H_{1/\cos t}(e^{i\lambda (\cdot)})(\check Z) = (\cos t)^{-1/2} e^{- i \check Z^2\, \frac{{\rm tan}(t)}{2}}\, \D(t) (e^{i\lambda (\cdot)})(\check Z),$$ and $\D$ acts continuously locally uniformly in $t$ from $A_1(\C)$ to $A_2(\C)$ thanks to Lemma \[sect2-lem2\], the sequence $$\Big\{\sum\limits_{j=0}^N C_j(N,a)\, (\varphi_{1-2j/N})_{|\mathscr U}\Big\}_{N\geq 1}$$ is, for any $a\in \R\setminus [-1,1]$, a supershift for the family $\mathscr F = \{(\varphi_\lambda)_{|\mathscr U}\,;\, \lambda \in \R\}$ (with $\mathscr F$-supershift domain $\mathscr U$). The last assertion follows from the same argument than that used for the last assertion in Proposition \[sect5-prop1\].
Singularities in the quantum harmonic oscillator evolution {#sect6}
==========================================================
This section is the natural continuation of subsection \[sect5-3\]. We continue to investigate with respect to the notion of supershift the evolution of initial data $x\in \R \mapsto e^{i\lambda x}$, when $\lambda \in \R$, through the Cauchy-Schrödinger problem for the quantum harmonic oscillator and focus now on singularities. In this section we keep the same notations as in Proposition \[sect5-prop2\] and fix a point $(t_0,x_0)$ in $\mathscr T
\setminus \mathscr U$. We will just consider the case $t_0=\pi/2$ since the situation is essentially identical at any point $((2k+1)\pi/2,x_0)$ with $k\in \N$ and $x_0\in \R$. Let, for $\lambda \in \R$, $\varphi_\lambda \in \mathscr D'(\mathscr T,\C)$ be the distribution evolved from the initial datum $x\mapsto e^{i\lambda x}$ through the Schrödinger operator for the quantum oscillator problem (with $\mathscr H~: x\in \R \mapsto (-\partial^2/\partial x^2 + x^2)/2$). Let $\theta \in \mathscr D(\mathscr T,\C)$ be a test-function with support in a small neighborhood of $(\pi/2,x_0)$ and $(t,x) \mapsto \xi(t,x) := \theta (t,x) \exp ((i x^2 {\rm cotan}\, t)/2)/\sqrt{2i\pi}$. One has (formally) for any $\lambda \in \R$, $$\begin{gathered}
\label{sect6-eq1}
\langle \varphi_\lambda,\theta \rangle
= - \int_{\R^2}
\Big[
\int_\R e^{i\frac{\sin u}{2}\, Z^2} e^{-i \check Z Z/\sqrt{\cos u}} (e^{i\lambda (\cdot)})\, dZ\Big]_{\check Z = x}\, \xi(\pi/2-u,x)\, du\, dx \\
= \int_{\R^2}
\Big[
\int_\R e^{i \frac{\sin u}{2}\, Z^2} e^{-i \check Z Z} (e^{i\lambda (\cdot)})\, dZ\Big]_{\check Z = x}\, \widetilde \xi(u,x)\, du\, dx,\end{gathered}$$ where $\widetilde \xi(u,x) = - \sqrt{\cos u}\, \xi(\pi/2-u,\sqrt{\cos u}\, x)$ is a test-function with support about $(0,x_0)$. The regularized integral is then $$\begin{gathered}
\lim\limits_{\varepsilon \rightarrow 0_+}
\int_{\R^2}
\Big[
\int_\R e^{-\varepsilon Z^2}\, e^{i \frac{\sin u}{2}\, Z^2} e^{-i \check Z Z} (e^{i\lambda (\cdot)})\, dZ\Big]_{\check Z = x}\, \widetilde \xi(u,x)\, du\, dx \\
= \lim\limits_{\varepsilon \rightarrow 0_+}
\int_{\R^2}
\Big[
\int_\R e^{-\varepsilon Z^2}
e^{i \frac{\sin u}{2}\, \big(Z^2 -2(\check Z -\lambda)/\sin u\big)} dZ\Big]_{\check Z=x} \, \widetilde \xi(u,x)\, du\, dx \\
= \int_{\R^2}
\Big[\exp \Big( \frac{2i}{\sin u}(\check Z -\lambda)^2\Big))\Big]_{\check Z=x}\,
\sqrt{\frac{2 i\pi}{\sin u}}\, \widetilde \xi(u,x)\, du\, dx \\
=
\int_{\R^2}
\Big[\exp \Big( \frac{i}{v}(\check Z -\lambda)^2\Big))\Big]_{\check Z=x}
\sqrt{\frac{1}{v}}\, \widetilde \theta(v,x)\, dv\, dx\end{gathered}$$ for some test-function $(v,x)\mapsto \widetilde \theta(v,x)$ with support about $(0,x_0)$ (one uses here Lebesgue domination theorem and the change of variables $(\sin u)/2\longleftrightarrow v$ about $u=0$). Though such expression makes sense when $\lambda \in \R$ (since $|\exp \big( i (x- \lambda)^2/v)\big)| =1$ for any point $(v,x)\in {\rm Supp}(\widetilde \xi)$), it does not make sense anymore when $\lambda \in \C$. In order to overcome this difficulty, one needs to formulate the following lemma.
\[sect6-lem1\] Let $\D(\check Z)$ ($\check Z\in \C$) be a differential operator of the form $$\label{sect6-eq2}
\sum\limits_{\kappa = 0}^\infty \Big[\frac{A_\kappa(\check Z,(d/dZ))}{\kappa!}(\cdot)\Big]_{Z=0}\, (d/dv)^\kappa,$$ (where $A_\kappa \in \C[[\check Z,d/dZ]]$ for any $\kappa \in \N$), considered as acting from the space of entire functions of the variable $Z$ to the space $\C[\check Z][[d/dv]]$. Suppose that there exist $p\geq 1$ and $\check p\geq 1$ and $B,\check B\geq 0$ such that $$\label{sect6-eq3}
\sup\limits_{\kappa \in \N,\check Z\in \C}
\big(|A_\kappa(\check Z,W)|\, \exp( - B\, |W|^p - \check B\, |W|^{\check p})\big)<+\infty.$$ Then, for any $b\geq 0$, there exists $A^{(b)}\geq 0$ such that $$\forall\, C\geq 0,\quad
\forall\, f\in A_1^{C,b}(\C), \quad
\sup\limits_{\kappa\in \N}
\big|A_\kappa (\check Z,(d/dZ))(f)(0)\big|
\leq C\, A^{(b)}\, e^{\check B\, |\check Z|^{\check p}}.$$ In particular, for any $f\in A_1^{C,b}(\C)$, $\D(\check Z)(f)$ remains an infinite order differential operator $\sum_{\kappa\geq 0} \alpha_\kappa(\check Z)(f)\, (d/dv)^\kappa$ with coefficients satisfying (independently of $f\in A_1^{C,b}(\C)$) $$\sum\limits_{\kappa \in \N}
k!\, |\alpha_\kappa (\check Z)(f)|\,
\exp(-B |\check Z|^{\check p}) = C\, A^{(b)} < +\infty.$$
The coefficients of $A_\kappa$ as a polynomial in $d/dZ$ satisfy $$\sum\limits_{\kappa,j\in \N,\ \check Z\in \C}
|a_{\kappa,j}(\check Z)| \leq C_0 \frac{b_0^j}{\Gamma(j/p)+1} e^{\check B |\check Z|^{\check p}}$$ for some absolute constants $C_0$ and $b_0$ (Lemma \[sect2-lem1\]). As in the proof of Lemma \[sect2-lem2\], one concludes that for any $f\in A_1^{C,b}(\C)$ and any $\kappa \in \N$, one has uniform estimates $|A_\kappa(\check Z,d/dZ)(f)|\leq C\, A^{(b)}\exp (b_0 b|Z|+ \check B
|\check Z|^{\check p})$ for some positive constant $A^{(b)}$. One gets the required estimates when evaluating at $Z=0$.
One can then complete Proposition \[sect5-prop2\] into the following companion proposition.
\[sect6-prop1\] Let $\mathscr T = ]0,+\infty[\times \R$ and $\mathscr H~: x\in \R \mapsto -(\partial^2/\partial x^2 + x^2)/2$. For any $\lambda \in \R$, let $\varphi_\lambda
\in \mathscr D'(\mathscr T,\C)$ be the evolved distribution from the initial datum $x\in \R \mapsto e^{i\lambda x}$ through the Cauchy-Kovalewski Schrödinger equation . Let $\mathscr F=\{\varphi_\lambda\,;\, \lambda \in \R\}$, where each $\varphi_\lambda$ is considered as a hyperfunction in $\mathscr T$. Then, for any $a\in \R\setminus [-1,1]$, the sequence $\big\{\sum_{j=0}^N C_j(N)\, \varphi_{1-2j/N}\big\}_{N\geq 1}$ is a $\mathscr F$-supershift of hyperfunctions over the $\mathscr F$-supershift domain $\mathscr T$.
Let $\theta \in \mathscr D(\R^2_{t,x},\C)$ with support a small neighborhood $V$ of the point $(\pi/2,x_0)$ ($x_0\in \R$) and $\widetilde \theta$ the test-function with support $V-(\pi/2,0) \ni (0,x_0)$ that corresponds to it through the successive transformations explicited previously. One has for any $\lambda \in \R$, $$\begin{gathered}
\label{sect6-eq4}
\langle \varphi_\lambda,\theta
\rangle = \int_\R \int_0^\infty
\Big[\Big(\exp \Big( \frac{i}{v}(\check Z -\lambda)^2\Big)\Big]_{\check Z = x}\, \frac{\widetilde \theta(v,x)}{\sqrt v}\, dv\, dx \\
- i \int_\R \int_0^\infty \Big[\Big(\exp \Big( -\frac{i}{v}(\check Z -\lambda)^2\Big)\Big]_{\check Z = x}\, \frac{\widetilde \theta(-v,x)}{\sqrt v}\, dv\, dx \\
\\
= \int_\R \int_0^\infty
\Big(
\Big[\sum\limits_{\kappa =0}^\infty
\frac{i^\kappa}{\kappa!}\, \frac{(\check Z- \lambda)^{2\kappa}}{v^{1/2 + \kappa}}\Big]_{\check Z =x}
\, \widetilde \theta(v,x) - i
\Big[\sum\limits_{\kappa =0}^\infty
\frac{(-i)^\kappa}{\kappa!}\, \frac{(\check Z- \lambda)^{2\kappa}}{v^{1/2 + \kappa}}\Big]_{\check Z =x}
\, \widetilde \theta(-v,x)\Big)\, dv\, dx.\end{gathered}$$ For any $\kappa \in \N$, the distribution $v_+^{-1/2-\kappa}
\in \mathscr D'([0,+\infty[,\R)$ can be expressed as $$v_+^{-1/2-\kappa} = \frac{2^\kappa}{\prod_{\ell=1}^\kappa
\big( 2(\kappa -\ell) +1\big)}\, (-d/dv)^\kappa (v_+^{-1/2})$$ in the sense of distributions in $\mathscr D'([0,+\infty[,\R)$. Then, one can reformulate formally as $$\begin{gathered}
\label{sect6-eq5}
\langle \varphi_\lambda,\theta \rangle
=
\sum\limits_{\kappa = 0}^\infty
\frac{(2i)^\kappa}{\kappa! \prod_{\ell=1}^\kappa
\big( 2(\kappa -\ell) +1\big)} \\
\int_{\R}
\Big\langle
\Big[\Big(\check Z + i\frac{d}{dZ}\Big)^{2\kappa}(e^{i\lambda (\cdot)})\Big]_{\check Z =x}(0)\,
\Big(\frac{d}{dv}\Big)^{\kappa} (
v_+^{-1/2}),\, \widetilde \theta(\cdot,x) - i(-1)^\kappa
\widetilde \theta(-\cdot,x) \Big\rangle\, dx.\end{gathered}$$ Lemma \[sect6-lem1\] applies to the two operators $$\begin{split}
& \D(\check Z) = \sum\limits_{\kappa =0}^\infty
\frac{1}{\kappa!}
\, \Big[
\frac{(2i)^\kappa \, (\check Z + i d/dZ)^{2\kappa}}
{\prod_{\ell=1}^\kappa
\big( 2(\kappa -\ell) +1\big)}(\cdot)\Big]_{Z=0}\, (d/dv)^\kappa \\
& \widetilde \D(\check Z) =
\sum\limits_{\kappa =0}^\infty
\frac{1}{\kappa!}
\, \Big[
\frac{(-i)^{\kappa+1} 2^\kappa \, (\check Z + i d/dZ)^{2\kappa}}
{2 \prod_{\ell=1}^\kappa
\big( (\kappa -\ell) +1\big)}(\cdot)\Big]_{Z=0}\, (d/dv)^\kappa
\end{split}$$ with $p=\check p =2$. These two operators act then continuously (locally uniformly with respect to the parameter $\check Z$) from $A_1(\C)$ into the space of infinite order differential operators in $d/dv$ (depending on the parameter $\check Z\in \C$). Such differential operators can be considered as hyperfunctions on $\R_v$ (elements of $\mathcal H(\R_v)$). Since $v_+^{-1/2}$ is a Fourier hyperfunction in the real line $\R$, the two $\mathcal H(\R)$-valued operators $f\in A_1(\C)
\longmapsto \D(\check Z)(f) \odot v_+^{-1/2}$ and $f\in A_1(\C) \longmapsto \widetilde \D(\check Z)(f) \odot v_+^{-1/2}$ are well defined (see [@kanbook Proposition 8.4.8 and Exercise 8.4.5]) and depend continuously (locally uniformly with respect to $\check Z$) on the entry $f$ in $A_1(\C)$. Proposition \[sect6-prop1\] follows then from Theorem \[sect2-thm1\] and from the expression (together with its formal reformulation ) for the evaluations $\langle \varphi_\lambda,\theta\rangle$ when $\lambda \in \R$ and $\varphi_\lambda$ is considered as an element in $\mathscr D'(\mathscr T,\C)$ (acting on $\theta \in \mathscr D(\mathscr T,\C)$) which can be also interpreted an a hyperfunction on $\mathscr T$.
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[^1]: Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9 20133 Milano, Italy
[^2]: The Donald Bren Presidential Chair in Mathematics, Chapman University, Orange, USA
[^3]: IMB, Université de Bordeaux, 33405, Talence, France
|
---
abstract: 'We show that the mass of the matter equal to the mass of the observable part of our Universe is reached at the Planck density in the volume which size is comparable with the nucleon size and is close to the pion Compton wavelength.'
author:
- 'N. A. Miskinova'
- 'B. N. Shvilkin'
title: |
A possible relation of the mass of the Universe\
with the characteristic sizes of elementary particles
---
The dimension analysis in physics often leads to the discovery of essentially important laws. Max Planck, for example, by the dimension analysis managed to introduce sizes, having dimensional lengths and time, the so-called Planck length and time [@r1], the extreme smallness of which have led to concept of discrete behavior of space and time. The Plank length ${\ell _{\rm{P}}} \sim {10^{ - 33}}\,\mbox{cm}$ defines the “quantum" of dimensional distance, and the time ${\tau _{\rm{P}}} \sim {10^{ - 44}}\,\mbox{s}$ defines the time “quantum". As a result from the dimension analysis the conclusion has been drawn on the necessity of building the quantum theory for discrete space-time [@r2] (see also [@r3] and the references therein).
Let’s carry out the analysis of dimensional Planck values in application to the expanding Universe. At the beginning of expansion of the Universe the substance was in a so-called vacuum state [@r4]. The matter density was extremely great and was sustained constant. This density called Planck density is possible to express through Planck dimensional values: $$\label{e1}
m_{\rm P} = \sqrt{\frac{\hbar c}{G}} \simeq 2.18 \times 10^{ - 5}\,\mbox{g},$$ Planck mass, and $$\label{e2}
\ell _{\rm P} = \frac{\hbar }{m_{\rm P}c} = \sqrt {\frac{\hbar G}{c^3}} \simeq 1.62 \times 10^{- 33}\,\mbox{cm},$$ Planck length, where $\hbar$ is Planck constant, $G$ is Newtonian gravitational constant, and $c$ is speed of light in vacuum (here and below values of physical and astrophysical constants are taken from [@r5]).
The substance in a vacuum state was characterized by gravitational repulsion (negative pressure), and it caused the powerful initial push, which caused almost instant expansion of material, so-called inflation [@r4].
At some instant of time the size of the Universe was insignificantly small, it was defined by the size of the Planck cell with volume $\ell _{\rm P}^3 \simeq 4.22 \times 10^{- 99}\,\mbox{cm}^3$. The density of matter in this cell was huge (see Eqs. (\[e1\]) and (\[e2\])): $$\label{e3}
\rho _{\rm P} = \frac{m_{\rm P}}{\ell_{\rm P}^3} = \frac{c^5}{\hbar G^2} \simeq 5.15 \times 10^{93}\,\mbox{g}\, \mbox{cm}^{- 3}.$$ And it was so despite of small mass of matter inside the Planck volume, of about $10^{-5}\,\mbox{g}$, and rather small number of nucleons ${m_{\rm{P}}}/{m_p} \simeq 1.30 \times {10^{19}}$, comparable with Loschmidt’s number ${N_{\rm{L}}} \simeq 2.69 \times {10^{19}}$ (number of molecules in $1\,\mbox{sm}^3$ of ideal gas under normal conditions).
It is considered that the Planck density dividing quantum and classical space-time defines a state of matter which conditionally is taken over for the “beginning" or the “birth" of our Universe [@r4].
While matter expanding, the volume occupied by it became more and more. Also the mass of substance, because of negative energy of gravitation, thus increased at almost constant density. So it proceeded until the vacuum substance through an insignificant instant had not turned in the quantum mode to the usual matter of the Universe. For this shortest time the Universe had swelled incredibly (see, for example, [@r4]).
Let’s estimate, what mass of material with the Planck density can be concentrated in the sphere volume with radius equal to the Compton wavelength of the $\pi$ meson ${\mathchar'26\mkern-10mu\lambda _\pi } = \hbar /({m_\pi }c)$. As it is known, this size defines the range of nuclear forces and is close to the nucleon size (the rms charge radius of the proton ${r_p} \simeq 0.88 \times {10^{ - 13}}\,{\mbox{cm}}$ [@r5]) and to the radius of confinement (holding quarks and gluons inside hadrons) [@r6]. In the case of the $\pi ^0$ meson we have length and the corresponding volume $$\label{e4}
{\mathchar'26\mkern-10mu\lambda _{{\pi ^0}}} \simeq 1.46 \times {10^{ - 13}}\,{\mbox{cm}},\quad {V_0} = \frac{{4\pi }}{3}\mathchar'26\mkern-10mu\lambda _{{\pi ^0}}^3 \simeq 3.12 \times {10^{ - 38}}\,{{\mbox{cm}}^3}.$$ For the $\pi^+$ meson, $$\label{e5}
{\mathchar'26\mkern-10mu\lambda _{{\pi ^ + }}} \simeq 1.41 \times {10^{ - 13}}\,\mbox{cm},\quad {V_ + } = \frac{{4\pi }}{3}\mathchar'26\mkern-10mu\lambda _{{\pi ^ + }}^3 \simeq 2.83 \times {10^{ - 38}}\,\mbox{cm}^3.$$
The mass of matter in volume ${V_0}$ at the Planck density (\[e3\]) is equal to $$\label{e6}
M_0 = \rho _{\rm P}V_0 = \frac{4\pi}{3}m_{\rm P}\left( \frac{\mathchar'26\mkern-10mu\lambda _{\pi ^0}}{\ell _{\rm P}} \right)^3 \simeq 6.75 \times 10^{55}\,\mbox{g}.$$ Similarly, $$\label{e7}
M_+ = \rho _{\rm P}V_ + = \frac{4\pi}{3}m_{\rm P}\left(\frac{\mathchar'26\mkern-10mu\lambda _{\pi ^ +}}{\ell _{\rm P}}\right)^3 \simeq 6.10 \times 10^{55}\,\mbox{g}.$$ Masses (\[e6\]) and (\[e7\]) are equivalent to the total mass of approximately $4\times10^{79}$ nucleons.
The obtained values of masses of $M_0$ and $M_+$ appear to be of the same order of magnitude with the mass of the observable part of our Universe $M_U$. According to modern knowledge, $$\label{e8}
M_U = \frac{4\pi}{3}\rho _c(ct_0)^3 \simeq 8.84 \times 10^{55}\,\mbox{g},$$ where $t_0 = 13.69(13) \times 10^9\,{\mbox{yr}}$ is the age of the Universe, ${\rho _c} = 1.87835(19) \times {10^{ - 29}}{h^2}\,{\mbox{g}}~\mbox{cm}^{ - 3}$ is the critical density of the Universe, $h = 0.72(3)$ is the present day normalized Hubble expansion rate.
Comparison of the masses (\[e6\])–(\[e8\]) shows that difference between them is rather small: $$\frac{M_U - M_0}{M_U} \simeq 0.24,\quad \frac{M_U - M_ + }{M_U} \simeq 0.31,\quad \frac{M_0 - M_ +}{M_ +} \simeq 0.10.$$
Let’s define also the effective size $r_c$ by a relation ${M_U} = (4\pi /3)\rho _{\rm P}r_c^3$, whence $$r_c = \left(\frac{3}{4\pi}\frac{M_U}{\rho _{\rm P}}\right)^{1/3} \simeq 1.60 \times {10^{ - 13}}\,\mbox{cm}.$$ This size differs from the pion Compton wavelength a little (see Eqs. (\[e4\]) and (\[e5\])): $$\frac{r_c - \mathchar'26\mkern-10mu\lambda _{\pi ^0}}{\mathchar'26\mkern-10mu\lambda _{\pi ^0}} \simeq 0.09,\quad \frac{r_c - \mathchar'26\mkern-10mu\lambda _{\pi ^ +}}{\mathchar'26\mkern-10mu\lambda _{\pi ^ +}} \simeq 0.13.$$
The presented data allow us to assume that the typical hadron size and the mass in the corresponding volume at the Planck density are the characteristic size and the critical mass at the Big Bang originated from the Planck length and mass at the birth of the Universe.
In conclusion, it is shown that the mass of a “Planck nucleon" (a sphere with a radius of order of the size of the nucleon filled with matter of the Planck density) is close to the mass of the observable part of our Universe.
The authors thank Prof. A. V. Borisov for his interest in this work and for discussion of the results.
[99]{}
M. Plank, [*Selected Works*]{} (Nauka, Moscow, 1975), p. 232 \[in Russian\].
V. Ambarzumian and D. Iwanenko, Z. Phys. [**64**]{}, 563 (1930).
A. N. Vyaltsev, [*Discrete Space-Time*]{} (KomKniga, Moscow, 2007) \[in Russian\].
I. D. Novikov, Vestnik RAN [**71**]{} (10), 886 (2001) \[in Russian\].
K. Nakamura et al. (Particle Data Group), J. Phys. G [**37**]{}, 075021 (2010).
L. B. Okun’, [*Elementary Particle Physics*]{} (Nauka, Moscow, 1986) \[in Russian\].
|
---
author:
- Jian Li
- Zhihong Jeff Xia
date: 'Received September 15, 1996; accepted March 16, 1997'
title: Mean plane of the Kuiper belt beyond 50 AU in the presence of Planet 9
---
[A recent observational census of Kuiper belt objects (KBOs) has unveiled anomalous orbital structures. This has led to the hypothesis that an additional $\sim5-10~m_{\oplus}$ planet exists. This planet, known as Planet 9, occupies an eccentric and inclined orbit at hundreds of astronomical units. However, the KBOs under consideration have the largest known semimajor axes at $a>250$ AU; thus they are very difficult to detect.]{} [In the context of the proposed Planet 9, we aim to measure the mean plane of the Kuiper belt at $a>50$ AU. In a comparison of the expected and observed mean planes, some constraints would be put on the mass and orbit of this undiscovered planet.]{} [We adopted and developed the theoretical approach of @volk17 to the relative angle $\delta$ between the expected mean plane of the Kuiper belt and the invariable plane determined by the eight known planets. Numerical simulations were constructed to validate our theoretical approach. Then similar to @volk17, we derived the angle $\delta$ for the real observed KBOs with $100<a<200$ AU, and the measurement uncertainties were also estimated. Finally, for comparison, maps of the theoretically expected $\delta$ were created for different combinations of possible Planet 9 parameters.]{} [The expected mean plane of the Kuiper belt nearly coincides with the said invariable plane interior to $a=90$ AU. But these two planes deviate noticeably from each other at $a>100$ AU owing to the presence of Planet 9 because the relative angle $\delta$ could be as large as $\sim10^{\circ}$. Using the $1\sigma$ upper limit of $\delta<5^{\circ}$ deduced from real KBO samples as a constraint, we present the most probable parameters of Planet 9: for mass $m_9=10~m_{\oplus}$, orbits with inclinations $i_9=30^{\circ}$, $20^{\circ}$, and $15^{\circ}$ should have semimajor axes $a_9>530$ AU, 450 AU, and 400 AU, respectively; for $m_9=5~m_{\oplus}$, the orbit is $i_9=30^{\circ}$ and $a_9>440$ AU, or $i_9<20^{\circ}$ and $a_9>400$ AU. In this work, the minimum $a_9$ increases with the eccentricity $e_9$ ($\in[0.2, 0.6]$) but not significantly.]{}
Introduction
============
The Kuiper belt is a disk of icy minor planets extending outward from the orbit of Neptune (i.e., $\sim30$ AU). At the time of writing, more than 600 distant Kuiper belt objects (KBOs hereafter) have been discovered at semimajor axes $a$ beyond 50 AU up to more than 1000 AU. These objects can provide new and rich information on the dynamical environment of the very edge of the solar system. Especially, the farthest components with $a>250$ AU, including Sedna and 2012 VP113, exhibit clustering in the argument of perihelion and the longitude of the ascending node [@truj14; @baty16]. One possible explanation for this orbital distributions is the existence of an additional $5-10~m_{\oplus}$ planet residing on an extremely wide orbit ($a_9=400-800$ AU), which has eccentricity $e_9=0.2-0.6$ and inclination $i_9=15^{\circ}-30^{\circ}$ [@baty16; @brow16; @baty19]. This hypothesized planet is currently referred to as Planet 9.
Moreover, @bail16, @lai16, and @gome17 found that the secular perturbation from Planet 9 could induce a slow precession of the invariable plane of the solar system. In this work, the invariable plane is defined as the plane perpendicular to the total angular momentum vector of the eight known planets, and hereafter is named IP8. Over a timespan of $\sim4.5$ Gyr, the proposed Planet 9 can yield the current $6^{\circ}$ tilt between the Sun’s equator and IP8. Then, we speculate that this mechanism could also cause the change of the apparent orbital planes of the distant KBOs.
The first attempt to determine the mean plane of the Kuiper belt was made more than a decade ago by @coll03 and @brow04. As a much larger number of KBOs have been discovered in recent years, @volk17 extended the study of the mean plane as a function of the semimajor axis. These authors found that, for the classical Kuiper belt between 42 AU $<a<$ 48 AU, the expected mean plane has an inclination $\lesssim2^{\circ}$ with respect to IP8 from secular theory; this value is consistent with the current observations. At a larger semimajor axis $a>50$ AU, the expected mean plane is very flat and close to IP8 in the unperturbed solar system model (i.e., without any additional planet). Even by including the possible Planet 9 on high-inclination orbit, for the Kuiper belt exterior to but not too far away from 50 AU, this plane would be nearly unaffected and still confined to the local Laplacian plane determined by the Jovian planets (almost equivalent to IP8). But if the KBOs are approaching the orbit of Planet 9 at hundreds of AU, the associated mean plane could substantially deviate from IP8. @laer19 found that from 50 AU to 150 AU the mean plane of the observed Kuiper belt is consistent with IP8, while @baty19 reported that the orbital planes of the 14 known KBOs with $a>250$ AU are clustered around a common plane (at the 96.5% confidence), which is inclined to the ecliptic by $\sim7^{\circ}$. In this paper, we aim to quantify exactly how far and how large the warp of the mean plane of the Kuiper belt can be produced by the perturbations from the unseen Planet 9. If such a warp is detectable, it could put constraints on the proposed mass and orbital elements of Planet 9.
The rest of this paper is organized as follows. In Section 2, we develop both the theoretical and numerical approaches to determine the expected mean plane of the Kuiper belt beyond 50 AU, in the presence of Planet 9. Accordingly, we calculate the relative angle $\delta$ between this plane and IP8. In Section 3, we measure the angle $\delta$ for the currently observed KBOs with semimajor axes in the range $100-200$ AU and the measurement uncertainty is also evaluated. In Section 4, based on the analysis of the expected and measured values of $\delta$, we present the most probable parameters of Planet 9. The conclusions are summarized in Section 5.
Expected mean plane {#EMP}
===================
Theoretical approach {#theory}
--------------------
To determine the expected mean plane of the Kuiper belt, we consider the gravitational perturbations from the eight known planets plus a hypothetical Planet 9. As for the motion of a test KBO, the inclination vector is defined by $(q, p)=(\sin i \cos \Omega, \sin i \sin \Omega)$, where $i$ and $\Omega$ are the inclination and the longitude of ascending node, respectively. Throughout this paper, the reference plane is given as IP8. Then, according to the classical Laplace-Lagrange secular theory, the forced inclination vector $(q_0, p_0)$ can be written as [see @murr99 chap. 7] $$\begin{aligned}
&&q_0=-\sum_{i=1}^{9}\frac{\mu_i}{B-f_i}\cos(f_i t + \gamma_i),\nonumber\\
&&p_0=-\sum_{i=1}^{9}\frac{\mu_i}{B-f_i}\sin(f_i t + \gamma_i),
\label{forced}\end{aligned}$$ where $f_i$ and $\gamma_i$ are the secular nodal eigenfrequency and associated phase, respectively; and $$\begin{aligned}
&&\mu_i=+n\frac{1}{4}\sum_{j=1}^{9}I_{ji}\frac{m_j}{m_{\odot}}{\alpha_j}{\bar{\alpha}_j}b_{3/2}^{(1)}({\alpha_j}),\nonumber\\
&&B=-n\frac{1}{4}\sum_{j=1}^{9}\frac{m_j}{m_{\odot}}{\alpha_j}{\bar{\alpha}_j}b_{3/2}^{(1)}({\alpha_j}),
\label{coefficient}\end{aligned}$$ where $m_{\odot}$ is the mass of the Sun, $m_j$ is the mass of the $j$-th planet, $n$ is the mean motion of the test KBO, $I_{ji}$ is the amplitude corresponding to $f_i$; and $$\alpha_j=\left\{
\begin{array}{lr}
a_j/a~~~~~~\mbox{if}~~a_j < a, & \\
a/a_j~~~~~~\mbox{if}~~a_j > a, &
\end{array}
\right.$$ $$\bar{\alpha}_j=\left\{
\begin{array}{lr}
1~~~~~~~~~~~\mbox{if}~~a_j < a, & \\
a/a_j~~~~~~\mbox{if}~~a_j > a, &
\end{array}
\right.$$ where $a$ and $a_j$ are the semimajor axes of the test KBO and the $j$-th planet, respectively. In Eq. (\[coefficient\]), the Laplace coefficient is given by $$b_{3/2}^{(1)}({\alpha})=\frac{1}{\pi}\int_{0}^{2\pi}\frac{\cos\psi \mbox{d}\psi}{(1-2\alpha\cos\psi+{\alpha}^2)^{3/2}}.
\label{laplace}$$
Since the secular modes of the known planets are supposed to be unaffected by the distant Planet 9, @volk17 express the forced inclination vector (Eq. (\[forced\])) in an approximate form $$\begin{aligned}
&&(q_0, p_0)\approx (q_0^0, p_0^0)+(q_0^1, p_0^1),\nonumber\\
&&(q_0^1, p_0^1)=\frac{\mu_9}{-B+f_9}\left(\cos(f_9 t + \gamma_9), \sin(f_9 t + \gamma_9)\right),
\label{approximate}\end{aligned}$$ where the subscript 9 refers to Planet 9; and $$\mu_9\approx +n\frac{1}{4}\frac{m_9}{m_{\odot}}{\alpha_9}{\bar{\alpha}_9}b_{3/2}^{(1)}({\alpha_9})\sin i_9,
\label{mu9}$$ $$f_9 \approx -n_9\frac{1}{4}\sum_{j=1}^{8}\frac{m_j}{m_{\odot}}\left(\frac{a_j}{a_9}\right)b_{3/2}^{(1)}\left(\frac{a_j}{a_9}\right),
\label{f9}$$ where $n_9$ is the mean motion of Planet 9. @volk17 also point out that, for $a>50$ AU, the vector $(q_0^0, p_0^0)$ represents the forced plane determined by the known plants, which should be nearly coincident with IP8. Therefore, the tilt of the forced plane of test KBOs in this $a$ region is solely induced by Planet 9 and can be measured by the forced inclination $$i_0=\sqrt{\left(q_0^1\right)^2+\left(p_0^1\right)^2}=\arcsin\left(\frac{\mu_9}{-B+f_9}\right).
\label{i0}$$ In this paper, we consider the Kuiper belt extended from 50 AU all the way out to the neighborhood of Planet 9 at hundreds of AU; that is, the semimajor axis ratio $\alpha_9=a/a_9$ could not always be close to 1. Then unlike in @volk17, for the calculation of the Laplace coefficient $b_{3/2}^{(1)}({\alpha})$, we evaluate the integral in Eq. (\[laplace\]) without any approximation.
Bearing in mind that, the proposed Planet 9 has a substantial eccentricity $e_9$. The value of $e_9$ is very important for determining the forced inclination $i_0$, as we see below. Borrowing the method from @gome06 to implement the averaged effect of the eccentric Planet 9, we assume a scaled semimajor axis $$\tilde{a}_9=a_9\sqrt{1-e_9^2}.$$ Substituting the expression of $\tilde{a}_9$ for $a_9$ into Eqs. (\[mu9\]) and (\[f9\]), we can finally obtain the intrigue value of $i_0$ from Eq. (\[i0\]).
![Relative angle $\delta$ between the expected mean plane of the Kuiper belt and IP8 in the presence of a $10~m_{\oplus}$ Planet 9 with orbital elements $a_9$ (top panel: $=400$ AU; bottom panel: $=600$ AU), $e_9$ ($=0.4-0.6$), and $i_9$ ($=20^{\circ}-30^{\circ}$), as a function of the semimajor axis $a$ ($>50$ AU). The curve denotes the prediction from secular theory and the dots indicate the results from numerical computations with no approximations. The figure indicates these two approaches show good agreement.[]{data-label="EMP"}](Fig1a.jpg){width="8.5cm"}
![Relative angle $\delta$ between the expected mean plane of the Kuiper belt and IP8 in the presence of a $10~m_{\oplus}$ Planet 9 with orbital elements $a_9$ (top panel: $=400$ AU; bottom panel: $=600$ AU), $e_9$ ($=0.4-0.6$), and $i_9$ ($=20^{\circ}-30^{\circ}$), as a function of the semimajor axis $a$ ($>50$ AU). The curve denotes the prediction from secular theory and the dots indicate the results from numerical computations with no approximations. The figure indicates these two approaches show good agreement.[]{data-label="EMP"}](Fig1b.jpg){width="8.5cm"}
For the KBOs with instantaneous locations in three-dimensional space, the pole of the mean plane should be aligned with the forced inclination vector [@chia08]. Equation (\[approximate\]) shows that the direction of $(q_0^1, p_0^1)$ depends on the secular mode of Planet 9 and varies with time. But as a magnitude, the forced inclination $i_0$ from Eq. (\[i0\]) remains constant. Therefore, $i_0$ measures the relative angle $\delta$ between the mean plane of the Kuiper belt and IP8. Since the masses and semimajor axes of the known planets are constant in calculating $B$, at a certain $a$, the tilt of the Kuiper belt’s mean plane (i.e., $\delta$) is completely determined by the mass ($m_9$) and orbital elements ($a_9$, $e_9$ and $i_9$) of Planet 9.
As described in the introduction, Planet 9 could possibly have $m_9=10~m_{\oplus}$, $a_9=400-600$ AU, $e_9=0.4-0.6$, and $i_9=20^{\circ}-30^{\circ}$. Within this parameter space, we calculated the tilt $\delta$ of the expected mean plane by our theoretical approach. Figure \[EMP\] shows that the value of $\delta$ is as small as $<1^{\circ}$ in the semimajor axis range of $a\sim50-90$ AU, indicating that the Kuiper belt’s mean plane should nearly coincide with IP8. But along with the increasing $a$, the mean plane would become more and more inclined relative to IP8, by up to the order of magnitude of $\delta\sim10^{\circ}$. As envisioned by @volk17, such a massive and distant Planet 9 has a negligible effect on the mean plane of the KBOs exterior to $a\sim100$ AU. We note that throughout this paper we do not consider the semimajor axis region of $q_9-a < 10$ AU ($q_9$ is the perihelion of Planet 9), where the KBOs could undergo strong gravitational interactions with Planet 9 and secular theory may be not available.
Numerical approach {#numerical}
------------------
For the eight known planets, their masses, initial positions, and velocities are adopted from DE405 with epoch 1969 June 28 [@stan98]. Then we calculate the total angular momentum vector $\vec{H}_8$ of these planets, by which the inclination and longitude of the ascending node of IP8 can be determined [@soua12]. Using the rotational transformation, we proceed to change the reference plane from the mean equator of J2000.0 to IP8. Finally, we introduce Planet 9 with $m_9$, $a_9$, $e_9$ and $i_9$ to our solar system model.
Considering a set of $N$ test particles, each having the position $\vec{r}_j$ and velocity $\vec{v}_j$, the total (unit) angular momentum vector is given by $$\vec{H}=\sum_{j=1}^{N}\vec{r}_i\times\vec{v}_i.$$ Then the relative angle $\delta$ between the vectors $\vec{H}$ and $\vec{H}_8$ is computed by [@li2019] $$\delta=\arccos\left(\frac{\vec{H}_8\cdot\vec{H}}{|\vec{H}_8|\cdot|\vec{H}|}\right).
\label{angle}$$ As long as the sample size $N$ is large enough, this angle could represent the deviation of the mean plane of the particles from IP8 [@camb18].
### Pre-runs
Given $m_9=10~m_{\oplus}$, firstly we chose the orbital elements of Planet 9 to be $a_9=400$ AU, $e_9=0.6$ and $i_9=30^{\circ}$. In this case, Planet 9 has the minimum perihelion $q_9=160$ AU and the highest inclination according to the proposed parameter space [@bail16; @baty19]. Consequently, such an additional planet would exert the strongest influence on the mean plane of the Kuiper belt.
The test particles were uniformly distributed in the $a$ space between 50 AU and 150 AU, where 10 AU interior to the perihelion of Planet 9. Within each $a$-bin of 10 AU (e.g., $a=50-60$ AU), there were 101 test particles with even separation $\Delta a=0.1$ AU. As for the “nominal” population, initial $e$-values were taken to be 0.01, and initial $i$-values were randomly sampled in the range $0^{\circ}-20^{\circ}$. The other three orbital elements were all chosen randomly between $0^{\circ}$ and $360^{\circ}$. For the sake of saving computation time, in all numerical simulations performed below, the four terrestrial plants are added to the Sun; and the angular momentum vector $\vec{H}_8$ is determined by the four Jovian planets. This simplicity has little influence on the measurement of the angle $\delta$. We then integrated the system consisting of the Sun, Jovian planets, Planet 9, and test particles over 4.5 Gyr.
We find that because of the perturbations from Planet 9, the direction of the vector $\vec{H}_8$ (i.e., the pole of IP8) is not steady but evolves with time. The precession of the inclination of IP8 with respect to its initial plane observed in our simulations is identical to that shown in Fig. 2 of @bail16. Such that, in the numerical approach, our reference plane is indeed the instantaneous IP8. Accompanying the evolution of $\vec{H}_8$, the total angular momentum vector $\vec{H}$ of test particles also keeps changing direction, but the relative angle $\delta$ between these two vectors would no longer remain around zero.
As we expected, at the end of the integration, the angle $\delta$ increases monotonously with increasing heliocentric distance, from $\sim1^{\circ}$ for the $a=50-60$ AU bin to $\sim6^{\circ}$ for the $a=120-130$ AU bin. For test particles farther than $a=130$ AU, they experienced stronger perturbations from Planet 9 and a small fraction survived, thus a meaningful measurement of the mean plane cannot be reached. The associated $\delta$-value is determined later from high $a$-resolution simulations with a larger sample size. Aside from the nominal population of test particles, we also carried out several additional simulations by varying either the initial $e$ or $i$, for example, $i=0.01^{\circ}$, $e=0-0.2,$ and $e=0.2-0.4$. These different inputs all reproduce nearly the same outcomes. The independence of the angle $\delta$ on particles $e$ and $i$ is easy to understand because these two orbital parameters are not visible in the calculation of the forced inclination $i_0$ (equivalent to $\delta$), as presented in Section \[theory\].
### High $a$-resolution runs
![Time evolution of the angle $\delta$ for the semimajor axis bin of $a=120-130$ AU due to the presence of a $10~m_{\oplus}$ Planet 9 with different orbit sets $(a_9, e_9, i_9)$. The figure shows that $\delta$ can nearly converge to the average value at the end of the 1.5 Gyr integration.[]{data-label="alphaT"}](Fig2.jpg){width="8.5cm"}
In order to refine the value of $\delta$, we employed a higher spatial resolution of $\Delta a=0.02$ AU, which yields 501 test particles in each $a$-bin with a width of 10 AU (starting from $a=50-60$ AU). The initial eccentricities and inclinations were adopted to be the same as those of the nominal population mentioned about above, i.e., $e=0.01$ and $i=0^{\circ}-20^{\circ}$. Then we performed the high $a$-resolution runs with six different sets of $(a_9, e_9, i_9)$ for a $10~m_{\oplus}$ Planet 9, corresponding to the theoretical study in Section 2.1. In this subsection, we chose a shorter integration timescale of 1.5 Gyr for the numerical simulations. Even so, each run for a specific orbit set of Planet 9 would take over ten days of computing time on our workstation.
As an example, Fig. \[alphaT\] shows the time evolution of the angle $\delta$ for the $a=120-130$ AU bin. We note that $\delta$ is osculating around the average value, and the amplitude could become rather small at the end of the integration. Thus for test particles in each semimajor axis bin of $[a^{(1)}, a^{(2)}]$, the angle $\delta$ is taken to be the average value during the integration, and it is assigned to the location of the median $a=(a^{(1)}+a^{(2)})/2$, as indicated by the dots in Fig. \[EMP\]. In comparison with the angle $\delta$ predicted by the secular theory, as plotted by the curves in Fig. \[EMP\], we find that the numerical results are in good agreement. This can nicely support the validity of our theoretical approach developed in Section \[theory\]. It should be noticed that, for the orbital elements of Planet 9, the theoretical approach only requires $a_9$, $e_9$, and $i_9$, but neither the argument of perihelion $\omega_9$ nor the longitude of ascending node $\Omega_9$. The parameters $\omega_9$ and $\Omega_9$ could control the orbital alignments of the 14 known KBOs with $a\ge250$ AU [@baty19], but they should not affect the tilt of the Kuiper belt’s mean plane.
Fig. \[EMP\] shows that, within the $a<100$ AU region, both the theoretical and numerical values of $\delta$ are below $\sim1^{\circ}-2^{\circ}$. Considering the observational bias, we do not think such a slight warp in the Kuiper belt could be detected by future surveys. As a result, we focus on the KBOs between $100 < a < 200$ AU. Even so, this population is much closer and more detectable than the extremely distant KBOs with $a > 250$ AU that are believed to be clustered in physical space by now. We intend to use the angle $\delta$ as a constraint on the mass and orbit of the possible Planet 9 in the solar system.
Kuiper belt observation
=======================
Real KBOs {#realMP}
---------
We selected the KBOs observed over multiple oppositions as of September 2019, taken from the Minor Planet Center[^1]. Among these, there are 46 objects with semimajor axes $100 < a <200$ AU and perihelion distances $q>30$ AU. These objects comprise our sample KBOs and were used to calculate the mean plane of such a truncated Kuiper belt. In @volk17, the resonant KBOs were excluded since Neptune’s mean motion resonances are not considered in secular theory. But in the direct N-body integrations performed in Section \[numerical\], we fully took the complete perturbations from the planets including Neptune into account. The agreement between our numerical and theoretical approaches suggests that the weak high-order resonances in the distant Kuiper belt could have little impact on the mean plane determination. As a matter of fact, @sail17 found that, for the majority of resonant KBOs with $a>50$ AU, their secular behaviors can hardly be affected by mean motion resonances. Up to now, only three resonant KBOs have been identified in the $a>100$ AU region, i.e., 2004 PB$_{112}$ in the 5:27 resonance [@sail17], 2015 KE$_{172}$, and 2007 TC$_{433}$ in the 1:9 resonance [@volk18]; these KBOs are in our sample.
Since the number density of our sample KBOs is very low in the wide $a$-space, the mean plane measured by their total angular momentum may suffer from severe observational bias. Instead, in this section we apply the alternative method employed in @brow04 and @volk17: if a plane can, on average, go through all the sky-plane velocity vectors of the considered objects, then it defines the associated mean plane. For the unit pole vector $\vec{n}$ perpendicular to this mean plane, it can be computed by minimizing the residual [@volk17] $$E=\sum_{i}|\vec{n}\cdot\vec{v_i}|,
\label{pole}$$ where $\vec{v_i}$ is the unit vector of the sky-plane velocity of a KBO. All the $\vec{v_i}$ used were evaluated at a common epoch of 2019 April 27. Consequently, the mean plane can be achieved in a manner almost regardless of the discovery positions of the KBOs. Detailed descriptions can be found in the two papers cited above. It must be noted that when the number density of sample particles is large enough (e.g., that used in Section 2.2.2), we would obtain exactly the same mean plane by applying either the velocity vector (Eq. (\[pole\])) or the angular momentum (Eq. (\[angle\])), while the latter approach is much computationally cheaper.
We find that, for the $100 < a < 200$ AU Kuiper belt, using the directional velocity $\vec{v_i}$ to determine the mean plane yields an overall inclination $\tilde{\delta}\approx13.1^{\circ}$. This value seems too large to be authentic according to our theoretical results in Section 2. We then realized that by minimizing the residual $E$ in Eq. (\[pole\]), such a large $\tilde{\delta}$ can result from the contamination of certain sample(s) with $\vec{v_i}$ deviated substantially from the others, especially when the sample size is very limited. By excluding a single KBO (2015 RQ$_{281}$), the angle $\tilde{\delta}$ can drop sharply to only $1.0^{\circ}$ and a comparable value could also be obtained even if we continue to remove some additional sample(s). With such a slight relative angle of $\tilde{\delta}\sim1.0^{\circ}$, the mean plane of the considered Kuiper belt could be regarded as nearly coincident with IP8. However, the census of the distant KBOs is far from observational completeness, thus the measurement error is clearly warranted.
Monte Carlo samples
-------------------
For the uncertainty of the derived mean plane due to the small number of real observed samples, following @li2019, Monte Carlo simulations were constructed to estimate the possible $\tilde{\delta}$ for the overall $100 < a < 200$ AU Kuiper belt. It is obvious that the measurement error strongly depends on the space dispersion of the inclined KBOs. By adopting an unbiased inclination distribution from @brow4b, as $$f(i)\propto \sin i \cdot \exp(-i^2/2\sigma^2),
\label{iDistribution}$$ where the Gaussian standard deviation $\sigma=20^{\circ}$, we first created 100,000 synthetic samples with random inclinations $i$ relative to IP8. For each synthetic sample with assigned $i$, we randomly chose $$0.95a_{real} \le a \le1.05a_{real},
\label{aDistribution}$$ and $$0.95e_{real}\le e \le1.05e_{real},
\label{eDistribution}$$ where $a_{real}$ is the semimajor axis of a random component from 568 real KBOs with multiple-opposition orbits and $a>50$ AU, and $e_{real}$ is the eccentricity of another random component [also see @volk17 appendix C]. The other three angles of an orbit were randomly selected in the range $0^{\circ}-360^{\circ}$.
Then, for each object in the catalog of these real observed KBOs, near the latitude and longitude where each object was discovered, we searched for a corresponding synthetic sample. In this way we obtained a set of 568 synthetic objects, among which those with $100 < a <200$ AU and $q>30$ AU were selected to be our Monte Carlo samples. Subsequently, we measured the mean plane of a Monte Carlo population via Eq. (\[pole\]). This procedure was repeated 10,000 times for the statistical analysis.
We find that the 10,000 Monte Carlo populations give the angle $\tilde{\delta}=2.8^{\circ}\pm1.8^{\circ}$ with a $1\sigma$ confidence level. This result indicates that, for the overall $a=100-200$ AU range, the measured mean plane of the real KBOs (i.e., having $\tilde{\delta}\sim1.0^{\circ}$) could be deemed within $1\sigma$ error of the true mean plane. Accordingly, the mean plane of the Kuiper belt for this semimajor axis bin probably deviates from IP8 by less than $5^{\circ}$. This critical value could be served as an upper limit of $\tilde{\delta}$ to put constraints on the parameter space of Planet 9.
Constraints for Planet 9
========================
With the addition of Planet 9 to the solar system, the expected mean plane of the Kuiper belt at semimajor axis $a>100$ AU would not remain in the vicinity of IP8, but can be substantially inclined by up to $\sim10^{\circ}$, as shown in Fig. \[EMP\]. While based on the observational data of the KBOs, the true mean plane is found to have an inclination of $<5^{\circ}$ at $1\sigma$ confidence. We thereby suppose that the existence of Planet 9 is possible if the relative angle $\delta$ between the expected mean plane and IP8 is smaller than or comparable to $5^{\circ}$. The larger $\delta$ may indicate that Planet 9 has produced a distinguishable discrepancy from the current observation, leading to the unlikelihood of certain mass ($m_9$) or orbital elements ($a_9$, $e_9$, $i_9$). Since the theoretical and numerical approaches agree nicely (see Fig. \[EMP\]), we used the former to make our prediction, while the latter is too computationally expensive to fulfill an extensive suite of calculations with various parameters of Plane 9.
To examine the possibility of Planet 9, we consider a combination of the proposed $m_9$, $a_9$, $e_9$, and $i_9$, as already presented in the introduction. Then the forced inclination $i_0$ in Eq. (\[i0\]), equivalent to the angle $\delta$, is solely a function of the location $a$ of the Kuiper belt. This is a specific calculation to every $a$, while the overall tilt of the expected mean plane for a wide semimajor axis range can be written as $$\tilde{\delta}=\frac{\int_{a_{in}}^{a_{out}}\delta(a)\mbox{d}a}{{a_{in}}-{a_{out}}},
\label{cumulative_alpha}$$ where ${a_{in}}=100$ AU and ${a_{out}}=200$ AU are the inner and outer edges of the considered Kuiper belt, respectively.
In Fig. \[zones\], we plot the maps of the angle $\tilde{\delta}$ for several representative values of $i_9$. The left-hand and right-hand columns are for the cases of $m_9=10~m_{\oplus}$ and $5~m_{\oplus}$, respectively. On the right side of the black curve, the colorful regions refer to the relatively small deviation (i.e., $\tilde{\delta}<5^{\circ}$) of the mean plane of the Kuiper belt from IP8. This is allowable according to the results obtained in Section 3, thus Planet 9 with the given parameters is most probable. The gray regions on the left side of the black curve (i.e., $\tilde{\delta}>5^{\circ}$) are considered to be the less likely zones, and the darker the color the lower the possibility. It seems that the most inclined Planet 9 with $i_9=30^{\circ}$ possibly has an orbit of $a_9\gtrsim440-530$ AU (see top panels). We also notice that the unlikely gray regions have significantly shrunk with decreasing $i_9$ and the effective constraints on the $(a_9, e_9)$ pair can only be found for $i_9\gtrsim20^{\circ}$. In Fig. \[zones\], the white zone on the top left of each panel corresponds to the unconsidered region of $q_9-a<10$ AU, where the KBOs may experience chaotic evolution due to strong perturbations from Planet 9 and thus secular theory would be not applicable.
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Conclusions
===========
The existence of an inclined Planet 9 can induce IP8 to evolve from its initial plane, yielding the current tilt of $\sim6^{\circ}$ relative to the Sun’s equator [@bail16; @lai16; @gome17]. As a natural extension, in this paper we explored the effect of this additional perturber on the mean plane of the Kuiper belt beyond 50 AU.
Firstly, in the context of the secular theory, we adopted and developed the theoretical approach of @volk17 to determine the relative angle $\delta$ between the expected mean plane of the Kuiper belt and IP8 at every specific semimajor axis $a$. We found that in the region of $a=50-90$ AU, the expected mean plane nearly coincides with IP8. But at $a>100$ AU, a noticeable deviation between these two planes appears because $\delta$ could become as large as $\sim10^{\circ}$. By taking into account the complete perturbations from the known planets and Planet 9, we also constructed numerical simulations to compute the angle $\delta$ for test KBOs with considerable space dispersion. The good agreement validates our theoretical approach, which allows us to explore a large suite of the mass ($m_9$) and orbital elements ($a_9, e_9, m_9$) of Planet 9 within a reasonable amount of computing time.
Next, for the real KBOs with semimajor axes $100 < a <200$ AU, we obtained an overall mean plane deviating from IP8 by a small angle of $\tilde{\delta}\sim1.0^{\circ}$. Considering the small number of such distant KBO samples at present, we carried out Monte Carlo simulations to evaluate the measurement uncertainty due to the observational incompleteness. The results show that the measured $\tilde{\delta}$ is just within $1\sigma$ limit of $1.0^{\circ}-4.6^{\circ}$. We then suppose that an upper limit of $\tilde{\delta}$, taken to be $5^{\circ}$, can be used as a constraint on the parameter space of Planet 9.
By integrating the angle $\delta$ as a function of the semimajor axis $a$ in our theoretical approach, we are able to obtain the overall tilt $\tilde{\delta}$ of the expected mean plane for a wide range, i.e., $100 < a <200$ AU. In our final results for the proposed Planet 9 with $m_9=5-10~m_{\oplus}$ and $i_9=15^{\circ}-30^{\circ}$, we plot the maps of the angle $\tilde{\delta}$ on the ($a_9, e_9$) plane. Confined by the prescribed constraint of $\tilde{\delta}<5^{\circ}$, we propose that Planet 9 has the most probable orbit depicted by the colorful region in Fig. \[zones\]:
\(1) Planet 9 could exist on a highly inclined orbit ($i_9=30^{\circ}$) in a more distant region beyond $a_9=530$ AU, or have moderate inclination $i_9=20^{\circ}$ ($15^{\circ}$) and smaller semimajor axis $a_9 > 450$ (400) AU.
\(2) Planet 9 is allowed to reside on a $i_9=30^{\circ}$ orbit with $a_9>440$ AU, or possibly any less inclined ($i_9<20^{\circ}$) orbit with $a_9>400$ AU.
[With increasing $e_9$, the deduced minimum $a_9$ would grow slightly but not significantly; this critical $a_9$ value at $e_9=0.6$ is on a level of 1.2 times larger than the corresponding value at $e_9=0.2$.]{}
The above results could help to reduce the uncertainty of the proposed Planet 9’s parameters. For instance, @bail16 showed a $10~m_{\oplus}$ planet on a $a_9=400$ AU, $e_9=0.4-0.6$, $i_9=20^{\circ}-30^{\circ}$ orbit is capable of inducing the observed solar obliquity. However, such combinations of the mass and orbital elements clearly correspond to the unlikely regions shown in Fig. \[zones\]. A less massive ($5~m_{\oplus}$) or more distant ($a_9=800$ AU) planet is suggested by @baty19 from the clustering of the orbital planes of KBOs. Furthermore, @kaib19 found that a much lower inclination ($i_9=5^{\circ}$) for Planet 9 seems favorable to replicate the inclination distribution of the observed scattering KBOs.
Since more and more faint KBOs will be discovered in the near future, for example, by the Large Synoptic Survey Telescope (LSST)[^2] [@jone16], they would help to further improve the measurement of the Kuiper belt’s mean plane. Especially, for the discussed KBOs with $100<a<200$ AU, a larger number of samples will allow us to examine smaller $a$-bin so that a finer profile of the mean plane can be drawn. Consequently, even tighter constraints would be put on the mass and orbit of yet undiscovered Planet 9.
This work was supported by the National Natural Science Foundation of China (Nos. 11973027 and 11933001), and National Key R&D Program of China (2019YFA0706601). We would also like to express our sincere thanks to the anonymous referee for the valuable comments.
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[^2]: The LSST is a project of the US National Science Foundation, which has recently been renamed to the NSF Vera C. Rubin Observatory.
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[Projet de note, par [**Joël MERKER${\:\!}^{*}$**]{} et [**Egmont PORTEN${\:\!}^{**}$**]{}]{}.
$(*)$ LATP, CMI, 39 rue Joliot Curie, F-13453 Marseille Cedex 13\
Courriel : merker@gyptis.univ-mrs.fr
$(**)$ Max-Planck-Gesellschaft, Humboldt-Universität zu Berlin\
Jägerstrasse, 10-11, D-10117 Berlin, Germany\
Courriel : egmont@mathematik.hu-berlin.de
[**Résumé.**]{}
[ Soient $M$ une variété CR localement plongeable et $\Phi\subset M$ un fermé. On donne des conditions suffisantes pour que les fonctions $L_{loc}^1$ qui sont CR sur $M\backslash \Phi$ le soient aussi sur $M$ tout entier.]{}
[**Abstract.**]{}
[ Let $M$ be a locally embeddable CR manifold and $\Phi\subset M$ be a closed set. We give sufficient conditions in order that $L^{1}_{loc}$ functions on $M$ which are CR on $M\backslash \Phi$ are CR on $M$. ]{}
[**Abridged English Version.**]{} Let $M$ be a locally embeddable CR manifold, ${\rm dim}_{CR} M=m$, ${\rm codim} \ M=n$, ${\rm dim} \ M=d=2m+n$ and $\Phi\subset M$ a closed set. We give various conditions in order that $\Phi$ is $L^1$-removable, [*i.e.*]{} $$L^1_{loc}(M) \cap L_{loc,CR}^1(M\backslash\Phi
)=L_{loc,CR}^1(M)$$ Let $H^{\kappa}$ denote $\kappa$-dimensional Hausdorff measure.
[Theorem 1.]{} – [*If $M$ is ${\cal C}^3$, a function $f\in L_{loc}^1(M)$ is CR if and only if $f|_{{\cal O}}$ belongs to $L_{loc,CR}^1({\cal O})$ for almost every CR orbit ${\cal
O}$.*]{}
[Corollary 1.]{} – [*If $\Phi=\cup_{a\in A}
{\cal O}_a$ is of zero $d$-dimensional measure, then $(1)$ is satisfied.*]{}
Theorem 1 reduces the problem to the case where $M$ is a single CR orbit, [*i.e.*]{} $M$ is [*globally minimal*]{} [@ME1].
[Theorem 2.]{} – [*Let $M$ be ${\cal
C}^{2,\alpha}$, $0\leq \alpha <1$, ${\rm dim}_{CR} M =m\geq 1$. Every closed subset $E$ of $M$ such that $M$ and $M\backslash E$ are globally minimal and such that $H_{loc}^{d-3}(E)<\infty$ is $L^1$-removable.*]{}
The notion of wedge (${\cal W}$-)-removability is defined here in higher codimension.
[Theorem 3.]{} – [*Let $M$ be ${\cal
C}^{\omega}$, ${\rm dim}_{CR}M=m\geq 1$. Every closed set $E\subset M$ such that $M$ and $M\backslash E$ are globally minimal and such that $H^{d-2}(E)=0$ is ${\cal W}$- and $L^1$-removable.*]{}
[Theorem 4.]{} –
*Let $M$ be ${\cal
C}^{2,\alpha}$, $m\geq 1$, and let $N$ be a ${\cal C}^2$ connected submanifold of $M$ such that $M$ and $M\backslash N$ are globally minimal.*
[(i)]{} If ${\rm codim}_M N \geq 3$, then $N$ is ${\cal
W}$- and $L^1$-removable[;]{}
[(ii)]{} Every closed set $\Phi \subset N$ is ${\cal
W}$- and $L^1$-removable if $\Phi \neq N$, ${\rm codim}_M N =2$ and $m\geq
1$[;]{}
[(iii)]{} $N$ is ${\cal W}$- and $L^1$-removable if $N$ is generic at one point, ${\rm codim}_M N =2$ and $m\geq 2$.
A set $S\subset M$ is called a ${\cal C}^{\lambda}$ peak set, $0<\lambda <1$, if there exists a [*nonconstant*]{} function $\varpi
\in {\cal C}^{\lambda}_{CR}(M)$ such that $S=\{\varpi=1\}$ and $|\varpi| \leq 1$.
[Theorem 5.]{} – [*Let $M$ be ${\cal
C}^{2,\alpha}$ globally minimal. Then every ${\cal C}^{\lambda}$ peak set $S$ satisfies $H^{d}(S)=0$ and is $L^1$-removable.*]{}
[Corollary 2.]{} – [*Let $M$ be ${\cal
C}^3$. Then a ${\cal C}^{\lambda}$ peak set $S$ is $L^1$-removable if $H^d(\cup_{{\cal O}\subset S} {\cal O})=0$.*]{}
Soient $M$ une variété CR localement plongeable, de dimension CR ${\rm dim}_{CR} M =m$, de codimension ${\rm codim} \ M= n$, de dimension ${\rm dim} \ M=
d=2m+n$ et soit $\Phi\subset M$ un fermé de $M$. Dans ce travail, on cherche des conditions, portant sur $M$ et $\Phi$, pour que l’on ait $$L_{loc}^1(M)\cap L_{loc,CR}^1(M
\backslash \Phi)=L_{loc}^1(M).$$ Si (1) est vérifiée, on dira que $\Phi$ est $L^1$-éliminable. D’après Trépreau [@TR], toute variété CR $M$ est réunion disjointe (en général transfinie) de sous-variétés CR immergées connexes $M=\cup_{i \in {\cal I}} {\cal O}_i$, ${\cal O}_i \subset M$, appelées [*orbites CR de $M$*]{}, qui sont [*caractéristiques*]{}, [*i.e.*]{} ${\rm dim}_{CR} {\cal O}_i={\rm dim}_{CR} M$ et minimales pour l’inclusion et cette propriété. Un premier résultat concerne ces variétés. Il a été démontré par B. Jöricke dans la classe ${\cal C}^0(M)$ [@JO1] et dans $L_{loc}^1(M)$ si $M$ est une hypersurface de classe ${\cal C}^2$ [@JO2]. Enfin, le second auteur l’a étendu à $L_{loc}^1(M)$ en codimension quelconque dans sa thèse [@PO].
[Théorème 1.]{} – [*Si $M$ est de classe ${\cal C}^3$, une fonction $f\in L_{loc}^1(M)$ est CR si et seulement si $f|_{{\cal O}_i}$ appartient à $L_{loc,CR}^1({\cal O}_i)$ pour presque toute orbite CR ${\cal O}_i$, au sens de la mesure sur $M$.*]{}
La restriction $f|_{{\cal O}_i}$ est bien définie et appartient à $L_{loc}^1({\cal O}_i)$ pour presque toute orbite ${\cal O}_i$. Nous renvoyons le lecteur à [@PO] ou [@MP1] pour une preuve complète.
[Corollaire 1.]{} – [*Si $\Phi=\cup_{a\in A} {\cal O}_a$ est de mesure $d$-dimensionnelle nulle, alors (1) est vérifiée.*]{}
Le théorème 1 réduit l’étude de (1) au cas où $M={\cal O}_i$ est une seule orbite. L’aspect central de notre travail consiste justement à replacer l’étude de (1) dans le contexte de la théorie des orbites CR et de l’extension des fonctions CR, bénéficiant en cela des travaux de Trépreau, Tumanov et Jöricke. Lorsque $M$ est une hypersurface, le problème (1) est traité par Jöricke et Chirka-Stout [@CS].
Bien entendu, les orbites sont aussi localement plongeables. Soit $M$ générique dans $\C^{m+n}$. Dans ce cas, un ouvert connexe ${\cal W}_0$ sera appelé [*wedge attaché*]{} à $M\backslash \Phi$ s’il existe une section continue $\eta: M\backslash \Phi \to T_M\C^{m+n}\backslash \{0\}$ du fibré normal à $M$ telle que ${\cal W}_0$ contient un wedge ${\cal W}_p$ d’edge $M$ en $(p,\eta(p))$ ([*cf.*]{} [@TU], p.3), pour tout point $p\in M\backslash \Phi$. Cette notion a un sens local lorsque $M$ est localement plongeable. Le problème (1) fait intervenir la géométrie des wedges attachés.
[Définition 1.]{} $\Phi$ est dit ${\cal W}$-éliminable si, pour tout wedge ${\cal W}_0$ attaché à $M\backslash \Phi$, il existe un wedge ${\cal W}$ attaché à $M$ tel que les fonctions holomorphes dans ${\cal W}_0$ se prolongent holomorphiquement à ${\cal W}$.
On note $H^{\kappa}(E)$ la mesure de Hausdorff $\kappa$-dimensionnelle de $E$, pour une métrique fixée sur $M$. $H^d$ s’identifie à la mesure de Lebesgue. Enfin, on dira qu’une variété CR est [*globalement minimale*]{} si elle consiste en une seule orbite CR. Le fermé $\Phi$ sera noté $E, N$ ou $S$, suivant le contexte.
[Théorème 2.]{} – [*Soit $M$ de classe ${\cal C}^{2,\alpha}$, $0<\alpha <1$, ${\rm dim}_{CR} M=m\geq 1$. Tout fermé $E$ de $M$ tel que $M$ et $M\backslash E$ sont globalement minimales et tel que $H_{loc}^{d-3}(E)<\infty$ est $L^1$-éliminable.*]{}
Par exemple, sous ces hypothèses, toute sous-variété $N$ de codimension au moins trois, est $L^1$-éliminable. Le résultat suivant ([@MP2]) s’applique à l’extension des fonctions CR méromorphes.
[Théorème 3.]{} – [*Soit $M$ de classe ${\cal C}^{\omega}$, ${\rm dim}_{CR}M= m\geq 1$. Tout fermé $E\subset M$ tel que $M$ et $M\backslash E$ sont globalement minimales et tel que $H^{d-2}(E)=0$ est ${\cal W}$- et $L^1$-éliminable.*]{}
Le cas de singularités plus massives est traité dans [@MP1]:
[Théorème 4.]{} –
*Soient $M$ ${\cal C}^{2,\alpha}$, $m\geq 1$. Soit $N$ une sous-variété connexe de $M$, de classe ${\cal C}^2$ telle que $M$ et $M\backslash N$ sont globalement minimales. Alors*
[(i)]{} $N$ est ${\cal W}$- et $L^1$-éliminable si ${\rm codim}_M N \geq 3$.[;]{}
[(ii)]{} Tout fermé $\Phi$ de $N$ est ${\cal W}$- et $L^1$-éliminable, si $\Phi\neq N$, ${\rm codim}_M N =2$ et $m\geq 1$[;]{}
[(iii)]{} $N$ est ${\cal W}$- et $L^1$-éliminable si $N$ est générique en au moins un point, ${\rm codim}_M N =2$ et $m\geq 2$.
L’élimination $L^1$ de compacts $K\subset\subset N$ de variétés $N\subset M$ génériques de codimension un apparaît dans les travaux de B. Jöricke [@JO2] pour $n=1$ et dans [@PO] pour $n\geq 2$.
Une version plus faible du théorème 4 est contenue dans [@ME2] dans le cas de l’élimination ${\cal W}$. L’hypothèse d’orbite sur $M$ et $M\backslash N$ est essentiellement nécessaire : si elle n’est pas satisfaite, il existe des exemples simples de $M$, $N$ et de distributions CR de support une sous-variété caractéristique fermée $S$ de $M\backslash N$ qui ne se prolongent pas holomorphiquement à un wedge au-dessus de $N$. Enfin, les hypothèses géométriques sur $N$ et $\Phi$ sont calquées sur celles qui rendent les théorèmes 2,3 et 4 connus lorsque $M$ est un ouvert de $\C^m$, [*i.e.*]{} $n=0$.
Maintenant, un sous-ensemble $S$ de $M$ est dit [*ensemble pic höldérien*]{} s’il a la forme $\{\varpi=1\}$, avec $\varpi\in {\cal C}^{\gamma}_{CR}(M)$ [*non constante*]{} pour un $\gamma$, $0<\gamma<1$ et $|\varpi|\leq 1$. Grâce au théorème 1 et aux techniques de déformation de disques, on généralise les résultats de [@KR].
[Théorème 5.]{} – [*Soient $M$ ${\cal C}^{2,\alpha}$ globalement minimale. Tout ensemble pic höldérien $S$ de $M$ vérifie $H^d(S)=0$ et est $L^1$-éliminable.*]{}
[Corollaire 2.]{}– [*Soit $M$ ${\cal C}^3$. Un ensemble pic höldérien $S$ est $L^1$-éliminable si $H^d(\bigcup_{{\cal O}_{\i} \subset S} {\cal O}_{\i})=0.$*]{}
Enfin, puisque $L^{\rm p}_{loc}$ se plonge dans $L_{loc}^1$ pour ${\rm p} \geq 1$, tous ces résultats sont valables dans $L^{\rm p}_{loc}$.
Nous allons donner ici un résumé des preuves des théorèmes 2,3,4 et 5. Les preuves rigoureuses sont contenues dans [@MP1], [@MP2], [@PO]. Le théorème 1 pour $f\in {\cal C}^0_{CR}(M)$ est démontré dans [@JO1] et dans [@PO] pour $f\in L_{loc,CR}^{\rm p}$.
C’est B. Jöricke qui a eu l’idée d’utiliser l’inégalité de Carleson sur des familles régulières de disques analytiques attachés à $M$ pour déduire l’élimination $L^1$ de l’élimination ${\cal W}$, dans le cas hypersurface. Dans ce travail et dans [@MP1], [@MP2], nous raffinons les résultats de [@ME2] pour la ${\cal W}$-élimination et les étendons à $L^1$ comme dans [@JO2]. Mentionnons enfin que la technique dite de <<balayage par des wedges>> utilisée dans [@JO2], [@CS], [@PO] ne s’appliquerait qu’en dimension CR $m\geq 2$; c’est pourquoi nous utilisons ici les déformations de disques analytiques et le principe de continuité.
On traite le cas $L^1$, le cas ${\cal W}$ sera démontré en cours. La technique consiste en un grand nombre de déformations de $M$ dans des ouverts obtenus en attachant des disques analytiques à $M$ et à ses déformations. En particulier, nous décrivons une partie de l’enveloppe d’holomorphie d’ouverts de type wedge attachés à $M\backslash \Phi$, assez étendue pour procéder ensuite à l’élimination $L^1$.
[*Étape 1.*]{} Grâce au théorème d’extension de Trépreau-Tumanov généralisé ([@ME1], [@ME2]) et au théorème de l’<<edge of the wedge>>, on peut prolonger holomorphiquement $f\in L_{loc,CR}^1(M\backslash \Phi)$ à un wedge ${\cal W}_0$ attaché à $M\backslash \Phi$. Pour le contrôle en norme $L^1$ de l’extension, on utilise de <<bonnes>> familles de disques analytiques attachés à $M$. Soit $\Delta$ le disque unité dans $\C$, $b\Delta$ son bord.
[Définition 2.]{} – On appelle [*famille régulière en $p$ de disques analytiques attachés à $M$*]{} une application ${\cal C}^{2,\beta}$, $\beta<\alpha$, $A: {\cal S} \times
{\cal V}\times \overline{\Delta} \to \C^{m+n}$, $(s,v,\zeta)\mapsto
A_{s,v}(\zeta)$, holomorphe en $\zeta$, où $(0\in)
{\cal S} \subset \R^{2m+n-1}$, $(0\in) {\cal V} \subset\R^{n-1}$ sont des ouverts, telle que $A_{0,v}(1)=p$ et que
1\) L’application ${\cal S} \times b\Delta \to M$, $(s,\zeta)\mapsto
A_{s,v}(\zeta)$ est un plongement, $\forall \ v\in {\cal V}$;
2\) Le vecteur $\eta:=-\partial A_{0,0}
/\partial\zeta (1) \not\in T_pM$ et ${\rm rang}(v\mapsto {\rm
pr}_{T_p\C^{m+n}/(T_pM\oplus\R\eta)}(-\partial A_{0,v} /\partial\zeta
(1)))=n-1$.
Toute famille régulière définit un wedge ${\cal
W}_{A,p}$ en $(p,\eta)$ par ${\cal W}_{A,p}:= \{A_{s,v}(\zeta)\in
\C^{m+n}: \ (s,v,\zeta)\in {\cal S}_1\times {\cal V}_1\times
\stackrel{\circ}{\Delta}_1\}$, où ${\cal S}_1 \subset {\cal S}$, ${\cal V}_1 \subset
{\cal V}$, $\Delta_1=\overline{\Delta}\cap \Delta(1,\rho_1)$, $\rho_1>0$. Soit ${\cal W}={\cal W}(U,C)=\{z+\eta: \ z\in U, \eta\in
C\}$ un wedge de base $U\subset M$ et de cône $C\subset \C^{m+n}$, [*e.g.*]{} ${\cal W}\approx {\cal W}_{A,p}$.
[Définition 3.]{} – Une fonction $f\in
L_{loc,CR}^1(M)$ est dite [*prolongeable dans*]{} ${\rm H}_{\rm
a}^1({\cal W})$ s’il existe $F\in {\cal H}({\cal W})$ telle que $F|_{U_{\eta}}\to f$ au sens $L^1$, où $U_{\eta}= U+\eta$, $\eta\in
C$, uniformément lorsque $\eta \to 0$.
Pour démontrer l’extension de $f$ dans la classe de Hardy ${\rm
H}_{\rm a}^1({\cal W}_{A,p})$, on utilise des familles régulières attachées à des déformations de $M$:
[Proposition 1]{} ([@PO]). –
*Soit $M$ globalement minimale, ${\cal C}^{2,\alpha}$. Pour tout $\varepsilon>0$, il existe une déformation ${\cal C}^{2,\beta}$ $(d,M^d)$ de $M$ à support compact avec $M^d\equiv M$ près de $p$ et $||M^d-M||_{{\cal
C}^{2,\beta}}<\varepsilon$, telle que*
1\) Il existe une famille régulière de disques $A_{s,v}$ attachés à $M^d$[;]{}
2\) Il existe un opérateur linéaire borné de prolongement $L_{loc,CR}^1(M)\to L^1_{loc,CR}(M^d)$, $f\mapsto f^d$, tel que $f^d\equiv f$ sur l’ensemble où $M^d$ coïncide avec $M$[;]{}
3\) Pour $f\in L_{loc,CR}^1(M)$ fixée, il existe une déformation $d$ telle que $||f^d-f||_{L^1}<\varepsilon$.
Cette construction demande l’emploi en détail des techniques de déformation de disques de Bishop élaborées par Tumanov ([@TU]) et l’existence de $M^d$ satisfaisant $1)$ et $2)$ équivaut à la globalité minimale ([@PO], [@MP1]).
[Proposition 2.]{} – [*$L_{loc,CR}^1(M)$ est prolongeable dans ${\rm H}_{\rm a}^1({\cal W}_{A,p})$.*]{}
En effet, sur chaque disque remplissant ${\cal W}_{A,p}$, on se ramène à l’estimée de Carleson sur $\Delta$: si $r(\theta)\in {\cal C}^1([-\pi,\pi],[0,1])$ telle que $r\equiv r_1$ sur $(-\theta_1,\theta_1)$, $0<r_1<1$ et $\hbox{supp} \ (1-r)\subset
(-\theta_0,\theta_0)$, $0<\theta_1 <\theta_0 <\pi$, alors il existe $C>0$ telle que $\forall \ u \in H^1_{\rm a}(\Delta), \
\int_{-\theta_0}^{\theta_0}
|u(re^{i\theta})| d\theta \leq C ||u||_{L^1(b\Delta)}$. $\square$
[*Étape 2.*]{} On note ${\cal H}({\cal U})$ l’anneau des fonctions holomorphes dans ${\cal U}$ et ${\cal V}(E)$ un voisinage ouvert arbitrairement petit d’un ensemble $E$ dans $\C^{m+n}$. La déformation suivante réduit la démarche au cas où $L^1_{loc,CR}(M\backslash \Phi)\cap L_{loc}^1(M)$ a été remplacé par ${\cal H}({\cal V}(M^d\backslash \Phi))\cap
L_{loc}^1(M^d)$.
[Proposition 3.]{} – [*Soient $M$ ${\cal C}^{2,\alpha}$, générique dans $\C^{m+n}$, $f\in L_{loc,CR}^1(M)$ et $U$ un ouvert de $M\backslash \Phi$ tel que $\overline{U}$ est compact, avec $M\backslash \Phi$ globalement minimale. Pour tout $\varepsilon>0$, il existe une déformation $M^d$ ${\cal C}^{2,\beta}$, $\beta<\alpha$, avec ${\rm supp} \ d =\overline{U}$, $M^d\supset \Phi$, $||M^d-M||_{{\cal C}^{2,\beta}}<\varepsilon$, telle qu’il existe une fonction $f^d\in L_{loc}^1(M^d)\cap
L_{loc,CR}^1(M^d\backslash \Phi)\cap
{\cal H}({\cal V}(U^d))$ coïncidant avec $f$ sur $M\backslash U$ et telle que $||f^d-f||_{L^1}<\varepsilon$.*]{}
La preuve utilise la Proposition 1 et l’inégalité de Carleson sur des déformations successives de $M$ à support la base $U_j\subset M$ de petits wedges ${\cal W}_{A_j,p_j}$ tels que $\bigcup_{j\in J} U_j =M$. Il suffira alors de démontrer que $L_{loc}^1(M^d) \cap {\cal H}({\cal V}(M^d\backslash \Phi))=
L_{loc,CR}^1(M^d)$. En effet, par $|\int_M (f^d-f)\overline{\partial}\varphi| \leq C_{\varphi} \varepsilon$ (les mesures sur $M$ et sur $M^d$ étant voisines, puisque $||M^d-M||_{{\cal C}^{2,\beta}}<\varepsilon$, une telle écriture à un sens), pour toute $(m+n,m-1)$-forme à support compact, l’égalité $\int_{M^d} f\overline{\partial} \varphi=0$ impliquera $\int_M f\overline{\partial} \varphi=0$, puisque $\varepsilon$ est arbitraire. $\square$
[*Étape 3: ${\cal W}$-élimination de la singularité.*]{} En utilisant l’hypothèse <<$M$ et $M\backslash \Phi$ globalement minimales>>, on élimine progressivement les points de $\Phi=N$, $E$ ou $K$ qui se trouvent à l’extrémité d’une courbe intégrale par morceaux de $T^cM$ issue d’un point de $M\backslash \Phi$ et on déforme ensuite $M$ dans le wedge obtenu au-dessus de chaque point qui a été éliminé. À chaque pas, sur une déformation de $M$ encore notée $M$, la situation se réduit à l’élimination d’un seul point $p$ de $\Phi$ disposé comme suit. Il existe un voisinage $U$ de $p$ dans $M$ et $M_1$ une hypersurface ${\cal C}^2$ dans $U$ qui partage $U$ en deux composantes fermées $M_1^-$ et $M_1^+$, $U=M_1^-\cup M_1^+$, $M_1^-\cap M_1^+=M_1$, telle que $\Phi \cap U \subset M_1^{-}$, et il existe un disque $A$ ${\cal C}^{2,\beta}$ attaché à $M_1^+$ avec $A(1)=p$, $dA/d\theta (1) \in T_pM_1$. Soient $\omega$ un voisinage de $U\backslash \Phi$ dans $\C^{m+n}$ et $f\in {\cal H}(\omega)$. Les déformations normales de Tumanov nous permettent de développer $A$ en une famille régulière en $p$ de disques analytiques $A_{s,v}$, $A_{0,0}=A$, qui engendre un wedge ${\cal W}_{A,p}$ :
[Lemme 1]{} ([@TU].) – [*Il existe une famille régulière $A_{s,v}$ attachée à $(M\cap U)\cup \omega$.*]{} $\square$
Cependant, le bord de ces disques peut toucher la singularité $\Phi$ (en fait, $A_{s,v}(b\Delta) \cap \Phi \subset A_{s,v}(b\Delta \cap
\Delta_1)$) et le théorème d’approximation de Baouendi-Treves n’est plus valable. Heureusement, le principe de continuité et une propriété d’isotopie des disques à un point nous permet de prolonger $f$ à ${\cal W}:={\cal W}_{A,p}$ moins un ensemble ${\cal
E}\subset {\cal W}$, [*i.e.*]{} $F\in {\cal H}(\omega \cup ({\cal W}
\backslash {\cal E}))$, comme suit:
[Définition 4.]{} Un disque plongé $A: \overline{\Delta}\to \C^{m+n}$ est dit [*$b$-isotope à un point dans $\omega$*]{} s’il existe une application ${\cal C}^1$ $[0,1]\times
\overline{\Delta} \ni (t,\zeta)\mapsto
A_t(\zeta)\in \C^{m+n}$ telle que $A_t(b\Delta)\subset\omega$, $A_0=A$, chaque $A_t$ est un disque analytique plongé pour $0\leq t< 1$ et $A_1$ est une application constante $\overline{\Delta} \to \{pt\}\in \omega$.
[Lemme 2.]{} – [*Sous les conditions des théorèmes 2, 3 et 4, tout disque $A_{s,v}$ tel que $A_{s,v}(b\Delta\cap \Delta_1) \cap \Phi =\emptyset$ est $b$-isotope à un point dans $\omega$.*]{} $\square$
[Lemme 3.]{} – [*Soit $A: b\Delta \to \omega$, $\overline{\Delta} \to \C^{m+n}$ $b$-isotope à un point dans $\omega$. Alors, pour toute fonction holomorphe $f\in {\cal H}(\omega)$, il existe $F\in {\cal H}(\omega \cup
{\cal V}(A(\overline{\Delta})))$ telle que $F\equiv f$ dans ${\cal V}(A(b\Delta))$.*]{} $\square$
L’ensemble ${\cal E}:=\{z_1=A_{s,v}(\zeta_1)\in \C^{m+n}: \ \zeta_1 \in
\stackrel{\circ}{\Delta}_1, A_{s,v}(b\Delta) \cap \Phi\neq \emptyset\}$ où l’on n’a pas prolongé est feuilleté par des courbes holomorphes. Grâce aux disques attachés à $M$, on se ramène donc à éliminer la singularité ${\cal E}\backslash \omega$ pour $F\in {\cal H}(\omega \cup ({\cal W} \backslash {\cal E}))=
{\cal H}(\omega \cup({\cal W} \backslash ({\cal E}\backslash
\omega)))$. À moins que ${\rm codim}_{\cal W} {\cal E}=
{\rm codim}_M \Phi$, le bord de presque tout disque $A_{s,v}$ ne touche en général $\Phi$ que sur un ensemble de mesure nulle de $b\Delta$. Le fait que de nombreux disques $A_{s,v}$ satisfont $A_{s,v}(b\Delta \cap \Delta_1) \not\subset {\cal E}$ près de $\zeta=1$, [*i.e.*]{} que ${\cal E}\backslash \omega\neq {\cal E}$, est crucial pour la suite.
La structure de ${\cal E}$ dépend des cas:
$\bullet$ Si ${\cal H}^{d-3}_{loc}(E)<\infty$, alors ${\cal H}_{loc}^{2m+2n-2}({\cal E})<\infty$. Soit $f\in L^1(U)$. Dans ce cas, $F\in L^1({\cal W})$ et on démontre que ${\cal E}\backslash \omega$ est une singularité éliminable pour $F$ grâce à un principe de méromorphie séparée dû à Shiffman. En effet, on a:
[Lemme 4.]{} – [*Soit $P\subset\subset
{\cal W}$ un polydisque. Alors pour presque tout disque de coordonnés $D\subset P$, $D\cap {\cal E}$ consiste en un nombre fini de points et $F|_D$ est méromorphe sur $D$ à pôles d’ordre au plus un.*]{}
Le lemme s’applique à $F\in L^1({\cal W})\cap {\cal H}(\omega \cup ({\cal W}
\backslash {\cal E}))$, d’où $F$ est méromorphe dans ${\cal W}$. Si l’ensemble polaire $P_F$ de $F$ est non vide, il ne peut pas contenir une courbe $A_{s,v}(\Delta)$ telle que $A_{s,v}(\stackrel{\circ}{\Delta}_1)\cap
\omega \neq \emptyset$. Donc $P_F$ est constitué des disques dont le bord est entièrement contenu, pour $\zeta$ près de $1$, dans $E$. L’ensemble ${\cal E}_1$ correspondant dans ${\cal W}$ satisfait maintenant $H^{2m+2n-3}({\cal E}_1)=0$, mais alors ${\cal H}({\cal W} \backslash {\cal E}_1) ={\cal H}({\cal W})$, donc $P_F=\emptyset$. $\square$
$\bullet$ Dans la situation du théorème 4, l’énoncé suivant s’applique à (i), (ii) et (iii) :
[Proposition 4.]{} – [*Soit $\Lambda$ une hypersurface d’un ouvert ${\cal U} \subset \C^{m+n}$ et $\Phi$ un fermé de $\Lambda$ qui ne contient pas d’orbite CR de $\Lambda$. Alors $\forall \ f\in {\cal H}({\cal U}
\backslash \Phi), \exists \ F\in {\cal H} ({\cal U})$ telle que $F|_{{\cal U} \backslash \Phi} =f$.*]{}
Lorsque ${\rm codim}_M \ N=2$ (Théorème 4), ${\cal E}$ est une hypersurface de ${\cal W}$ feuilletée par les $A_{s,v}(\stackrel{\circ}{\Delta}_1)$, mais tout bord de disque $A_{s,v}(b\Delta\cap \Delta_1)$ ne rencontre $N$ qu’en au plus un point. Donc ${\cal E}\backslash \omega$ ne contient pas d’orbite CR de ${\cal E}$: la proposition 4 s’applique. $\square$
$\bullet$ Lorsque $M$ est ${\cal C}^{\omega}$ et $H^{d-2}({\cal E})=0$, ${\cal E}$ satisfait $H^{2m+2n-1}(E)=0$. Dans ce cas, le feuilletage de ${\cal E}$ est analytique réel et on applique près d’un point de $b\omega\cap {\cal E}$ l’énoncé ([@MP2]):
[Proposition 5.]{} – [*Soient ${\cal U}\subset \C^{m+n}$ un ouvert ${\cal C}^{\omega}$-feuilleté par des courbes holomorphes $A_{\vartheta}, \vartheta \in D$, ${\cal U}=\cup_{\vartheta} A_{\vartheta}$, $0\in {\cal U}$, $D\subset \R^{2m+2n-2}$ un ouvert, $0\in D$, ${\cal G}\subset D$ un fermé avec ${\cal H}^{2m+2n-3}({\cal G})=0$ et $M_1$ une hypersurface ${\cal C}^1$ dans ${\cal U}$, $0\in M_1$, $T_0M_1+T_0A_0=T_0\C^{m+n}$, et posons ${\cal E}:= (\cup_{\vartheta\in
{\cal G}} A_{\vartheta}) \cap M_1^-$. Alors ${\cal H}({\cal U}
\backslash {\cal E})={\cal H}({\cal U})$.*]{}
Ici, l’argument utilise une complexification de courbes réelles du feuilletage pour avoir la $b$-isotopie. $\square$
[*Étape 4: valeurs au bord dans $L^1$.*]{} Pour conclure, Il reste à estimer l’extension $F$ dans le wedge ${\cal W}$ en norme $L^1$. On applique à $\Phi\subset M_1^-$ l’énoncé:
[Proposition 6.]{} – [*Si ${\cal H}_{loc}^{d-2}(\Phi)<\infty$ et si ${\cal H}(\omega)$ se prolonge holomorphiquement à un wedge en $p\in M_1^-$, alors $p$ est $L^1$-éliminable, [*i.e.*]{} <<$\Phi$ ${\cal W}$-éliminable>> entraîne <<$\Phi$ $L^1$-éliminable>>.*]{}
Les disques $A_{s,v}$ et l’inégalité de Carleson donnent un contrôle en norme $L^1$ pour les disques ne touchant pas $\Phi$ ([*i.e.*]{} presque tout disque, parce que ${\cal H}^{d-2}_{loc}(\Phi) < \infty$), donc de l’extension dans le wedge. L’extension appartient enfin à ${\rm H}_{\rm a}^1({\cal W})$, ce qui achève la preuve. $\square$
[*Preuve du Théorème 5.*]{} Compte tenu de l’existence des familles régulières de disques analytiques et de la Proposition 2, en suivant la démonstration du Théorème 1 dans Kytmanov-Rea [@KR], on obtient le Théorème 5. $\square$
[1]{}
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, , Preprint 1996, à paraître J. Diff. Equ., 82pp.
and [ C. Rea]{}, Ann. Scuola Norm. Sup. Pisa, Classe di Scienze, [**22**]{} (1995), 211-226.
, , Int. Math. Res. Not. [**8**]{} (1994), 329-342.
, , Int. Math. Res. Not. [**1**]{} (1997), 21-56.
and [ E. Porten]{}, Preprint 1997, à paraître, 32pp., http://www.dmi.ens.fr/EDITION/preprints.
and [ E. Porten]{}, Ann. Polon. Math. [**70**]{} (1998), 163-193.
, Preprint 1998, à paraître.
, , Bull. Soc. Math. Fr. [**118**]{} (1990), 403-450.
, , Duke Math. J. [**73**]{} (1994), 1-24.
|
---
abstract: 'We consider gapped fractional quantum Hall states on the lowest Landau level when the Coulomb energy is much smaller than the cyclotron energy. We introduce two spectral densities, $\rho_T(\omega)$ and $\bar\rho_T(\omega)$, which are proportional to the probabilities of absorption of circularly polarized gravitons by the quantum Hall system. We prove three sum rules relating these spectral densities with the shift $\mathcal S$, the $q^4$ coefficient of the static structure factor $S_4$, and the high-frequency shear modulus of the ground state $\mu_\infty$, which is precisely defined. We confirm an inequality, first suggested by Haldane, that $S_4$ is bounded from below by $|\mathcal S-1|/8$. The Laughlin wavefunction saturates this bound, which we argue to imply that systems with ground state wavefunctions close to Laughlin’s absorb gravitons of predominantly one circular polarization. We consider a nonlinear model where the sum rules are saturated by a single magneto-roton mode. In this model, the magneto-roton arises from the mixing between oscillations of an internal metric and the hydrodynamic motion. Implications for experiments are briefly discussed.'
author:
- 'Siavash Golkar, Dung X. Nguyen, and Dam T. Son'
title: 'Spectral Sum Rules and Magneto-Roton as Emergent Graviton in Fractional Quantum Hall Effect'
---
Introduction
============
Fractional quantum Hall (FQH) systems represent the paradigm for interacting topological states of matter. Much attention has been concentrated on the topological properties of the quantum Hall states, encoded in the ground state wavefunction and the statistics of quasiparticle excitations. On the other hand, FQH systems also have a neutral collective excitation known as the magneto-roton. The magneto-roton is an excitation within one Landau level and hence has energy at the interaction (Coulomb) energy scale. The existence of the magneto-roton was suggested in the classic work of Girvin, MacDonald, and Platzman [@Girvin:1986zz], in which Feynman’s theory of the roton in superfluid helium is extended to the FQH case. In this picture the magneto-roton is visualized as a long-wavelength density fluctuation. Within the composite fermion theory, the magneto-roton is interpreted as a particle-hole bound state, in which the particle lies in an empty composite-fermion Landau level, and the hole lies in a filled one. In the composite-boson approach, the magneto-roton is interpreted as vortex-antivortex bound state. The magneto-roton has been observed in Raman scattering experiments [@Pinczuk; @Pinczuk:2000] and more recently, in experiments using surface acoustic waves [@Kukushkin]. At nonzero wavenumbers, there is some evidence in favor of more than one magneto-roton mode [@Pinczuk:2005].
Very recently, Haldane has proposed a drastically different interpretation of the magneto-roton [@Haldane:2009; @Haldane:2011; @Haldane-model-wf]. He argues that there is a dynamic degree of freedom in FQH systems which can be interpreted as an internal metric. In this picture, the magneto-roton at long wavelength is the quantum of the fluctuations of this metric.
In this paper, we derive some new results related to the physics of a gapped FQH system at the lowest Landau level ($\nu<1$) at the interaction energy scale. We assume that the Coulomb energy is much smaller than the cyclotron energy. First, we derive new, exact sum rules involving the spectral densities of the traceless part of the stress tensor. The two components of the traceless part of the stress tensor are $T_{zz}=\frac14(T_{xx}-T_{yy}-2iT_{xy})$, $T_{\bar
z\bar z}=\frac14(T_{xx}-T_{yy}+2iT_{xy})$ (here $z=x+iy$), hence we can define two spectral densities: $$\begin{aligned}
\rho_T(\omega) &= \frac1N \sum_n\big|{\langle}n | \int\!d^2\!x\, T_{zz}({\bf x}) |0 {\rangle}\big|^2
\delta(\omega-E_n), \label{rho}\\
\bar\rho_T(\omega) &= \frac1N \sum_n|\big{\langle}n | \int\!d^2\!x\,
T_{\bar z\bar z}({\bf x}) |0 {\rangle}\big|^2
\delta(\omega-E_n), \label{rhobar}\end{aligned}$$ where $N$ is the total number of particles in the system, $|0\rangle$ is the ground state and the sums are taken over all excited states $|n\rangle$. Physically, these spectral densities are proportional to the probability that a circularly polarized graviton with energy $\omega$ falling perpendicularly to the system is absorbed. The two functions correspond to the two circular polarizations of the graviton. Without a complete solution to the FQH problem, we do not know $\rho_T(\omega)$ and $\bar\rho_T(\omega)$, but for a gapped FQH system we expect these functions to be zero below a gap $\Delta_0$ and to fall to zero when $\omega$ increases far above $\Delta_0$. If there is a well-defined magneto-roton at $q=0$, we expect it to appear as peaks in the spectral densities.
We will show that the spectral densities satisfy three sum rules, $$\begin{aligned}
\int_0^\infty\!\frac{d\omega}{\omega^2}
[\rho_T(\omega) - \bar \rho_T(\omega)]& = \frac{\mathcal S-1}8\,,
\label{shift-sr-intro}\\
\int\limits_0^\infty\!\frac{d\omega}{\omega^2}\,
[\rho_T(\omega) + \bar \rho_T(\omega)] &= S_4,
\label{S4-sr-intro}\\
\int_0^\infty\!\frac{d\omega}{\omega}
[\rho_T(\omega) + \bar \rho_T(\omega)] &= \frac{\mu_\infty}{\rho_0}\,.
\label{elasticity-sr-intro}\end{aligned}$$ In Eq. , $\mathcal S$ is the shift of the ground state, defined as the offset in the relationship between the number of magnetic flux quanta $N_\phi$, the filling factor $\nu$ and the number of electrons $Q$ when the latter are put on a sphere: $Q=\nu(N_\phi+\cal S)$ [@Wen:1992ej]. In Eq. , $S_4$ is the coefficient governing the low-momentum behavior of the projected structure factor [@Girvin:1986zz]: $\bar
s(q)=S_4(q\ell_B)^4$, where $\ell_B$ is the magnetic length. Finally, in Eq. , $\mu_\infty$ is the high-frequency elastic modulus, which will be defined exactly later in the text \[see Eqs. and \], and $\rho_0$ is the particle number density in the ground state. In all sum rules, the limit $m\to0$ of the spectral densities is taken first, before the upper limit of integration is taken to infinity. In this order of limits, the integral in each sum rule is dominated by $\omega$ of the order of the Coulomb energy.
The sum rule is particularly interesting, as it establishes a connection between a topological characteristic of the ground state (the shift) and dynamic information (the spectral densities). In the $\nu=1$ integer quantum Hall state, the sum rule becomes trivial, as there is no degree of freedom at the interaction energy scale (hence $\rho_T=\bar\rho_T=0$), and the shift is $\mathcal
S=1$, so both sides of the sum rule vanish.
Using the three sum rules, we derive some inequalities between different observables in the FQH states. One of these inequalities, previously derived by Haldane [@Haldane:2009], places a lower bound on the coefficient of the $q^4$ asymptotics of the projected static structure factor, which is saturated by the Laughlin’s trial wavefunction.
We will also consider a simple model where the sum rules are saturated by one magneto-roton mode, which manifests as the oscillation of the internal metric of the fluid mixed with the hydrodynamic fluid motion. The model provides a concrete realization of Haldane’s idea of an internal metric degree of freedom in FQH systems [@Haldane:2009]. This model is not meant to be exact, however it does exhibit some of the characteristic properties of the observed magneto-roton modes. We discuss the polarization properties of the magneto-roton in this model, which may be measurable in future experiments.
The effective action
====================
Review of the Newton-Cartan formalism
-------------------------------------
Recently, one of the authors has proposed the use of nonrelativistic general coordinate invariance as a way to constrain the dynamics of quantum Hall systems [@Son:2013rqa] (for related work, see Refs. [@Wiegmann; @Abanov]). Although the method can be thought of as “gauging” the Galilean invariance, the local symmetry remains nontrivial in the limit of zero bare electron mass, and hence is a symmetry intrinsic to the physics of electrons at the lowest Landau level, coupled to electromagnetism and gravity.
In Ref. [@Son:2013rqa], the attention was focused on the regime of long wavelengths (much larger than the magnetic length) and low frequencies (much smaller than the gap). In this paper we will relax the latter condition, allowing for energies comparable to the gap. In this regime, terms with arbitrary number of time derivatives must be taken into account in the effective action. However, as we will demonstrate, we can still obtain nontrivial relationships by expanding in the number of spatial derivatives.
We briefly review the main result of Ref. [@Son:2013rqa] here. The effective Lagrangian describing the response of a gapped quantum Hall state to external electromagnetic ($A_0$, $A_i$) and gravitational perturbations ($h_{ij}$), in the massless limit, is: $$\label{L-NC}
\mathcal L = \frac\nu{4\pi}\varepsilon^{\mu\nu\lambda} a_\mu{\partial}_\nu
a_\lambda - \rho v^\mu({\partial}_\mu\varphi-\tilde A_\mu -s\omega_\mu +a_\mu)
+ {\mathcal L}_0[\rho, v^i, h_{ij}].$$ Here $v^\mu=(1,v^i)$; $a_\mu$, $\rho$ and $v^i$ are dynamical fields, with respect to which one should extremize the action; and $\tilde A_\mu$ is related to the external electromagnetic potential by: $$\tilde A_0 = A_0 - \frac12\varepsilon^{ij}{\partial}_i(h_{jk}v^k), \qquad
\tilde A_i = A_i,$$ $\omega_\mu$ is the spin connection of the Newton-Cartan space $(h_{ij}, v^i)$, defined through the derivatives of the vielbein $e^a_i$ ($h_{ij}=e^a_i e^a_j$): $$\omega_ 0 = \frac12 \epsilon^{ab} e^{aj}{\partial}_0 e^b_j
+ \frac12 \varepsilon^{ij}{\partial}_i(h_{jk}v^k), \qquad
\omega_i = \frac12 \epsilon^{ab} e^{aj}\nabla_{\!i} e^b_j.$$ Finally, $\mathcal L_0$ contains all “non-universal” terms, i.e., terms that cannot be fixed by symmetry arguments alone.
There are two parameters that enter the Lagrangian : $\nu$, which is identified with the filling factor, and $s$, identified with the orbital spin (per particle) and is related to the shift by $s=\mathcal S/2$. In Ref. [@Son:2013rqa] it was found that these two parameters control some quantities, most notably the $q^2$ part of the Hall conductivity at zero frequency ($q$ being the wavenumber of the perturbation).
Physics at the Coulomb energy scale
-----------------------------------
It was also found in Ref. [@Son:2013rqa] that most physical quantities, e.g., the same $q^2$ part of the Hall conductivity, but calculated at nonzero frequency, are not fixed by $\nu$ and $s$ alone. The same is true for the $q^4$ term in the density-density correlation function. Physically, these quantities depend crucially on the physics happening at the Coulomb energy scale $\Delta$. The physics of the gapped excitations is contained in the non-universal part of the Lagrangian $\mathcal L_0$.
Terms in $\mathcal L_0$ can be organized in a series over powers of derivatives. We will be interested in the physics at long wavelengths, $q\ell_B\ll1$. The expansion parameter in frequency would be $\omega/\Delta$, however since we are interested in physical phenomena at the scale $\Delta$, we need to keep terms to all orders in time derivatives.
A consistent power counting scheme is to consider fluctuations of the metric $h_{ij}$ and the gauge potentials $A_0$, $A_i$ as $O(1)$, and expand in powers of the spatial derivatives. In this work, we will be interested only in the response of the quantum Hall systems to *unimodular* metric perturbations, i.e., those in which the perturbed metric $h_{ij}$ has determinant equal to one, and to perturbations of the scalar potential $A_0$, i.e., perturbations corresponding to a longitudinal electric field, without changing the magnetic field. In this case, the lowest non-trivial terms entering $\mathcal L_0$ are $O(q^4)$. These we parameterize, without loss of generality[^1], with two functions $F(\omega)$ and $G(\omega)$. In the following equation the frequency $\omega$ is replaced by $iv\cdot\nabla$ where $\nabla$ is the Newton-Cartan covariant derivative [@Son:2013rqa], $$\label{L0-FG}
\mathcal L_0 = -\frac\rho 4 \left[\sigma^{\mu\nu} F\big(iv\cdot\nabla\big)
\sigma_{\mu\nu} + \tilde\sigma^{\mu\nu}G\big(iv\cdot\nabla\big) (v\cdot\nabla
\sigma_{\mu\nu})
\right].$$ Here $\sigma_{\mu\nu}$ is the traceless part of shear tensor (see [@Son:2013rqa] for precise definition) and $\tilde\sigma^{\mu\nu}$ is defined as: $$\tilde\sigma^{\mu\nu}=
\frac12 (\varepsilon^{\mu\alpha\gamma} h^{\nu\beta}
+ \varepsilon^{\nu\alpha\gamma} h^{\mu\beta}) \sigma_{\alpha\beta}
n_\gamma,$$ where $n_\mu=(1,\mathbf 0)$.
We will work only to quadratic order, hence we only need to know the leading terms in the spatial components of the shear tensor: $$\sigma_{ij} = {\partial}_i v_j + {\partial}_j v_i + \dot h_{ij}
- \delta_{ij} ({\partial}_k v_k + \frac12 \dot h).$$ The quadratic part of the Lagrangian is then: $$\label{L0-quad}
\mathcal L_0 = -\frac{\rho_0}4\left[ \sigma_{ij} F(i{\partial}_t) \sigma_{ij}
+ \tilde \sigma_{ij} G(i{\partial}_t) \dot\sigma_{ij} \right].$$
Here, for any symmetric traceless tensors $A_{ij}$ we define $\tilde A_{ij} = \frac12 ( \epsilon_{ik}A_{kj} + \epsilon_{jk}A_{ki} )$ which is again a symmetric traceless tensor. It is also easy to show ${\tilde {\tilde A}}_{ij} = - A_{ij}$ and $\tilde A_{ij} B_{ij} = - A_{ij}\tilde B_{ij}$.
Gravitational response, spectral representations and shift sum rule
-------------------------------------------------------------------
We now relate the two functions $F(\omega)$ and $G(\omega)$ to the spectral densities of the stress tensor $\rho_T$ and $\bar\rho_T$. The two-point function of the stress tensor can be read directly from the action, and is simplest for the traceless components at zero spatial momentum. After a simple calculation we get: $$\begin{aligned}
{\langle}\bar T T{\rangle}_\omega &= -\frac\omega 4 s\rho_0 + \frac{\omega^3}2 \rho_0 G
+ \frac{\omega^2}2 \rho_0 F,\\
{\langle}T\bar T{\rangle}_\omega &= \phantom{+}\frac\omega 4 s\rho_0
- \frac{\omega^3}2 \rho_0 G
+ \frac{\omega^2}2 \rho_0 F.\end{aligned}$$ The spectral densities defined in Eqs. and are related to $F$ and $G$ by: $$\begin{aligned}
\rho_T(\omega) &= -\frac{\omega^2}{2\pi}{\mathop{\mathrm{Im}}}(F+\omega G),\\
\bar \rho_T(\omega) &=
-\frac{\omega^2}{2\pi} {\mathop{\mathrm{Im}}}(F-\omega G),\end{aligned}$$ where we have extended the definition of the spectral densities to negative $\omega$’s by requiring $\rho_T(-\omega)=\bar\rho_T(\omega)$. The functions $F(\omega)$ and $G(\omega)$ are regular in the $\omega\to0$ limits, and as we shall explain below \[see Eqs. (\[omega2G\]) and (\[omega2F\])\] they should both fall as $1/\omega^2$ when $\omega\gg\Delta$. From these behaviors we can write down the spectral representations of $F$ and $G$: $$\begin{aligned}
F(\omega) &= 2 \int\limits_0^\infty\!\frac{d\omega'}{\omega'} \,
\frac{\rho_T(\omega') + \bar\rho_T(\omega')}{\omega^2-\omega'^2+i\epsilon},
\label{F-disp}
\\
G(\omega) &= 2 \int\limits_0^\infty\!\frac{d\omega'}{\omega'^2} \,
\frac{\rho_T(\omega')-\bar\rho_T(\omega')}{\omega^2-\omega'^2+i\epsilon}.
\label{G-disp}\end{aligned}$$
The Hall viscosity, as a function of frequency, can be related through a Kubo’s formula to the parity-odd part of the two-point function of the stress tensor [@Bradlyn:2012ea]. We find: $$\eta_{\rm H}(\omega) = \rho_0\left( \frac s2 - \omega^2 G(\omega)\right).$$ At $\omega\to0$ this equation gives the relationship between the (zero-frequency) Hall viscosity and the shift: $\eta_{\rm
H}(0)=\rho_0\mathcal S/4$. At frequencies much larger than $\Delta$, interactions can be neglected and the Hall viscosity is determined completely by the Berry phase of each orbital under homogeneous metric deformation. The computation of the high-frequency Hall viscosity (where “high” means frequencies much larger than the Coulomb energy scale, but still much smaller than the cyclotron energy) proceeds in exactly the same way as in the integer quantum Hall case [@Avron:1995fg], and the result is $\eta_\mathrm{H}(\infty)=\rho_0/4$. Thus we find: $$\label{omega2G}
\lim_{\omega\to\infty}\omega^2 G(\omega) = \frac{\mathcal S-1}4,$$ and using Eq. (\[G-disp\]) we derive our first sum rule: $$\label{shift-sr}
\int_0^\infty\!\frac{d\omega}{\omega^2}
[\rho_T(\omega) - \bar \rho_T(\omega)] = \frac{\mathcal S-1}8.$$
Static structure factor and high-frequency shear modulus {#sec:shear_mod}
--------------------------------------------------------
We now derive two sum rules involving the sum of the two spectral densities $\rho_T(\omega)+\bar\rho_T(\omega)$. Computing the two-point function of the density from the action from and , we find: $$\label{sbar}
\int\!d^3x\, e^{i\omega t - i{\bf q}\cdot {\bf x}}
{\langle}T \rho(t,{\mathbf{x}}) \rho(0,{\bf 0}){\rangle}= i\rho_0 (q\ell_B)^4 F(\omega),$$ Integrating both sides over $\omega$, we get: $$\label{sbar1}
\bar s(q)
= i(q\ell_B)^4\!\! \int\limits_{-\infty}^\infty\!
\frac{d\omega}{2\pi} \, F(\omega),$$ where $\bar s(q)$ is the projected static structure factor [@Girvin:1986zz]. The reason we get the projected structure factor instead of the unprojected one is that we are working in the limit of zero band mass, and so when we took the integral over $\omega$, implicitly we have assumed that upper limit of integration is still much smaller than the cyclotron frequency $B/m$. By using Eq. (\[F-disp\]), we find the second sum rule: $$\label{S4-sr}
\int\limits_0^\infty\!\frac{d\omega}{\omega^2}\,
[\rho_T(\omega) + \bar \rho_T(\omega)] = S_4,$$ where $S_4=\lim\limits_{q\to0}\bar s(q)/(q\ell_B)^4$.
The third sum rule again comes from the stress response at large $\omega$. For this purpose, it is convenient to describe the motion of the fluid in terms of the displacement $u^i$, which is related to the velocity by $v^i=\dot u^i$. The $F$ term in the action now reads: $$- \frac{\rho_0}4\int\!d^2x\, \omega^2 F(\omega)
\left[{\partial}_i u_j + {\partial}_j u_i+h_{ij}- \delta_{ij}
\left({\partial}\cdot u + \frac h2\right)\right]^2.$$ In the limit $\omega\to\infty$, the $G$ term, having an extra time derivative, does not contribute to the energy. Hence, we are left with only the $F$ contribution above, which takes the exact same form as the the deformation energy of a solid, with the shear modulus $\mu_\infty$ given by: $$\label{omega2F}
\mu_\infty = \frac{\rho_0}2
\lim_{\omega\to\infty} \omega^2F(\omega) .$$ Using the spectral representation of $F(\omega)$ in , we find: $$\label{elasticity-sr}
\int_0^\infty\!\frac{d\omega}{\omega}
[\rho_T(\omega) + \bar \rho_T(\omega)] = \frac{\mu_\infty}{\rho_0}\,.$$
The high-frequency shear modulus $\mu_\infty$ was introduced in the phenomenological model of Refs. [@Tokatly:2006; @TokatlyVignale:2007]. We now give the precise meaning of this constant. From our discussion, we know that $\mu_\infty$ characterizes the stress response of the system under uniform metric perturbations with frequencies much higher than the Coulomb energy scale, but much lower than the cyclotron energy. Since at these frequencies the Coulomb interaction between electrons can be ignored, each particle evolves independently under such a perturbation. The orbital of each electron is continuously deformed and projected down to the lowest Landau level. In this way, we can completely determine the wavefunction of the deformed state from that of the the ground state. For example, consider a metric perturbation in which the $x$ coordinate is stretched by a factor of $e^{\alpha/2}$ while the $y$ coordinate compressed by $e^{-\alpha/2}$. If we denote the ground state wave function as: $$\Psi(z_i) = f(z_i) \exp\Bigl(-\sum_i |z_i|^2/4\ell_B^2\Bigr),$$ the deformed state $|\Psi_\alpha{\rangle}$ is obtained by replacing $f(z_i)$ by $f_\alpha(z_i)$, $$\label{falpha}
\Psi_\alpha(z_i) = f_\alpha(z_i) \exp\Bigl(-\sum_i |z_i|^2/4\ell_B^2\Bigr),
\qquad
f_\alpha(z_i)= \exp\Bigl[ \frac\alpha2 \sum_i
\Bigl( \ell_B^2 \frac{{\partial}^2}{{\partial}z_i^2}
- \frac{z_i^2}{4\ell_B^2}\Bigr)\Bigr] f(z_i).$$ If we now use Laughlin’s wavefunction to substitute for $f(z_i)$ above, the deformed states that we obtain coincide exactly with the ones recently considered in Ref. [@Haldane-Yang].
The energy of these states is a function of $\alpha$ with minimum at $\alpha=0$, and the high-frequency shear modulus is simply the curvature of this function at the minimum: $$\label{muinf-def}
\mu_\infty = \frac1A \frac{{\partial}^2}{{\partial}\alpha^2} {\langle}\Psi_\alpha|\hat H
| \Psi_\alpha{\rangle}|_{\alpha=0}.$$ where $A$ is the total area of the system. Equations (\[muinf-def\]) and (\[falpha\]) define the constant $\mu_\infty$ appearing in the sum rule (\[elasticity-sr\]).
Inequalities following from the sum rules
-----------------------------------------
The sum rules have important implications. First, since $\rho_T$ and $\bar\rho_T$ are non-negative spectral densities, comparing eqs. and , we obtain the following inequality between $S_4$ and $\mathcal S$: $$\label{S4-lowerbound}
S_4 \ge \frac{|\mathcal S-1|}8 \,.$$ This inequality has been previously derived by Haldane [@Haldane:2011; @Haldane-selfdual]. For Laughlin’s fractions $\nu=$ $1/(2p+1)$, $\mathcal S = 1/\nu$, and the inequality becomes $S_4\ge (1-\nu)/8\nu$. Remarkably, the Laughlin wavefunction has $S_4$ saturating the lower bound. Hence, if the Laughlin wavefunction was the true wavefunction of the ground state, that would imply $\bar\rho_T=0$ for all $\omega$.
Read and Rezayi [@ReadRezayi:2011] argued that the inequality (\[S4-lowerbound\]) is actually an equality for all lowest-Landau-level ground states with rotational invariance. From our derivation, we do not expect the equality to hold generally: the spectral density $\bar\rho_T$ need not necessarily vanish. Nevertheless, the Laughlin wavefunction seems to be a very good approximation to the true wavefunction of the Coulomb potential, thus it is possible that for the true ground state of the Coulomb problem, $\bar\rho_T$ is numerically much smaller than $\rho_T$.
Finally, we can also put a lower bound on the energy gap $\Delta_0$ at $q=0$. The energy gap may correspond not to a single quasiparticle, but, for example, to a pair of magneto-rotons, in which case $\Delta_0$ is the start of a continuum. The inequality that follows from comparing the sum rules and is: $$\label{ineq2}
\Delta_0 \le \frac{\mu_\infty}{\rho_0 S_4},$$ where equality would be achieved only when $\rho_T$ and $\bar\rho_T$ are proportional to $\delta(\omega-\Delta_0)$. The equality in this case has the same form as the Girvin-MacDonald-Platzman variational formula for the magneto-roton energy, but in contrast to the latter, both the numerator and the denominator in our formula are finite in the limit $q\to0$.
By combining these two inequalities we can also write: $$\label{ineq3}
\Delta_0 \le \frac{8\mu_\infty}{\rho_0 |\mathcal S-1|},$$ which saturates under the conditions $\bar\rho_T=0$ and $\rho_T(\omega)\sim\delta(\omega-\Delta_0)$.
A gravitational model of the magneto-roton
==========================================
We now present a simple model where the sum rules are satisfied by construction and are dominated by one single mode which is identified with the magneto-roton. To start, we adapt the Lagrangian formulation of fluid dynamics [@Dubovsky:2005xd], in which the degrees of freedom of the quantum Hall fluids are the Lagrangian coordinates $X^I(t,{\mathbf{x}})$, $I=1,2$. The density and velocity of the fluid are given by: $$\rho v^\mu=\rho_0\varepsilon^{\mu\nu\lambda}\epsilon_{IJ}{\partial}_\nu
X^I{\partial}_\lambda X^J,$$ such that the divergence of the current vanishes identically. The theory is required to be invariant under volume-preserving diffeomorphisms in the $X^I$ space. Imposing this condition sets the shape modulus to zero, thereby ensuring that our action describes a fluid and not a solid.
The degree of freedom saturating the sum rules, is assumed to be a unimodular metric tensor $G_{I\!J}$. Physically, one should think of $G_{I\!J}$ as parameterizing the anisotropic deformation of the ground state, as constructed in section \[sec:shear\_mod\]. In this model, $G_{I\!J}$ is the only dynamical degree of freedom at the Coulomb energy scale.
The theory can either be written in $x$ space, treating $X^I$ as functions of $t$ and $x^i$, or in $X$ space, where the dynamical fields are $x^i=x^i(t,X^I)$. The action of the model is the sum of three parts $S=S_1+S_2+S_3$, where the first part is written in $x$ space: $$\mathcal L_1 = \frac\nu{4\pi}\epsilon^{\mu\nu\lambda} a_\mu{\partial}_\nu a_\lambda
+ \rho v^\mu\Bigl({\partial}_\mu\varphi - \tilde A_\mu - \frac12 \omega_\mu + a_\mu
\Bigr).$$ This has the same form as the the universal part of the action derived in Ref. [@Son:2013rqa], but with $s$ replaced by $1/2$. The reason for this replacement is that we expect $L_1$ to encode the Hall viscosity at high frequency, but not at low frequency.
The second part of the action is written in $X$ space. It is a Wess-Zumino-Witten action: $$S_2 = \frac{\alpha\rho_0}2\!
\int_0^1\!d\tau\!\int\!dt\,d^2X\, {\partial}_\tau G_{I\!J}\, G^{J\!K}\, {\partial}_t G_{K\!L}\, \epsilon^{LI},$$ where as a function of $\tau$, $G_{I\!J}(0,t,X)=\delta_{I\!J}$ and $G_{I\!J}(1,t,X)=G_{I\!J}(t,X)$, and $\alpha$ is a parameter that will be fixed later. Although the action is written as an integral in $\tau$ space, one can check that the action depends only on the boundary value at $\tau=1$, but is independent of the interpolation between $\tau=0$ and $\tau=1$. It is identical to the action considered in Ref. [@Sondhi].
Finally, in $S_3$ we include the potential energy, which depends on the density and one parameter characterizing the eccentricity of the deformation: $$\mathcal L_3 = \mathcal L_3( \varepsilon^{ij}\epsilon_{IJ}{\partial}_i X^I {\partial}_j X^J ,
G_{IJ} h^{ij}{\partial}_i X^I {\partial}_j X^J ).$$ We only consider small perturbations around the ground state; we take: $X^I = x^I-u^I$ and $G_{IJ}=\delta_{IJ}+H_{IJ}$. Ignoring the constant term, total derivatives and terms proportional to squares of ${\partial}_i u^i$ and $h_{ii}$, which are small in the regime we are considering, we have: $$\mathcal L_3 = -\frac{\alpha\rho_0\Delta}4
({\partial}_i u_j + {\partial}_j u_i + h_{ij} - H_{ij}
)^2.$$ Now we introduce the variable $\gamma_{ij}$ as: $$\gamma_{ij} = {\partial}_i u_j + {\partial}_j u_i + h_{ij} - H_{ij} .
$$ Using $\gamma$ we can rewrite the quadratic action in the form of eq. where $\mathcal{L}_0$ is given by: $$\label{L-gamma2}
\mathcal L_0 = \frac{\alpha\rho_0}2 \left(
\tilde\sigma_{ij}\gamma_{ij} + \frac12 \tilde\gamma_{ij}\dot\gamma_{ij}
- \frac\Delta 2 \gamma_{ij}^2
\right).$$ After integrating out $\gamma_{ij}$, $\mathcal L_0$ reduces to the form (\[L0-FG\]), with functions $F$ and $G$ given by: $$\label{F-G}
F(\omega) = \frac{\alpha \Delta}{\omega^2-\Delta^2+i\epsilon}\,,\quad
G(\omega) = \frac\alpha{\omega^2-\Delta^2+i\epsilon} \, .$$ This corresponds exactly to the spectral functions $\rho_T(\omega)\sim
\delta(\omega-\Delta)$ and $\bar\rho_T=0$. In particular, $\alpha=(\mathcal S-1)/4$ and the inequality (\[S4-lowerbound\]) becomes an equality in this model.
Dispersion relation for the magneto-roton
-----------------------------------------
We reiterate the form of the effective Lagrangian in flat space-time and in the massless limit: $$\begin{gathered}
\label{L-Eff}
\mathcal L = \frac\nu{4\pi}\varepsilon^{\mu\nu\lambda} a_\mu{\partial}_\nu
a_\lambda - \rho v^\mu({\partial}_\mu\varphi- A_\mu +a_\mu)+\rho\frac{s-1}{2}\epsilon^{ij}\partial_i v_j\\
- \frac\rho 4\left( \sigma_{ij} F(\omega) \sigma_{ij}
+ \tilde \sigma_{ij} G(\omega) \dot\sigma_{ij} \right) -\epsilon_i(\rho),\end{gathered}$$ where $F$ and $G$ are given in eq. and the function $\epsilon_i(\rho)$ denotes interaction energy of the Hall state and depends only on the particle density.
We are interested in the dispersion relation of the magneto-roton excitations. To this end, we linearize the equations of motion and turn on perturbations about the ground state: $$\begin{aligned}
a_\mu=A_\mu+\tilde{a}_\mu,\quad
\rho=\rho_0+\tilde{\rho},\quad
v_i=0+v_i,\end{aligned}$$ where $\rho_0=\frac{\nu}{2\pi}\epsilon^{ij}\partial_i A_j=\frac{\nu}{2\pi}B$ is the ground state electron density and $\tilde{a}_\mu,\tilde{\rho},v_i$ are small perturbations. We further assume that the energy arises purely from pairwise interactions with the form: $$\epsilon_i(\rho)=\frac{1}{2}\int{\int{d^2x d^2y (\rho(x)-\rho_0)V(|\mathbf{x}-\mathbf{y}|)(\rho(y)-\rho_0)}}.$$ For the sake of definiteness, we work with the Coulomb potential with a strength parameter $\lambda$ defined as: $$\frac{\lambda}{q}=\int{d^2(\mathbf{x}-\mathbf{y})V_c(|\mathbf{x}-\mathbf{y}|)e^{i(\mathbf{q(x-y)})}}.$$ It should be noted that even though the specifics of the calculation depend on the exact form of the chosen potential, the qualitative behavior that we derive here is independent of such details, so long as the potential remains repulsive.
In what follows we set $p=k l_B = k/\sqrt{B}$ and $\sigma_s=\frac12 (1-s)$. We note that the parameter $\alpha$ used in the non-universal functions $F$ and $G$ (see eq. ) is fixed by our first sum rule given in eq. : $\alpha=2S_4$. Also, as noted previously, the inequality is saturated in this model.
We find the dispersion relation of the magneto-roton mode to be: $$\label{Dispersion1}
\omega(p)=\Delta\frac{\sqrt{1+2\sigma_s p^2+\dfrac{\alpha \lambda}{\Delta l_B}p^3+\sigma_s^2p^4}}{1+\big(\alpha +\sigma_s\big)p^2} \, ,$$ which exhibits the properties of the magneto-roton with a downward slope at small $p$ which turns around after a characteristic minimum (Fig \[Dis\]).
![Dispersion relation of the collective mode with $\nu=1/3$ and $\lambda/ (l_B\Delta)=0.3$. Minimum dispersion appears at $k_{min} \approx 1.69 \,l^{-1}_B $.[]{data-label="Dis"}](Dis.pdf)
For the purpose of comparison, we also report this this dispersion relation up to fourth order in momentum expansion: $$\label{Dispersion2}
\omega(p)=\Delta\left[1-2S_4 p^2+\frac{\lambda S_4}{l_B\Delta}p^3+(4S^2_4+2S_4\sigma_s)p^4+\mathcal{O}(p^5)\right].$$ Note that the coefficient in front of $p^2$ is, for Laughlin’s fractions, $-(1-\nu)/4\nu$, which is also seen in the model of Ref. [@Tokatly:2006]. However, we cannot argue that this coefficient is universal. For example, the coefficient will change if we add to the action (\[L-gamma2\]) a term proportional to the square of the spatial gradients of $\gamma_{ij}$.
There is one more interesting feature of this mode that reveals itself under closer inspection. From the equations of motion, we can derive that the eigenmodes satisfy: $$\mathbf{p\times v}=if(p)\mathbf{p\cdot v}, \qquad
f(p)=\frac{1+p^2 \sigma_s +{\omega_p}^2p^2 G(\omega_p)}{\omega_p \, p^2 F(\omega_p)} \,,$$ where $\mathbf{k\times v}=\epsilon^{ij}k_j v_j$ and $\omega_p=\omega(p)$ given in .
If we decompose $\mathbf{v}$ into a parallel component $v_\|$ and a perpendicular component $v_\bot$ to the direction of momentum $\mathbf{p}$, we find that $v_\bot=if(p)v_\|$. Using the dispersion relation , and the explicit form of $G(\omega),F(\omega)$, we see that the value of $f(k)$ evolves from $f(0)=-1$ to $f(k\approx k_{min})=0$ and finally to $f(k=\infty)=1$ (Fig \[Pol\]).
This implies that, from the point of view of current pattern, the excitations exhibit counterclockwise rotation at small momenta, which turns into a linear oscillation in direction of $\mathbf p$ in the vicinity of the magneto-roton minimum and finally develops into clockwise rotation at large values of the momentum. It would be interesting to understand if this feature of the magneto-roton may be detected experimentally.
![Polarization function $f(k)$, with $\nu=1/3$ , $\lambda/ (l_B\Delta)=0.3$. Linear polarization appears at $k\approx 1.49 l^{-1}_B$.[]{data-label="Pol"}](Pol.pdf)
Here we only briefly discuss the implication of the model for the observation of the magneto-roton at low momentum in Raman scattering experiments. In previous theoretical treatments [@PlatzmanHe], it was assumed that the magneto-roton is excited chiefly through the coupling of electric field to density: $\rho E^2$. This coupling, however, implies that the intensity of the magneto-roton peak should scale as the fourth power of the magneto-roton momentum, $q^4$. In experiments, magneto-roton was seen down to even the lowest momenta [@Pinczuk], a fact that may be attributed to disorders violating momentum conservation. However, we cannot rule out a coupling of the type $T_{ij}E_i E_j$ from symmetry consideration, with $T_{ij}$ being the stress tensor. Even if the coefficient in front of this term is small, it would dominate the intensity of magneto-roton peak in the limit $q\to0$, since the residue at the pole in in ${\langle}TT{\rangle}$ correlators remains finite in this limit. This coupling thus provides an alternative explanation of the observation of the magneto-roton at lowest momenta in Raman scattering experiments.
Moreover, in our model at $q=0$ the magneto-roton is circularly polarized with angular momentum 2. We suggest that the polarization of the magneto-roton at $q=0$ may be detectable by Raman scattering with polarized light.
Conclusion
==========
In this paper we looked at gapped fractional quantum Hall states with filling factors $\nu<1$ in the regime where the Coulomb energy is much smaller than the cyclotron energy, however with energies comparable to that of the gap. We developed three sum rules involving the spectral densities of the stress tensor which we then used to verify Haldane’s conjectured lower bound on the quartic coefficient of the structure factor $\mathcal S_4$, as well as introduce other inequalities.
We also introduced a simple model that saturates these inequalities via a mode which arises from the mixing between the oscillations of an internal metric and the hydrodynamic excitations. We identifed this mode as the magneto-roton and calculated its dispersion relation. We argued that the intensity of the magneto-roton line in Raman scattering experiments should not vanish at zero momentum, and that the magneto-roton at $q=0$ is a spin-2 object. Finally, we suggest that the spin of the magneto-roton can be determined by Raman scattering with polarized light.
The authors thank Ilya Gruzberg, Michael Levin, Emil Martinec, Aron Pinczuk, and Paul Wiegmann for discussion. This work is supported, in part, by NSF MRSEC grant DMR-0820054. D.T.S. is supported, in part, by DOE grant DE-FG02-90ER-40560 and a Simons Investigator grant from the Simons Foundation.
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[^1]: As an example, the term $({\partial}_i \rho)^2$ is of higher order, because fluctuations of $\rho$ are of order $q^4$, as evident from Eq. (\[sbar\]) below. The same is true for $({\partial}_i v^i)^2$: because of charge conservation ${\partial}_i v^i \sim {\partial}_t\rho \sim q^4$.
|
---
abstract: 'This study concerns online inference (i.e., filtering) on the state of reaction networks, conditioned on noisy and partial measurements. The difficulty in deriving the equation that the conditional probability distribution of the state satisfies stems from the fact that the master equation, which governs the evolution of the reaction networks, is analytically intractable. The linear noise approximation (LNA) technique, which is widely used in the analysis of reaction networks, has recently been applied to develop approximate inference. Here, we apply the projection method to derive approximate filters, and compare them to a filter based on the LNA numerically in their filtering performance. We also contrast the projection method with moment-closure techniques in terms of approximating the evolution of stochastic reaction networks.'
address: 'Department of Statistical Modeling, The Institute of Statistical Mathematics, Tokyo, Japan'
author:
- Shinsuke Koyama
bibliography:
- 'mybib.bib'
title: 'Projection-based filtering for stochastic reaction networks'
---
February 2016
Introduction
============
Stochastic reaction networks provide probabilistic descriptions of the evolution of interacting species. They are used for modeling phenomena in a wide range of disciplines; those species can represent molecules in chemical reactions [@Higham08; @Thattai01; @Shahrezaei08], animal species in ecology [@Spencer05], susceptibles and infectives in epidemic models [@Pastor-Satorras15], and information packets in telecommunication networks [@Adas97].
The evolution of a network is modeled by a continuous-time Markov jump process, for which the probability distribution of the number of individuals of each species obeys the master equation [@Gardiner85; @Kampen92]. Here, we consider a situation wherein only noisy and partial measurements of underlying reaction networks are available. Our objective is to infer the number of individuals of species from the observations obtained up to the current time. In the literature on signal processing, this problem is called [*filtering*]{} [@Jazwinski70].
The filtering equation, which governs the posterior distribution conditioned on the observations, is not analytically obtainable due to the intractability of the master equation. It is possible to perform exact numerical simulation and obtain samples from the Markov jump processes using a stochastic simulation algorithm (SSA) [@Gillespie07]. Simulating many “particles" with the SSA and sampling the weighted particles in the favor of the observations, we could obtain samples from the posterior distribution. This technique is known as the [*sequential Monte Carlo method*]{} or [*particle filtering*]{} [@Doucet01]. However, the SSA is often too slow. Moreover, particle filtering sufficiently requires many particles to obtain precise posterior expectations. Thus, particle filtering might not be efficient for performing online inference.
An alternative approach is to consider the suitable approximations of the Markov jump processes. In the linear noise approximation (LNA), which is most widely used in such analysis, a Gaussian process whose mean obeys the deterministic rate equation approximates a Markov jump process [@Kampen92]. The LNA is valid under the assumption that the number of individuals of a species is large [@Kurtz71]. It is also exact for all systems with affine propensities as well as for some systems with nonlinear propensities [@Grima15]. As the Gaussian process is tractable, The LNA allows us to derive an analytical expression of the approximate filtering equation [@Fearnhead14]. In addition to the LNA, a number of approximation techniques have been proposed such as system-size expansions [@Kampen92], moment-closure approximations [@Grima12] and conditional moment equations [@Hasenauer14], and have been applied to inference of model parameters [@Milner13; @Frohlich16].
In this study, we propose applying the projection method [@Brigo98; @Brigo99] to derive approximate filters. In this method, the evolution of the probability distributions is constrained on a finite-dimensional family of densities through orthogonal projection onto the tangent space with respect to the Fisher metric. We derive the projection-based filter for stochastic reaction networks, and compare it to an approximate filter based on the LNA numerically in their filtering performance. We also contrast between the projection method and moment-closure techniques in terms of approximating the master equation.
Method
======
Reaction networks
-----------------
Throughout the study, the transpose of a matrix $B$ is written $B^T$. Let $\mathcal{X}=\{X_1,\ldots,X_n\}$ be $n$ species, and consider $m$ reactions among these species described by $$\sum_{i=1}^n \nu_{ij}^{-}X_i \autorightarrow{$k_j$}{} \sum_{i=1}^n \nu_{ij}^{+}X_i
\qquad j=1,\ldots,m
\label{eq:reacnet}$$ where $\nu_{ij}^{-}$ and $\nu_{ij}^{+}$ are stoichiometric coefficients of reactants and products, respectively, and $k_j$ is the reaction rate constant. We denote by $x=(x_1,\ldots,x_n)^T$ the discrete composition vector whose $i$th component, $x_i$, is the number of individuals of species $X_i$. Let $A=(\Delta x_{ij})$ be an $n\times m$ matrix, called the [*net effect matrix*]{}, whose $(i,j)$ element, $\Delta x_{ij} = \nu^+_{ij}-\nu^-_{ij}$, is the change in the number of individuals of the $i$th species after one step of the $j$th reaction. Let $h(x)=(h_1(x),\ldots,h_m(x))^T$ be the vector whose $j$th component, $h_j(x)$, is the rate of $j$th reaction, given as $$h_j(x) = k_j \prod_{i=1}^n
\Bigg(
\begin{array}{c}
x_i \\
\nu^-_{ij}
\end{array}
\Bigg)~.$$ From the Markov property, it follows that the probability distribution over $x$ at time $t$, $P(x,t)$, is governed by the master equation [@Gillespie92; @Gadgil05]: $$\frac{dP(x,t)}{dt} = \sum_{j=1}^m h_j(x-\Delta x_{\cdot j})P(x-\Delta x_{\cdot j},t)
- \sum_{j=1}^m h_j(x)P(x,t).
\label{eq:stochastic}$$
Stochastic processes described by Eq. (\[eq:stochastic\]) are related to an ordinary differential equation (ODE), called the [*rate equation*]{}, via the thermodynamic limit. To see this, we introduce a scale factor $\Omega$ (typically taken to be “volume"), and rescale the composition vector and the reaction rate as $$z = \frac{x}{\Omega},
\label{eq:concentration}$$ $$\tilde{h}(z) = \frac{h(\Omega z)}{\Omega}.
\label{eq:rate}$$ Accordingly, the reaction rate constants are rescaled as $$\tilde{k}_j = V^{\sum_{i=1}^n \nu^-_{ij}-1} k_j
\qquad j=1,\ldots,m.$$ With these rescaled parameters, it has been proved in [@Kurtz70] that $z\to\phi$ as $\Omega\to\infty$ in probability, where $\phi$ satisfies the rate equation: $$\frac{d\phi}{dt} = A\tilde{h}(\phi).
\label{eq:deterministic}$$
State space model and filtering
-------------------------------
We consider a situation wherein the system of interest is given by a stochastic reaction network, whose state is not directly observable, but instead, we have noisy and partial measurements at discrete time points [@Fearnhead14; @Golightly06; @Golightly11; @Komorowski09; @Finkenstadt13]; this situation is formulated within the framework of state space models. In state space modeling, the state process, $x(t)$, is given by the master equation (\[eq:stochastic\]), and the measurement model is assumed to be $$y_i = Gx(t_i) + \xi_i
\qquad i=1,\ldots N,
\label{eq:observation}$$ where $y_i\in\mathbb{R}^d$ $(d\le n)$, $G\in\mathbb{R}^{d\times n}$, and $\xi_i$ is a $d$-dimensional Gaussian random variable with zero mean and covariance matrix $V$. The goal of a filtering problem is to compute the posterior probability of the state $x$ at time $t_i$, when the observations $y_1,\ldots,y_i$ are given.
Projection-based filter
-----------------------
### Projection method
We apply the projection method proposed in [@Brigo98; @Brigo99] to derive approximate filters. To apply the projection method, we need a Fokker-Planck equation derived from the master equation (\[eq:stochastic\]). By taking up to the second-order terms in the Kramers-Moyal expansion of the master equation, a Fokker-Planck equation is obtained as $$\begin{aligned}
\frac{\partial p(x,t)}{\partial t}
&=&
\mathcal{L}^*p(x,t) \nonumber\\
&:=&
-\sum_{i=1}^n \frac{\partial}{\partial x_i}\Bigg[\sum_{k=1}^m\Delta x_{ik}h_k(x)p(x,t)\Bigg]
\nonumber\\
& & { } + \frac{1}{2}\sum_{i,j=1}^n\frac{\partial^2}{\partial x_i\partial x_j}
\Bigg[
\sum_{k=1}^m \Delta x_{ik}h_k(x)\Delta x_{jk}
p(x,t)\Bigg]~,
\label{eq:Fokker-Planck}\end{aligned}$$ where $p(x,t)$ is the probability density of $x$ at time $t$ [@Kampen92]. We apply the projection method to Eq. (\[eq:Fokker-Planck\]). The key idea is to introduce a finite-dimensional family of probability densities $p(x,\theta)$, where $\theta=(\theta_1,\ldots,\theta_r)\in \Theta \subseteq \mathbb{R}^r$ is the parameter characterizing the probability distributions, and to project the evolution of the probability density $p(x,t)$ onto the space of $p(x,\theta)$; the resulting ODE for $\theta$ approximates the master equation.
Let $L_2$ be a space of square-integrable functions, and consider the square roots of the probability densities, $S^{1/2}=\{p(x,\theta)^{1/2}, \theta\in\Theta \} \subset L_2$. The tangent space of $S^{1/2}$ at $p(x,\theta)^{1/2}$ is given by $$T_{p(x,\theta)^{1/2}} S^{1/2} =
\mathrm{span}
\Bigg\{
\frac{\partial p(x,\theta)^{1/2}}{\partial\theta_1}, \ldots,
\frac{\partial p(x,\theta)^{1/2}}{\partial\theta_r}
\Bigg\}~.$$ The $L_2$ inner product of any two bases of $S^{1/2}$ is defined as $$\begin{aligned}
\fl
\bigg\langle
\frac{\partial p(x,\theta)^{1/2}}{\partial\theta_i},
\frac{\partial p(x,\theta)^{1/2}}{\partial\theta_j}
\bigg\rangle &:=&
\int
\frac{\partial p(x,\theta)^{1/2}}{\partial\theta_i}
\frac{\partial p(x,\theta)^{1/2}}{\partial\theta_j}
dx \nonumber\\
&=&
\frac{1}{4}\int \frac{\partial\log p(x,\theta)}{\partial\theta_i}
\frac{\partial\log p(x,\theta)}{\partial\theta_j} p(x,\theta)dx \nonumber\\
&=&
\frac{1}{4} g_{ij}(\theta),\end{aligned}$$ where $(g_{ij}(\theta))$ is the Fisher information matrix. Then, the orthogonal projection of $q\in L_2$ onto $T_{p(x,\theta)^{1/2}} S^{1/2}$ is given by $$q \mapsto
\sum_{i=1}^r
\Bigg(
\sum_{j=1}^r 4g^{ij}(\theta)
\bigg\langle
q, \frac{\partial p(x,\theta)^{1/2}}{\partial\theta_j}
\bigg\rangle
\Bigg)
\frac{\partial p(x,\theta)^{1/2}}{\partial\theta_i} ~,
\label{eq:projection}$$ where $(g^{ij})$ is the inverse of the Fisher information matrix.
Using Eq. (\[eq:projection\]), we project the Fokker-Plank equation (\[eq:Fokker-Planck\]) onto $S^{1/2}$ as follows: Using the chain rule, we obtain the equation for $p(x,\theta)^{1/2}$ as $$\label{eq:squareroot}
\frac{\partial p(x,\theta)^{1/2}}{\partial t} = \frac{p(x,\theta)^{1/2}\mathcal{L}^*p(x,\theta)}{2p(x,\theta)}.$$ Applying the orthogonal projection (\[eq:projection\]) to Eq. (\[eq:squareroot\]), we obtain an ODE for $\theta$ as $$\label{eq:projectionFP}
\frac{d\theta_i}{dt} =
\sum_{j=1}^r g^{ij}(\theta) \mathrm{E}\Bigg[
\frac{\mathcal{L}^*p(x,\theta)}{p(x,\theta)}
\frac{\partial \log p(x,\theta)}{\partial \theta_j}
\Bigg]
\qquad i=1,\ldots, r,$$ where $\mathrm{E}[\cdot]$ is the expectation of $x(t)$ with respect to $p(x,\theta)$. We further assume that $p(x,\theta)$ is an exponential family of probability densities [@Amari01]: $$p(x,\theta) = \exp[\theta^Tc(x) - \psi(\theta)],
\label{eq:expfamily}$$ where $\theta = (\theta_1,\ldots,\theta_r)^T$ is the natural parameter, $c(x) = (c_1(x), \ldots,c_r(x))^T$ is the sufficient statistic for $\theta$ and $\exp[-\psi(\theta)]$ is the normalization factor. Substituting Eq. (\[eq:expfamily\]) into Eq. (\[eq:projectionFP\]) leads to the projection approximation onto the exponential family: $$\frac{d\theta}{dt} = g^{-1}(\theta)\mathrm{E}[\mathcal{L}c],
\label{eq:expfilter}$$ where $\mathcal{L}$ is the backward diffusion operator: $$\fl
\mathcal{L} = \sum_{i=1}^n \Bigg[\sum_{k=1}^m\Delta_{ik}h_k(x)\Bigg] \frac{\partial}{\partial x_i} + \frac{1}{2}
\sum_{i,j=1}^n
\Bigg[\sum_{k=1}^m \Delta x_{ik}h_k(x)\Delta x_{jk}\Bigg]
\frac{\partial^2}{\partial x_i \partial x_j}.
\label{eq:backward}$$
### Bayesian update
Let $\theta(t_i)$ be the solution of Eq. (\[eq:expfilter\]) at time $t_i$. At time $t_i$, the observation $y_i$ is combined with $p(x,\theta(t_i))$ through Bayes’ rule, leading to the posterior probability density of $x$: $$p^+(x,t_i) = \frac{p(y_i|x)p(x,\theta(t_i))}{\int p(y_i|x)p(x,\theta(t_i))dx},
\label{eq:Bayes}$$ where $p(y_i|x)$ is the likelihood function of the observation model (\[eq:observation\]). If $p(x,\theta)$ is a conjugate family for $p(y_i|x)$, then the posterior probability density is in the same exponential family (\[eq:expfamily\]): $$p^+(x,t_i) = \exp[\theta^+(t_i)^Tc(x)- \phi(\theta^+(t_i))],$$ where $\theta^+(t_i)$ is the parameter updated by Bayes’ rule.
The filtering algorithm is summarized in the following two steps:
1. (Prediction step) Solve the ODE (\[eq:expfilter\]) from time $t_{i-1}$ to $t_i$ with initial conditions $\theta^+(t_{i-1})$ to obtain $\theta(t_i)$.
2. (Correction step) Update the parameter $\theta(t_i)$ to $\theta^+(t_i)$ by Bayes’ rule (\[eq:Bayes\]).
Filtering is performed by executing these two steps recursively from time $t_1$ to $t_N$.
Choice of probability distributions
-----------------------------------
We use two specific probability distributions for $p(x,\theta)$ to illustrate our method.
### Gaussian distribution {#sec:Gaussian}
Consider a multi-dimensional Gaussian distribution with mean vector $\mu$ and covariance matrix $Q$: $$p(x, \mu,Q) =
(2\pi)^{-n/2}|Q|^{-1/2}\exp \left[ -\frac{1}{2}(x-\mu)^TQ^{-1}(x-\mu) \right].
\label{eq:gausspdf}$$ It is easily confirmed that the Gaussian distribution belongs to the exponential families (\[eq:expfamily\]). The projection approximation (\[eq:expfilter\]) is obtained as (see \[appendix:GP\]) $$\begin{aligned}
\frac{d\mu}{dt} &=& A \mathrm{E}[h(x)], \label{eq:GP_mean} \\
\frac{dQ}{dt} &=& Q\mathrm{E}[J_h(x)]^TA^T + A\mathrm{E}[J_h(x)]Q +
A\mathrm{E}[H(x)]A^T, \label{eq:GP_cov}\end{aligned}$$ where $$J_h(x) := \frac{\partial h(x)}{\partial x}$$ is the Jacobian matrix of $h(x)$. Note that Eqs. (\[eq:GP\_mean\])-(\[eq:GP\_cov\]) are expressed with $(\mu,Q)$ instead of the natural parameter $\theta$ of the exponential family. For systems with reactions of order three or higher, $h(x)$ contains polynomials in the variables of order three or higher, so that Eqs. (\[eq:GP\_mean\])-(\[eq:GP\_cov\]) depend on moments of order three or larger; these moments can be computed with $\mu$ and $Q$ due to the Gaussian assumption, and therefore Eqs. (\[eq:GP\_mean\]) and (\[eq:GP\_cov\]) are closed for such systems. We also point out that the Gaussian projection is equivalent to the normal moment-closure approximation (see \[appendix:normal\_moment\_closure\] for proof).
Since both $p(x,\theta(t_i))$ and $p(y_i|x)$ in Eq. (\[eq:Bayes\]) are Gaussian distributions, $p^+(x, t_i)$ is also Gaussian, and its mean vector $\mu^+(t_i)$ and covariance matrix $Q^+(t_i)$ are computed using the standard Kalman filter recursion as $$\begin{aligned}
\mu^+(t_i) &=& \mu(t_i) + K_i\{y_i - G\mu(t_i)\}, \label{eq:post_mean} \\
Q^+(t_i) &=& Q(t_i) - K_iGQ(t_i), \label{eq:post_cov}\end{aligned}$$ where $$K_i = Q(t_i)G^T\{GQ(t_i)G^T+V\}^{-1}
\label{eq:post_gain}$$ is the Kalman gain [@Sarkka13].
### Quartic polynomial {#sec:quartic}
Another example is an exponential family of probability distributions with quartic polynomials in the exponent: $c(x) = (x,x^2,x^3,x^4)^T$ $(x\in\mathbb{R}^1)$ and $\theta = (\theta_1,\theta_2,\theta_3,\theta_4)^T$. A characteristic of this exponential family is that it allows bimodality. We briefly summarize how to compute the Fisher information matrix $g(\theta)$ and the moments $\eta_i:=\mathrm{E}[x^i]$ $(i=1,2,\ldots)$ that are required to solve the ODE (\[eq:expfilter\]) (see [@Brigo99] for details).
1. For $i=0,1,2$, compute the following integral numerically: $$I_i(\theta) = \int_{-\infty}^{\infty}x^i\exp(\theta_1x+\theta_2x^2+\theta_3x^3+\theta_4x^4)dx$$ and $\eta_i = I_i(\theta)/I_0(\theta)$.
2. Compute recursively the higher-order moments $\eta_i(\theta)$, $i\ge3$ by $$\fl
\eta_i(\theta) = -\frac{1}{4\theta_4}\{ (i-3)\eta_{i-4}(\theta) + \theta_1\eta_{i-3}(\theta)
+ 2\theta_2\eta_{i-2}(\theta) + 3\theta_3\eta_{i-1}(\theta) \}.$$
3. Compute the Fisher information matrix $g(\theta) = (g_{ij}(\theta))$ where $$g_{ij}(\theta) = \eta_{i+j}(\theta) - \eta_i(\theta)\eta_j(\theta).$$
For this exponential family distribution, the parameter update through Bayes’ rule (\[eq:Bayes\]) becomes $$\begin{aligned}
\left( \begin{array}{c}
\theta_1^+(t_i) \\
\theta_2^+(t_i) \\
\theta_3^+(t_i) \\
\theta_4^+(t_i)
\end{array} \right)
=
\left( \begin{array}{c}
\theta_1(t_i) + \frac{Gy_i}{V} \\
\theta_2(t_i) - \frac{G^2}{2V} \\
\theta_3(t_i) \\
\theta_4(t_i)
\end{array} \right).\end{aligned}$$
Results
=======
We illustrate our method on two reaction networks, and compare it to an approximate filter based on the LNA in their filtering performances. The LNA-based filter is briefly summarized in \[appendix:LNA\]. Hereafter, we label the projection-based filter onto Gaussian distributions “GPF" and that onto quartic polynomial exponential distributions “QPF".
Bistable system
---------------
We first consider the following reaction network consisting of a single species [@Erban09]: $$\begin{aligned}
\emptyset \autorightleftharpoons{$k_1$}{$k_2$} X, \quad
2X \autorightleftharpoons{$k_3$}{$k_4$} 3X. \nonumber\end{aligned}$$ The net effect matrix and the reaction rate vector, respectively, are given by $$A = (1, -1, 1 ,-1),
\label{eq:neteffect_bistable}$$ and $$h(x) = (k_1, k_2x, k_3x(x-1), k_4x(x-1)(x-2))^T.
\label{eq:propensity_bistable}$$ The rate equation (\[eq:deterministic\]) for $z\Omega\gg1$ is given by $$\frac{dz}{dt} = -\frac{dU(z)}{dz},$$ where $U(z)$ is the potential: $$U(z) = -\tilde{k}_1z + \frac{\tilde{k}_2}{2}z^2 - \frac{\tilde{k}_3}{3}z^3 + \frac{\tilde{k}_4}{4}z^4,
\label{eq:potential}$$ with the rescaled rate constants: $$\tilde{k}_1 = \frac{k_1}{\Omega}, \
\tilde{k}_2 = k_2, \
\tilde{k}_3 = \Omega k_3, \
\tilde{k}_4 = \Omega^2k_4.$$ The parameter values were considered to be $\tilde{k}_1 = 22.5$, $\tilde{k}_2 = 37.5$, $\tilde{k}_3 = 18$ and $\tilde{k}_4 = 2.5$, with which the potential (\[eq:potential\]) has two local minima (Figure \[fig:bistable\]a). The stochastic version of the reaction network with $\Omega=100$ was simulated using the SSA. A sample path is shown in Figure \[fig:bistable\]b (gray line) wherein we see that the reaction network exhibits stochastic switching between the two states that correspond to the two local minima of the potential.
For this reaction network, we applied the GPF, QPF and LNA. A numerical study was conducted using the following steps: First, the reaction network was simulated with the SSA in a time interval $T=100$ to generate a sample path, $\{x(t), 0\le t \le T\}$ (Figure \[fig:bistable\]b, gray line). The observations, $\{y_i, i=1,\ldots,N\}$, were simulated using Eq. (\[eq:observation\]), where we set $G=1$. The inter-observation interval, $\Delta:=t_i-t_{i-1}$, ranged from $0.1$ to $1$, and the variance of the observation noise, $V$, ranged from $500$ to $5,000$ (Figure \[fig:bistable\]b; crosses represent the observations with $\Delta=1$ and $V=500$). The three approximate filters were then performed to estimate the simulated path from the observations.
![ (a) Potential $U(z)$ has two local minima at $z=1.06$ and $z=4.04$. (b) Gray line represents a sample path of $x(t)$ simulated with the stochastic simulation algorithm, and crosses represent observations with the noise variance $V=500$. []{data-label="fig:bistable"}](fig_bistable.eps){width="15cm"}
To quantify the extent to which the approximate filters estimate the true path, we employed a [*maximum a posteriori*]{} (MAP) estimate, $\hat{x}(t)$, for each filter, and computed the mean squared error (MSE) between the true and estimated paths: $$\mathrm{MSE} = \frac{1}{T}\int_0^T | x(t)-\hat{x}(t) |^2dt.$$ We plotted the MSE for the three approximate filters as a function of $V$ (Figure \[fig:bistable\_mse\]a) and as a function of $\Delta$ (Figure \[fig:bistable\_mse\]b). The difference in the MSE among the three filters is small when $V$ or $\Delta$ is small. The MSE for the LNA increases more than that for the GPF and QPF as $V$ or $\Delta$ is increased. In particular, the MSE for the QPF remains relatively small over the range of $V$ and $\Delta$. Figure \[fig:bistable\_est\] depicts sample paths estimated by the three filters for $V=3,000$ and $\Delta=1$; as seen in this figure, while the QPF can capture the sharp transitions from one local equilibrium state to the other, the GPF and LNA fail, resulting in the large estimation error. These results suggest that for the reaction network with bistability, the QPF performs better that the GPF and LNA; the superiority of the QPF over the others stands out for noisy and sparse observations.
![ Mean squared error (MSE) between true and estimated paths (a) as a function of noise variance $V$ with $\Delta=1$ and (b) as a function of interval $\Delta$ with $V=2,000$. Solid, dashed and dotted lines represent MSE for QPF, GPF and LNA, respectively. Mean squared errors at each point were calculated with 20 repetitions. MSE for QPF is smaller than that for LNA and GPF over the range of $V$ and $\Delta$. []{data-label="fig:bistable_mse"}](fig_bistable_mse.eps)
![ Sample simulated paths for $V=3,000$ and $\Delta=1$. Gray line represents true path, and solid, dashed and dotted lines represent paths estimated by QPF, GPF and LNA, respectively. While QPF captures the abrupt jumps, GPF and LNA fail, resulting in the large estimation error. []{data-label="fig:bistable_est"}](fig_bistable_path_sg3000dt1.eps)
Reaction network with limit cycle
---------------------------------
![ The phase space $(z_1, z_2, z_3)$. Black line represents a solution of ordinary differential equations (\[eq:limcyc\_rate\_1\])-(\[eq:limcyc\_rate\_3\]), and gray line represents a sample path of stochastic model. []{data-label="fig:limcyc"}](fig_limcyc.eps)
Next, we consider a reaction network consisting of three species, $X = (X_1,X_2,X_3)$, which follow a set of five reactions [@Wilhelm95]: $$\begin{aligned}
X_1 \autorightarrow{$k_1$}{} 2X_1, \quad
X_1 + X_2 \autorightarrow{$k_2$}{} X_2, \quad
X_2 \autorightarrow{$k_3$}{} \emptyset, \nonumber\\
X_1 \autorightarrow{$k_4$}{} X_3, \quad
X_3 \autorightarrow{$k_5$}{} X_2. \nonumber\end{aligned}$$ The net effect matrix and the reaction rate vector, respectively, are given by $$A = \left(
\begin{array}{rrrrrrr}
1 & -1 & 0 & -1 & 0 \\
0 & 0 & -1 & 0 & 1 \\
0 & 0 & 0 & 1 & -1 \\
\end{array}
\right),$$ $$h(x) = (k_1x_1, k_2x_1x_2, k_3x_2, k_4x_1, k_5x_3)^T.$$ The rate equation (\[eq:deterministic\]) is derived as $$\begin{aligned}
\frac{dz_1}{dt} = (\tilde{k}_1 - \tilde{k}_4)z_1 - \tilde{k}_2z_1z_2, \label{eq:limcyc_rate_1} \\
\frac{dz_2}{dt} = -\tilde{k}_3z_2 + \tilde{k}_5z_3, \label{eq:limcyc_rate_2} \\
\frac{dz_3}{dt} = \tilde{k}_4z_1 - \tilde{k}_5z_3, \label{eq:limcyc_rate_3}\end{aligned}$$ where the reaction rate constants are rescaled as $$\tilde{k}_1 = k_1, \
\tilde{k}_2 = \Omega k_2, \
\tilde{k}_3 = k_3, \
\tilde{k}_4 = k_4, \
\tilde{k}_5 = k_5.$$ The values of the rate constants were chosen as $\tilde{k}_1 = 3.1$, $\tilde{k}_2 = 1$, $\tilde{k}_3 = 1$, $\tilde{k}_4 = 1$ and $\tilde{k}_5 = 1$. Figure \[fig:limcyc\] depicts the phase space $(z_1,z_2,z_3)$ wherein an illustrative path of the rate equation is plotted (black line), showing that it converges to the limit cycle. The stochastic version of the reaction network with $\Omega=100$ was simulated with the SSA. A sample path of the rescaled variable $x/\Omega$ was also plotted in Figure \[fig:limcyc\] (gray line).
We applied the GPF and the LNA for this reaction network. A numerical study for this reaction network was performed using the same procedure as for the bistable system. The duration of the simulation interval was chosen as $T=30$. The parameter of the observation model (\[eq:observation\]) was considered to be $G=(1,0,0)$. The inter-observation interval, $\Delta:=t_i-t_{i-1}$, ranged from $0.1$ to $0.5$, and the variance of the observation noise, $V$, ranged from $5,000$ to $50,000$. We plotted the MSE between the true and estimated paths as a function of $V$ (Figure \[fig:limcyc\_mse\]a) and as a function of $\Delta$ (Figure \[fig:limcyc\_mse\]b) for the GPF (solid line) and for the LNA (dashed line). We see that the MSE for the GPF is smaller than that for the LNA. However, a very little difference in the MSE between these two methods is observed.
![ Mean squared error (MSE) between true and estimated paths (a) as a function of noise variance $V$ with $\Delta=0.1$ and (b) as a function of interval $\Delta$ with $V=2,500$ for GPF (solid line) and for LNA (dashed line). Mean squared errors at each point were calculated with 20 repetitions. MSE for GPF is slightly smaller than that for LNA. []{data-label="fig:limcyc_mse"}](fig_limcyc_mse.eps)
Discussion
==========
In this section, we compared between the projection and moment-closure approximations. As seen in the section \[sec:Gaussian\] and \[appendix:normal\_moment\_closure\], the projection approximation onto Gaussian distributions is equivalent to the moment-closure approximation based on the same Gaussian distributions. However, the projection approximation does not always coincide with moment-closure approximations even if these share a common probability distribution. A difference between the two approximation techniques is that while moment-closures yield ODEs for the moments $\mathrm{E}(x^i)$, the projection method produces ODEs for the natural parameter $\theta$ of exponential family distributions, which is related to the expectation of the sufficient statistic $c(x)$ [@Amari01].
We illustrate this difference using a reaction network consisting of single species and at most bimolecular reactions: $$A=(a_1, a_2), \quad h(x)=(k_1x, k_2x(x-1))^T,$$ and using gamma distributions for the base probability distributions. The probability density of a gamma distribution is given by $$p(x,\mu,\kappa) = \frac{\kappa^{\kappa}x^{\kappa-1}}{\mu^{\kappa}\Gamma(\kappa)}e^{-\frac{\kappa x}{\mu}},
\label{eq:gammapdf}$$ whose mean and variance are $\mathrm{E}(x)=\mu$ and $\mathrm{Var}(x)=\mu^2/\kappa$, respectively. Eq. (\[eq:gammapdf\]) can be rewritten in the form (\[eq:expfamily\]) with the natural parameter $\theta = (-\kappa/\mu, \kappa-1)$ and the sufficient statistic $c(x) = (x, \log x)$. The expectations of $c(x)$ is expressed with $(\mu,\kappa)$ as $$\mathrm{E}[c(x)]
=
(\mathrm{E}[x], \mathrm{E}[\log x])
=
(\mu, \varphi(\kappa) - \log\kappa + \log\mu),$$ where $\varphi(\kappa):=\frac{d}{d\kappa}\log\Gamma(\kappa)$ is the digamma function. The Fisher information matrix of the gamma distribution with respect to $(\mu,\kappa)$ is given by $$g(\mu,\kappa) =
\left(\begin{array}{cc}
\kappa/\mu^2 & 0 \\
0 & \dot{\varphi}(\kappa) - \kappa^{-1}
\end{array}\right).$$ Using these quantities, the projection approximation of the reaction network onto the gamma distributions is derived as $$\begin{aligned}
\fl
\frac{d\mu}{dt} = (a_1k_1-a_2k_2)\mu + a_2k_2\mu^2 + \frac{a_2k_2\mu^2}{\kappa},
\label{eq:pa_gamma_mu} \\
\fl
\frac{d\kappa}{dt} = \frac{1}{1-\kappa\dot{\varphi}(\kappa)}
\Bigg\{ a_2k_2\mu + \frac{a_2^2k_2\kappa}{2} + \frac{(a_1^2k_1-a_2^2k_2)\kappa^2}{2\mu(\kappa-1)} \Bigg\}. \label{eq:pa_gamma_kp}\end{aligned}$$ On the other hand, the moment-closure approximation based on the gamma distributions yields a set of ODEs for $\mu$ and $\sigma^2:=\mathrm{Var}(x)$: $$\begin{aligned}
\fl
\frac{d\mu}{dt} = (a_1k_1-a_2k_2)\mu + a_2k_2\mu^2 + a_2k_2\sigma^2,
\label{eq:mc_gamma_mu} \\
\fl
\frac{d\sigma^2}{dt} = 2(a_1k_1-a_2k_2)\sigma^2 + \frac{4a_2k_2(\sigma^2+\mu^2)\sigma^2}{\mu} + (a_1^2k_1-a_2^2k_2)\mu + a_2^2k_2(\sigma^2+\mu^2), \label{eq:mc_gamma_var}\end{aligned}$$ where we used $\mathrm{E}(x^3)=(\mu^2+2\sigma^2)(\mu^2+\sigma^2)/\mu$ to derive Eq. (\[eq:mc\_gamma\_var\]).
Conclusion
==========
This study concerned the filtering problem for stochastic reaction networks. The difficulty in deriving filtering algorithms stems from the analytical intractability of the master equation. We applied the projection method to derive approximate filters.
The projection method provides a flexible framework for approximating reaction networks, as any probability distribution in exponential families fits this method. We demonstrated it on the two reaction networks. In particular, the projection-based filter with quartic polynomials exhibited much better performance than the other methods for the reaction system with bistability (Figure \[fig:bistable\_mse\]), due to its capability to accommodate bimodal distributions.
We note that numerical methods based on particle filtering have been proposed for the inference of reaction networks [@6161329], which would be applicable for the considered molecule numbers. It would be interesting to compare the projection-based filter with these methods in terms of the balance between accuracy and computational time of estimation.
We considered the filtering problem wherein the objective is to estimate the state paths from the observations obtained up to the current time; another related problem is [*smoothing*]{}, which aims to estimate the state paths from the whole observations [@Sarkka13; @Anderson72]. The smoothing equation is not analytically tractable except in the case of linear Gaussian systems, hence approximate methods must be developed along the same line.
It is also an important issue to infer the model parameters [@Golightly06; @Golightly11]. Methods for estimating the reaction rate constants have been developed using the LNA, the system-size expansion and moment-closure approximations [@Milner13; @Frohlich16; @Komorowski09; @Finkenstadt13; @Ruttor09; @Stathopoulos13]. In addition, it is difficult to distinguish between process and measurement noise; the simultaneous estimation of the noise parameters would render the problem substantially more challenging. We leave it for future research.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank Prof. Shinomoto for valuable comments. The author would also like to thank the reviewers for their comments that help improve the manuscript.
Derivation of the Gaussian projection {#appendix:GP}
=====================================
The probability density of the multi-dimensional Gaussian distribution (\[eq:gausspdf\]) is rewritten in the form of (\[eq:expfamily\]) with $$\psi(\theta) = \frac{1}{2} (\mu^TQ^{-1}\mu + n\log 2\pi + \log|Q|),$$ $$c(x) =
\left(\begin{array}{c}
x \\ \mathrm{col}(xx^T)
\end{array}\right),
\label{eq:suffstat}$$ and $$\theta^T
=
(\theta_1,\theta_2,\ldots,\theta_n, \mathrm{col}(\Phi)^T),$$ where $\Phi =(\phi_{ij}) := -\frac{1}{2}Q^{-1}$ and $$\theta_i = -\sum_{j=1}^n(\phi_{ij}+\phi_{ji})\mu_j, \quad i=1,\ldots,n.$$ Here, for a $n\times n$ matrix $B$ we defined the [*column*]{} operation as $$\mathrm{col}(B) =
\left(\begin{array}{c}
B(1) \\ B(2) \\ \vdots \\ B(n)
\end{array}\right),$$ where $B(i)$ is the $i$th column of $B$.
We introduce the following two parameterizations: $$\zeta =
\left( \begin{array}{c}
\mu \\ \mathrm{col}(\Phi)
\end{array} \right),
\quad
\eta =
\left( \begin{array}{c}
\mu \\ \mathrm{col}(Q)
\end{array} \right),$$ and consider the transformations of parameters, $\theta \mapsto \zeta \mapsto \eta$. The Jacobian matrices of these transformations, $J_{\theta}(\zeta):=\partial \theta/\partial \zeta$ and $J_{\zeta}(\eta):=\partial \zeta/\partial \eta$, are given by $$\label{eq:Jacobians_GP}
J_{\theta}(\zeta) =
\left( \begin{array}{cc}
Q^{-1} & M^T \\
\mathbf{0}_{n^2\times n} & \mathbf{1}_{n^2}
\end{array} \right),
\quad
J_{\zeta}(\eta) =
\left( \begin{array}{cc}
\mathbf{1}_n & \mathbf{0}_{n\times n^2} \\
\mathbf{0}_{n^2\times n} & J_{\Phi}(Q)
\end{array} \right),$$ where $J_{\Phi}(Q):= \partial \mathrm{col}(\Phi)/\partial \mathrm{col}(Q)$ is the Jacobian matrix of $\mathrm{col}(\Phi)$, and $M$ is a $n^2\times n$ matrix given by $$M = -\mu \otimes \mathbf{1}_n - \mathbf{1}_n \otimes \mu,$$ where $\otimes$ is the tensor product for two matrices $B=(b_{ij})$ and C defined by $$B \otimes C =
\left( \begin{array}{cccc}
b_{11}C & b_{12}C & \cdots & b_{1n}C \\
b_{21}C & b_{22}C & \cdots & b_{2n}C \\
\vdots & \vdots & \ddots & \vdots \\
b_{n1}C & b_{n2}C & \cdots & b_{nn}C \\
\end{array}\right).$$ By transforming the parameters as $\theta \mapsto \zeta \mapsto \eta$, we can express Eq. (\[eq:expfilter\]) as $$\frac{d\eta}{dt} = g^{-1}(\eta)J_{\zeta}(\eta)^TJ_{\theta}(\zeta)^T\mathrm{E}[\mathcal{L}c],
\label{eq:expfilter_eta}$$ where $g(\eta)$ is the Fisher information matrix of $\eta$, given by $$\label{eq:g_eta}
g(\eta) =
\left( \begin{array}{cc}
Q^{-1} & \mathbf{0}_{n\times n^2} \\
\mathbf{0}_{n^2\times n} & \mathcal{I}(Q)
\end{array} \right).$$ In Eq. (\[eq:g\_eta\]), $\mathcal{I}(Q)$ is the Fisher information matrix of $\mathrm{col}(Q)$, which is expressed by the change of parameter as $$\mathcal{I}(Q) = J_{\Phi}(Q)^T\mathcal{I}(\Phi)J_{\Phi}(Q).$$ Since $\mathrm{col}({\Phi})$ is the natural parameter of the Gaussian distribution (\[eq:gausspdf\]), and $\mathrm{col}({Q})$ is the corresponding expectation parameter, $\mathcal{I}(\Phi)$ is given by the Jacobian matrix $\partial \mathrm{col}(Q)/\partial \mathrm{col}(\Phi) = J_{\Phi}^{-1}(Q)$ [@Amari01]. Thus, we obtain $$\label{eq:I_Q}
\mathcal{I}(Q) = J_{\Phi}(Q)^T.$$ The factor $\mathrm{E}[\mathcal{L}c]$ in Eq. (\[eq:expfilter\_eta\]) is obtained from Eqs. (\[eq:backward\]) and (\[eq:suffstat\]) as $$\label{eq:EL_c}
\fl
\mathrm{E}[\mathcal{L}c] =
\left(\begin{array}{cc}
A\mathrm{E}[h(x)] \\
\mathrm{col}\{ A\mathrm{E}[h(x)x^T] + \mathrm{E}[xh(x)^T]A^T + A\mathrm{E}[H(x)]A^T \}
\end{array}\right).$$ Substituting Eqs. (\[eq:Jacobians\_GP\]), (\[eq:g\_eta\]), (\[eq:I\_Q\]) and (\[eq:EL\_c\]) into Eq. (\[eq:expfilter\_eta\]) leads to $$\begin{aligned}
\fl
\frac{d\eta}{dt}
=
\left(\begin{array}{c}
A\mathrm{E}[h(x)] \\
MA\mathrm{E}[h(x)] +
\mathrm{col}\{ A\mathrm{E}[h(x)x^T] + \mathrm{E}[xh(x)^T]A^T + A\mathrm{E}[H(x)]A^T \}
\end{array}\right).
\label{eq:GP_A1}\end{aligned}$$ Using the following equality, $$\begin{aligned}
MA\mathrm{E}[h(x)]
&=&
-(\mu \otimes \mathbf{1}_{n})A\mathrm{E}[h(x)] - (\mathbf{1}_{n} \otimes \mu) A\mathrm{E}[h(x)] \nonumber\\
&=&
-\mathrm{col}\{A\mathrm{E}[h(x)]\mu^T)\} - \mathrm{col}\{\mu\mathrm{E}[h(x)]^TA^T\}, \end{aligned}$$ the second row of Eq. (\[eq:GP\_A1\]) can be rewritten as $$\begin{aligned}
\fl
\lefteqn{
\mathrm{col}\{
A\mathrm{E}[h(x)(x-\mu)^T] + \mathrm{E}[(x-\mu)h(x)^T]A^T + A\mathrm{E}[H(x)]A^T \}
}\hspace{1cm}\nonumber\\
&=&
\mathrm{col}\{ A\mathrm{E}[J_h(x)]Q + Q\mathrm{E}[J_h(x)]^TA^T + A\mathrm{E}[H(x)]A^T \},
\label{eq:GP_A2}\end{aligned}$$ where the equality follows from the Gaussian assumption. From Eqs. (\[eq:GP\_A1\]) and (\[eq:GP\_A2\]), we obtain the Gaussian projection (\[eq:GP\_mean\])-(\[eq:GP\_cov\]).
Derivation of the normal moment-closure approximation {#appendix:normal_moment_closure}
=====================================================
In this appendix, we derive the normal moment-closure approximation for the stochastic reaction networks [@Milner13; @Goodman53; @Gomez-Uribe07; @Cseke15], and show that it is equivalent to the Gaussian projection approximation.
The mean of $x$ is defined by $\mu = \sum_{x=0}^{\infty}P(x,t)x$, where $\sum_{x=0}^{\infty} := \sum_{x_1=0}^{\infty} \sum_{x_2=0}^{\infty}\cdots \sum_{x_n=0}^{\infty}$. Then, from Eq. (\[eq:stochastic\]) we obtain $$\fl
\frac{d\mu}{dt}
=
\sum_{x=0}^{\infty} \sum_{j=1}^m h_j(x-\Delta x_{\cdot j})P(x-\Delta x_{\cdot j},t) x
- \sum_{x=0}^{\infty} \sum_{j=1}^m h_j(x)P(x,t)x.
\label{eq:mc_mean_1}$$ For each $j=1,\ldots,m$, it follows that $$\begin{aligned}
\lefteqn{
\sum_{x=0}^{\infty}h_j(x-\Delta x_{\cdot j})P(x-\Delta x_{\cdot j},t) x
}\hspace{1cm}\nonumber\\
&=&
\sum_{y=0}^{\infty}h_j(y)P(y,t)(y+\Delta x_{\cdot j}) \nonumber\\
&=&
\sum_{x=0}^{\infty}h_j(x)P(x,t)x + \sum_{x=0}^{\infty}h_j(x)P(x,t)\Delta x_{\cdot j},
\label{eq:mc_mean_2}\end{aligned}$$ where we used the fact that $P(x,t)=0$ and $h(x)=0$ if there exists $i\in\{1,\ldots,n\}$ such that $x_i<0$. Putting Eq. (\[eq:mc\_mean\_2\]) back into Eq. (\[eq:mc\_mean\_1\]) leads to $$\begin{aligned}
\frac{d\mu}{dt}
=
\sum_{j=1}^m \sum_{x=0}^{\infty} h_j(x)P(x,t)\Delta x_{\cdot j}
=
A\mathrm{E}[h(x)].
\label{eq:mc_mean}\end{aligned}$$
Next, we consider the second moment, $\mathrm{E}(xx^T) = \sum_{x=0}^{\infty}P(x,t)xx^T$. From Eq. (\[eq:stochastic\]), the equation for the second moment reads $$\fl
\frac{d\mathrm{E}(xx^T)}{dt}
=
\sum_{x=0}^{\infty} \sum_{j=1}^m h_j(x-\Delta x_{\cdot j})P(x-\Delta x_{\cdot j},t) xx^T
- \sum_{x=0}^{\infty} \sum_{j=1}^m h_j(x)P(x,t)xx^T.
\label{eq:mc_2nd_1}$$ In the same manner as Eq. (\[eq:mc\_mean\_2\]), we obtain $$\begin{aligned}
\fl
\sum_{x=0}^{\infty}h_j(x-\Delta x_{\cdot j})P(x-\Delta x_{\cdot j},t) xx^T
\nonumber\\
=
\sum_{y=0}^{\infty}h_j(y)P(y,t)(y+ \Delta x_{\cdot j})(y+ \Delta x_{\cdot j})^T
\nonumber\\
=
\sum_{x=0}^{\infty} h_j(x)P(x,t)
\{
xx^T + x(\Delta x_{\cdot j})^T + (\Delta x_{\cdot j})x^T + (\Delta x_{\cdot j})(\Delta x_{\cdot j})^T
\}.
\label{eq:mc_2nd_2}\end{aligned}$$ Substituting Eq. (\[eq:mc\_2nd\_2\]) into Eq. (\[eq:mc\_2nd\_1\]) yields $$\begin{aligned}
\fl
\frac{d\mathrm{E}(xx^T)}{dt}
&=
\sum_{j=1}^m \sum_{x=0}^{\infty} h_j(x)P(x,t)
\{
x(\Delta x_{\cdot j})^T + (\Delta x_{\cdot j})x^T + (\Delta x_{\cdot j})(\Delta x_{\cdot j})^T
\} \nonumber\\
\fl
&=
\mathrm{E}[xh(x)^T]A^T + A\mathrm{E}[h(x)x^T] + A\mathrm{E}[H(x)]A^T.
\label{eq:mc_2nd}\end{aligned}$$
Taking the derivative of the covariance of $x$, $Q:=\mathrm{Cov}(x) = \mathrm{E}(xx^T) - \mu\mu^T$, with respect to $t$, and using Eqs. (\[eq:mc\_mean\]) and Eq. (\[eq:mc\_2nd\]) leads to the equation for $Q$ as $$\begin{aligned}
\fl
\frac{dQ}{dt}
&=
\frac{d\mathrm{E}(xx^T)}{dt} - \frac{d\mu}{dt}\mu^T - \mu\frac{d\mu^T}{dt} \nonumber\\
\fl
&=
\mathrm{E}[(x-\mu)h(x)^T]A^T + A\mathrm{E}[h(x)(x-\mu)] + A\mathrm{E}[H(x)]A^T \nonumber\\
\fl
&=
Q\mathrm{E}[J_h(x)]^TA^T + A\mathrm{E}[J_h(x)]Q + A\mathrm{E}[H(x)]A^T,
\label{eq:mc_cov}\end{aligned}$$ where the last equality follows from the Gaussian assumption. Thus, we show that the normal moment-closure approximation, (\[eq:mc\_mean\]) and (\[eq:mc\_cov\]), is equivalent to the Gaussian projection approximation, (\[eq:GP\_mean\]) and (\[eq:GP\_cov\]).
Approximate filter based on the LNA {#appendix:LNA}
===================================
In this appendix, we derive an approximate filter based on the LNA. The LNA, which is the leading-order term in the system size expansion, is given by a Gaussian process, $x(t) \sim \mathcal{N}(\Omega\phi(t) + \sqrt{\Omega}m(t), \Omega \Psi(t))$, where $\phi(t)$, $m(t)$ and $\Psi(t)$ are obtained by solving the following ODEs: $$\begin{aligned}
\frac{d\phi}{dt} &=& A\tilde{h}(\phi), \label{eq:LNA1} \\
\frac{dm}{dt} &=& AJ_{\tilde{h}}(\phi)m, \label{eq:LNA2} \\
\frac{d\Psi}{dt} &=& \Psi J_{\tilde{h}}(\phi)^TA^T + AJ_{\tilde{h}}(\phi)\Psi + A\tilde{H}(\phi)A^T, \label{eq:LNA3}\end{aligned}$$ with a set of initial conditions, $\phi_0$, $m_0$ and $\Psi_0$ [@Kampen92]. Suppose that in solving Eq. (\[eq:LNA1\])-(\[eq:LNA3\]), the initial distribution of $x$ is given by $\mathcal{N}(\mu_0^*,Q_0^*)$. Then, we may take an arbitrary $\phi_0$, and set $m_0 = \sqrt{\Omega}(\mu_0^*/\Omega-\phi_0)$ and $\Psi_0=Q_0^*/\Omega$. The arbitrariness of initial condition can be resolved by choosing $\phi_0 = \mu_0^*/\Omega$, which makes a relative difference of order $\Omega^{-1/2}$ in $x(t)$. This initial condition leads to $m(t)=0$ for all $t$ as $m_0=0$, and thus $m(t)$ can be omitted from the LNA.
We can construct an approximate filter by using the above LNA for the prediction step [@Fearnhead14]. Since the approximate state $x(t_i)$ and the observations $y_i$ follow Gaussian distributions, the correction step can be implemented with the standard Kalman recursions (\[eq:post\_mean\])-(\[eq:post\_gain\]). To summarize, the filtering algorithm consists of the following two steps:
1. (Prediction step) Solve the ODEs: $$\begin{aligned}
\fl
\frac{d\mu(t)}{dt} = Ah(\mu(t)), \label{eq:LNA_mean} \\
\fl
\frac{dQ(t)}{dt} = Q(t)J_h(\mu(t))^TA^T + AJ_h(\mu(t))Q(t) +
AH(\mu(t))A^T. \label{eq:LNA_cov}\end{aligned}$$ from time $t_{i-1}$ to $t_i$ with initial conditions $\mu^+(t_{i-1})$ and $Q^+(t_i)$ to obtain $\mu(t_i)$ and $Q(t_i)$.
2. (Collection step) Compute the posterior mean $\mu^+(t_i)$ and covariance matrix $Q^+(t_i)$ at time $t_i$ by Eqs. (\[eq:post\_mean\])-(\[eq:post\_gain\]).
Eqs. (\[eq:LNA\_mean\]) and (\[eq:LNA\_cov\]) are obtained by rescaling Eqs. (\[eq:LNA1\]) and (\[eq:LNA3\]) with $\mu(t)=\Omega\phi(t)$ and $Q(t) = \Omega\Psi(t)$. Notice the difference between the Gaussian projection (\[eq:GP\_mean\])-(\[eq:GP\_cov\]) and LNA (\[eq:LNA\_mean\])-(\[eq:LNA\_cov\]). In the Gaussian projection, the expectation of $x(t)$ is taken outside of $h(x(t))$ and $J_h(x(t))$, while it is taken inside of these functions in the LNA. Hence, these two approximations are equivalent for first-order reactions; they differ for second- and higher-order reactions.
References {#references .unnumbered}
==========
|
[**Multivariate normal approximation using Stein’s method and Malliavin calculus**]{}\
\
Ivan Nourdin[^1], Giovanni Peccati[^2] and Anthony Réveillac[^3]\
[*Université Paris VI, Université Paris Ouest and Humboldt-Universität zu Berlin*]{}\
\
\
[**Abstract:** We combine Stein’s method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional Brownian motion.\
]{}
[*Key words:* Breuer-Major CLT, fractional Brownian motion, Gaussian processes, Malliavin calculus, Normal approximation, Stein’s method, Wasserstein distance. ]{}
[**Résumé:** Nous expliquons comment combiner la méthode de Stein avec les outils du calcul de Malliavin pour majorer, de manière explicite, la distance de Wasserstein entre une fonctionnelle d’un champs gaussien donnée et son approximation normale multidimensionnelle. Notre travail généralise et affine des résultats antérieurs prouvés par Peccati et Tudor (2005), Nualart et Ortiz-Latorre (2007), Peccati (2007) et Nourdin et Peccati (2007b, 2008). Entre autres exemples, nous associons des bornes à la version fonctionnelle du théorème de la limite centrale de Breuer-Major dans le cas du mouvement brownien fractionnaire.\
]{}
[*Mots clés:* théorème de la limite centrale de Breuer et Major, mouvement brownien fractionnaire, processus gaussiens, calcul de Malliavin, approximation normale, méthode de Stein, distance de Wasserstein. ]{}
[*This revised version*: November 2008]{}
Introduction
============
Let $Z\sim\mathscr{N}(0,1)$ be a standard Gaussian random variable on some probability space $(\Omega,\mathcal{F},P)$, and let $F$ be a real-valued functional of an infinite-dimensional Gaussian field. In the papers [@NourdinPeccati1; @NourdinPeccati2] it is shown that one can combine Stein’s method (see e.g. [@Chen_Shao_sur], [@Stein1] or [@Stein2]) with Malliavin calculus (see e.g. [@Nualart]), in order to deduce explicit (and, sometimes, optimal) bounds for quantities of the type $d(F,Z)$, where $d$ stands for some distance between the law of $F$ and the law of $Z$ (e.g., $d$ can be the Kolmogorov or the Wasserstein distance). The aim of this paper is to extend the results of [@NourdinPeccati1; @NourdinPeccati2] to the framework of the [*multidimensional*]{} Gaussian approximation in the Wasserstein distance. Once again, our techniques hinge upon the use of infinite-dimensional operators on Gaussian spaces (like the *divergence operator* or the *Ornstein-Uhlenbeck generator*) and upon an appropriate multidimensional version of Stein’s method (in a form close to Chatterjee and Meckes [@ChatterjeeMeckes], but see also Reinert and Röllin [@ReinertRollin]). As a result, we will obtain explicit bounds, both in terms of Malliavin derivatives and contraction operators, thus providing a substantial refinement of the main findings by Nualart and Ortiz-Latorre [@NualartOrtizLatorre] and Peccati and Tudor [@PeccatiTudor]. Note that an important part of our computations (see e.g. Lemma \[lm-control\]) are directly inspired by those contained in [@NualartOrtizLatorre]: we shall indeed stress that this last reference contains a fundamental methodological breakthrough, showing that one can deal with (possibly multidimensional) weak convergence on a Gaussian space, by means of Malliavin-type operators and “characterizing” differential equations. See [@NouPec07] for an application of these techniques to non-central limit theorems. Incidentally, observe that the paper [@Pecc], which is mainly based on martingale-type techniques, also uses distances between probability measures (such as the Prokhorov distance) to deal with multidimensional Gaussian approximations on Wiener space, but without giving explicit bounds.
The rationale behind Stein’s method is better understood in dimension one. In this framework, the starting point is the following crucial result, proved e.g. in [@Stein1].
\[lemma:SteinIBP1\] A random variable $Y$ is such that $Y
\overset{\rm Law}{=}Z\sim \mathscr{N}(0,1)$ if and only if, for every continuous and piecewise continuously differentiable function $f:\mathbb{R}\to\mathbb{R}$ such that $E\big|f'(Z)\big|<\infty$, one has $$\label{urStein}
E[f'(Y)-Yf(Y)]=0.$$
The fact that a random variable $Y$ satisfying (\[urStein\]) is necessarily Gaussian can be proved by several routes: for instance, by taking $f$ to be a complex exponential, one can show that the characteristic function of $Y$, say $\psi(t)$, is necessarily a solution to the differential equation $\psi'(t)+t\psi(t)=0$, and therefore $\psi(t)=\exp(-t^2/2)$; alternatively, one can set $f(x)=x^n$, $n=1,2,...$, and observe that (\[urStein\]) implies that, for every $n$, one must have $E(Y^n)=E(Z^n)$, where $Z\sim \mathscr{N}(0,1)$ (note that the law of $Z$ is determined by its moments).
Heuristically, Lemma \[lemma:SteinIBP1\] suggests that the distance $d(Y,Z)$, between the law of a random variable $Y$ and that of $Z\sim \mathscr{N}(0,1)$, must be “small” whenever $ E[f'(Y)-Yf(Y)] \simeq 0, $ for a sufficiently large class of functions $f$. In the seminal works [@Stein1; @Stein2], Stein proved that this somewhat imprecise argument can be made rigorous by means of the use of differential equations. To see this, for a given function $g:\real\to\real$, define the *Stein equation* associated with $g$ as $$\label{eq:SteinEquation1}
g(x)-E[g(Z)] = h'(x)-x h(x), \quad \forall x\in \real,$$ (we recall that $Z\sim\mathscr{N}(0,1)$). A solution to (\[eq:SteinEquation1\]) is a function $h$ which is Lebesgue-almost everywhere differentiable, and such that there exists a version of $h'$ satisfying (\[eq:SteinEquation1\]) for every $x\in\real$. If one assumes that $g\in {\rm Lip}(1)$ (that is, if $\|g\|_{Lip}\leq 1$, where $\|\cdot\|_{Lip}$ stands for the usual Lipschitz seminorm), then a standard result (see e.g. [@Stein2]) yields that (\[eq:SteinEquation1\]) admits a solution $h$ such that $\|h'\|_\infty \leq 1$ and $\|h''\|_\infty
\leq 2$. Now recall that the *Wasserstein distance* between the laws of two real-valued random variables $Y$ and $X$ is defined as $$d_{\rm W}(Y,X) = \sup_{g\in {\rm Lip}(1)}
\left\vert E[g(Y)]-E[g(X)]\right\vert,$$ and introduce the notation $\mathscr{F}_{\rm W} = \{ f:
\|f'\|_\infty \leq 1, \, \|f''\|_\infty \leq 2\}$. By taking expectations on the two sides of (\[eq:SteinEquation1\]), one obtains finally that, for $Z\sim\mathscr{N}(0,1)$ and for a generic random variable $Y$, $$\label{Stbound}
d_{\rm W}(Y,Z) \leq \sup_{f\in \mathscr{F}_{\rm W}} \left\vert
E[f'(Y)-Yf(Y)]\right\vert,$$ thus giving a precise meaning to the heuristic argument sketched above (note that an analogous conclusion can be obtained for other distances, such as the total variation distance or the Kolmogorov distance – see e.g. [@Chen_Shao_sur] for a discussion of this point). We stress that the topology induced by $d_{\rm W}$, on probability measures on $\real$, is stronger than the topology induced by weak convergence.
The starting point of [@NourdinPeccati1; @NourdinPeccati2] is that a relation such as (\[Stbound\]) can be very effectively combined with Malliavin calculus, whenever $Y$ is a centered regular functional of some infinite dimensional Gaussian field. To see this, denote by $DY$ the Malliavin derivative of $Y$ (observe that $DY$ is a random element with values in some adequate Hilbert space $\EuFrak{H}$), and write $L$ to indicate the (infinite-dimensional) Ornstein-Uhlenbeck generator (see Section \[section:Malliavin\] below for precise definitions). One crucial relation proved in [@NourdinPeccati1], and then further exploited in [@NourdinPeccati2], is the upper bound $$\label{GioIvan}
d_{\rm W}(Y,Z) \leq E|1-\langle DY,
-DL^{-1}Y\rangle_\EuFrak{H}|.$$
As shown in [@NourdinPeccati1], when specialized to the case of $Y$ being equal to a multiple Wiener-Itô integral, relation (\[GioIvan\]) yields bounds that are intimately related with the CLTs proved in [@NualartOrtizLatorre] and [@NP]. See [@NourdinPeccati2] for a characterization of the optimality of these bounds; see again [@NourdinPeccati1] for extensions to non-Gaussian approximations and for applications to the Breuer-Major CLT (stated and proved in [@BM]) for functionals of a fractional Brownian motion.
The principal contribution of the present paper (see e.g. the statement of Theorem \[theo:majDist\] below) consists in showing that a relation similar to (\[GioIvan\]) continues to hold when $Z$ is replaced by a $d$-dimensional ($d\geq2$) Gaussian vector $F=(F_1,...,F_d)$ of smooth functionals of a Gaussian field, and $d_{\rm W}$ is the Wasserstein distance between probability laws on $\real^d$ (see Definition \[DefWass\] below). Our results apply to Gaussian approximations by means of Gaussian vectors with arbitrary positive definite covariance matrices. The proofs rely on a multidimensional version of the Stein equation (\[eq:SteinEquation1\]), that we combine with standard integration by parts formulae on an infinite-dimensional Gaussian space. Our approach bears some connections with the paper by Hsu [@Hsu], where the author proves an hybrid Stein/semimartingale characterization of Brownian motions on manifolds, via Malliavin-type operators.
The paper is organized as follows. In Section \[section:Malliavin\] we provide some preliminaries on Malliavin calculus. Section \[S : Main\] contains our main results, concerning Gaussian approximations by means of vectors of Gaussian random variables with positive definite covariance matrices. Finally, Section \[section:applications\] deals with two applications: (i) to a functional version of the Breuer-Major CLT (see [@BM]), and (ii) to Gaussian approximations of functionals of finite normal vectors, providing a generalization of a technical result proved by Chatterjee in [@Chatterjee_ptrf].
Preliminaries and notation {#section:Malliavin}
==========================
In this section, we recall some basic elements of Malliavin calculus for Gaussian processes. The reader is referred to [@Nualart] for a complete discussion of this subject. Let $X=\{X(h),\; h\in \EuFrak{H}\}$ be an *isonormal Gaussian process* on a probability space $(\Omega,\mathcal{F},P)$. This means that $X$ is a centered Gaussian family indexed by the elements of an Hilbert space $\EuFrak{H}$, such that, for every pair $h,g\in\EuFrak{H}$ one has that $ E[X(h) X(g)]=\langle h, g\rangle_{\EuFrak{H}}.$
We let $L^2(X)$ be shorthand for the space $L^2(\Omega,\sigma(X),P)$. It is well known that every random variable $F \in L^2(X)$ admits the chaotic expansion $ F=E(F)+\sum_{n=1}^\infty I_n(f_n) $ where the deterministic kernels $f_n$, $n\geq 1$, belong to $\EuFrak{H}^{\odot n}$ and the convergence of the series holds in $L^2(X)$. One sometimes uses the notation $I_0(f_0)=E[F]$. In the particular case where $\EuFrak{H}:=L^2(T,\mathcal{A},\mu)$, with $(T,\mathcal{A})$ a measurable space and $\mu$ is a $\sigma$-finite measure without atoms, the random variable $I_n(f_n)$ coincides with the *multiple Wiener-Itô integral* (of order $n$) of $f_n$ with respect to $X$ (see [@Nualart Section 1.1.2.]).
Let $f\in\EuFrak{H}^{\odot p}$, $g\in\EuFrak{H}^{\odot q}$ and $0 \leq r \leq p\wedge q$. We define the $r$th *contraction* $f\otimes_r g$ of $f$ and $g$ as the element of $\EuFrak{H}^{\otimes (p+q-2r)}$ given by $$f\otimes_r g :=\sum_{i_1,\ldots,i_r=1}^\infty \langle f, e_{i_1}\otimes
\ldots\otimes e_{i_r} \rangle_{\EuFrak{H}^{\otimes r}} \otimes
\langle g, e_{i_1} \otimes \ldots\otimes e_{i_r}
\rangle_{\EuFrak{H}^{\otimes r}},$$ where $\{e_k,\; k\geq 1\}$ is a complete orthonormal system in $\EuFrak{H}$. Note that $f\otimes_0 g=f\otimes g$; also, if $p=q$, then $f\otimes_p g=\langle f, g \rangle_{\EuFrak{H}^{\otimes
p}}$. Note that, in general, $f\otimes_r g$ is not a symmetric element of $\EuFrak{H}^{\otimes (p+q-2r)}$; the canonical symmetrization of $f\otimes_r g$ is denoted by $f
\widetilde{\otimes}_r g$. We recall the product formula for multiple stochastic integrals:
$$I_p(f)I_q(g)=\sum_{r=0}^{p\wedge q}
r! \binom{p}{r}\binom{p}{q}
I_{p+q-2r}(f\widetilde{\otimes}_r g).$$
Now, let $\mathscr{S}$ be the set of cylindrical functionals $F$ of the form $$\label{eq:cylindrical}
F=\varphi(X(h_1),\ldots,X(h_n)),$$ where $n\geq 1$, $h_i \in \EuFrak{H}$ and the function $\varphi\in
\mathscr{C}^{\infty}(\real^n)$ is such that its partial derivatives have polynomial growth. The *Malliavin derivative* $DF$ of a functional $F$ of the form (\[eq:cylindrical\]) is the square integrable $\EuFrak{H}$-valued random variable defined as $$\label{baldassarre}
DF=\sum_{i=1}^n \partial_i \varphi(X(h_1),\ldots,X(h_n)) h_i,$$ where $\partial_i \varphi$ denotes the $i$th partial derivative of $\varphi$. In particular, one has that $DX(h)=h$ for every $h$ in $\EuFrak{H}$. By iteration, one can define the $m$th derivative $D^m F$ of $F\in\mathscr{S}$, which is an element of $L^2(\Omega;\EuFrak{H}^{\odot m})$, for $m\geq 2$. As usual $\mathbb{D}^{m,2}$ denotes the closure of $\mathscr{S}$ with respect to the norm $\|\cdot\|_{m,2}$ defined by the relation $ \|F\|_{m,2}^2=E[F^2] +\sum_{i=1}^m E[\|D^iF\|^2_{\EuFrak{H}^{\otimes i}}].$
Note that every finite sum of Wiener-Itô integrals always belongs to $\mathbb{D}^{m,2}$ ($\forall m\geq 1$). The Malliavin derivative $D$ satisfies the following *chain rule formula*: if $\varphi:\real^n\to \real$ is in $\mathscr{C}_b^1$ (defined as the set of continuously differentiable functions with bounded partial derivatives) and if $(F_1,\ldots,F_n)$ is a random vector such that each component belongs to $\mathbb{D}^{1,2}$, then $\varphi(F_1,\ldots,F_n)$ is itself an element of $\mathbb{D}^{1,2}$, and moreover $$\label{eq:ChainRule}
D\varphi(F_1,\ldots,F_n)=\sum_{i=1}^n \partial_i\varphi(F_1,\ldots,F_n) DF_i.$$
The *divergence operator* $\delta$ is defined as the dual operator of $D$. Precisely, a random element $u$ of $L^2(\Omega;\EuFrak{H})$ belongs to the domain of $\delta$ (denoted by ${\rm Dom} \delta$) if there exists a constant $c_u$ satisfying $\left\vert E[\langle DF, u \rangle_{\EuFrak{H}}]\right\vert \leq c_u \|F\|_{L^2(\Omega)}$ for every $F \in \mathscr{S};$ in this case, the divergence of $u$, written $\delta(u)$, is defined by the following duality property: $$\label{eq:MalliavinIBP}
E[F \delta(u)]=E[\langle DF, u
\rangle_{\EuFrak{H}}], \quad \forall F \in \mathbb{D}^{1,2}.$$ The crucial relation (\[eq:MalliavinIBP\]) is customarily called the (Malliavin) *integration by parts formula*.
In what follows, we shall denote by $T = \{T_t: t\geq 0\}$ the *Ornstein-Uhlenbeck semigroup*. We recall that, for every $t\geq 0$ and every $F\in L^2(X)$, $$\label{OrnstU}
T_t(F) = \sum_{n=0}^\infty e^{-nt}J_n(F),$$ where, for every $n\geq0$ and for the rest of the paper, the symbol $J_n$ denotes the projection operator onto the $n$th Wiener chaos, that is onto the closed linear subspace of $L^2(X)$ generated by the random variables of the form $H_n(X(h))$ with $h\in\EuFrak{H}$ such that $\|h\|_\EuFrak{H}=1$, and $H_n$ the $n$th Hermite polynomial defined by (\[her-pol\]). Note that $T$ is indeed the semigroup associated with an infinite-dimensional stationary Gaussian process with values in $\real^{\HH}$, having the law of $X$ as an invariant distribution (see e.g. [@Nualart Section 1.4] for a more detailed discussion of the Ornstein-Uhlenbeck semigroup in the context of Malliavin calculus; see Barbour [@Barbour] for a version of Stein’s method involving Ornstein-Uhlenbeck semigroups on infinite-dimensional spaces; see Götze [@Goetze] for a version of Stein’s method based on multi-dimensional Ornstein-Uhlenbeck semigroups). The *infinitesimal generator of the Ornstein-Uhlenbeck semigroup* is noted $L$. A square integrable random variable $F$ is in the domain of $L$ (noted ${\Dom}L$) if $F$ belongs to the domain of $\delta D$ (that is, if $F$ is in $\mathbb{D}^{1,2}$ and $DF \in {\rm Dom}\delta$) and, in this case, $L F =- \delta D F .$ One can prove that $LF$ is such that $ LF=- \sum_{n=0}^\infty n J_n(F).$ As an example, if $F=I_q(f_q)$, with $f_q \in \EuFrak{H}^{\odot
q}$, then $LF=-q F$. Note that, for every $F\in {\Dom}L$, one has $E(LF)=0$. The inverse $L^{-1}$ of the operator $L$ acts on zero-mean random variables $F\in L^2(X)$ as $ L^{-1}F=-\sum_{n=1}^\infty \frac1n J_n(F).$ In particular, for every $q\geq 1$ and every $F=I_q(f_q)$ with $f_q \in \EuFrak{H}^{\odot q}$, one has that $L^{-1}F=-\frac1q F$.
We conclude this section by recalling two important characterizations of the Ornstein-Uhlenbeck semigroup and its generator.
i\) Let $F$ be an element of $L^2(X)$, so that $F$ can be represented as an application from $\real ^ \HH$ into $\real$. Then, an alternative representation (due to Mehler) of the action of the Ornstein-Uhlenbeck semigroup $T$ (as defined in (\[OrnstU\])) on $F$, is the following: $$\label{MehlerF}
T_t(F) =E[F(e^{-t} a + \sqrt{1-e^{-2t}} X)] \mid _{a = X}, \quad
t\geq 0,$$ where $a$ designs a generic element of $\real ^ \HH$. See Nualart [@Nualart Section 1.4.1] for more details on this and other characterizations of $T$.
ii\) Let $F\in L^2(X)$ have the form $
F = f(X(h_1),...,X(h_d)),
$ where $f \in \mathscr{C}^2(\real ^d)$ has bounded first and second derivatives, and $h_i \in \HH$, $i =
1,...,d$. Then, $$\begin{aligned}
\label{DiffELLE}
LF& =& \sum_{i,j=1}^d \frac{\partial^2 f}{\partial x_i \partial
x_j}(X(h_1),...,X(h_d))\langle h_i , h_j \rangle_\HH
-\sum_{i=1}^d \frac{\partial f}{\partial
x_i}(X(h_1),...,X(h_d))X(h_i). \quad\quad\end{aligned}$$ See Propositions 1.4.4 and 1.4.5 in [@Nualart] for a proof and some generalizations of (\[DiffELLE\]).
Stein’s method and Gaussian vectors {#S : Main}
===================================
We start by giving a definition of the Wasserstein distance, as well as by introducing some useful norms over classes of real-valued matrices.
\[DefWass\]
- The [*Wasserstein distance*]{} between the laws of two $\real^d$-valued random vectors $X$ and $Y$, noted $d_{\rm W}(X,Y)$, is given by $$d_{\rm W}(X,Y):=\sup_{g\in\mathscr{H}; \|g\|_{Lip}\leq 1} \big\vert E[g(X)]-E[g(Y)] \big\vert,$$ where $\mathscr{H}$ indicates the class of Lipschitz functions, that is, the collection of all functions $g:\real^d\to\real$ such that $\displaystyle{\|g\|_{Lip}:=\sup_{x\neq y}\frac{\vert
g(x)-g(y) \vert}{\| x-y \|_{\real^d}}<\infty}$ (with $\|\cdot\|_{\real^d}$ the usual Euclidian norm on $\real^d$).
- The [*Hilbert-Schmidt inner product*]{} and the [*Hilbert-Schmidt norm*]{} on the class of $d\times d$ real matrices, denoted respectively by $\langle \cdot, \cdot \rangle_{H.S.}$ and $\|\cdot\|_{H.S.}$, are defined as follows: for every pair of matrices $A$ and $B$, $ \langle A, B \rangle_{H.S.}:={\rm Tr}(A B^{T})$ and $\|A\|_{H.S.}:=\sqrt{\langle A, A \rangle_{H.S.}}.$
- The [*operator norm*]{} of a $ d\times d$ matrix $A$ over $\real$ is given by $ \|A\|_{op} :=\sup_{\|x\|_{\real^d}=1}\|A
x\|_{\real^d}.$
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- For every $d\geq 1$ the topology induced by $d_{\rm
W}$, on the class of all probability measures on $\real^d$, is strictly stronger than the topology induced by weak convergence (see e.g. Dudley [@Dudley; @book Chapter 11]).
- The reason why we focus on the Wasserstein distance is nested in the statement of the forthcoming Lemma \[lemma:SteinIBP2\]. Indeed, according to relation (\[SteinEstimate\]), in order to control the second derivatives of the solution of the Stein equation (\[eq:SteinEquation2\]) associated with $g$, one must use the fact that $g$ is Lipschitz.
- According to the notation introduced in Definition \[DefWass\](ii), relation (\[DiffELLE\]) can be rewritten as $$\label{compactDIFF}
LF = \langle C , {\rm Hess} f(Z) \rangle_{H.S.} - \langle Z ,
\nabla f (Z) \rangle_{\real^d},$$ where $Z =(X(h_1),...,X(h_d))$, and $C = \{C(i,j) : i,j=1,...,d\}$ is the $d \times d$ covariance matrix such that $C(i,j)
=E(X(h_i)X(h_j))=\langle h_i , h_j \rangle_\HH$.
Given a $d\times d$ positive definite symmetric matrix $C$, we use the notation $\mathscr{N}_d(0,C)$ to indicate the law of a $d$-dimensional Gaussian vector with zero mean and covariance $C$. The following result, which is basically known (see e.g. [@ChatterjeeMeckes] or [@ReinertRollin]), is the $d$-dimensional counterpart of Stein’s Lemma \[lemma:SteinIBP1\]. In what follows, we provide a new proof which is almost exclusively based on the use of Malliavin operators.
\[lemma:SteinIBP2\] Fix an integer $d\geq 2$ and let $C=\{C(i,j) : i,j=1,...,d\}$ be a $d \times d$ positive definite symmetric real matrix.
- Let $Y$ be a random variable with values in $\real^d$. Then $Y\sim\mathscr{N}_d(0,C)$ if and only if, for every twice differentiable function $f:\real^d\to\real$ such that $ E\vert
\langle C , {\rm Hess} f(Y) \rangle_{H.S.}\vert+E\vert \langle Y, \nabla f(Y)
\rangle_{\real^d}\vert <\infty$, it holds that $$\label{SteinCharD}
E[
\langle Y, \nabla f(Y)
\rangle_{\real^d}
-\langle C,
{\rm Hess} f(Y) \rangle_{H.S.}
]=0.$$
- Consider a Gaussian random vector $Z\sim \mathscr{N}_d(0,C)$. Let $g:\real^d\to\real$ belong to $\mathscr{C}^2(\real^d)$ with first and second bounded derivatives. Then, the function $U_0(g)$ defined by $$U_0g(x):=\int_0^1 \frac{1}{2 t}
E[g(\sqrt{t}x+\sqrt{1-t}Z)-g(Z)] dt$$ is a solution to the following differential equation (with unknown function $f$): $$\label{eq:SteinEquation2}
g(x)-E[g(Z)]=
\langle x, \nabla f(x) \rangle_{\real^d}
-\langle C , {\rm Hess} f(x)
\rangle_{H.S.}
, \quad
x\in\real^d.$$ Moreover, one has that $$\label{SteinEstimate}
\sup_{x\in \real^d} \| {\rm Hess}\,U_0g(x) \|_{H.S.}\leq \|
C^{-1}\|_{op} \,\, \| C\|_{op}^{1/2} \,\, \|g\|_{Lip}.$$
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- If $C = \sigma^2 \textbf{I}_d$ for some $\sigma >0$ (that is, if $Z$ is composed of i.i.d. centered Gaussian random variables with common variance equal to $\sigma^2$), then $$\| C^{-1}\|_{op} \,\, \| C\|_{op}^{1/2} = \| \sigma^{-2}
\textbf{I}_d\|_{op} \,\, \| \sigma^2 \textbf{I}_d \|_{op}^{1/2} =
\sigma^{-1}.$$
- Unlike formulae (\[urStein\]) and (\[eq:SteinEquation1\]) (associated with one-dimensional Gaussian approximations) the relation (\[SteinCharD\]) and the Stein equation (\[eq:SteinEquation2\]) involve second-order differential operators. A discussion of this fact is detailed e.g. in [@ChatterjeeMeckes Theorem 4].
[*Proof of Lemma \[lemma:SteinIBP2\].*]{} We start by proving Point (ii). First observe that, without loss of generality, we can suppose that $Z=(Z_1,...,Z_d):=(X(h_1),...X(h_d))$, where $X$ is an isonormal Gaussian process over $\HH=\real^d$, the kernels $h_i $ belong to $\HH$ ($i=1,...,d$), and $\langle h_i ,
h_j \rangle_\HH = E(X(h_i)X(h_j))= E(Z_i Z_j) = C(i,j)$. By using the change of variable $2u =-\log t$, one can rewrite $U_0 g(x)$ as follows $$U_0 g(x) = \int_0^\infty \{E[g(e^{-u}
x+\sqrt{1-e^{-2u}}Z)]-E[g(Z)]\} du.$$ Now define $\widetilde{g}(Z) := g(Z)-E[g(Z)]$, and observe that $\widetilde{g}(Z)$ is by assumption a centered element of ${\rm
L}^2(X)$. For $q\geq 0$, denote by $J_q(\widetilde{g}(Z))$ the projection of $\widetilde{g}(Z)$ on the $q$th Wiener chaos, so that $J_0(\widetilde{g}(Z))=0$. According to Mehler’s formula (\[MehlerF\]), $$E[g(e^{-u} x+\sqrt{1-e^{-2u}}Z)]|_{x = Z}-E[g(Z)] =
E[\widetilde{g}(e^{-u} x+\sqrt{1-e^{-2u}}Z)]|_{x = Z} = T_u
\widetilde{g}(Z),$$ where $x$ denotes a generic element of $\real
^d$. In view of (\[OrnstU\]), it follows that $$U_0 g(Z) = \int_0^\infty T_u \widetilde{g}(Z) du = \int_0^\infty
\sum_{q\geq 1} e^{-qu} J_q(\widetilde{g}(Z)) du = \sum_{q\geq 1}
\frac1q J_q(\widetilde{g}(Z)) = -L^{-1}\widetilde{g}(Z).$$ Since $g$ belongs to $\mathscr{C}^2(\real^d)$ with bounded first and second derivatives, it is easily seen that the same holds for $U_0 g$. By exploiting the differential representation (\[compactDIFF\]), one deduces that $$\langle Z ,
\nabla U_0 g (Z) \rangle_{\real^d}
-
\langle C , {\rm Hess} U_0 g(Z) \rangle_{H.S.}
= -L U_0 g(Z) =
LL^{-1}\widetilde{g}(Z) = g(Z) - E[g(Z)].$$ Since the matrix $C$ is positive definite, we infer that the support of the law of $Z$ coincides with $\real ^d$, and therefore (e.g. by a continuity argument) we obtain that $$\langle x ,
\nabla U_0 g (x) \rangle_{\real^d}
-
\langle C , {\rm Hess}\, U_0 g(x) \rangle_{H.S.}
= g(x) - E[g(Z)],$$ for every $x\in\real ^d$. This yields that the function $U_0g$ solves the Stein’s equation (\[eq:SteinEquation2\]).
To prove the estimate (\[SteinEstimate\]), we first recall that there exists a unique non-singular *symmetric* matrix $A$ such that $A^2 = C$, and that one has that $A^{-1} Z
\sim \mathscr{N}_d(0,\mathbf{I}_d)$. Now write $U_0g(x) =
h(A^{-1}x)$, where $$h(x)= \int_0^1 \frac{1}{2 t}
E[g_A(\sqrt{t}x+\sqrt{1-t}A^{-1}Z)-g_A(A^{-1}Z)] dt,$$ and $g_A(x)=g(Ax)$. Note that, since $A^{-1} Z \sim
\mathscr{N}_d(0,\mathbf{I}_d)$, the function $h$ solves the Stein’s equation $
\langle x , \nabla h (x) \rangle_{\real^d}-\Delta h(x) = g_A(x)
- E[g_A(Y)],
$ where $Y\sim \mathscr{N}_d(0,\mathbf{I}_d)$. We can now use the same arguments as in the proof of Lemma 3 in [@ChatterjeeMeckes] to deduce that $$\label{MeckessMeckess}
\sup_{x\in \real^d} \| {\rm Hess}\,h(x) \|_{H.S.}\leq
\|g_A\|_{Lip}\leq \|A\|_{op}\|g\|_{Lip}.$$ On the other hand, by noting $h_{A^{-1}}(x) = h(A^{-1}x)$, one obtains by standard computations (recall that $A$ is symmetric) that $
{\rm Hess}\,U_0g(x)={\rm Hess}\,h_{A^{-1}}(x) = A^{-1}{\rm
Hess}\,h(A^{-1}x) A^{-1},
$ yielding $$\begin{aligned}
\sup_{x\in \real^d} \| {\rm Hess}\,U_0g(x) \|_{H.S.}&=& \sup_{x\in
\real^d} \|A^{-1} {\rm Hess}\,h(A^{-1}x) A^{-1} \|_{H.S.} \notag \\
&=& \sup_{x\in
\real^d} \|A^{-1} {\rm Hess}\,h(x) A^{-1} \|_{H.S.} \notag \\
& \leq & \|A^{-1}\|_{op}^2 \sup_{x\in \real^d} \| {\rm
Hess}\,h(x) \|_{H.S.} \label{Sing1}\\
& \leq &\|A^{-1}\|_{op}^2 \,\, \| A \|_{op} \,\, \|g\|_{Lip} \label{Sing2} \\
& \leq &\| C^{-1}\|_{op} \,\, \| C\|_{op}^{1/2} \,\, \|g\|_{Lip}.
\label{Sing3}\end{aligned}$$ The chain of inequalities appearing in formulae (\[Sing1\])–(\[Sing3\]) are mainly a consequence of the usual properties of the Hilbert-Schmidt and operator norms. Indeed, to prove inequality (\[Sing1\]) we used the relations $$\begin{aligned}
\|A^{-1} {\rm Hess}\,h(x) A^{-1}
\|_{H.S.}& \leq &
\|A^{-1}\|_{op} \,\, \| {\rm Hess}\,h(x) A^{-1}\|_{H.S.} \\
&\leq & \|A^{-1}\|_{op}\,\, \| {\rm Hess}\,h(x)
\|_{H.S.}\,\,\|A^{-1}\|_{op}\,\,;\end{aligned}$$ relation (\[Sing2\]) is a consequence of (\[MeckessMeckess\]); finally, to show the inequality (\[Sing3\]), one uses the fact that $$\begin{aligned}
\|A^{-1}\|_{op} \leq \sqrt{\|A^{-1} A^{-1}\|_{op}}=
\sqrt{\|C^{-1}\|_{op}} \quad\mbox{and}\quad
\|A\|_{op} \leq \sqrt{\|A A\|_{op}}= \sqrt{\|C\|_{op}} \,\,.\end{aligned}$$ We are now left with the proof of Point (i) in the statement. The fact that a vector $Y\sim\mathscr{N}_d(0,C)$ necessarily verifies (\[SteinCharD\]) can be proved by standard integration by parts. On the other hand, suppose that $Y$ verifies (\[SteinCharD\]). Then, according to Point (ii), for every $g\in
\mathscr{C}^2(\real^d)$ with bounded first and second derivatives, $$E(g(Y)) - E(g(Z)) =E(\langle Y, \nabla U_0g(Y) \rangle_{\real^d} - \langle C , {\rm Hess}\, U_0g(Y)
\rangle_{H.S.}) = 0,$$ where $Z\sim \mathscr{N}_d(0,C)$. Since the collection of all such functions $g$ generates the Borel $\sigma$-field on $\real^d$, this implies that $Y \stackrel{\rm Law}{=} Z$, thus yielding the desired conclusion.\
The following statement is the main result of this paper. Its proof makes a crucial use of the integration by parts formula (\[eq:MalliavinIBP\]) discussed in Section \[section:Malliavin\].
\[theo:majDist\] Fix $d\geq 2$ and let $C = \{C(i,j) :
i,j=1,...,d\}$ be a $d\times d$ positive definite matrix. Suppose that $Z\sim\mathscr{N}_d(0,C)$ and that $F=(F_1,\ldots,F_d)$ is a $\real^d$-valued random vector such that $E[F_i]=0$ and $F_i \in
\mathbb{D}^{1,2}$ for every $i=1,\ldots,d$. Then, $$\begin{aligned}
d_{\rm W}(F,Z)&\leq &\| C^{-1}\|_{op} \,\,
\| C\|_{op}^{1/2} \sqrt{E\| C - \Phi(D F) \|_{H.S}^2}\label{WassONE}
\\
&=& \| C^{-1}\|_{op} \,\, \| C\|_{op}^{1/2} \sqrt{\sum_{i,j=1}^d
E[( C(i,j) - \langle DF_i, -DL^{-1}F_j \rangle_{\EuFrak H})^2] },
\label{WassTWO}\end{aligned}$$ where we write $\Phi(DF)$ to indicate the matrix $\Phi(DF):=\{\langle DF_i, -DL^{-1}F_j \rangle_{\EuFrak H}:1\leq
i,j \leq d \}.$
We start by proving that, for every $g\in\mathscr{C}^2(\real^d)$ with bounded first and second derivatives, $$|E[g(F)] - E[g(Z)]| \leq \| C^{-1}\|_{op} \,\, \|
C\|_{op}^{1/2}\,\, \|g\|_{Lip} \sqrt{E\| C - \Phi(D F)
\|_{H.S}^2}.$$ To prove such a claim, observe that, according to Point (ii) in Lemma \[lemma:SteinIBP2\], $E[g(F)] - E[g(Z)] =
E[
\langle F , \nabla U_0 g (F)
\rangle_{\real^d}
-
\langle C ,
{\rm Hess} U_0 g(F) \rangle_{H.S.}
]$. Moreover, $$\begin{aligned}
&&\displaystyle{ \big\vert E[\langle C , {\rm Hess} U_0 g(F)
\rangle_{H.S.} - \langle F , \nabla U_0 g (F)
\rangle_{\real^d}] \big\vert}\\
&=&\displaystyle{\left\vert
E\left[ \sum_{i,j=1}^d C(i,j)\partial^2_{ij} U_0g(F) - \sum_{i=1}^d F_i \partial_i U_0g(F)\right] \right\vert}\\
&=&\displaystyle{\left\vert\sum_{i,j=1}^d E\left[
C(i,j)\partial^2_{ij} U_0g(F)\right] -
\sum_{i=1}^d E\left[\big(LL^{-1}F_i\big) \partial_i U_0g(F)\right] \right\vert} \,\,\, (\textrm{since } E(F_i)=0) \\
&=&\displaystyle{\left\vert \sum_{i,j=1}^d E\left[C(i,j)
\partial^2_{ij} U_0 g(F)\right] +
\sum_{i=1}^d E\left[\delta(DL^{-1}F_i) \partial_i U_0g(F)\right] \right\vert\,\,\, (\textrm{since } \delta D=-L})\\
&=&\displaystyle{ \left\vert\sum_{i,j=1}^d E\left[C(i,j)
\partial^2_{ij} U_0g(F)\right]\!\!- \!\!\sum_{i=1}^d E\left[\langle
D(\partial_i U_0g(F)),-DL^{-1}F_i\rangle_\EuFrak{H}\right] \right\vert \,\,\,\, (\textrm{by } (\ref{eq:MalliavinIBP}) })\\
&=&\displaystyle{ \left\vert\sum_{i,j=1}^d E\left[C(i,j)
\partial^2_{ij} U_0g(F)\right] - \sum_{i,j=1}^d E\left[\partial^2_{ji}
U_0g(F) \langle DF_j,-DL^{-1}F_i \rangle_\EuFrak{H}\right] \right\vert}\,\,\,\, (\textrm{by }(\ref{eq:ChainRule})) \\
&=&\displaystyle{ \left\vert\sum_{i,j=1}^d E\left[\partial^2_{ij}
U_0g(F) \big( C(i,j)-
\langle DF_i,-DL^{-1}F_j \rangle_\EuFrak{H} \big)\right] \right\vert }\\
&=&\displaystyle{ \big\vert E\langle {\rm Hess}\, U_0g(F),C- \Phi(DF)\rangle_{H.S.} \big\vert}\\
&\leq& \displaystyle{\sqrt{E\|{\rm Hess} \,U_0g(F)
\|_{H.S}^2} \sqrt{E\| C - \Phi(D F) \|_{H.S}^2}\,\,\, (\textrm{by the Cauchy-Schwarz inequality}})\\
&\leq& \| C^{-1}\|_{op} \,\, \| C\|_{op}^{1/2}\,\, \|g\|_{Lip}
\sqrt{E\| C - \Phi(D F) \|_{H.S}^2} \,\,\,(\textrm{by
(\ref{SteinEstimate})}).\end{aligned}$$ To prove the Wasserstein estimate (\[WassONE\]), it is sufficient to observe that, for every globally Lipschitz function $g$ such that $\|g\|_{Lip}\leq 1$, there exists a family $\{g_{\varepsilon}: {\varepsilon}> 0\}$ such that:
- for each ${\varepsilon}>0$, the first and second derivatives of $g_{\varepsilon}$ are bounded;
- for each ${\varepsilon}>0$, one has that $\|g_{\varepsilon}\|_{Lip} \leq \|g\|_{Lip}$;
- as ${\varepsilon}\rightarrow 0$, $\|g_{\varepsilon}- g\|_\infty \downarrow 0$.
For instance, we can choose $g_{\varepsilon}(x)=E\big[g(x+\sqrt{{\varepsilon}}N)\big]$ with $N\sim\mathscr{N}_d(0,{\bf I}_d)$.
Observe that Theorem \[theo:majDist\] generalizes relation (\[GioIvan\]) (that was proved in [@NourdinPeccati1 Theorem 3.1]). We now aim at applying Theorem \[theo:majDist\] to vectors of multiple stochastic integrals.
\[prop:Chaos\] Fix $d\geq 2$ and $1\leq q_1\leq\ldots\leq
q_d$. Consider a vector $F:=(F_1,\ldots,F_d)=(I_{q_1}(f_1),\ldots,I_{q_d}(f_d))$ with $f_{i}\in \EuFrak{H}^{\odot q_i}$ for any $i=1\ldots,d$. Let $Z\sim\mathscr{N}_d(0,C)$, with $C$ positive definite. Then, $$\label{star}
d_{\rm W} (F,Z) \leq \| C^{-1}\|_{op} \,\, \| C\|_{op}^{1/2}
\sqrt{ \! \sum_{1\le i, j\le d} E\left[\left( C(i,j)\!-\!
\frac{1}{q_j} \langle DF_i,DF_j \rangle_{\EuFrak{H}}\right)^2
\right]}.$$
We have $-L^{-1}F_j=\frac{1}{q_j}\,F_j$ so that the desired conclusion follows from (\[WassTWO\]).
When one applies Corollary \[prop:Chaos\] in concrete situations (see e.g. Section \[section:applications\] below), one can use the following result in order to evaluate the right-hand side of (\[star\]).
\[lm-control\] Let $F=I_p(f)$ and $G=I_q(g)$, with $f\in\HH^{\odot p}$ and $g\in\HH^{\odot q}$ ($p,q\geq 1$). Let $a$ be a real constant. If $p=q$, one has the estimate: $$\begin{aligned}
&&E\left[\left(a-\frac1p\left\langle DF,DG\right\rangle_\HH\right)^2\right] \leq (a-p!\langle f,g\rangle_{\HH^{\otimes p}})^2\\
&&\hskip2cm+
\frac{p^2}{2}\sum_{r=1}^{p-1}(r-1)!^2\binom{p-1}{r-1}^4(2p-2r)!
\big( \|f\otimes_{p-r}f\|^2_{\HH^{\otimes
2r}}+\|g\otimes_{p-r}g\|^2_{\HH^{\otimes 2r}}\big).\end{aligned}$$ On the other hand, if $p< q$, one has that $$\begin{aligned}
&&E\left[\left(a-\frac1q\left\langle DF,DG\right\rangle_\HH
\right)^2\right]
\leq a^2+p!^2\binom{q-1}{p-1}^2(q-p)!\|f\|^2_{\HH^{\otimes p}}\|g\otimes_{q-p}g\|_{\HH^{\otimes 2p}}\\
&&+
\frac{p^2}{2}\sum_{r=1}^{p-1}(r-1)!^2\binom{p-1}{r-1}^2\binom{q-1}{r-1}^2(p+q-2r)!\big(
\|f\otimes_{p-r}f\|^2_{\HH^{\otimes
2r}}+\|g\otimes_{q-r}g\|^2_{\HH^{\otimes 2r}}\big).\end{aligned}$$
\[r\]
1. Recall that $E\big(I_p(f)I_q(g)\big)=\left\{\begin{array}{lll}
p!\langle f,g\rangle_{\HH^{\otimes p}}&\quad\mbox{if $p=q$},\\
0&\quad\mbox{otherwise}.
\end{array}\right.$
2. In order to estimate the right-hand side of (\[star\]), we see that it is sufficient to asses the quantity $\|f_i\otimes_r
f_i\|_{\HH^{\otimes2(q_i-r)}}$ for any $i\in\{1,\ldots,d\}$ and $r\in\{1,\ldots,q_i-1\}$ on the one hand, and $\langle
f_i,f_j\rangle_{\HH^{\otimes q_i}}$ for any $1\leq i,j\leq d$ such that $q_i=q_j$ on the other hand.
[*Proof of Lemma \[lm-control\]*]{} (see also [@NualartOrtizLatorre Lemma 2]). Without loss of generality, we can assume that $\HH=L^{2}(A,\mathscr{A},\mu)$, where $(A,\mathscr{A})$ is a measurable space, and $\mu$ is a $\sigma$-finite and non-atomic measure. Thus, we can write $$\begin{aligned}
\langle DF,DG\rangle_\HH &=&p\,q\left\langle I_{p-1}(f),I_{q-1}(g)\right\rangle_\HH
=p\,q\int_A I_{p-1}\big(f(\cdot,t)\big)I_{q-1}\big(g(\cdot,t)\big)\mu(dt)\\
&=&p\,q\int_A \sum_{r=0}^{p\wedge q-1} r!\binom{p-1}{r} \binom{q-1}{r} I_{p+q-2-2r}\big(f(\cdot,t)\widetilde{\otimes}_r g(\cdot,t)\big)\mu(dt)\\
&=&p\,q\sum_{r=0}^{p\wedge q-1} r!\binom{p-1}{r}\binom{q-1}{r} I_{p+q-2-2r}(f\widetilde{\otimes}_{r+1}g)\\
&=&p\,q \sum_{r=1}^{p\wedge q} (r-1)!\binom{p-1}{r-1}\binom{q-1}{r-1} I_{p+q-2r}(f\widetilde{\otimes}_r g).\end{aligned}$$ It follows that $$\begin{aligned}
&&E\left[\left(a-\frac1q\left\langle DF,DG\right\rangle_\HH\right)^2\right] \label{Murray}\\
&=&\left\lbrace
\begin{array}{l}
a^2+p^2\sum_{r=1}^{p}(r-1)!^2
\binom{p-1}{r-1}^2\binom{q-1}{r-1}^2 (p+q-2r)!
\|f\widetilde{\otimes}_r g\|^2_{\HH^{\otimes (p+q-2r)}} \textrm{ if } p< q,\\\\
(a-p!\langle f, g\rangle_{\EuFrak{H}^{\otimes
p}})^2+p^2\sum_{r=1}^{p-1}(r-1)!^2 \binom{p-1}{r-1}^4 (2p-2r)!
\|f\widetilde{\otimes}_r g\|^2_{\HH^{\otimes (2p-2r)}} \textrm{ if
} p=q.
\end{array}\notag
\right.\end{aligned}$$ If $r<p\leq q$ then $$\begin{aligned}
\|f\widetilde{\otimes}_r g\|^2_{\HH^{\otimes (p+q-2r)}}
&\leq& \|f\otimes_r g\|^2_{\HH^{\otimes (p+q-2r)}}
=\langle f\otimes_{p-r} f, g\otimes_{q-r}g\rangle_{\HH^{\otimes 2r}}\\
&\leq&
\|f\otimes_{p-r}f\|_{\HH^{\otimes 2r}}\|g\otimes_{q-r}g\|_{\HH^{\otimes 2r}}\\
&\leq&\frac12\left(
\|f\otimes_{p-r}f\|_{\HH^{\otimes 2r}}^2+\|g\otimes_{q-r}g\|_{\HH^{\otimes 2r}}^2
\right).\end{aligned}$$ If $r=p<q$, then $$\|f\widetilde{\otimes}_p\, g\|^2_{\HH^{\otimes (q-p)}} \leq
\|f\otimes_p \,g\|^2_{\HH^{\otimes (q-p)}} \leq
\|f\|^2_{\HH^{\otimes p}}\|g\otimes_{q-p}g\|_{\HH^{\otimes 2p}}.$$ If $r=p=q$, then $ f\widetilde{\otimes}_p g =\langle
f,g\rangle_{\HH^{\otimes p}}. $ By plugging these last expressions into (\[Murray\]), we deduce immediately the desired conclusion.\
Let us now recall the following result, which is a collection of some of the findings contained in the papers by Peccati and Tudor [@PeccatiTudor] and Nualart and Ortiz-Latorre [@NualartOrtizLatorre].
\[theo:NOL-PT\] Fix $d\geq2$ and let $C=\{C(i,j):i,j=1,...,d\}$ be a $d\times d$ positive definite matrix. Fix integers $1\leq q_1\leq\ldots\leq q_d$. For any $n\geq
1$ and $i=1,\ldots,d$, let $f_{i}^{(n)}$ belong to $
\EuFrak{H}^{\odot q_i}$. Assume that $$F^{(n)}=(F^{(n)}_1,\ldots,F^{(n)}_d):=(I_{q_1}(f_{1}^{(n)}),
\ldots,I_{q_d}(f_{d}^{(n)}))\quad n\geq 1,$$ is such that $$\label{eq:asympcov}
\lim_{n\to\infty}
E[F_i^{(n)}F_j^{(n)}]=C(i,j),\quad 1\leq i,j\leq d.$$ Then, as $n\to\infty$, the following four assertions are equivalent:
- For every $1\leq i\leq d$, $F_i^{(n)}$ converges in distribution to a centered Gaussian random variable with variance $C(i,i)$.
- For every $1\leq i\leq d$, $E\left[(F_i^{(n)})^4\right]\rightarrow3C(i,i)^2$.
- For every $1\leq i\leq d$ and every $1\leq r \leq q_i-1$, $\|f_{i}^{(n)} \otimes_r f_{i}^{(n)}\|_{\EuFrak{H}^{\otimes
2(q_i-r)}} \rightarrow 0$.
- The vector $F^{(n)}$ converges in distribution to a $d$-dimensional Gaussian vector $\mathscr{N}_d(0,C)$.
Moreover, if $C(i,j)= \delta_{ij}$, where $\delta_{ij}$ is the Kronecker symbol, then either one of conditions (i)–(iv) above is equivalent to the following:
- For every $1\leq i\leq d$, $\|DF_i^{(n)}\|_{\EuFrak{H}}^2 \overset{L^2}{\longrightarrow} q_i$.
We conclude this section by pointing out the remarkable fact that, for vectors of multiple Wiener-Itô integrals of arbitrary length, *the Wasserstein distance metrizes the weak convergence towards a Gaussian vector with positive definite covariance*. Note that the next statement also contains a generalization of the multidimensional results proved in [@NualartOrtizLatorre] to the case of an arbitrary covariance.
Fix $d\geq 2$, let $C$ be a positive definite $d\times d$ symmetric matrix, and let $1\leq q_1\leq\ldots\leq q_d$. Consider vectors $$F^{(n)}:=(F_1^{(n)},\ldots,F_d^{(n)})=
(I_{q_1}(f_1^{(n)}),\ldots,I_{q_d}(f^{(n)}_d)), \quad n\geq 1,$$ with $f^{(n)}_{i}\in \EuFrak{H}^{\odot q_i}$ for every $i=1\ldots,d$. Assume moreover that $F^{(n)}$ satisfies condition (\[eq:asympcov\]). Then, as $n\to\infty$, the following three conditions are equivalent:
- $d_{\rm W}(F^{(n)},Z)\rightarrow 0$.
- For every $1\leq i\leq d$, $q_i^{-1}\|DF_i^{(n)}\|_{\EuFrak{H}}^2
\overset{\rm L^2}{\longrightarrow}C(i,i)$ and, for every $1\leq
i\neq j \leq d$, $\langle DF^{(n)}_i,-DL^{-1}F^{(n)}_j \rangle_{\HH} =
q_j^{-1}\langle DF^{(n)}_i,DF^{(n)}_j \rangle_{\HH} \overset{\rm
L^2}{\longrightarrow}C(i,j)$.
- $F^{(n)}$ converges in distribution to $Z\sim\mathscr{N}_d(0,C)$.
Since convergence in the Wasserstein distance implies convergence in distribution, the implication (a) $\rightarrow$ (c) is trivial. The implication (b) $\rightarrow$ (a) is a consequence of relation (\[star\]). Now assume that (c) is verified, that is, $F^{(n)}$ converges in law to $Z\sim\mathscr{N}_d(0,C)$ as $n$ goes to infinity. By Theorem \[theo:NOL-PT\] we have that, for any $i\in\{1,\ldots,d\}$ and $r\in\{1,\ldots,q_i-1\}$, $$\|f_i^{(n)}\otimes_r f_i^{(n)}\|_{\HH^{\otimes2(q_i-r)}} \underset{n\to\infty}{\longrightarrow} 0.$$ By combining Corollary \[prop:Chaos\] with Lemma \[lm-control\] (see also Remark \[r\](2)), one therefore easily deduces that, since (\[eq:asympcov\]) is in order, condition (b) must necessarily be satisfied.
Applications {#section:applications}
============
Convergence of marginal distributions in the functional Breuer-Major CLT {#SS : BreuMaj}
------------------------------------------------------------------------
In this section, we use our main results in order to derive an explicit bound for the celebrated *Breuer-Major CLT* for fractional Brownian motion (fBm). We recall that a fBm $B=\{B_t:t\geq 0\}$, with Hurst index $H\in(0,1)$, is a centered Gaussian process, started from zero and with covariance function $E(B_{s}B_{t})=R(s,t)$, where $$R(s,t)=\frac{1}{2}\left( t^{2H}+s^{2H}-|t-s|^{2H}\right);\quad
s,t\geq 0.$$ For any choice of the Hurst parameter $H\in(0,1)$, the Gaussian space generated by $B$ can be identified with an isonormal Gaussian process of the type $X=\{X(h):h\in\HH\}$, where the real and separable Hilbert space $\EuFrak H$ is defined as follows: (i) denote by $\mathscr{E}$ the set of all $\mathbb{R}$-valued step functions on $[0,\infty)$, (ii) define $\EuFrak H$ as the Hilbert space obtained by closing $\mathscr{E}$ with respect to the scalar product $
\left\langle
{\mathbf{1}}_{[0,t]},{\mathbf{1}}_{[0,s]}\right\rangle _{\EuFrak
H}=R(t,s).
$ In particular, with such a notation, one has that $B_t=X(\mathbf{1}_{[0,t]})$. The reader is referred e.g. to [@Nualart] for more details on fBm, including crucial connections with fractional operators. We also define $\rho(\cdot)$ to be the covariance function associated with the stationary process $x\mapsto B_{x+1}-B_x$ ($x\in\mathbb{R}$), that is $$\rho(x):=\frac12\big(|x+1|^{2H}+|x-1|^{2H}-2|x|^{2H}\big)\underset{|x|\to\infty}{\sim}
H|2H-1|\,|x|^{2H-2}.$$ Now fix an integer $q\geq 2$, assume that $H<1-\frac1{2q}$ and set $$S_n(t)=\frac{1}{\sigma\sqrt{n}}\sum_{k=0}^{\lfloor nt\rfloor -1}H_q(B_{k+1}-B_k),\quad
t\geq 0,$$ where $H_q$ is the $q$th Hermite polynomial defined as $$\label{her-pol}
H_q(x)=(-1)^q e^{x^2/2} \frac{d^q}{dx^q}e^{-x^2/2},\quad x\in\mathbb{R},$$ and where $
\sigma=\sqrt{q!\sum_{r\in\mathbb{Z}}\rho^2(r)}.
$ According e.g. to the main results in [@BM] or [@GS], one has the following CLT: $$\{S_n(t),\,t\geq 0\} \,\,\,\xrightarrow[n\to\infty]{{\rm
f.d.d.}}\,\,\, \mbox{standard Brownian motion},$$ where ‘f.d.d.’ indicates convergence in the sense of finite-dimensional distributions. To our knowledge, the following statement contains the first multidimensional bound for the Wasserstein distance ever proved for $\{S_n(t),\,t\geq 0\}$.
\[BM-rev\] For any fixed $d\geq 1$ and $0=t_0<t_1<\ldots<t_d$, there exists a constant $c$, (depending only on $d$, $H$ and $(t_0,
t_1,\ldots,t_d)$, on $n$) such that, for every $n\geq 1$: $$\begin{aligned}
d_{\rm W}\left(
\left(\frac{S_n(t_i)-S_n(t_{i-1})}{\sqrt{t_i-t_{i-1}}}\right)_{1\leq
i \leq d};\mathscr{N}_d(0,{\bf I}_d) \right)&\leq& c\times
\left\{\begin{array}{lll}
n^{-\frac12}&\,\,\mbox{if $H\in (0,\frac12]$}\\
\\
n^{H-1}&\,\,\mbox{if $H\in (\frac12,\frac{2q-3}{2q-2}]$}\\
\\
n^{qH-q+\frac12}&\,\,\mbox{if $H\in (\frac{2q-3}{2q-2},\frac{2q-1}{2q})$}\\
\end{array}\right..\end{aligned}$$
Fix $d\geq 1$ and $t_0=0<t_1<\ldots<t_d$. In the sequel, $c$ will denote a constant independent of $n$, which can differ from one line to another.
First, observe that $$\frac{S_n(t_i)-S_n(t_{i-1})}{\sqrt{t_i-t_{i-1}}}=I_q(f_i^{(n)})\quad\mbox{with}\quad
f_i^{(n)}=\frac{1}{\sigma\sqrt{n}
\sqrt{t_i-t_{i-1}}
}
\sum_{k=\lfloor nt_{i-1}\rfloor}^{\lfloor nt_i\rfloor -1}
{\bf 1}_{[k,k+1]}^{\otimes q}.$$
In [@NourdinPeccati1], proof of Theorem 4.1, it is shown that, for any $i\in\{1,\ldots,d\}$ and $r\in\{1,\ldots,q_i-1\}$: $$\label{bound1}
\|f_i^{(n)}\otimes_r f_i^{(n)}\|_{\HH^{\otimes 2(q_i-r)}}\leq c\times
\left\{\begin{array}{lll}
n^{-\frac12}&\,\,\mbox{if $H\in (0,\frac12]$}\\
\\
n^{H-1}&\,\,\mbox{if $H\in (\frac12,\frac{2q-3}{2q-2}]$}\\
\\
n^{qH-q+\frac12}&\,\,\mbox{if $H\in (\frac{2q-3}{2q-2},\frac{2q-1}{2q})$}\\
\end{array}\right..$$ Moreover, when $1\leq i<j\leq d$, we have: $$\begin{aligned}
\notag
&&\big|\langle f_i^{(n)},f_j^{(n)}\rangle_{\HH^{\otimes q}}\big|\\
&=&\left|
\frac{1}{\sigma^2\,n
\sqrt{t_i-t_{i-1}}\sqrt{t_j-t_{j-1}}
}
\sum_{k=\lfloor nt_{i-1}\rfloor}^{\lfloor nt_i\rfloor -1}
\sum_{l=\lfloor nt_{j-1}\rfloor}^{\lfloor nt_j\rfloor -1}
\rho^q(l-k)\right|\notag\\
&=& \frac{c}{n}\left|
\sum_{|r|=\lfloor nt_{j-1}\rfloor-\lfloor nt_{i}\rfloor+1}
^{\lfloor nt_{j}\rfloor-\lfloor nt_{i-1}\rfloor-1}
\big[ ( \lfloor nt_{j}\rfloor -1 -r)\wedge( \lfloor nt_{i}\rfloor -1)
-( \lfloor nt_{j-1}\rfloor -r)\vee( \lfloor nt_{i-1}\rfloor )
\big]
\rho^q(r)\right|\notag\\
&\leq& c\,\,\frac{\lfloor nt_{i}\rfloor -\lfloor nt_{i-1}\rfloor -1}{n}
\sum_{|r|\geq\lfloor nt_{j-1}\rfloor-\lfloor nt_{i}\rfloor+1}
\big|\rho(r)\big|^q
=O\big(n^{2qH-2q+1}\big),\quad\mbox{as $n\to\infty$},\label{bound2}\end{aligned}$$ the last equality coming from $$\sum_{|r|\geq N}\big|\rho(r)\big|^q=O(\sum_{|r|\geq N}|r|^{2qH-2q})=O(N^{2qH-2q+1}),\quad\mbox{as $N\to\infty$}.$$
Finally, by combining (\[bound1\]), (\[bound2\]), Corollary \[prop:Chaos\] and Lemma \[lm-control\], we obtain the desired conclusion.
Vector-valued functionals of finite Gaussian sequences {#SS : Chatterjee}
------------------------------------------------------
Let $Y=(Y_1,...,Y_n)\sim \mathscr{N}_n(0, {\bf I}_n)$, and let $f:\real^n \rightarrow \real$ be an absolutely continuous function such that $f$ and its partial derivatives have subexponential growth at infinity. The following result has been proved by Chatterjee in [@Chatterjee_ptrf], in the context of limit theorems for linear statistics of eigenvalues of random matrices. We use the notation $d_{\rm TV}$ to indicate the *total variation distance* between laws of real valued random variables.
\[Chatterbox\] Assume that the random variable $W=f(Y)$ has zero mean and unit variance, and denote by $Z \sim \mathscr{N}(0,1)$ a standard Gaussian random variable. Then $
d_{\rm TV}(W,Z)\leq 2 {\rm Var}(T(Y))^{1/2},
$ where the function $T(\cdot)$ is defined as $$T(y) = \int_0^1 \frac{1}{2\sqrt{t}}\sum_{i=1}^{n}E\big
[\frac{\partial f}{\partial y_i}(y)\frac{\partial f}{\partial
y_i}(\sqrt{t}\,y+\sqrt{1-t}\,\,Y)\big ]dt.$$
In what follows, we shall use Theorem \[theo:majDist\] in order to deduce a multidimensional generalization of Proposition \[Chatterbox\] (with the Wasserstein distance replacing total variation).
\[MultiChatterbox\] Let $Y\sim \mathscr{N}_n(0, K)$, where $K=\{K(i,l):i,l=1,...,n\}$ is a $n\times n$ positive definite matrix. Consider absolutely continuous functions $f_j : \real^n \rightarrow \real$, $j=1,...,d$. Assume that each random variable $f_j(Y)$ has zero mean, and also that each function $f_j$ and its partial derivatives have subexponential growth at infinity. Denote by $Z
\sim \mathscr{N}_d(0,C)$ a Gaussian vector with values in $\real^d$ and with positive definite covariance matrix $C=\{C(a,b)
: a,b=1,...,d\}$. Finally, write $W =(W_1,...,W_d) =
(f_1(Y),...,f_d(Y))$. Then, $$d_{\rm W}(W,Z)\leq \| C^{-1}\|_{op} \,\, \|
C\|_{op}^{1/2}\sqrt{\sum_{a,b=1}^d E[(C(a,b)-T_{ab}(Y))^2]}$$ where the functions $T_{ab}(\cdot)$ are defined as $$T_{ab}(y) = \int_0^1 \frac{1}{2\sqrt{t}}\sum_{i,j=1}^{n}K(i,j)
E\big [\frac{\partial f_a}{\partial
y_i}(y)\frac{\partial f_b}{\partial
y_j}(\sqrt{t}\,y+\sqrt{1-t}\,\,Y)\big ]dt.$$
Without loss of generality, we can assume that $Y=(Y_1,...,Y_n) = (X(h_1),...,X(h_n))$, where $X$ is an isonormal Gaussian process over some Hilbert space $\HH$, and $\langle h_i ,
h_l\rangle_\HH = E(X(h_i)X(h_l))=K(i,l)$. According to Theorem \[theo:majDist\], it is therefore sufficient to show that, for every $a,b=1,...,d$, $
T_{ab}(Y) = \langle DW_a , -DL^{-1}W_b \rangle_{\HH}.
$ To prove this last claim, introduce the two $\HH$-valued functions $\Theta_a(y)$ and $\Theta_b(y)$, defined for $y\in\real^d$ as follows: $$\Theta_a(y) =\sum_{i=1}^n \frac{\partial f_a}{\partial
y_i}(y)h_i
\quad\mbox{and}\quad
\Theta_b(y) = \int_0^1 \frac{1}{2\sqrt{t}}\sum_{j=1}^{n}\left\{
E\big [\frac{\partial f_b}{\partial
y_j}(\sqrt{t}\,y+\sqrt{1-t}\,\,Y)\big ]h_j \right\} dt.$$ By using (\[baldassarre\]), it is easily seen that $\Theta_a(Y)
= DW_a$. Moreover, by using e.g. formula (3.46) in [@NourdinPeccati1], one deduces that $\Theta_b(Y) =
-DL^{-1}W_b$. Since $T_{ab}(Y) =\langle \Theta_a(Y) , \Theta_b(Y)
\rangle_{\HH}$, the conclusion is immediately obtained.
By specializing the previous statement to the case $n=d$ and $f_j(y) = y_j$, $j=1,...,d$, one obtains the following simple bound on the Wasserstein distance between Gaussian vectors of the same dimension (the proof is straightforward and omitted).
Let $Y\sim \mathscr{N}_d(0, K)$ and $Z\sim \mathscr{N}_d(0, C)$, where $K$ and $C$ are two positive definite covariance matrices. Then $
d_{\rm W}(Y,Z)\leq Q(C,K)\times \|C-K\|_{H.S.},
$ where $$Q(C,K) :=\min\{\| C^{-1}\|_{op} \,\, \| C\|_{op}^{1/2}, \|
K^{-1}\|_{op} \,\, \| K\|_{op}^{1/2}\}.$$
**Acknowledgments.** We thank an anonymous referee for insightful remarks. Part of this paper has been written while I. Nourdin and G. Peccati were visiting the Institute of Mathematical Sciences of the National University of Singapore, in the occasion of the “Workshop on Stein’s Method”, from March 31 until April 4, 2008. These authors heartily thank A. Barbour, L. Chen, K.-P. Choi and A. Xia for their kind hospitality and support.
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[^1]: LPMA, Université Paris VI, Boîte courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France. Email: `ivan.nourdin@upmc.fr`
[^2]: Équipe Modal’X, Université Paris Ouest - Nanterre la Défense, 200 avenue de la République, 92000 Nanterre and LSTA, Université Paris VI, France. Email: `giovanni.peccati@gmail.com`
[^3]: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. Email: `areveill@mathematik.hu-berlin.de`
|
---
abstract: |
The management of security credentials (e.g., passwords, secret keys) for computational science workflows is a burden for scientists and information security officers. Problems with credentials (e.g., expiration, privilege mismatch) cause workflows to fail to fetch needed input data or store valuable scientific results, distracting scientists from their research by requiring them to diagnose the problems, re-run their computations, and wait longer for their results. SciTokens introduces a capabilities-based authorization infrastructure for distributed scientific computing, to help scientists manage their security credentials more reliably and securely. SciTokens uses IETF-standard OAuth JSON Web Tokens for capability-based secure access to remote scientific data. These access tokens convey the specific authorizations needed by the workflows, rather than general-purpose authentication impersonation credentials, to address the risks of scientific workflows running on distributed infrastructure including NSF resources (e.g., LIGO Data Grid, Open Science Grid, XSEDE) and public clouds (e.g., Amazon Web Services, Google Cloud, Microsoft Azure). By improving the interoperability and security of scientific workflows, SciTokens 1) enables use of distributed computing for scientific domains that require greater data protection and 2) enables use of more widely distributed computing resources by reducing the risk of credential abuse on remote systems.
In this extended abstract, we present the results over the past year of our open source implementation of the SciTokens model and its deployment in the Open Science Grid, including new OAuth support added in the HTCondor 8.8 release series.
author:
- Alex Withers
- Brian Bockelman
- Derek Weitzel
- Duncan Brown
- Jason Patton
- Jeff Gaynor
- Jim Basney
- Todd Tannenbaum
- You Alex Gao
- Zach Miller
bibliography:
- 'scitokens.bib'
title: 'SciTokens: Demonstrating Capability-Based Access to Remote Scientific Data using HTCondor'
---
<ccs2012> <concept> <concept\_id>10002978.10002991.10010839</concept\_id> <concept\_desc>Security and privacy Authorization</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Introduction
============
We introduced the SciTokens model last year in [@SciTokensPEARC18]. In this extended abstract, we present the implementation and deployment experience we have gained since then, including new OAuth support added in the HTCondor 8.8 release series.
SciTokens applies the well-known principle of capability-based security to remote data access. Rather than sending unconstrained identity tokens with compute jobs, we send capability-based access tokens. These access tokens grant the specific data access rights needed by the jobs, limiting exposure to abuse. These tokens comply with the IETF OAuth standard [@RFC6749], enabling interoperability with the many public cloud storage and computing services that have adopted this standard. By improving the interoperability and security of scientific workflows, we 1) enable use of distributed computing for scientific domains that require greater data protection and 2) enable use of more widely distributed computing resources by reducing the risk of credential abuse on remote systems.
As illustrated in Figure \[fig:model\] from [@SciTokensPEARC18], our SciTokens model applies capability-based security to three common domains in the computational science environment: Submit (where the researcher submits and manages scientific workflows), Execute (where the computational jobs run), and Data (where remote read/write access to scientific data is provided). The Submit domain obtains the needed access tokens for the researcher’s jobs and forwards the tokens to the jobs when they run, so the jobs can perform the needed remote data access. The Scheduler and Token Manager work together in the Submit domain to ensure that running jobs have the tokens they need (e.g., by refreshing tokens when they expire) and handle any errors (e.g., by putting jobs on hold until needed access tokens are acquired). The Data domain contains Token Servers that issue access tokens for access to Data Servers. Thus, there is a strong policy and trust relationship between Token Servers and Data Servers. In the Execute domain, the job Launcher delivers access tokens to the job’s environment, enabling it to access remote data.
![The SciTokens Model[]{data-label="fig:model"}](scitokens_model.png)
The SciTokens model adopts token types from OAuth (see Figure \[fig:token\_types\]). Users authenticate with identity tokens to submit jobs (workflows), but identity tokens do not travel along with the jobs. Instead, at job submission time the Token Manager obtains OAuth refresh tokens with needed data access privileges from Token Servers. The Token Manager securely stores these relatively long-lived refresh tokens locally, then uses them to obtain short-lived access tokens from the Token Server when needed (e.g., when jobs start or when access tokens for running jobs near expiration). The Scheduler then sends the short-lived access tokens to the jobs, which the jobs use to access remote data.
![Different Token Types[]{data-label="fig:token_types"}](token_types.png)
The remainder of our extended abstract is organized as follows. In Section \[sec:htcondor\], we provide an overview of HTCondor’s support for SciTokens. In Section \[sec:htcondorupdates\], we describe recent HTCondor updates related to SciTokens. In Section \[sec:osg\], we describe our Open Science Grid deployment experience. In Section \[sec:c\], we describe our new C/C++ implementation. In Section \[sec:oauth\], we describe our OAuth server updates. Lastly, in Section \[sec:relatedwork\] we describe our interoperability efforts with related work, and we conclude in Section \[sec:conclusions\].
\[sec:htcondor\]HTCondor Implementation
=======================================
As the component that actually executes a scientific workflow, HTCondor serves as the linchpin that ties together all the SciTokens components. To best communicate our HTCondor approach, we first present a walk-thru of how HTCondor orchestrates the component interactions upon submission of a job, followed by a discussion of integration points.
As illustrated in Figure \[fig:arch\] from [@SciTokensPEARC18], the process begins when the researcher submits the computational job using the condor\_submit command (or more likely using Pegasus or similar workflow front-end that then runs condor\_submit). As part of the submission, the researcher specifies required scientific input data and locations for output data storage in the condor\_submit input file. For example, in a LIGO PyCBC [@PyCBC16] submission, the researcher will specify a set of data “frames” from the LIGO instrument that are the subject of the analysis. Then condor\_submit authenticates the researcher to the token\_server(s) to obtain the tokens needed for the job’s data access; as an optimization, condor\_submit may first check for any locally cached tokens from the researcher’s prior job submissions. The token\_server determines if the researcher is authorized for the requested data access, based on the researcher’s identity and/or group memberships or other researcher attributes. If the authorization check succeeds, the Token Server issues an OAuth refresh token back to condor\_submit, which stores the refresh token securely in the condor\_credd, and sends the job information to the condor\_schedd. Since condor\_submit gathers all the needed data access tokens, there is no need to store any identity credentials (e.g., passwords, X.509 certificates, etc.) with the job submission, thereby achieving our goal of a capability-based approach.
![The SciTokens System Architecture[]{data-label="fig:arch"}](arch_diagram2.png)
The next phase of the process begins when the condor\_schedd has scheduled the job on a remote execution site. The condor\_schedd communicates with the condor\_startd to launch the job, establishing a secure communication channel between the condor\_shadow on the submission side and the condor\_starter on the remote execution side. The condor\_starter then requests access tokens from the condor\_shadow for the job’s input data. The condor\_shadow forwards the access token requests to the condor\_credd, which uses its stored refresh tokens to obtain fresh access tokens from the token\_server. The condor\_credd returns the access tokens to the condor\_shadow which forwards them securely to the condor\_starter which provides them to the researcher’s job. Note that only access tokens are sent to the remote execution environment; the longer-lived refresh tokens remain secured in the submission environment which typically resides at the researcher’s home institution. Lastly, the job uses the access tokens to mount CVMFS filesystem(s) to access scientific data. CVMFS verifies each access token to confirm that the token was issued by its trusted token\_server and that the token’s scope includes access to the scientific data being requested. If verification succeeds, CVMFS grants the requested data access. If the access token needs to be refreshed, the condor\_starter makes another request back to the condor\_shadow.
Note that SciTokens can leverage additional aspects of HTCondor, such as the fact that the condor\_shadow can be made explicitly aware if the job is staging input data, accessing data online while the job is running, or staging output data. We allow the job submission to state three different sets of access tokens, which will only be instantiated at file stage-in, execution, and file stage-out, respectively. This enables long running jobs, for instance, to fetch a very short-lived write token for output that will only be instantiated once processing has completed. We also adjust the granularity of access token restrictions; for instance, the condor\_shadow may request fresh access tokens for each job instance, allowing the token to be restricted in origin to a specific execution node. Alternatively, for greater scalability, access tokens can be cached at the credd and shared across all condor\_shadow processes serving jobs that need the same data sets. Finally, we are investigating scenarios in which the data service and its accompanying token service is not fixed infrastructure, but instead is dynamically deployed upon execute nodes, perhaps by the workflow itself. In this scenario, the token service could be instantiated with a set of recognized refresh tokens a priori.
\[sec:htcondorupdates\]HTCondor Updates
=======================================
In the past year our OAuth-enabled CredMon progressed from a research prototype to a supported component of HTCondor 8.8 [@HTCondor882]. The CredMon is a plug-in to the condor\_credd that implements support for OAuth credentials. Since we use the OAuth standard, it was relatively straightforward to support both SciTokens and Box.com credentials in the same CredMon, so HTCondor jobs can access files both in CVMFS (using SciTokens) and in Box.com folders (using Box tokens).
OAuth support in HTCondor requires a lightweight web server on the submission node, to enable users to authorize credential issuance from the token server (e.g., SciTokens or Box.com) to the condor\_credd. Packaging and configuring this web server component for successful deployment by HTCondor users was a source of multiple lessons learned over the past year. Of special note is the challenge of transferring credentials from the web application to the condor\_credd, which are running under different service accounts for proper isolation. When the OAuth protocol delivers temporary credentials (the OAuth “code”) to the web application, the web application writes the credentials to a roundevous directory that the condor\_credd can read from to obtain longer-lived refresh tokens and access tokens. Thus, the more sensitive credentials are not exposed to the web application.
\[sec:osg\]OSG Deployment Experience
====================================
Open Science Grid (OSG) has been an early adopter of the SciTokens model. Over the past year, 13 OSG users have used SciTokens credentials to secure almost two million StashCP uploads across over two thousand servers at 60 unique sites.
OSG currently uses SciTokens with HTCondor in “Local CredMon Mode”. In contrast to HTCondor’s “OAuth CredMon Mode”, the “Local CredMon Mode” configuration uses a SciTokens credential issuer that’s local to the HTCondor submit node, that issues credentials according to project-specific policies set by the submit node administrator. Since this mode does not use OAuth, the submitter does not see an OAuth consent screen, but instead HTCondor transparently adds the needed SciToken credentials to the user’s job environment to enable project-specific StashCache access. This mode still relies on HTCondor’s end-to-end credential management capabilities, for sending access tokens along with the jobs and refreshing tokens as needed.
This early OSG deployment experience has been especially helpful for working out SciTokens packaging and upgrade path details, since we have gone through upgrades of the SciTokens software (including file-server plug-ins) and credential profiles. Rolling out these updates across the distributed OSG infrastructure has required the SciTokens project to think about compatibility and versioning from the start.
\[sec:c\]C/C++ Implementation
=============================
In addition to our Java [@scitokens-java] and Python [@derek_weitzel_2018_1187173] SciTokens implementations, which we described in [@SciTokensPEARC18], we have added a C/C++ implementation [@scitokenscpp], which enables improved performance for our CVMFS and XrootD integrations.
It has also enabled development of a SciTokens Apache module, which implements an Apache user authentication type. The module is developed using the SciTokens authorization helper for CVMFS and the JWT Apache authentication module. In the module, the token is retrieved from the current request with the Apache Portable Runtime (APR) library and verified with functions in the SciTokens C/C++ library.
The SciTokens C/C++ implementation has also enabled support for SciTokens as a native HTCondor authorization method (e.g., for authorizing access to the condor\_schedd). SciTokens C++ RPMs will soon be included in the OSG software distribution.
\[sec:oauth\]OAuth Server Improvements
======================================
We have updated our Token Server to support a more flexible policy language, including per-client policies based on SAML attributes and LDAP queries. We have also added support for OAuth token revocation [@RFC7009], dynamic client registration [@RFC7591; @RFC7592], and mobile clients [@RFC8252].
\[sec:relatedwork\]Related Work
===============================
The SciTokens project has benefited from participation in the WLCG Authorization working group, including involvement in the development of a WLCG profile for JSON Web Tokens [@WLCGJWTProfile] that is compatible with SciTokens.
The SciTokens project has also engaged in interoperability testing with other JWT implementations in the scientific community, including INFN IAM[^1] and dCache[^2] services.
\[sec:conclusions\]Conclusions
==============================
The JSON Web Token and OAuth standards provide a solid foundation for distributed, capability-based authorization for scientific workflows. By enhancing existing components (CILogon, CVMFS, HTCondor, XrootD) to support the SciTokens model, we have provided a migration path from X.509 identity-based delegation to OAuth capability-based delegation for existing scientific infrastructures.
All SciTokens code is open source and published at <https://github.com/scitokens>. The HTCondor CredMon is also open source, published at <https://github.com/htcondor/scitokens-credmon>. Visit <https://scitokens.org/> for the latest information about the SciTokens project.
This material is based upon work supported by the National Science Foundation under Grant No. 1738962.
[^1]: <https://iam.infn.it/>
[^2]: <https://www.dcache.org/>
|
---
abstract: 'We propose a method to exponentially speed up computation of various fingerprints, such as the ones used to compute similarity and rarity in massive data sets. Rather then maintaining the full stream of $b$ items of a universe $[u]$, such methods only maintain a concise fingerprint of the stream, and perform computations using the fingerprints. The computations are done approximately, and the required fingerprint size $k$ depends on the desired accuracy $\epsilon$ and confidence $\delta$. Our technique maintains a single bit per hash function, rather than a single integer, thus requiring a fingerprint of length $k = O(\frac{\ln \frac{1}{\delta}}{\epsilon^2})$ bits, rather than $O(\log u \cdot \frac{\ln \frac{1}{\delta}}{\epsilon^2})$ bits required by previous approaches. The main advantage of the fingerprints we propose is that rather than computing the fingerprint of a stream of $b$ items in time of $O(b \cdot k)$, we can compute it in time $O(b \log k)$. Thus this allows an exponential speedup for the fingerprint construction, or alternatively allows achieving a much higher accuracy while preserving computation time. Our methods rely on a specific family of pseudo-random hashes for which we can quickly locate hashes resulting in small values.'
author:
- 'Yoram Bachrach, Ely Porat'
bibliography:
- 'fprf.bib'
title: 'Fast Pseudo-Random Fingerprints'
---
Introduction {#sec:intro}
============
Hashing is a key tool in processing massive data sets. Many uses of hashing in various applications require computing many hash functions in parallel. In this paper we present a technique that “ties together” many hashes in a novel way, which enables us to speed up such algorithms by an *exponential factor*. Our method also works for some complicated hash function such as min-wise independent families of hashes. In this paper we focus on producing an optimal similarity fingerprint using this method, but our technique is *general*, as it is easy to use our approach to speed up other hash intensive computations. One easy example where our technique applies is approximating the number of distinct elements from [@alon1999space]. A another example, which requires a slightly stronger analysis, is computing of $L_p$ sketches [@indyk2006stable] for $0\le p\le 2$ .
Min-wise independent families of hash functions, which we call *MWIFs* for short, were introduced in [@mulmuley1996randomized; @broder2000min]. Computations using MWIFs have been used in many algorithms for processing massive data streams. The properties of MWIFs allow maintaining concise descriptions of massive streams. These descriptions, called “fingerprints” or “sketches”, allow computing properties of these streams and relations between them. Examples of such “fingerprint” computations include data summerization and subpopulation-size queries [@cohen2007summarizing; @cohen2007sketching], greedy list intersection [@krauthgamer2008greedy], approximating rarity and similarity for data streams [@datar2002estimating], collaborative filtering fingerprints [@bachrach2009sketching; @bachrach2009spire; @bachrach-fingerprinting] and estimating frequency moments [@alon1999space]. Another motivation for studying MWIFs is reducing the amount of randomness used by algorithms [@broder2003derandomization; @mulmuley1996randomized; @broder2000min].
Recent research reduced the amount of information stored, while accurately computing properties data streams. Such techniques improve the *space complexity*, but much less attention has been given to *computation complexity*. For example, many streaming algorithms compute huge amounts of hashes, as they apply *many* hashes to each element in a very long stream of elements. This leads to a high computation time, not always tractable for many applications.
Our main contribution is a method allowing an *exponential* speedup in *computation time* for constructing fingerprints of massive data streams. Our technique is *general*, and can speed up many processes that apply many random hashes. The heart of the method lies in using a specific family of pseudo-random hashes shown to be approximately-MWIF [@indyk2001small], and for which we can quickly locate the hashes resulting in a small value of an element under the hash. Similarly to [@patrascu:k] we use the fact that members of the family are pairwise independent between themselves. We also extend the technique and show one can maintain just a *single* bit rather than the full element IDs, thus improving the fingerprint size. Independently of us [@li2010b] also considered storing few bits per hash function, but focused only on minimizing storage rather than computation time.
Preliminaries {#sec:perlim}
-------------
Let $H$ be a family of functions over the same source $X$ and target $Y$, so each $h \in H$ is a function $h : X \rightarrow Y$, where $Y$ is a completely ordered set. We say that $H$ is min-wise independent if, when randomly choosing a function $h \in H$, for any subset $C \subseteq X$, any $x \in C$ has an equal probability of being the minimal after applying $h$.
[$H$ is min-wise independent]{} (MWIF), if for all $C \subseteq X$, for any $x \in C$, $Pr_{h \in H}[h(x) = min_{a \in C}h(a)] = \frac{1}{|C|}$
[$H$ is a $\gamma$-approximately min-wise independent]{} ($\gamma$-MWIF), if for all $C \subseteq X$, for any $x \in C$, $ \left| Pr_{h \in H}[h(x) = min_{a \in C}h(a)] - \frac{1}{|C|} \right| \leq \frac{\gamma}{|C|}$
[$H$ is $k$-wise independent]{}, if for all $x_1,x_2,\ldots,x_k,y_1,y_2,\ldots,y_k \subseteq X$, $Pr_{h \in H}[ (h(x_1) = y_1) \wedge \ldots \wedge (h(x_k) = y_k) ] = \frac{1}{|X|^k}$
Pseudo-Random Family of Hashes {#l_sect_hash_family}
==============================
We describe the hashes we use.Given the universe of item IDs $[u]$, consider a big prime $p$, such that $p>u$. Consider taking random coefficients for a $d$-degree polynomial in ${\mathbb{Z}_p}$. Let $a_0, a_1, \ldots, a_d \in [p]$ be chosen uniformly at random from $[p]$, and the following polynomial in ${\mathbb{Z}_p}$: $f(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_d x^d$. We denote by $F_d$ the family of all $d$-degree polynomials in ${\mathbb{Z}_p}$ with coefficients in ${\mathbb{Z}_p}$, and later choose members of this family uniformly at random. Indyk [@indyk2001small] shows that choosing a function $f$ from $F_d$ uniformly at random results in $F_d$ being a $\gamma$-MWIF for $d = O(\log\frac{1}{\gamma})$. Randomly choosing $a_0,\ldots,a_d$ is equivalent to choosing a member of $F_d$ uniformly at random, so $f(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_d x^d$ is a hash chosen at random from the $\gamma$-MWIF $F_d$. Similarly, consider $b_0, b_1, \ldots, b_d \in [p]$ be chosen uniformly at random from $[p]$, and $g(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_d x^d$, which is also a hash chosen at random from the $\gamma$-MWIF $F_d$. Now consider the hashes $h_0(x) = f(x), h_1(x) = f(x) + g(x), h_2(x) = f(x) + 2 g(x), \ldots, h_i(x) = f(x) + i g(x), \ldots, h_{k-1}(x) = f(x) + (k-1) g(x)$. We call this random construction procedure for $f(x),g(x)$ the *base random construction*, and the construction of $h_i$ the *composition construction*. We prove properties of such hashes. We denote the probability of an event $E$ when the hash $h$ is constructed by choosing $f,g$ using the base random construction and composing $h(x) = f(x) + i \cdot g(x)$ (for some $i \in [p]$) as $Pr_{h}(E)$.
Let $f,g$ be constructed using the base random construction, using $d = O(\log\frac{1}{\gamma})$. For any $z \in [u]$, any $X \subseteq [u]$ and any value $i$ used to compose $h(x)=f(x)+i \cdot g(x)$: $Pr_{h} [h(z) < min_{y\in X} (h(y)] = (1\pm \gamma) \frac{1}{|X|}$.
Fix $i$, $z \in [u]$ and $X \subseteq [u]$, construct $f,g$ using the base random construction, and compose $h(x)=f(x)+i \cdot g(x)$. Note in $i\cdot g(x) = i \cdot (b_0 + b_1 x + \ldots b_d x^d)$, the coefficient of $x^j$ is $q = (i \cdot b_j) \mod p$. Given a value $s \in [p]$ There is exactly one value in $r \in [p]$ such that $(q+r) \mod p = s$. Thus, for any $s \in [p]$, the probability that the coefficient of $x^j$ in $h(x)$ is $s$ is $\frac{1}{p}$. Therefor $Pr_h[h(x)\equiv p(x)]=\frac{1}{p^{d+1}}=\frac{1}{F_d}$. We have: $Pr_{h} [h(z) < min_{y\in X} h(y)]=
\sum_{p(x) \in F_d} Pr_{h} [ h(z) < min_{y\in X} h(y) | h(x) \equiv f(x) + i \cdot g(x)\equiv p(x)] \cdot Pr_{h}[h(x) \equiv p(x)] =
\sum_{p(x) \in F_d} \frac{Pr_{h} [h(z) < min_{y\in X} h(y) | h(x) \equiv p(x)] }{|F_d|}= \sum_{p(x) \in F_d} \frac{Pr[p(z) < min_{y\in X} (p(y))] }{|F_d|}=
Pr_{p(x) \in F_d} [p(z) < min_{y\in X} (p(y))]=(1\pm \gamma) \frac{1}{|X|}$. If $p(x)$ is a polynom such that for any $z \in {\mathbb{Z}_p}$ we have $p(z) < min_{y\in X} (p(y))$, then we have $Pr[p(z) < min_{y\in X} (p(y))]=1$, and otherwise $Pr[p(z) < min_{y\in X} (p(y))]=0$. Thus we get $\sum_{p(x) \in F_d} \frac{Pr[p(z) < min_{y\in X} (p(y))] }{|F_d|}= Pr_{p(x) \in F_d} [p(z) < min_{y\in X} (p(y))]$. The last transition uses the fact that $F_d$ is an $\gamma$-MWIF, which requires $d = O(\log\frac{1}{\gamma})$.
\[l\_lem\_pairwise\_interact\] Let $f,g$ be constructed using the base random construction, using $d = O(\log\frac{1}{\gamma})$. For all $x_1,x_2 \in [u]$ and all $X_1,X_2 \subseteq [u]$, and all $i \neq j$ used to compose $h_i(x)=f(x)+i \cdot g(x)$ and $h_j(x)=f(x)+j\cdot g(x)$: $$Pr_{f,g \in F_d} [ (h_i(x_1) < min_{y\in X_1} h_i(y)) \wedge (h_j(x_2) < min_{y\in X_2} h_i(y)) ] =(1\pm \gamma)^2\frac{1}{|X_1| \cdot |X_2|}$$
Given $p_1(x) \in F_d = u_0 + u_1 x + \ldots + u_d x^d$ and $p_2(x) \in F_d = v_0 + v_1 x + \ldots + v_d x^d$, there is *exactly one* pair of polynoms $f(x),g(x) \in F_d$ such that both $f(x)+i \cdot g(x) = p_1(x)$ and $f(x) + j \cdot g(x) = p_2(x)$. Each coefficient location $l \in [d]$ results in two equations with two unknowns in ${\mathbb{Z}_p}$, with a single solution $(a_l,b_l)$ (where $a_l$ is the coefficient of $x^l$ in $f(x)$, and $b_l$ is the coefficient of $x^l$ in $g(x)$.
Fix $i \neq j$, $x_1,x_2 \in [u]$ and $X_1,X_2 \subseteq [u]$, construct $f,g$ using the base random construction, and compose $h_i(x)=f(x)+i \cdot g(x)$, $h_j(x)=f(x)+j \cdot g(x)$. For brevity, denote $m^i_1 = \min_{y\in X_1} h_i(y)$. Similarly, denote $m^j_2 = \min_{y\in X_2} h_j(y)$. We have: $Pr_{f,g \in F_d} [ (h_i(x_1) < m^i_1) \wedge (h_j(x_2) < m^j_2)] =
\sum_{p_1,p_2 \in F_d} Pr [ (h_i(x_1) < m^i_1) \wedge (h_j(x_2) < m^j_2) | (h_i(x)\equiv p_1(x) \wedge h_j(x)\equiv p_2(x))] \cdot Pr [(h_i(x)\equiv p_1(x) \wedge h_j(x)\equiv p_2(x))] =
\sum_{p_1,p_2 \in F_d} \frac{Pr[(h_i(x_1) < m^i_1) \wedge (h_j(x_2) < m^j_2) | (h_i(x)\equiv p_1(x) \wedge h_j(x)\equiv p_2(x))]}{|F_d|^2}
$. Thus, $Pr_{f,g \in F_d} [ (h_i(x_1) < m^i_1) \wedge (h_j(x_2) < m^j_2)] =
\sum_{p_1,p_2 \in F_d} \frac{Pr[(p_1(x_1) < m^i_1) \wedge (p_2(x_2) < m^j_2)]}{|F_d|^2} =\sum_{p_1,p_2 \in F_d} \frac{Pr[ p_1(x_1) < m^i_1 ] \cdot Pr[p_2(x_2) < m^j_2]}{|F_d|^2} =
\sum_{p_1\in F_d} \sum_{p_2\in F_d} \frac{Pr[ p_1(x_1) < m^i_1 ]}{|F_d|} \cdot \frac{Pr[p_2(x_2) < m^j_2]}{|F_d|} =
(1\pm \gamma)^2\frac{1}{|X_1| \cdot |X_2|} $
Fingerprinting Using Pseudo-Random Hashes {#l_sect_pairwise_block_const_conf}
=========================================
Several methods were suggested for building fingerprints for approximating relations between massive datasets, such as the Jackard similarity (see [@broder2000min] for example). Given a universe $U$, where $|U|=u$, consider $C_1,C_2$, where each $C_i \subseteq U$ is described as a set $|C_i|$ integers in $[u]$ (we use $[u]$ to denote $\{1,2,\ldots,u\})$. The Jackard similarity is $J_{1,2}=\frac{|C_1 \cap C_2|}{|C_1 \cup C_2|}$. Many fingerprints rely on applying many hashes to each elements in the long streams. We use a the hashes of Section \[l\_sect\_hash\_family\] to exponentially speed up such computations. We use pseudo-random effects in this hash, so we must relax the MWIF requirement to a pairwise independence requirement (2-wise independence). For completeness, we briefly consider previously suggested approaches for approximating Jackard similarity [@broder2000min]. Let $h \in H$ be a randomly chosen function from a MWIF $H$. We can apply $h$ on all elements $C_1$ and examine the minimal integer we get, $m^h_1 = \arg \min_{x \in C_1} h(x)$. We can do the same to $C_2$ and examine $m^h_2 = \arg \min_{x \in C_2} h(x)$. Fingerprints for estimating the Jackard similarity are based on computing the probability that $m_1 = m_2$: $Pr_{h \in H} [m^h_1 = m^h_2] = Pr_{h \in H} [\arg \min_{x \in C_1} h(x) = \arg \min_{x \in C_2} h(x)]$.
$Pr_{h \in H} [m^h_i = m^h_j] = J_{i,j}$. \[l\_thm\_prob\_collision\_jackard\] The proof is given in [@broder2000min], and in the appendix for completeness.
Similarly, regarding a hash $h$ from a $\gamma$-MWIF, [@broder1998resemblance; @broder2000min] shows that:
$|Pr_{h \in H} [m^h_i = m^h_j] - J_{i,j}| \leq \gamma$. \[l\_thm\_prob\_collision\_jackard\_approx\_mwif\]
Rather than maintaining the full $C_i$’s, previous approaches [@broder1998resemblance; @broder2000min] suggest maintaining their fingerprints. Given $k$ hashes $h_1,\ldots,h_k$ randomly chosen from an $\gamma$-MWIF, we can maintain $m^{h_1}_i,\ldots,m^{h_k}_i$. Given $C_i, C_j$, for any $x \in [k]$, the probability that $m^{h_x}_i = m^{h_x}_j$ is $J_{i,j} \pm \gamma$. A hash $h_x$ where we have $m^{h_x}_i = m^{h_x}_j$ is called a hash collision. We can thus estimate $J$ by counting the proportion of collision hashes out of all the chosen hashes. In this approach, the fingerprint contains $k$ item identities in $[u]$, since for any $x$, $m^{h_x}_i$ is in $[u]$. Thus, such a fingerprint requires $k \log u$ bits. To achieve an accuracy $\epsilon$ and confidence $\delta$, such approaches require $k = O ( \frac{\ln \frac{1}{\delta}}{\epsilon^2} )$. Our basis for the fingerprint is a “block fingerprint” which allows approximating $J_{i,j}$ with a given accuracy $\epsilon$ and a confidence of $\frac{7}{8}$. This block fingerprint maintains only a *single bit* per hash, as opposed to previous approaches which maintain $\log u$ bits per hash. Later we show how to achieve a given accuracy $\epsilon$ with a given confidence $\delta$, by combining several block fingerprints, and creating a full fingerprint.
To shorten the fingerprints using a single bit per hash, we use a hash mapping elements in $[u]$ to a single bit — $\phi : [u] \rightarrow \{0,1\}$, taken from a pairwise independent family (PWIF for short) of such hashes. Rather than defining $m^h_i = \arg \min_{x \in C_1} h(x)$ we define $m^{\phi,h}_i = \phi(\arg \min_{x \in C_1} h(x))$. Maintaining $m^{\phi,h}_i$ rather than $m^\phi_i$ shortens the fingerprint by a factor of $\log u$. We examine the resulting accuracy and confidence.
$Pr_{h \in H} [m^{\phi,h}_i = m^{\phi,h}_j] = \frac{J_{i,j}}{2} + \frac{1}{2} \pm \frac{\gamma}{2}$. \[l\_thm\_prob\_collision\_jackard\_approx\_mwif\_single\_bit\]
$Pr_{h\in H,\phi\in H'}[m_i^{\phi,h} = m_j^{\phi,h}] =
Pr[m_i^{\phi,h} = m_j^{\phi,h} | m_i^h = m_j^h] \cdot Pr_{h\in H}[m_i^{h} = m_j^{h}]+
Pr[m_i^{\phi,h} = m_j^{\phi,h} | m_i^h \neq m_j^h] \cdot Pr_{h\in H}[m_i^{h} \neq m_j^{h}]=
1 \cdot Pr_{h\in H}[m_i^{h} = m_j^{h}]+\frac{1}{2}\cdot(1-Pr_{h\in H}[m_i^{h} = m_j^{h}])=\frac{1+J_{i,j}\pm\gamma}{2}$
The purpose of the fingerprint block is to provide an approximation of $J$ with accuracy $\epsilon$. We use $k$ hashes, and choose $k=\frac{8.02}{\epsilon^2}$. Denote $\alpha = \frac{2^{10}-1}{2^{10}}$, and let $\gamma = (1-\alpha) \cdot \epsilon = \frac{1}{2^{10}} \epsilon$. We construct a $\gamma$-MWIF [^1]. To construct the family, consider choosing $a_0,\ldots,a_d$ and $b_0, b_1, \ldots, b_d$ uniformly at random from $[p]$, constructing the polynomials $f(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_d x^d$, $g(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_d x^d$, and using the $k$ hashes $h_i(x) = f(x) + i g(x)$, where $i \in \{0,1,\ldots,k-1\}$. We also use a hash $\phi : [u] \rightarrow \{0,1\}$ chosen from the PWIF of such hashes. We say there is a collision on $h_l$ if $m^{\phi,h_l}_i = m^{\phi,h_l}_j$, and denote the random variable $Z_l$ where $Z_l = 1$ if there is a collision on $h_l$ for users $i,j$ and $Z_l=0$ if there is no such collision. $Z_l = 1$ with probability $\frac{1}{2} + \frac{J}{2} \pm \frac{\gamma}{2}$ and $Z_l = 0$ with probability $\frac{1}{2} - \frac{J}{2} \pm \frac{\gamma}{2}$. Thus $E(Z_l) = \frac{1}{2} + \frac{J}{2} \pm \frac{\gamma}{2}$. Denote $X_l = 2 Z_l - 1$. $E(X_l) = 2 E(Z_l) - 1 = J \pm \gamma$. $X_l$ can take two values, $-1$ when $Z_l=0$, and $1$ when $Z_l=1$. Thus $X_l^2$ always takes the value of $1$, so $E(X_l^2)=1$. Consider $X=\sum_{l=1}^k X_l$, and take $Y=\hat{J}=\frac{X}{k}$ as an estimator for $J$. We show that for the above choice of $k$, $Y$ is accurate up to $\epsilon$ with probability of at least $\frac{7}{8}$.
$Pr(|Y-J| \leq \epsilon) \geq \frac{7}{8}$. Proof given in appendix. \[l\_thm\_jackard\_estimator\_single\_bit\]
Due to Theorem \[l\_thm\_jackard\_estimator\_single\_bit\], we can approximate $J$ with accuracy $\epsilon$ and confidence $\frac{7}{8}$ using a “block fingerprint” for $C_i$, composed of $m^{h_1, \phi_1}_i, \ldots, m^{h_k, \phi_k}_i$, where $h_1,\ldots,h_k$ are randomly constructed members of a $\gamma$-MWIF and $\phi_1,\ldots,\phi_k$ are chosen from the PWIF of hashes $\phi : [u] \rightarrow \{0,1\}$. We shows that it suffices to take $k=O(\frac{1}{\epsilon^2})$ to achieve this. Constructing each $h_i$ can be done by choosing $f,g$ using the base random construction and composing $h_i(x) = f(x) + i \cdot g(x)$. The base random construction chooses $f,g$ uniformly at random from $F_d$, the family of $d$-degree polynoms in ${\mathbb{Z}_p}$, where $d = O(\log \frac{1}{\epsilon})$. This achieves a $\gamma$-MWIF where $\gamma = (1-\alpha) \cdot \epsilon = \frac{1}{2^{10}} \epsilon$.
#### [Achieving a Desired Confidence]{}
We combine several *independent* fingerprints to increase the confidence to a desired level $\delta$. Section \[l\_sect\_pairwise\_block\_const\_conf\] used a fingerprint of length $k$ to achieve a confidence of $\frac{7}{8}$. Consider taking $m$ fingerprints for each stream, each of length $k$. Given two streams, $i,j$, we have $m$ pairs of fingerprints, each approximating $J$ with accuracy $\epsilon$, and confidence $\frac{7}{8}$. Denote the estimators we obtain as $\hat{J}_1,\hat{J}_2,\ldots,\hat{J}_m$, and denote the *median* of these values as $\hat{J}$. Consider using $m > \frac{32}{9} \ln \frac{1}{\delta}$ “blocks”.
$Pr(|\hat{J}-J| \leq \epsilon) \geq 1-\delta$. Proof given in appendix. \[l\_thm\_jackard\_median\_estimator\]
Due to Theorem \[l\_thm\_jackard\_median\_estimator\] to make sure that $|\hat{J} - J| \leq \epsilon$ it suffices to take $m > \frac{32}{9} \ln \frac{1}{\delta}$ fingerprints, each with $k=\frac{8.02}{\epsilon^2}$ hashes. In total, it is enough to take $\frac{32}{9} \ln \frac{1}{\delta} \cdot \frac{8.02}{\epsilon^2} \leq \frac{28.45 \ln \frac{1}{\delta}}{\epsilon^2}$ hashes. Thus, we use $O(\frac{\ln \frac{1}{\delta}}{\epsilon^2})$ hashes, storing a single bit per hash.
Fast Method for Computing the Fingerprint {#l_sect_fast_compute}
=========================================
We discuss speeding up the fingerprint computation. Consider computing the fingerprint for a set of $b$ items $X = \{x_1,\ldots,x_b\}$ where $x_i \in [u]$. The fingerprint is composed of $m$ “block fingerprints”, where block $r$ is constructed using $k$ hashes $h^r_1,\ldots,h^r_k$, built using $2 \cdot d$ random coefficients in ${\mathbb{Z}_p}$. The $i$’th location in the block is the minimal item in $X$ under $h_i$: $m_i = \arg \min_{x \in X} h_i(x)$, which is then hashed through a hash $\phi$ mapping elements in $[u]$ to a single bit. We show how to quickly compute the block fingerprint $(m_1,\ldots,m_k)$. A naive way to do this is applying $k \cdot b$ hashes to compute $h_i(x_j)$ for $i \in [k], j \in [b]$. The values $h_i(x_i)$ where $i \in [k], j \in [b]$ form a matrix, where row $i$ has the values $(h_i(x_1), \ldots, h_i(x_b))$, illustrated in Figure \[fig:fingerprint-chunk\].
![A fingerprint “chunk” for a stream.[]{data-label="fig:fingerprint-chunk"}](block-full.jpg){width="110mm" height="50mm"}
Once all $h_i(x_j)$ values are computed for $i \in [k], j \in [b]$ , for each row $i$ we check for which column $j$ the row’s minimal value occurs, and store $m_i = x_j$, as illustrated in the left of Figure \[fig:fingerprint-min-row\]. Thus, computing the fingerprint requires finding the minimal value across the rows (or more precisely, the value $x_j$ for the column $j$ where this minimal value occurs). To speed up the process, we use a method similar to the one discussed in [@pavan2008range] as a building block. Recall the hashes $h_i$ were defined as $h_i(x) = f(x) + i g(x)$ where $f(x),g(x)$ are $d$-degree polynomials with random coefficients in ${\mathbb{Z}_p}$. Our algorithm is based on a procedure that gets a value $x \in[u]$ and a threshold $t$, and returns all elements in $(h_0(x),h_1(x),\ldots,h_{k-1}(x))$ which are smaller than $t$, as well as their locations. Formally, the method returns the index list $I_t = \{ i | h_i(x) \leq t \}$ and the value list $V_t = \{ h_i(x) | i \in I_t \}$ (note these are lists, so the $j$’th location in $V_t$, $V_t[j]$, contains $h_{I_t[j]}(x)$). We call this the *column procedure*, and denote by $pr-small-loc(f(x),g(x),k,x,t)$ the function that returns $I_t$, and by $pr-small-val(f(x),g(x),k,x,t)$ the function that returns $V_t$ . We describe a certain implementation of these operations in Section \[fast\_pr\_compute\]. The running time of this implementation is $O(\log k + |I_t|)$, rather than the naive algorithm which evaluates $O(k)$ hashes. Thus, this procedure quickly finds small elements across columns (where by “small” we mean smaller than $t$). This is illustrated on the right of Figure \[fig:fingerprint-min-row\].
![Finding small elements across columns rather than minimal elements across rows[]{data-label="fig:fingerprint-min-row"}](block-row-col.jpg){width="110mm" height="40mm"}
Roughly speaking, our algorithm maintains a bound for the minimal value for each row, and operates by going through the columns, finding the small values in each of them, and updating the bounds for the rows where these occur.
$block-update( (x_1,\ldots,x_b), f(x), g(x), k, t):$
1. Let $m_i = \infty$ for $i \in [k]$
2. Let $p_i = 0$ for $i \in [k]$
3. For $j = 1$ to $b$:
1. Let $I_t = pr-small-val (f(x),g(x),k,x_j,t)$
2. Let $V_t = pr-small-loc (f(x),g(x),k,x_j,t)$
3. For $y\in I_t$: // Indices of the small elements
1. If $m_{I_t[y]} > V_t[y]$ // Update to row $x$ required
1. $m_{I_t[y]} = V_t[y]$
2. $p_{I_t[y]} = x_j$
If our method updates $m_i, p_i$ for row $i$, once the procedure is done, $m_i$ indeed contains the minimal value in that row, and $p_i$ the column where this minimal value occurs, since if even a single update occurred then the row indeed contains an item that is smaller than $t$, so the minimal item in that row is smaller than $t$ and an update would occur for that item. On the other hand, if all the items in a row are bigger than $t$, an update would not occur for that row. The running time of the column procedure is $O(\log k + |I_t|)$, which is a random variable, that depends on the number of elements returned for that column, $|I_t|$. Denote by $L_j$ the number of elements returned for column $j$ (i.e. $|I_t|$ for column $j$). Since we have $b$ columns, the running time of the block update is $O(b \log k) + O(\sum_{j=1}^b L_j)$. The total number of returned elements is $\sum_{j=1}^b L_j$, which is the total number of elements that are smaller than $t$. We denote by $Y_t = \sum_{j=1}^b L_j$ the random variable which is the number of all elements in the block that are smaller than $t$. The running time of our block update is thus $O(b \log k + Y_t)$.
The random variable $Y_t$ depends on $t$, since the smaller $t$ is the less elements are returned and the faster the column procedure runs. On the other hand, we only update rows whose minimal value is below $t$, so if $t$ is too low we have a high probability of having rows which are not updated correctly. We show that a certain compromise $t$ value allows achieving both a good running time of the block update, with a good probability of correctly computing the values for all the rows.
Given the threshold $t=\frac{12 \cdot p \cdot l'}{b}$, where $l' = 80 + 2 \log \frac{1}{\epsilon}$ (so $l'=O(\log \frac{1}{\epsilon})$), the runtime of the $block-update$ procedure is $O(b\log\frac{1}{\epsilon} + \frac{1}{\epsilon^2}\log\frac{1}{\epsilon})$. \[l\_thm\_runtime\_block\_good\_threshold\]
Recall that to get a $\gamma$-MWIF (for $\gamma = \frac{1}{2^{10}} \epsilon$) we used $d = O(\log\frac{1}{\gamma})$ as the degree of the random polynoms $f,g$ in the base random construction, used to compose the $h_1,\ldots,h_k$ hashes. Examining the constant in the work of Indyk [@indyk2001small] shows that the requirement is $d>80+2\log\frac{1}{\epsilon}$. Denote $l' = 80 + 2 \log \frac{1}{\epsilon}$. Due to our choice of $d$ we have $d > l'$, so the hashes $h_1,\ldots,h_k$ were effectively chosen at random from an $l'$-wise independent family. Let $H$ be an $l'-wise$ independent family of hashes. Consider the following equation from [@indyk2001small], regarding $E_t$, the expected number of elements $x \in X$ such that $h(x) \leq t$ (i.e. elements that are smaller than $t$ under $h$ chosen at random from $H$): $Pr[min_{x \in X} h(x) > t] \leq 48 \left( \frac{6 \cdot l'}{E_t} \right) ^{(l'-1)/2}$. When computing the fingerprint for the elements in $X$, we know $|X|$[^2] and denoted $|X|=b$. Each $h_i$ is $\gamma$-MWIF, so $E_t = \frac{tb}{p}$. Now consider choosing $t=\frac{12 \cdot p \cdot l'}{b}$. Under this choice[^3] of $t=\frac{12l'\cdot p}{b}$ we have $E_t = \frac{tb}{p} = 12l'$ and using the fact that $l' = 80 + 2 \log \frac{1}{\epsilon}$ the above lemma can be rewritten as: $Pr[min_{x \in X} h(x) > t]<48 \left( \frac{6l'}{E_t} \right) ^{(l'-1)/2} = 48 \cdot \left( \frac{1}{2} \right) ^ {\frac{79}{2}} \cdot \left( \frac{1}{2} \right) ^ {2\log \frac{1}{\epsilon}} < \frac{1}{2^{33}} \cdot \epsilon^2 $. There are $k$ rows, and by applying the union bound we obtain: $Pr[\exists i \in [k] ( min_{x \in X} h_i(x) > t] < \frac{k \cdot \epsilon }{2^{33}} = \frac{8.02 \cdot \epsilon^2 }{2^{8.9} \cdot \epsilon^2} < \frac{1}{2^{29}} $.
We prove our algorithm runs in time $O(b\log\frac{1}{\epsilon} + \frac{1}{\epsilon^2}\log\frac{1}{\epsilon})$ with high probability. We have $kb$ random values, $h_1(x_1),\ldots, h_{k-1}(x_b)$, which are (at least) pairwise independent. Denote $Y_{i,j}$ the indicator variable of the event that $h_j(x_i) < t = \frac{12pl'}{b}$, and so $Pr[Y_{i,j}=1]=\frac{12l'}{b}$ and $E[Y_{i,j}=1]=\frac{12l'}{b}$. Then $Y=\sum_{i=0}^b\sum_{j=0}^{k-1} Y_{i,j}$. The running time of the algorithm is $O(b\log\frac{1}{\epsilon} + Y)$. We show that $Y=O(\frac{1}{\epsilon^2}\log\frac{1}{\epsilon})$ with high probability[^4]. We obtain: $E[Y]=E[\sum_{i=0}^b\sum_{j=0}^{k-1} Y_{i,j}]=\sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}]=12 \cdot l' \cdot k$. We use the following lemma, proven in the appendix: $Var(Y) \leq E(Y)$, and using Chebychev’s inequality obtain: $Pr[Y>11E(Y)]\le Pr[|Y-E(Y)|>10Var(Y)]<\frac{1}{100}$. To guarantee the required run time in a worst case analysis, we can drop all the blocks which require too long to compute. This reduces our probability of success in each block from $\frac{7}{8}$ to at least $\frac{7}{8}-2^{-29}-\frac{1}{100}$ (The $2^{-29}$ factor is due to the probability that there exists a hash that gets a minimum value higher than $t$). Taking $4\log\frac{1}{\delta}$ blocks still obtains this probability. Overall the algorithm runs in time $O(b\log\frac{1}{\epsilon} + \frac{1}{\epsilon^2}\log\frac{1}{\epsilon})$ per block, or $O(\log\frac{1}{\delta}(b\log\frac{1}{\epsilon} + \frac{1}{\epsilon^2}\log\frac{1}{\epsilon}))$ for all blocks.
Computing The Minimal Elements of the Pseudo-Random Series {#fast_pr_compute}
----------------------------------------------------------
We give a recursive implementation of $pr-small-loc(f(x),g(x),k,x,t)$ and $pr-small-val(f(x),g(x),k,x,t)$, the procedures for computing $V_t$ and $I_t$. Recall the hashes $h_i$ were defined as $h_i(x) = f(x) + i g(x)$ where $f(x),g(x)$ are $d$-degree polynomials with random coefficients in ${\mathbb{Z}_p}$. Consider a given element $x \in {\mathbb{Z}_p}$ for which we attempt to find all the values (and indices) in $(h_0(x),h_2(x),\ldots,h_{k-1}(x))$ smaller than $t$. Given $x$, we can evaluate $f(x), g(x)$ in time $O(d) = O(\log\frac{1}{\gamma})$[^5], and denote $a = f(x) \in {\mathbb{Z}_p}$ and $b=g(x) \in {\mathbb{Z}_p}$. Thus, we are seek all values in $\{ a \mod p,(a+b) \mod p, (a+2b) \mod p,\ldots, (a+(k-1) b) \mod p \}$ smaller than $t$, and the indices $i$ where they occur. Consider the series $S = (s_1,\ldots,s_k)$ where $s_i = (a+ib) \mod p$ and $i = \{0,1,\ldots,k-1 \}$. We denote the arithmetic series $a+bi \mod p$ for $i \in \{0,1,\ldots,k-1\}$ as $S(a,b,k,p)$, so under this notation $S = S(a,b,k,p)$.
Given a value we can find the index where it occurs, and vice versa. To compute the value for index $i$, we compute $(a+ib) \mod p$. To compute the index $i$ where a value $v$ occurs, we solve $v=a+ib$ in ${\mathbb{Z}_p}$ (i.e. $i=\frac{v-a}{b}\mod p$). This can be done in $O(\log p)$ time using Euclid’s algorithm. Note we compute $b^{-1}$ in ${\mathbb{Z}_p}$ only once to transform *all* values to generating indices[^6]. We call a location $i$ where $s_i < s_{i-1}$ a *flip location*. The first index is a flip location if $a-b \mod p > a$. First, consider the case $b < \frac{p}{2}$. If $s_i$ is a flip location, we have $s_{i-1} < p$ but $s_{i-1} + b > p$, so $s_i < b$. Also, since $b < \frac{p}{2}$ there is at least one location which is *not* a flip location between any two flip locations. Given $S=S(a,b,k,p)$, denote by $f(S)$ the flip locations in $S$.
\[l\_lem\_flip\_small\] When $b < \frac{p}{2}$, at most $\frac{k}{2}$ elements are flip locations, and all elements that are smaller than $b$ are flip locations.
Note that the non-flip locations between any two flip locations are monotonically increasing. Any flip location has a value of at most $b$, since the element before a flip location is smaller than $p$ (modulo $p$), and adding $b$ to it exceeds $p$, but through this addition it is impossible to exceed $p$ by more than $b$.
We denoted by $f(S)$ the flip locations of $S$. Denote $f_0(S)=f(S)$. Denote by $f_1(S)$ all elements that occur directly after a flip location, $f_2(S)$ all elements that occur exactly two places after the closest flip locations (i.e they cannot be flip locations) and by $f_i(S)$ all elements that occur $i$ places after the closest flip location.
\[l\_lem\_element\_comp\] When $b < \frac{p}{2}$, if $x \in f_i(S)$ and $y \in f_j(S)$ where $i > j$, then $x > y$.
All flip locations have a value of at most $b$. Due to Lemma \[l\_lem\_flip\_small\], a location directly after a flip location is not a flip location, and is thus bigger than the flip location before it by exactly $b$, and is thus greater than $b$. Thus any element in $f_1(S)$ must be greater than any element in $f_0(S)$. Using the same argument, we see that any element in $f_2(S)$ is greater than any element in $f_1(S)$ and so on. A simple induction completes the proof.
The first flip location is ${\lceil{\frac{p-a}{b}}\rceil}$, as to exceed $p$ we add $b$ ${\lceil{\frac{p-a}{b}}\rceil}$ times. Also, the number of flip locations is ${\lfloor{\frac{a+bk}{p}}\rfloor}$. Denote the first flip location as $j={\lceil{\frac{p-a}{b}}\rceil}$, with value $a' = (a+jb) \mod p$. Denote $b' = (b-p) \mod b$ and the number of flip locations as $k' = {\lfloor{\frac{(a+bk)}{p}}\rfloor}$. The flip locations are known to also be an arithmetic progression [@pavan2008range] [^7].
\[l\_lem\_flip\_arit\_prog\] The flip locations of $S=S(a,b,k,p)$ are also an arithmetic progression $S'=(a',b',k', b)$.
Given the above lemmas, we can search for the elements smaller than $t$, by examining the flip locations series in recursion. If case $b<t$, given $q={\lceil{t}\rceil}{b}$, due to Lemma \[l\_lem\_element\_comp\] $f(S),f_1(S),\ldots f_{q-1}(S)$ are smaller then $t$, and all of their elements must be returned. We must also scan $f_q(S)$ and also return all the elements of $f_q(S)$ which are smaller then $t$. This additional scan requires $O(|f_q(S)|)$ time $|f_q(S)|\leq |f(S)|$. Thus this case of $b < t$ examines $O(|I_t|)$ elements. Due to Lemma \[l\_lem\_flip\_small\], if $b > t$, all non-flip locations are bigger than $b$ and thus bigger than $t$, and thus we must only consider the flip-locations as candidates. Using Lemma \[l\_lem\_flip\_arit\_prog\] we can scan the flip locations recursively by examining the arithmetic series of the flip locations. If at most half of the elements in each recursion are flip locations, this results in a logarithmic running time. However, if $b$ is high more than half the elements are flip locations. For the case where $b > \frac{p}{2}$ we can examine the same flip-location series $S'$, in reverse order. The first element in the reversed series would be the last element of the current series, and rather than progressing in steps of $b$, we progress in steps of $p-b$. This way we obtain exactly the same elements, but in reverse order. However, in this reversed series, at most half the elements are flip locations. The following procedure implements the above method. It finds elements smaller then $t$ in time $O(\log k) = O(\log\frac{1}{\epsilon}+|I_t|)$ where $|I_t|$ is the number of such values. Given the returned indices, we get the values in them. We use the same $b$ for all $|I_t|$, so this can be done in time $O(c\log c + |I_t|)$ (Usually $c$ is a constant). $ps-min(a,b,p,k,t):$
1. if $b<t$:
1. $V_t=[]$
2. if $a < t$ then $V_t=V_t+[a+ib \text{ for i in range } ({\lceil{\frac{t-a}{b}}\rceil})]$
3. $j={\lceil{\frac{p-a}{b}}\rceil}$ // First flip (excluding first location)
4. while $j<k$:
1. $v=(a+jb) \mod p$
2. while $j<k$ and $v<t$:
1. $V_t$.append(v)
2. $j=j+1$
3. $v=v+b$
3. $j=j+{\lceil{\frac{p-v}{b}}\rceil}$ //next flip location
4. return list1
5. if $b>\frac{p}{2}$ then return $f((a + (k-1) \cdot b) \mod p, p-b, p, k, t)$
6. $j={\lceil{\frac{p-a}{b}}\rceil}$
7. $new_k={\lfloor{\frac{a+bk}{p}}\rfloor}$
8. if $a<b$ then $j=0$ and $new_k=new_k+1$// calculate the first flip location and the number of flip locations
9. return $f( (a+jb) \mod p, -p \mod b, b, new_k, t )$
Conclusions {#sec:conclusions}
===========
We have presented a fast method for computing fingerprints of massive datasets, based on pseudo-random hashes. We note that although we have examined the Jackard similarity in detail, the exact same technique can be used for any fingerprint which is based on minimal elements under several hashes. Thus we have described a general technique for exponentially speeding up computation of such fingerprints. Our analysis has used fingerprints using a single bit per hash. We have shown that even for these small fingerprints which can be quickly computed, the required number of hashes is asymptotically similar to previously known methods, and is logarithmic in the required confidence and polynomial in the required accuracy. Several directions remain open for future research. Can we speed up the fingerprint computation even further? Can similar techniques be used for computing fingerprints that are not based on minimal elements under hashes?
Appendix: Proofs {#l_sect_proofs}
================
The proof of Theorem \[l\_thm\_prob\_collision\_jackard\]: $Pr_{h \in H} [m^h_i = m^h_j] = J_{i,j}$.
Denote $x=J_{1,2}$. The set $C_i \cup C_j$ contains three types of items: items that appear *only* in $C_i$, items that appear *only* in $C_j$, and items that appear in $C_i \cap C_j$. When an item in $C_i \cap C_j$ is minimal under $h$, i.e., for some $a \in C_i \cap C_j$ we have $h(a) = min_{x \in C_1 \cup C_2} h(x)$, we get that $min_{x \in C_i} h(x) = min_{x \in C_j} h(x)$. On the other hand, if for some $a \in C_i \cup C_j$ such that $a \notin C_i \cap C_j$ we have $h(a) = min_{x \in C_1 \cup C_2} h(x)$, the probability that $min_{x \in C_i} h(x) = min_{x \in C_j} h(x)$ is negligible [^8]. Since $H$ is MWIF, any element in $C = C_i \cup C_j$ is equally likely to be minimal under $h$. However, only elements in $I = C_i \cap C_j$ would result in $m^h_i = m^h_j$. Thus $Pr_{h \in H} [m^h_i = m^h_j] = \frac{1}{|C_i \cup C_j|} \cdot |C_i \cap C_j| = \frac{|C_i \cap C_j|}{|C_i \cup C_j|} = J_{i,j}$.
The proof of Theorem \[l\_thm\_jackard\_estimator\_single\_bit\] (Simple Estimator for Jackard With Single Bit Per Hash): $Pr(|Y-J| \leq \epsilon) \geq \frac{7}{8}$.
Our proof uses Chebychev’s inequality: $$Pr(|X-E(X)| \geq \epsilon) \leq \frac{Var(X)}{\epsilon^2}$$
We have: $$E(X)=E(\sum_{l=1}^k X_l)=\sum_{l=1}^k E(X_l) = k \cdot (J \pm \gamma)$$ $$(J-\gamma) \leq E(Y) \leq (J+\gamma)$$ We now bound $Var(X)$: $$\begin{split}
Var(X) &= E(X^2) - E^2(X) \\
&=E((\sum_{l=1}^k X_l)^2)-E^2(\sum_{l=1}^k X_l)\\
&=E(\sum_{l=1}^k X_l^2+2\sum_{i\neq j} X_i X_j) -(E(\sum_{l=1}^k X_l))^2\\
&=\sum_{l=1}^k E(X_l^2)+ 2\sum_{i\neq j} E(X_i X_j)-(\sum_{l=1}^k E(X_l))^2\\
&= \sum_{l=1}^k E(X_l^2)+ 2\sum_{i\neq j} E(X_iX_j)-(\sum_{l=1}^k E(X_l)^2 +2\sum_{i\neq j} E(X_i)E(X_j))\\
&=\sum_{l=1}^k E(X_l^2)+ 2\sum_{i\neq j} E(X_i)E(X_j)-(\sum_{l=1}^k E(X_l)^2 +2\sum_{i\neq j} E(X_i)(X_j))\\
&=\sum_{l=1}^k E(X_l^2)-\sum_{l=1}^k E(X_l)^2 \leq k
\end{split}$$
We use this to bound $Var(Y)$: $$Var(Y)=Var(\frac{1}{k} \cdot X)=\frac{1}{k^2} Var(X) \leq \frac{1}{k^2} \cdot k \leq = \frac{1}{k}$$
Using Chebychev’s inequality we get that: $$Pr(|Y-E(Y) | > \beta ) \leq \frac{Var(Y)}{\beta^2} \leq \frac{1}{k \cdot \beta^2}$$
Denote $\alpha = \frac{2^{10}-1}{2^{10}}$. Let $\beta = \alpha \cdot \epsilon$, so we obtain:
Thus using our choice of $k=\frac{8.02}{\epsilon^2}$ and $\beta = \alpha \cdot \epsilon$ (and noting that $J \leq 1, \epsilon \leq 1$) we have:
$$Pr(|Y-E(Y) | > \beta ) \leq \frac{1}{k {\beta}^2} =
\frac{1}{k \cdot \alpha^2 \cdot \epsilon^2} =
\frac{1}{8.0001} \leq \frac{1}{8}$$
Proof of Theorem \[l\_thm\_jackard\_median\_estimator\] (Median Estimator for Jackard): $Pr(|\hat{J}-J| \leq \epsilon) \geq 1-\delta$.
We use Hoeffding’s inequality [@hoeffding:1963]. Let $X_1,\ldots , X_n$ be independent random variables, where all $X_i$ are bounded so that $X_i \in [a_i, b_i]$, and let $X = \sum_{i=1}^n X_i$. Hoeffding’s inequality states that: $$\Pr(X - \mathrm{E}[X] \geq n \epsilon) \leq \exp \left( -\frac{2\,n^2\,\epsilon^2}{\sum_{i=1}^n (b_i - a_i)^2} \right)$$
We say that the estimator $\hat{J}_l$ is *good* if $|\hat{J}_l-J| \leq \epsilon$ and that $\hat{J}_l$ is *bad* if $|\hat{J}_l-J| > \epsilon$. Each estimator $\hat{J}_l$ is bad with probability of $p \leq \frac{1}{8}$. Consider the random variable $X_l$ where $X_l=1$ if $\hat{J}_l$ is bad, and $X_l=0$ if $\hat{J}_l$ is good. We have $Pr(X_l=1) = p \leq \frac{1}{8}$, so $E(X_l) = p \leq \frac{1}{8}$. Denote $X=\sum_{l=1}^m X_l$, so $E(X)=m \cdot p \leq \cdot m \cdot \frac{1}{8}$. We now note that the $\hat{J}$ can be bad only if at least half the estimators $\hat{J}_1,\ldots,\hat{J}_m$ are bad, or in other words, when $X \geq \frac{m}{2}$.
The $X_l$’s are independent, since for any $x,y$ the hashes used to obtain the $\hat{J}_x$ are independent of the hashes used to obtain the $\hat{J}_y$. Since $p \leq \frac{1}{8}$ we have: $$\Pr(X \geq \frac{m}{2}) \leq \Pr(X \geq (\frac{3}{8} + p) \cdot m) = \Pr(X - mp \geq \frac{3}{8} m)$$ However, $E(X) = mp$, so using Hoeffding’s inequality, we require that $\Pr(X \geq \frac{m}{2}) \leq \delta$: $$\Pr(X \geq \frac{m}{2}) \leq \Pr(X-mp \geq \frac{3}{8} m) \leq \exp (-2m \cdot \frac{9}{64}) \leq \delta$$
Extracting $m$ we obtain that we require: $$m > \frac{32}{9} \ln \frac{1}{\delta}$$
Proof of the lemma in Theorem \[l\_thm\_runtime\_block\_good\_threshold\]:
Let $Y=\sum_{i=0}^b\sum_{j=0}^{k-1} Y_{i,j}$ in Theorem \[l\_thm\_runtime\_block\_good\_threshold\]. Then $Var[Y] \leq E[Y]$.
$$\begin{split}
Var[Y] &= E[Y^2]-E^2[Y] = E[(\sum_{i=0}^b\sum_{j=0}^{k-1} Y_{i,j})^2]-E^2[\sum_{i=0}^b\sum_{j=0}^{k-1} Y_{i,j}] \\
&= E[\sum_{i=0}^b\sum_{j=0}^{k-1} Y_{i,j}^2+2\sum_{i'\neq i}\sum_{j'\neq j} Y_{i,j} Y_{i',j'}] -(\sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}])^2 \\
&= \sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}^2]+2\sum_{i'\neq i}\sum_{j'\neq j} E[Y_{i,j} Y_{i',j'}] -(\sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}]^2+2\sum_{i'\neq i}\sum_{j'\neq j} E[Y_{i,j}][Y_{i',j'}]) \\
&= \sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}^2]+2\sum_{i'\neq i}\sum_{j'\neq j} E[Y_{i,j}][Y_{i',j'}] -(\sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}]^2+2\sum_{i'\neq i}\sum_{j'\neq j} E[Y_{i,j}][Y_{i',j'}]) \\
&= \sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}^2]-\sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}]^2 = \sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}]-\sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}]^2 \\
&= E[Y]-\sum_{i=0}^b\sum_{j=0}^{k-1} E[Y_{i,j}]^2\le E[Y]
\end{split}$$
[^1]: The accuracy $\gamma$ is much stronger than the overall accuracy $\epsilon$ required of the full fingerprint, for reasons to be later examined
[^2]: We use this assumption for simplicity. If we don’t know $|X|$, we can update the threshold $t$ online. We store all elements until we have $\frac{\log\frac{1}{\delta}}{\epsilon^2}$ elements. Then we set $t$ according to $b=2\frac{\log\frac{1}{\delta}}{\epsilon^2}$. We double $b$ by $2$ each time $|X|>b$ and update $t$ according to the new $b$.
[^3]: Notice that this constant is only to bound the worst case usually in a block the maximum between the minimal values is about $l'$ moreover we can improve the running time if we drop from the sketch all the hash functions which there minimal value is to big.
[^4]: We base our calculation on the pairwise independence of $Y_{i,j}$. Notice that $Y_{i,j}$ is more independent when running over $i$. Therefor in practice the constants are smaller.
[^5]: Using multipoint evaluation we can calculate it in amortized time $O(\log^2\log\frac{1}{\gamma})$. Moreover we can use other constructions for $d$-wise independent which can be evaluate in $O(1)$ time in the cost of using more space.
[^6]: We can store a table of inverse to further reduce processing time. If the required memory for the table is unavailable, we can do the computation in $F_{p^c}$ for smaller $p$ and store table of size $p$ and then calculating the inverse requires $O(c\log c)$ time. Notice that we can easily take $c<\log_{\frac{\log\frac{1}{\delta}}{\epsilon^2}} u$ which will probably be less then $\log\frac{1}{\epsilon}$
[^7]: See Lemma 2 page 11.
[^8]: Such an event requires that two *different* items, $x_i \in C_i$ and $x_j \in C_j$ would be mapped to the same value $h^* = h(x_i) = h(x_j)$, and that this value would also be the minimal value obtained when applying $h$ to both all the items in $C_i$ and in $C_j$. As discussed in [@indyk2001small], the probability for this is negligible when the range of $h$ is large enough.
|
---
abstract: |
It is shown that if $w(z)$ is a finite-order meromorphic solution of the equation $$H(z,w) P(z,w)=Q(z,w),$$ where $P(z,w)=P(z,w(z),w(z+c_1),\ldots,w(z+c_n))$, $c_1,\ldots,c_n\in{\mathbb{C}}$, is a homogeneous difference polynomial with meromorphic coefficients, and $H(z,w)=H(z,w(z))$ and $Q(z,w)=Q(z,w(z))$ are polynomials in $w(z)$ with meromorphic coefficients having no common factors such that $$\begin{split}
&\max\{\deg_w(H),\deg_w(Q)-\deg_w(P)\}> \min\{\deg_w(P),{\textrm{ord}}_0(Q)\}-{\textrm{ord}}_0(P),
\end{split}$$ where ${\textrm{ord}}_0(P)$ denotes the order of zero of $P(z,x_0,x_1,\ldots,x_n)$ at $x_0=0$ with respect to the variable $x_0$, then the Nevanlinna counting function $N(r,w)$ satisfies $N(r,w)\not=S(r,w)$. This implies that $w(z)$ has a relatively large number of poles. For a smaller class of equations a stronger assertion $N(r,w)=T(r,w)+S(r,w)$ is obtained, which means that the pole density of $w(z)$ is essentially as high as the growth of $w(z)$ allows. As an application, a simple necessary and sufficient condition is given in terms of the value distribution pattern of the solution, which can be used as a tool in ruling out the possible existence of special finite-order Riccati solutions within a large class of difference equations containing several known difference equations considered to be of Painlevé type.
address: 'Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland'
author:
- Risto Korhonen
title: A new Clunie type theorem for difference polynomials
---
Introduction and main results
=============================
According to Clunie’s theorem [@clunie:62], if a meromorphic function $f$ satisfies the differential equation $$\label{clunieeq}
f^nP(z,f)=Q(z,f),$$ where $n\in{\mathbb{N}}$, and $P(z,f)$ and $Q(z,f)$ are differential polynomials in $f$ with meromorphic coefficients such that $\deg_f
Q(z,f)\leq n$, then the Nevanlinna proximity function $m(r,\cdot\,)$ satisfies $$m(r,P(z,f))=O(\log r + \log T(r,f)+ \mathcal{T}(r))\nonumber$$ where $r$ approaches infinity outside of a set of finite linear measure, and $\mathcal{T}(r)$ is the maximum of the Nevanlinna characteristics of the coefficients of $P(z,f)$ and $Q(z,f)$. Originally Clunie used his result to consider certain properties of entire and meromorphic functions, and later on Clunie’s theorem and its subsequent generalizations, see, e.g., [@doeringer:82; @lahirib:04; @yangy:07], have proven to be valuable tools in the study of value distribution of meromorphic solutions of Painlevé, and other non-linear differential equations, see, e.g., [@laine:93; @gromakls:02].
Ablowitz, Halburd and Herbst suggested that the existence of sufficiently many finite-order meromorphic solutions is a unique characteristic of a Painlevé type difference equation [@ablowitzhh:00]. In [@halburdk:07PLMS] it was shown that the existence of one finite-order meromorphic solution is enough to reduce the second order difference equation $$\label{class}
w(z+1)+w(z-1)=R(z,w),$$ where $R(z,w)$ is rational in $w(z)$ with meromorphic coefficients, into a list of equations consisting only of difference Painlevé equations and linear equations within the class (\[class\]), provided that the finite-order solution $w(z)$ does not simultaneously satisfy a difference Riccati equation $$w(z+1)=\frac{a_1(z)w(z)+a_0(z)}{b_1(z)w(z)+b_0(z)}$$ where the coefficients are meromorphic functions (small with respect to $w$ in the sense of Nevanlinna theory) such that $a_1b_0\not\equiv a_0b_1$. An essential part of the method used in this classification is based on a local analysis of the behavior of a meromorphic solution near its poles, which can only be performed non-vacuously if there are sufficiently many poles to begin with. High pole density of solutions can be verified by applying a direct difference analogue of Clunie’s theorem [@halburdk:06JMAA] (concerning the equation $f^nP(z,f)=Q(z,f)$ where $P(z,f)$ and $Q(z,f)$ are difference polynomials, and $f$ is of finite order).
The class (\[class\]) contains many equations considered to be of Painlevé type, including three alternate versions of difference Painlevé I, and a difference Painlevé II. Nevertheless, most of the difference Painlevé equations fall outside of class (\[class\]) [@grammaticosnr:99]. For instance, a known discretization of the Painlevé III equation, $$\label{dpiii}
w(z+1)w(z-1)=\frac{\gamma(z+1)w(z)^2+\zeta(z)\lambda^zw(z)+\mu(z)\lambda^{2z}}{(w(z)-1)(w(z)-\gamma(z))},$$ where $\lambda\in{\mathbb{C}}$, $\gamma$ and $\zeta$ are periodic meromorphic functions with period two, and $\mu$ is a period one meromorphic function, not only lies outside of the class (\[class\]) but the Clunie difference analogue in [@halburdk:06JMAA] is inapplicable for this equation. This causes a difficulty in making sure that solutions have enough poles so that the local analysis needed for the classification can be performed. Theorem \[laineyangthm\] below by I. Laine and C. C. Yang is a generalization of [@halburdk:06JMAA Theorem 3.1] and applicaple for the equation (\[dpiii\]).
At this point we pause briefly to introduce the notation used in this paper. Let $c_j\in{\mathbb{C}}$ for $j=1,\ldots,n$ and let $I$ be a finite set of multi-indexes $\lambda=(\lambda_0,\ldots,\lambda_n)$. A difference polynomial of a meromorphic function $w(z)$ is defined as $$\label{Pzw}
\begin{split}
P(z,w)&=P(z,w(z),w(z+c_1),\ldots,w(z+c_n))\\ & =\sum_{\lambda\in I}
a_\lambda(z)w(z)^{\lambda_0} w(z+c_1)^{\lambda_1} \cdots
w(z+c_n)^{\lambda_n},
\end{split}$$ where the coefficients $a_\lambda(z)$ are *small* with respect to $w(z)$ in the sense that $T(r,a_\lambda)=o(T(r,w))$ as $r$ tends to infinity outside of an exceptional set $E$ of finite logarithmic measure $$\label{finlogmeas}
\lim_{r\to\infty}\int_{E\cap[1,r)} \frac{dt}{t}<\infty.$$ From now on we use an abbreviated notation $\int_E dt/t<\infty$ instead of to denote finite logarithmic measure. The notation $T(r,a_\lambda)=S(r,w)$ is also used to indicate that the characteristic function of $a_\lambda(z)$ is small with respect to the characteristic of $w(z)$. The *total degree* of $P(z,w)$ in $w(z)$ and the shifts of $w(z)$ is denoted by $\deg_w(P)$, and the *order of a zero* of $P(z,x_0,x_1,\ldots,x_n)$, as a function of $x_0$ at $x_0=0$, is denoted by ${\textrm{ord}}_0(P)$. (For instance, if $P(z,w)=w(z)^2+w(z+1)w(z)$, then $\deg_w(P)=2$ and ${\textrm{ord}}_0(P)=1$). Moreover, the *weight* of a difference polynomial is defined by $$\kappa(P)=\max_{\lambda\in I}\left\{\sum_{j=1}^n \lambda_j\right\},$$ where $\lambda=(\lambda_0,\ldots,\lambda_n)$, and the set $I$ is the same as in (\[Pzw\]) above. By this definition the weight of the polynomial $w(z+1)w(z-1)+w(z)w(z-1)+w(z)w(z+1)$ is two, for instance. The difference polynomial $P(z,w)$ is said to be *homogeneous* with respect to $w(z)$ if the degree $d_\lambda=\lambda_0+\cdots+\lambda_n$ of each term in the sum (\[Pzw\]) is non-zero and the same for all $\lambda\in I$. Finally, the *order of growth* of a meromorphic function $w$ is defined by $$\rho(w)=\limsup_{r\to\infty}\frac{\log T(r,w)}{\log r}.$$ Notation and fundamental results from Nevanlinna theory are frequently used throughout this paper, see, e.g., [@cherryy:01; @goldbergo:08; @hayman:64].
\[laineyangthm\] Let $f$ be a transcendental meromorphic solution of finite order $\rho$ of a difference equation of the form $$U(z, f)P(z, f) = Q(z, f),\nonumber$$ where $U(z, f)$, $P(z, f)$ and $Q(z, f)$ are difference polynomials such that the total degree $\deg_f U(z, f) = n$ in $f(z)$ and its shifts, and $\deg_fQ(z, f) \leq n$. If $U(z, f)$ contains just one term of maximal total degree in $f(z)$ and its shifts, then, for each $\varepsilon>0$, $$m(r, P(z, f)) = O(r^{\rho-1+\varepsilon}) + S(r, f),\nonumber$$ possibly outside of an exceptional set of finite logarithmic measure.
With the help of Theorem \[laineyangthm\] the full classification of $$\label{class2}
w(z+1)w(z-1)=R(z,w)$$ containing (\[dpiii\]) has been completed in [@ronkainen:09]. Although Theorem \[laineyangthm\] covers a large class of equations, the equation known as the difference Painlevé IV (d-P$_{IV}$) is not one of them (see [@ramanigh:91] for a discretization of the Painlevé IV equation). One of the aims of this paper is to prove the following alternative version of the difference Clunie lemma for a class of difference equations which includes d-P$_{IV}$.
\[1stthm\] Let $w(z)$ be a finite-order meromorphic solution of $$\label{cleq0}
H(z,w) P(z,w)=Q(z,w),$$ where $P(z,w)$ is a homogeneous difference polynomial with meromorphic coefficients, and $H(z,w)$ and $Q(z,w)$ are polynomials in $w(z)$ with meromorphic coefficients having no common factors. If $$\label{cleqassumpt}
\begin{split}
&\max\{\deg_w(H),\deg_w(Q)-\deg_w(P)\}> \min\{\deg_w(P),\emph{{\textrm{ord}}}_0(Q)\}-\emph{{\textrm{ord}}}_0(P),
\end{split}$$ then $N(r,w)\not=S(r,w)$.
The expression $N(r,w)\not=S(r,w)$ means that the pole counting function of $w$ is not small with respect to the characteristic function of $w$. In other words, there exists an absolute constant $K\in (0,1]$ and a set $E$ of infinite logarithmic measure, such that $N(r,w)\geq K\,T(r,w)$ for all $r\in E$.
The class of equations (\[cleq0\]) contains many difference equations considered to be of Painlevé type, including equations known as difference Painlevé I–IV, for suitable choices of the polynomials $H$, $P$ and $Q$. In Section \[sec2\] below we will consider a class of equations within (\[cleq0\]) containing d-P$_{IV}$ as an example of an application of Theorem \[1stthm\].
By adding a constraint to the degrees and weights of the difference polynomials in Theorem \[1stthm\], the following stronger assertion is obtained.
\[2ndthm\] Let $w(z)$ be a finite-order meromorphic solution of $$\label{cleq2}
H(z,w) P(z,w)=Q(z,w),$$ where $P(z,w)$ is a homogeneous difference polynomial with meromorphic coefficients, and $H(z,w)$ and $Q(z,w)$ are polynomials in $w(z)$ with meromorphic coefficients having no common factors. If $$\label{condcleq2}
2\kappa(P) \leq \max\{\deg_w(Q),\deg_w(H)+\deg_w(P)\}-\min\{\deg_w(P),\emph{{\textrm{ord}}}_0(Q)\},$$ then, for any $\delta\in(0,1)$, $$\label{mvsT}
m(r,w)=o\left(\frac{T(r,w)}{r^\delta}\right) + O(\mathcal{T}(r)),$$ where $r$ runs to infinity outside of an exceptional set of finite logarithmic measure, and $\mathcal{T}(r)$ is the maximum of the Nevanlinna characteristics of the coefficients of $P(z,w)$, $Q(z,w)$ and $H(z,w)$.
Theorems \[1stthm\] and \[2ndthm\] provide some of the necessary tools needed to single out Painlevé type equations out of the difference equation $$\label{exampleeqIV}
\overline{w}\underline{w}+\overline{w}w+w\underline{w}=\frac{a_3w^3+a_2w^2+a_1w+a_0}{(w-b)(w-c)}$$ where the coefficients are rational functions, $a_0\not\equiv0$, and we have suppressed the $z$-dependence of $w(z)$ by writing $\overline{w}\equiv w(z+1)$, $\underline{w}\equiv w(z-1)$ and $w\equiv w(z)$. Namely, by taking $H(z,w)=(w-b)(w-c)$, $Q(z,w)=a_3w^3+a_2w^2+a_1w+a_0$ and $P(z,w)=\overline{w}\underline{w}+\overline{w}w+w\underline{w}$ in Theorem \[2ndthm\], it follows that $\kappa(P)=2$, $\deg_w(Q)=3$, $\deg_w(P)=2$, $\deg_w(H)=2$ and ${\textrm{ord}}_0(Q)=0$. Hence, the assumption is satisfied, and so $m(r,w)=S(r,w)$ by Theorem \[2ndthm\]. Therefore, $N(r,w)=T(r,w)+S(r,w)$, and all non-rational finite-order meromorphic solutions of have nearly as many poles as their growth enables. In particular, this is true for all non-rational finite-order meromorphic solutions of d-P$_{IV}$, since this equation is a special case of .
Applications to difference Riccati equation {#sec2}
===========================================
The Painlevé property has proved to be a good detector of integrability in differential equations [@ablowitzc:91]. In the beginning of the $20^\textrm{th}$ century, Painlevé, Gambier and Fuchs identified all those equations that possess this property out of a large class of second-order ordinary differential equations [@fuchs:05; @gambier:10; @painleve:00; @painleve:02]. All of the equations could be solved in terms of previously known functions, solutions of linear equations, or in terms of solutions of one of six new equations, now known as the Painlevé equations. The Painlevé equations were later on integrated by using inverse scattering transform techniques, see, e.g., [@ablowitzs:77]. In the first order case Malmquist [@malmquist:13] has shown that the existence of one meromorphic solution of the differential equation $$\label{malmq}
w'=R(z,w),$$ where $R(z,w)$ is rational in both arguments, reduces into a Riccati equation. A simple proof of this fact was given later on by Yosida [@yosida:33] using techniques from Nevanlinna theory.
There are several candidates for the discrete Painlevé property, including the singularity confinement by Grammaticos, Ramani and Papageorgiou [@grammaticosrp:91], algebraic entropy by Hietarinta and Viallet [@hietarintav:98], the existence of sufficiently many finite-order meromorphic solutions by Ablowitz, Halburd and Herbst [@ablowitzhh:00], and Diophantine integrability by Halburd [@halburd:05]. As mentioned in the introduction, Halburd and the author [@halburdk:07PLMS] showed that the existence of one finite-order meromorphic solution growing faster than the coefficients is sufficient to reduce the second order difference equation (\[class\]), where $R(z,w)$ is rational in $w(z)$ with meromorphic coefficients, into a list of equations consisting of difference Painlevé equations and linear equations within the class (\[class\]), provided that the finite-order solution $w(z)$ does not simultaneously satisfy a difference Riccati equation. Most of the equations of Painlevé type - both continuous and discrete - are known to possess special solutions satisfying a first order Riccati equation for particular choices of their parameter values [@gromakls:02; @tamizhmanitgr:04]. Although this property is considered to be one of the typical characteristics of Painlevé equations, Riccati type solutions also appear as special solutions of non-integrable equations. Therefore, in the Ablowitz-Halburd-Herbst approach, the existence of any number of finite-order meromorphic solutions is insufficient to indicate integrability of a difference equation, if these solutions happen to be simultaneously solutions to a Riccati equation. At the same time, the existence of already one non-Riccati finite-order solution appears to indicate a Painlevé type difference equation.
The purpose of this section is to give a simple necessary and sufficient condition which can be used to rule out the possible existence of special finite-order Riccati type solutions within a large class of difference equations. The condition is formulated in terms of the value distribution pattern of the considered meromorphic solution near its poles, which is straightforward to work out for most difference equations.
\[mainthm\] Let $a(z)$ and $c(z)$ be rational functions, and let $w(z)$ be a non-rational finite-order meromorphic solution of $$\label{cleq}
H(z,w) P(z,w)=Q(z,w),$$ where $P(z,w)$ is a homogeneous difference polynomial with respect to $w(z)$ having rational coefficients, and $H(z,w)$ and $Q(z,w)$ are polynomials in $w(z)$ with rational coefficients having no common factors. If $$\label{cond2}
2\kappa(P) \leq \max\{\deg_w(Q),\deg_w(H)+\deg_w(P)\}-\min\{\deg_w(P),\emph{{\textrm{ord}}}_0(Q)\},$$ then the following statements are equivalent:
- There exists a positive integer $k_{\hat{z}}$ and complex constants $\alpha_{\hat{z}}$, $\beta_{\hat{z}}\not=0$ and $\gamma_{\hat{z}}$ such that, at all except at most finitely many poles $\hat{z}$ of $w(z)$, $$\begin{aligned}
w(z-1)&=&c(z-1)+\alpha_{\hat{z}}(z-\hat{z})^{k_{\hat{z}}}+O((z-\hat{z})^{k_{\hat{z}}+1})\\
w(z)&=&\beta_{\hat{z}}(z-\hat{z})^{-k_{\hat{z}}}+O((z-\hat{z})^{1-k_{\hat{z}}})\\
w(z+1)&=&a(z)+\gamma_{\hat{z}}(z-\hat{z})^{k_{\hat{z}}}+O((z-\hat{z})^{k_{\hat{z}}+1})
\end{aligned}$$ for all $z$ in a neighborhood of $\hat{z}$.
- The function $w(z)$ is a solution of the difference Riccati equation $$\label{Riccati0}
w(z+1)=\frac{a(z)w(z)+b(z)}{w(z)-c(z)},$$ where $b(z)$ is a meromorphic function having at most finitely many poles, and satisfying $\rho(b)\leq\max\{0,\rho(w)-1\}$.
To demonstrate the use of Theorem \[mainthm\], we consider the (in general non-integrable) non-autonomous difference equation $$\label{exampleeq0}
\overline{w}+\underline{w}=\frac{a_2 w^2+a_1 w +a_0}{w^2}$$ which contains a known discrete form of the Painlevé I (see, e.g., [@fokasgr:93]) as a special case. Assuming that $w$ is a non-rational finite-order meromorphic solution of (\[exampleeq0\]) where the coefficients are rational functions and $a_0\not\equiv0$, it follows by applying Theorem \[2ndthm\] with $P(z,w)=\overline{w}+\underline{w}$, $H(z,w)=w^2$ and $Q(z,w)=a_2w^2+a_1w+a_0$ that $m(r,w)=S(r,w)$. Suppose that $w$ has only finitely many poles. Then, since $N(r,w)=O(\log r)$ and $T(r,w)=N(r,w)+S(r,w)$, it follows that $T(r,w)=S(r,w)+O(\log r)$. But this is impossible due to the fact that $w$ is non-rational, and so $w$ has infinitely many poles. Suppose now that $w$ has finitely many zeros. Then $N(r,1/w)=O(\log r)$, and, by Lemma \[logdiff\] below, $$\begin{split}
m(r,\overline{w}+\underline{w})&=m\left(r,\frac{w(\overline{w}+\underline{w})}{w}\right)
\leq m(r,w)+m\left(r,\frac{\overline{w}+\underline{w}}{w}\right)\\ &= m(r,w)+S(r,w)=S(r,w).
\end{split}$$ Therefore, by the Valiron-Mohon’ko identity [@valiron:31; @mohonko:71] (see also [@goldbergo:08 Theorem 6.5] and [@laine:93 Theorem 2.2.5]), it follows that $$\begin{split}
2T(r,w)&=T(r,\overline{w}+\underline{w})+O(\log r)\\
&=m(r,\overline{w}+\underline{w})+N(r,\overline{w}+\underline{w}) +O(\log r)\\
&=N(r,\overline{w}+\underline{w})+S(r,w)+O(\log r).\\
\end{split}$$ Since by , $$N(r,\overline{w}+\underline{w})=N\left(r,\frac{1}{w^2}\right)+O(\log r)
=2N\left(r,\frac{1}{w}\right)+O(\log r)=O(\log r),$$ it follows that $T(r,w)=S(r,w)+O(\log r)$ which is impossible. Therefore $w$ has also infinitely many zeros. By equation (\[exampleeq0\]) it follows that whenever $w$ has a zero of multiplicity $k_0$ at $z=z_0$, then $w$ has a pole at least of multiplicity $2k_0$ at $z=z_0+1$ or $z=z_0-1$. By Theorem \[mainthm\] these poles are not of the type allowed for the solution of a Riccati difference equation. Since we have shown that $w$ has infinitely many such poles, it follows by Theorem \[mainthm\] that $w$ cannot be a solution of the Riccati difference equation (\[Riccati0\]).
As another example we consider the equation $$\label{exampleeq}
\overline{w}\underline{w}+\overline{w}w+w\underline{w}=\frac{a_3w^3+a_2w^2+a_1w+a_0}{(w-b)(w-c)}$$ where the coefficients are rational functions, and $a_0\not\equiv0$. Equation (\[exampleeq\]) contains a known discrete form of the Painlevé IV (see, e.g., [@ramanigh:91]) as a special case, which is known to have special difference Riccati solutions. If $w$ is a non-rational finite-order solution of (\[exampleeq\]), then, similarly as above, it follows by applying Theorem \[2ndthm\] with $P(z,w)=\overline{w}\underline{w}+\overline{w}w+w\underline{w}$, $H(z,w)=(w-b)(w-c)$ and $Q(z,w)=a_3w^3+a_2w^2+a_1w+a_0$ that there are infinitely many points $\hat{z}$ where $w(\hat{z})=\infty$. By substituting a suitable Laurent series expansion into (\[exampleeq\]) it follows that all except possibly finitely many of these poles appear as a part of one of the following sequences: $$\begin{aligned}
&& w(\hat{z}-1)=b(\hat{z}-1),\quad w(\hat{z})=\infty, \quad w(\hat{z}+1)=a_3(\hat{z})-b(\hat{z}-1) \label{seq1}\\
&& w(\tilde{z}-1)=c(\tilde{z}-1),\quad w(\tilde{z})=\infty, \quad w(\tilde{z}+1)=a_3(\tilde{z})-c(\tilde{z}-1) \label{seq2}\\
&& w(\breve{z}-1)=a_3(\breve{z})-b(\breve{z}+1),\quad w(\breve{z})=\infty, \quad w(\breve{z}+1)=b(\breve{z}+1) \label{seq3}\\
&& w(\acute{z}-1)=a_3(\acute{z})-c(\acute{z}+1),\quad w(\acute{z})=\infty, \quad w(\acute{z}+1)=c(\acute{z}+1) \label{seq4}\\
&& w(\grave{z}-1)=\infty,\quad w(\grave{z})=K_{\grave{z}},\quad w(\grave{z}+1)=\infty
\label{seq5},
\end{aligned}$$ where $\hat{z}, \tilde{z}, \breve{z}, \acute{z}, \grave{z} \in {\mathbb{C}}$ and $K_{\grave{z}}\in{\mathbb{C}}\cup \{\infty\}$. (The possible finitely many exceptional poles which are not one of the types (\[seq1\])–(\[seq4\]) arise from the poles of the coefficients.) According to Theorem \[mainthm\], the function $w(z)$ is a special Riccati solution of (\[exampleeq\]) if and only if all except possibly finitely many poles of $w(z)$ are of exactly one of the types (\[seq1\])–(\[seq4\]).
Proofs of theorems {#proof1}
==================
We begin by stating a known difference analogue of the lemma on the logarithmic derivative.
\[logdiff\] Let $f$ be a non-constant meromorphic function of finite order, $c\in{\mathbb{C}}$ and $\delta\in(0,1)$. Then $$\label{diff}
m\left(r,\frac{f(z+c)}{f(z)}\right)= o\left(\frac{T(r,f)}{r^\delta}\right)$$ where $r$ approaches infinity outside of a possible exceptional set $E$ with finite logarithmic measure $\int_E\frac{dr}{r}<\infty$.
See [@chiangf:08 Corollary 2.5] for an alternative version of Lemma \[logdiff\]. The following generalization of [@halburdk:07PLMS Lemma 2.1] is needed in the proof Theorem \[2ndthm\].
\[technical\] Let $T:[0,+\infty)\to[0,+\infty)$ be a non-decreasing continuous function, let $\delta\in(0,1)$, and let $s\in(0,\infty)$. If $T$ is of finite order, i.e., $$\label{assu}
\limsup_{r\to\infty}\frac{\log T(r)}{\log r}<\infty,$$ then $$T(r+s) = T(r)+ o\left(\frac{T(r)}{r^\delta}\right)\nonumber$$ where $r$ runs to infinity outside of a set of finite logarithmic measure.
Note that by using [@chiangf:08 Theorem 2.2 and Corollary 2.5] instead of Lemma \[logdiff\] the proof of Theorem \[2ndthm\] could be simplified in the sense that Lemma \[technical\] would no longer be required. However, this would change the assertion of Theorem \[2ndthm\] into $$\label{cf}
m(r,w)=O(r^{\rho(w)-1+\varepsilon}) + O(\mathcal{T}(r)),$$ where $\rho(w)$ is the order of $w$, $\varepsilon>0$ and $r>0$. Unfortunately does not necessarily imply that $m(r,w)$ is small compared to $T(r,w)$ for all, or even most values of $r$. Namely, if the *lower order* of $w$, defined by $$\mu(w)=\liminf_{r\to\infty}\frac{\log T(r,w)}{\log r}$$ satisfies $\mu(w)<\rho(w)-1$, then $T(r,w)<r^{\rho(w)-1}$ in a significant (and in some cases the largest) part of $\mathbb{R}_+$. Therefore, for these particular values of $r$, equation gives no information on the relative size of $m(r,w)$ compared to $T(r,w)$, and so the set where $m(r,w)$ may not be small compared to $T(r,w)$ in can be much larger than the exceptional set in . On the other hand, if the growth of $w$ is assumed to be sufficiently regular in the sense that $\mu(w)>\rho(w)-1$, then implies that $m(r,w)=o(T(r,w)) + O(\mathcal{T}(r))$ without an exceptional set.
*Proof of Lemma \[technical\]:* Denote $\nu(r)=1-r^{-\delta}$, and assume conversely to the assertion that the set $F\subset{\mathbb{R}}^{+}$ of all $r$ such that $$T(r) \leq \nu(r) T(r+s)\nonumber$$ is of infinite logarithmic measure. Set $r_n=\min\{F\cap
[r_{n-1}+s,\infty)\}$ for all $n\in{\mathbb{N}}$, where $r_0$ is the smallest element of $F$. Then the sequence $\{r_n\}_{n\in{\mathbb{N}}}$ satisfies $r_{n+1}-r_n\geq s$ for all $n\in{\mathbb{N}}$, $F\subset
\bigcup_{n=0}^\infty [r_n,r_n+s]$ and $$\label{assuinpr}
T(r_n) \leq \nu(r_n) T(r_{n+1})$$ for all $n\in{\mathbb{N}}$.
Let $\varepsilon\in(0,\delta^{-1}-1)$, and suppose that there exist an $m\in{\mathbb{N}}$ such that $r_n\geq n^{1+\varepsilon}$ for all $r_n\geq m$. But then, $$\begin{aligned}
\int_F\frac{dt}{t} &\leq& \sum_{n=0}^\infty
\int_{r_n}^{r_n+s}\frac{dt}{t}
\leq \int_1^{m} \frac{dt}{t} + \sum_{n=1}^\infty
\log\left(1+\frac{s}{r_n}\right)\\
&\leq& \sum_{n=1}^\infty
\log\left(1+s n^{-(1+\varepsilon)}\right) +O(1) <\infty
\end{aligned}$$ which contradicts the assumption $\int_F\frac{dt}{t}=\infty$. Therefore the sequence $\{r_n\}_{n\in{\mathbb{N}}}$ has a subsequence $\{r_{n_j}\}_{j\in{\mathbb{N}}}$ such that $r_{n_j}\leq n_j^{1+\varepsilon}$ for all $j\in{\mathbb{N}}$. By iterating (\[assuinpr\]) along the sequence $\{r_{n_j}\}$ and using the fact that $\nu(r)$ is an increasing function, it follows that $$T(r_{n_j}) \geq \frac{1}{\nu(r_j)^{n_j}} T(r_0)\nonumber$$ for all $j\in {\mathbb{N}}$, and hence $$\begin{aligned}
\limsup_{r\to\infty}\frac{\log T(r)}{\log r}&\geq& \limsup_{j\rightarrow\infty}
\frac{\log T(r_{n_j})}{\log r_{n_j}}\\
&\geq & \limsup_{j\rightarrow\infty} \frac{-n_j\log \nu(r_j)+\log T(r_0)}
{\log r_{n_j}}\\
&\geq &\limsup_{j\rightarrow\infty} \frac{-n_j\log (1-n_j^{-(1+\varepsilon)\delta})+\log T(r_0)}
{(1+\varepsilon)\log n_j}=\infty
\end{aligned}$$ since $(1+\varepsilon)\delta<1$. This contradicts (\[assu\]), and so the assertion follows. $\Box$
*Proof of Theorem \[1stthm\]:* Taking into account the fact that $P(z,w)$ is homogeneous, it follows by Lemma \[logdiff\] that $$\label{th1eq}
m\left(r,\frac{P(z,w)}{w^{\deg_w(P)}}\right)=o\left(\frac{T(r,f)}{r^\delta}\right)$$ for any $\delta\in(0,1)$, and for all $r$ outside of an exceptional set of finite logarithmic measure. Moreover, by applying an identity due to Valiron [@valiron:31] and Mohon’ko [@mohonko:71] (see also [@goldbergo:08 Theorem 6.5 and Appendix B, p. 453]) to (\[cleq0\]), it follows that $$\begin{split}
T\left(r,\frac{P(z,w)}{w^{\deg_w(P)}}\right)
&= d_w T(r,w) + O(\mathcal{T}(r)),
\label{T}
\end{split}$$ where $$\label{dw}
d_w=\max\{\deg_w(Q),\deg_w(H)+\deg_w(P)\}-\min\{\deg_w(P),{\textrm{ord}}_0(Q)\}$$ and $r$ approaches infinity outside of an exceptional set of finite logarithmic measure. By combining (\[th1eq\]), (\[T\]) and (\[cleqassumpt\]) it follows that $$\label{th1eq2}
N\left(r,\frac{P(z,w)}{w^{\deg_w(P)}}\right)\geq (1+\deg_w(P)-{\textrm{ord}}_0(P))T(r,w)+S(r,w).$$
Suppose now on the contrary to the assertion of Theorem \[1stthm\] that $N(r,w)=S(r,w)$. Therefore, denoting $C=\max_{j=1,\ldots,n}\{|c_j|\}$ in (\[Pzw\]), it follows from Lemma \[technical\] that $$\begin{split}
N\left(r,\frac{P(z,w)}{w^{{\textrm{ord}}_0(P)}}\right)&\leq (\deg_w(P)-{\textrm{ord}}_0(P)) N(r+C,w)\\
&=(\deg_w(P)-{\textrm{ord}}_0(P)) N(r,w)+S(r,w)\\&=S(r,w).
\end{split}$$ Thus, $$\begin{split}
N\left(r,\frac{P(z,w)}{w^{\deg_w(P)}}\right)&\leq N\left(r,\frac{P(z,w)}{w^{{\textrm{ord}}_0(P)}}\right)
+N\left(r,\frac{1}{w^{\deg_w(P)-{\textrm{ord}}_0(P)}}\right)\\
&= N\left(r,\frac{1}{w^{\deg_w(P)-{\textrm{ord}}_0(P)}}\right) + S(r,w)\\
&\leq T\left(r,\frac{1}{w^{\deg_w(P)-{\textrm{ord}}_0(P)}}\right)+S(r,w).\nonumber
\end{split}$$ Hence, by the first main theorem of Nevanlinna theory, it follows that $$N\left(r,\frac{P(z,w)}{w^{\deg_w(P)}}\right)\leq (\deg_w(P)-{\textrm{ord}}_0(P))T(r,w)+S(r,w)\nonumber$$ which contradicts (\[th1eq2\]). We conclude that $N(r,w)\not=S(r,w)$. $\Box$
*Proof of Theorem \[2ndthm\]:* Suppose now that $w(z)$ is a finite-order meromorphic solution of (\[cleq2\]) such that (\[condcleq2\]) holds. By denoting $C=\max_{j=1,\ldots,n}\{|c_j|\}$ in (\[Pzw\]), it follows that $$\label{aux1}
N\left(r,\frac{P(z,w)}{w^{\deg_w(P)}}\right)\leq
\kappa(P)\left(N(r+C,w)+N\left(r,\frac{1}{w}\right)\right)+O(\mathcal{T}(r)).$$ Since by Lemma \[technical\] $$N(r+C,w)=N(r,w)+o\left(\frac{N(r,w)}{r^\delta}\right)\nonumber$$ for all $r$ outside of a $E$ set of finite logarithmic measure, inequality (\[aux1\]) yields $$\label{aux2}
N\left(r,\frac{P(z,w)}{w^{\deg_w(P)}}\right)\leq \kappa(P)(2T(r,w)-m(r,w))+o\left(\frac{T(r,w)}{r^\delta}\right) +
O(\mathcal{T}(r))$$ for all $r\not\in E$. On the other hand, by (\[th1eq\]) and (\[T\]), $$\label{aux3}
\begin{split}
N\left(r,\frac{P(z,w)}{w^{\deg_w(P)}}\right)= d_w T(r,w)
+o\left(\frac{T(r,w)}{r^\delta}\right) + O(\mathcal{T}(r)),
\end{split}$$ where $r$ lies outside of a set $F$ of finite logarithmic measure, and $d_w$ is as in . By combining inequalities (\[aux2\]) and (\[aux3\]) with the assumption (\[condcleq2\]), it follows that $$m(r,w)=o\left(\frac{T(r,w)}{r^\delta}\right) + O(\mathcal{T}(r))$$ for all $r\not\in E\cup F$. $\Box$
*Proof of Theorem \[mainthm\]:* Suppose first that all except finitely many poles of $w(z)$ are in a sequence of the type (i). Recall that we have adopted the short notation $w=w(z)$, $\overline{w}=w(z+1)$ and $\underline{w}=w(z-1)$. The auxiliary function $g$ defined by $$\label{g}
g=(\overline{w}-a)(w-c)$$ is meromorphic, of finite order, and, by Lemma \[logdiff\] and Theorem \[2ndthm\], it satisfies $$\begin{split}
m(r,g)&\leq m(r,\overline{w})+m(r,w)+m(r,a)+m(r,c)+O(1) \\
&\leq 2m(r,w)+m\left(r,\frac{\overline{w}}{w}\right) +O(\log r)
\\
&= o\left(\frac{T(r,w)}{r^\delta}\right)+O(\log r)
\label{mrg}
\end{split}$$ as $r\to\infty$ outside of a set of finite logarithmic measure. Moreover, all possible poles of $g$, with at most finitely many exceptions, arise from poles of $w$ or $\overline{w}$ and therefore are part of the sequence in (i). Suppose first that $w(\hat{z})=\infty$ with multiplicity $k_{\hat{z}}$. Then by the sequence in (i), $w(\hat{z}+1)=a(\hat{z})$ with multiplicity no less than $k_{\hat{z}}$, and so $g$ assumes a finite value at $z=\hat{z}$. Similarly, if $w(\hat{z}+1)=\infty$ with multiplicity $k_{\hat{z}+1}$, then $w(\hat{z})=c(\hat{z})$ with multiplicity no less than $k_{\hat{z}+1}$, and so $g$ is again finite at $z=\hat{z}$. Hence $g$ has only finitely many poles. By combining this fact with (\[mrg\]), it follows that $$T(r,g)=o\left(\frac{T(r,w)}{r^\delta}\right)+O(\log r)\nonumber$$ for all $r$ outside of an exceptional set of finite logarithmic measure. Therefore, $$\label{estim}
T(r,g)\leq r^{\rho(w)+\varepsilon-\delta}+K\log r$$ where $K>0$ is an absolute constant, $\varepsilon>0$, and $r$ lies outside an exceptional set of finite logarithmic measure. By [@laine:93 Lemma 1.1.2] the exceptional set can be removed if $r$ is replaced by $r^{1+\varepsilon}$ on the right side of (\[estim\]). Then (\[estim\]) becomes $$T(r,g)\leq r^{(1+\varepsilon)(\rho(w)+\varepsilon-\delta)}+K(1+\varepsilon)\log r\nonumber$$ for all $r$ sufficiently large. Since $\varepsilon>0$ and $\delta\in(0,1)$ are arbitrary, it follows that $\rho(g)\leq\max\{0,\rho(w)-1\}$. The first part of the assertion follows by choosing $b=g-ac$.
Assume now that $w(z)$ is a finite-order meromorphic solution of the Riccati equation in (ii), and suppose that $w(z)$ has a pole of order $k_{\hat{z}}$ at $z=\hat{z}$. Then, by writing the Riccati equation in the form $$w(z+1)=a(z)+\frac{a(z)c(z)+b(z)}{w(z)-c(z)}$$ and substituting $z=\hat{z}$, it follows that either $$w(z+1)=a(z)+\gamma_{\hat{z}}(z-\hat{z})^{k_{\hat{z}}}+O((z-\hat{z})^{k_{\hat{z}}+1})
\nonumber$$ for all $z$ in a small enough neighborhood of $\hat{z}$, or $a(z)c(z)+b(z)$ and/or $c(z)$ has a pole at $z=\hat{z}$. The former case is, as required, one of the entries of the sequence in (i), while the latter case can occur only at most finitely many times.
By writing the difference Riccati equation in (ii) as $$w(z-1)=\frac{c(z-1)w(z)+b(z-1)}{w(z)-a(z-1)}\nonumber$$ it follows, similarly as above, that either $$\label{finalentry}
w(z-1)=c(z-1)+\alpha_{\hat{z}}(z-\hat{z})^{k_{\hat{z}}}+O((z-\hat{z})^{k_{\hat{z}}+1})$$ for all $z$ in a small enough neighborhood of $\hat{z}$, or $a(z)c(z)+b(z)$ and/or $a(z)$ has a pole at $z=\hat{z}-1$. As above we conclude that (\[finalentry\]) holds for all except finitely many poles of $w$. $\Box$
Acknowledgements {#acknowledgements .unnumbered}
================
The research reported in this paper was supported in part by the Academy of Finland grant \#118314 and \#210245. We would like to thank the anonymous referees for their helpful comments on the paper.
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abstract: 'Can we use mathematics, and in particular the abstract branch of category theory, to describe some basics of dance, and to highlight structural similarities between music and dance? We first summarize recent studies between mathematics and dance, and between music and categories. Then, we extend this formalism and diagrammatic thinking style to dance.'
author:
- Maria Mannone
- Luca Turchet
title: 'Shall We (Math and) Dance? '
---
3em
Introduction: Why mathematics for dance
=======================================
Joining the abstraction of mathematics with expressivity and passion in dance is possible. It means rationally exploring basic features of dance, and appreciating the flexibility of modern mathematics, for speculative investigation and practical purposes. A joint approach between music, dance, and mathematics would involve rational thinking upon the arts, as well as the ‘translations’ of ideas and transformational mechanisms between disciplines in a STEAM (Science, Technology, Engineering, Art, and Math) attitude.
‘Concrete’ applications can also involve pedagogy in one or more of these disciplines, giving amusing examples of applications of mathematical concepts and procedures, and investigating some hidden, theoretical roots of artistic practices. In particular, in the case of math and dance, an interplay between rationality and intuition may even help people learn dancing. This is the case of a software that helps tango learners to find the main pulse via a mathematical analysis of the underlying musical structures [@amiot]. A way to compare underlying rhythms for dance is suggested in E. Amiot’s book [@amiot_book], where Fourier discrete transforms are used to give an idea of the distribution of durations of notes and rhythm patterns.
The world of dance includes different styles and techniques, each of them with inner symmetries. These symmetries may be thought of throught the lens of mathematics. An overview of mathematics for different dance styles is proposed in [@wasilewska], with a focus on the geometry of figures (‘poses’). The geometry of choreographies also includes connecting movements between poses of the group of dancers. One can analyze the geometry of poses of a single dancer, but a more complete investigation should also involve the analysis of the geometry generated by the entire group of dancers on stage. The audience of a ballet, and, in general, of a dance show, pays attention differently to dancers, dancers’ groups, and parts of them. To investigate the relationship between attention and single dancers the concept of the [*center of attention*]{} has been used, an element already used in visual arts. It can be studied geometrically and statistically as a trajectory [@schaffer]. In analogy with the center of mass that connects form and mass, the center of attention has been connected with the ideal ‘mass’ of dancers, with the concept of ‘center of attention mass.’ The center of attention mass of an ensemble connects physics, group dancing schemes, and rhythm patterns. It is evaluated assigning weights to dancers’ bodies based “on the type of movement performed and how likely the moves are to attract the audience’s attention” [@wasilewska]. A scientific perspective on choreography does not involve only analysis and understanding of dance practice: it can suggest new ideas sparkling creativity. Furthermore, specific geometric forms can be used as bases for choreography and other artistic applications; this is the case of the truncated octahedron[^1] cited in [@borkovitz], and the ‘Apollonian circles’[^2] from [*The Daughters of Hypatia*]{} by Karl Schaffer [@schaffer].
Dance involves figures as events in time continuously connected, and time – and expressivity – is shaped by music. Music does also influence not only the [*when*]{} of figures and movements but also their [*how*]{}, their style. Thus, a complete study of dance cannot prescind from the role of music. In fact, music and dance are tightly connected in several cultures [@patel].
One of the elements connecting music and dance is [*gesture*]{}, which influences expressive parameters of music. Gestures also contain information about pulse. The conductor communicates pulse and rhythm to orchestral musicians, as well as overall style and expressivity. Each musician performs specific gestures, according to the technique of the specific instrument he or she plays. Dancers do not take pulse directly from the conductor, but from performed music, via a ‘filtering’ operation; see [@amiot]. Dancers’ movements and gestures, especially in ballet, are linked with music and thus, indirectly, with musicians’ gestures that produce that music.
We can wonder if it is possible to describe within some unitary vision the gestures of conductor, musicians, and dancers. To the best of our knowledge, there is not such a theory yet, but only a collection of experiences and case studies. A recent mathematical overview of the similarities between conducting and orchestral musicians’ gestures uses the formalism of category theory [@mannone_jmm]. In fact, the power of abstraction of categories allows for the schematization of similarities and similar transformations between different objects, relaxing the condition of ‘equality’ in favor of ‘up to an isomorphism.’
The flexibility of categories and, more in general, of diagrammatic thinking is exemplified by the mathematical definition of [*gesture*]{}. While trying to sketch a general theory of music and dance, it is helpful to highlight common elements between different dance styles. Arrows and diagrams in categories can connect the discussed existing studies with current research in mathematical music theory. Figure \[ballerine\] shows an intuitive and yet precise application of categorical formalism to dance. This schematization is inspired by the definition of musical gestures proposed in [@mazzola_andreatta], that makes use of dance as a metaphor to understand music.
In this paper, we briefly summarize mathematical theory of musical gestures [@mazzola_andreatta; @mannone_jmm; @mannone_knots; @arias] and basics of categories, and we show their possible application to dance. This approach may be useful to investigate formal and cognitive studies about dance [@patel].
![A mapping from a simple diagram (with two points and an arrow) representing points in space and time touched by the foot of a dancer, and two different realizations of this scheme. The arrows between the two images of each pair of dancers represent the continuous movements to connect the two poses. The mapping consists of two (topological) functors from points in space-time (first category) to dance figures and movements (second category). This is a category of gestures, where objects are gestures, and morphisms are hypergestures. We are working with curves of gestures, that is, a particular instance of hypergestures. In the second category, the points are the figures (poses), and the arrows are the movements. We are using functors and not only functions because we need to map not only objects from a domain to a codomain but also the transformations between them, and functors map objects into objects, and morphisms into morphisms. Dancers are not usually aware of this, but the mathematics behind their art is the definition of gestures on topological categories [@mazzola_andreatta; @arias]. Drawings by Maria Mannone. []{data-label="ballerine"}](ballerine){width="12cm"}
Categories, Music, and Dance {#categories_dance}
============================
Category theory can be applied to music [@mazzola_topos; @fiore_noll], included musical gestures [@mazzola_andreatta; @arias; @mazzola_topos; @jed]. A category is given by objects (points or, to avoid a set-theoretic feeling, [*vertices*]{}) and morphisms between them (arrows), verifying associative and identity properties. We can define arrows between categories ([*functors*]{}) bringing objects of a category into objects of the other one, and morphisms of a category into morphisms of the other one. Considering categories themselves as vertices (to be connected via functors), we can easily build nested structures.[^3] In [@mazzola_andreatta], a gesture is defined as a collection of vertices in space and time connected by a system of continuous curves.[^4] Musical notes (vertices) are like ‘the points touched by a dancer while moving continuously in the space’ (arrows) [@mazzola_andreatta]. This definition can be extended to visual arts, $n$-categories, and recursive gestures [@mannone_knots]. Musical gestures allow one to shape physical sound parameters for expressive reasons: a purely ‘technical’ gesture such as pressing a piano key can be shaped into, let us say, a delicate, caressing gesture.[^5]
The most basic gesture is probably [*breathing*]{}, that can be seen in general as a couple of (inverted) curves between two points — a ‘tense’ state and a ‘relaxed’ state. In piano playing, we have the general idea of preparation and key-pressing. Breathing, in music and dance, is the starting point of both basic technique and expressive motion.[^6] From conductor’s hands to dancers’ feet, all feelings are conveyed into muscular tension/release and drawing curves in space and time. ‘Breathing’ reminds of diaphragm movements for singers and wind players. More generally, breathing reminds of arsis and thesis, preparation and beat (conducting), inspiration and exhalation, dissonance and resolution in harmony. The dualism tension-relaxation is also relevant for dance [@charnavel]. In general, we can find [*similarities*]{} of gestures on different instruments — e.g., the necessary variations of movements to play a [*forte*]{} on violin and on piano [@mannone_jmm]; these similarities also deform the basic metric gestures of a conductor. In dance, we have poses/figures (vertices) and movements (arrows) connecting them. Movements can be seen as vertices, and transformations of these movements as arrows. Thus, both in music and dance we can categorically investigate basic technical gestures and their expressive deformations, as well as their compositions and groupings. Groupings and hierarchies of musical structures [@lerdhal; @mazzola_topos] and gestures [@mannone_knots] have a correspondence for dance: basic feelings represented via expressive gestures can be connected within a whole story, letting dance acquire a narrative dimension.
Diagrammatic Details {#diagrammatic_details}
====================
Gestures[^7] in music and dance can be seen as two categories. In dance, the objects are the positions, and the morphisms are the movements to reach them.[^8] We can define a functor from music to dance, involving metrical aspects in music to be connected with metrical aspects in dance, and expressive aspects of music with expressive aspects in dance. The ‘vertices’ would be isolated, remarkable points of the score or short sequences in music and figures in dance, and the ‘arrows’ would be the connecting passages in music and connecting gestures in dance.
The category [*dance*]{} is given by positions of the human body in space and time, called [*figures*]{}, connected via continuous movements:
$\begin{diagram}\mbox{dance figure 1} & \rTo^{movement} & \mbox{dance figure 2}\end{diagram}$.
We can easily introduce 2-categories if we consider different ways to connect two figures, whose variation, represented by the double arrow, can embody changes in speed, amplitude, and expressivity; see diagram \[dance\_cat2\].
$$\label{dance_cat2}
\begin{tikzcd}
\mbox{dance figure 1}
\arrow[r,bend left=30, "movement\,1"{name=U, above}]
\arrow[r,bend right=30, "movement\,2"{name=D, below}]
&
\mbox{dance figure 2}
\arrow[Rightarrow, from=U, to=D]
\end{tikzcd}$$
The identity corresponds to the absence of movement – keeping a static position. Horizontal composition is well-defined: the composition of two dance movements is another dance movement. Vertical composition is also well-defined: we can transform a movement into another, and the composition of two movement transformations leads to another movement. Vertical associativity is verified if we consider equivalence classes of movements connecting dance figures.[^9] Horizontal associativity is also verified: similarly with (musical or generic) gestures, the composition of dance hypergestures (paths) is associative up to a path of paths. Diagrams help us compare figures and movements of different styles; see diagram \[dance\_functor\]. In fact, we can define, within the category ‘dance,’ a subcategory for dance-style 1, and another subcategory for dance-style 2. For sake of simplicity, we can just define a category for style 1 and a category for style 2. $$\label{dance_functor}
\begin{footnotesize}
\begin{diagram}
\mbox{dance figure 1 (style 1)} & \rTo^{movement\,(style\,1)} & \mbox{dance figure 2 (style 1)}
\\ \dMapsto^{figure\,mapping} & \dMapsto^{gesture\,mapping} & \dMapsto^{figure\,mapping} % \dMapsto^{figure\,mapping} & \dImplies^{gesture\,mapping} & \dMapsto^{figure\,mapping}
\\ \mbox{dance figure 1 (style 2)} & \rTo^{movement\,(style\,2)} & \mbox{dance figure 2 (style 2)}
\end{diagram}
\end{footnotesize}$$
$$\label{dance_functor_2} % change as dance_functor and then with the following diagram names
\begin{tikzcd}[swap,bend angle=45]
\mbox{dance figure 1 (style 1)}
\arrow[mapsto]{dddd}{figure\,mapping} % maps to
\arrow[r,bend right=20, "movement\,2\,(style\,1)"{name=D}]
%\arrow[Rightarrow, from=D, to=Z, "gesture\,mapping"]
\arrow[r,bend left=20, "movement\,1\,(style\,1)"{name=U, above}]
&
\arrow[Rightarrow, from=U, to=D, " "] % Rightarrow
\mbox{dance figure 2 (style 1)}
\arrow[mapsto]{dddd}{figure\,mapping} % maps to
%\arrow[Rightarrow, from=D, to=V, "gesture\,mapping"]
\\ %\arrow[mapsto]{ddd}{gesture\,mapping} &
\\
\\ & & & &
\\
\mbox{dance figure 1 (style 2)}
\arrow[r,bend right=20, "movement\,2\,(style\,2)"{name=Z}]
\arrow[r,bend left=20, "movement\,1\,(style\,2)"{name=V, above}]
&
\mbox{dance figure 2 (style 2)} % maps to
\arrow[Rightarrow, from=V, to=Z, " "] % Rightarrow
\end{tikzcd}$$
We can group movements in two, three, and so on, according to tempo. Dance teachers can ask their students to start clapping hands, letting them think of musical rhythm. Then, students will start making movements with their bodies according to the given rhythm. Mathematically, this is a mapping from the category of pulses to the category of basic dance movements; thus, we have a functor. In the category of pulses, objects are accents (beats), and arrows are time intervals between them. Beats are mapped into positions of the body in space and time, and time intervals are mapped into body movements to reach these positions; see diagram \[pulse\_dance\_2\],[^10] enriched with $2$-categories. Diagram \[dance\_functor\] is enriched in terms of $2$-categories in diagram \[dance\_functor\_2\]. In diagram \[dance\_functor\_2\], additional arrows, not shown for reasons of graphic clarity, map $movement\,1\,(style\,1)$ into $movement\,1\,(style\,2)$, $movement\,2\,(style\,1)$ into $movement\,2\,(style\,2)$, and double arrows into double arrows. The same applies to diagram \[pulse\_dance\_2\], where $2$-cells of $2$-category pulse may represent the time variation between two consecutive pulses. We are using the formalism of $2$-functors, that are morphisms between $2$-cells[^11] [@maclane]. In fact, $2$-cells give a metaphorical idea of the different ways to connect figures within the same style, to move hands in conducting between two pulses (diagram \[music-to-dance\_2\]) for technical and expressive reasons, and so on.
$$\label{pulse_dance_2}
\begin{tikzcd}[swap,bend angle=45]
\mbox{pulse 1}
\arrow[mapsto]{dd}{} % maps to
\arrow[r,bend right=20, "time\,interval\,2"{name=D}]
\arrow[r,bend left=20, "time\,interval\,1"{name=U, above}]
&
\arrow[Rightarrow, from=U, to=D, " "]
\mbox{pulse 2}
\arrow[mapsto]{dd}{} % maps to
\\ & & & &
\\
\mbox{dance figure 1}
\arrow[r,bend right=20, "movement\,2"{name=Z}]
\arrow[r,bend left=20, "movement\,1"{name=V, above}]
&
\mbox{dance figure 2} % maps to
\arrow[Rightarrow, from=V, to=Z, " "]
\end{tikzcd}$$
Figures in dance can be different for women and men, for single or couple dance, for groups of three or more people, and so on. Thus, there will be different mappings from pulse to dance figures, but dance figures of all dancers, and their connecting movements, have to be coordinated. Temporal consistency is guaranteed by the action of the functor from time pulse to dance. The specific connection between dance figures and movements depends on the style of dance and the role of each dancer. The personal contribution of individual dancers is mathematically represented by additional arrows that modify dance figures and their connecting movements. Transformations between poses (i.e., movements) should take into account the expression of physical constraints: for example, the distance between dancers’ bodies. Tempo transformations play an important role: for example, a basic ternary movement can be transformed into a ‘valzer’ movement by prolonging the first pulse.
The upper side of diagram \[music-to-dance\_2\] represents the mapping from rhythmic pulse to conducting gestures (r. h. indicates right hand), where we consider the $2$-cell given by two different right-hand’s movements connecting two right-hand’s positions; the lower side, the mapping from the same rhythmic pulse to basic dance movements (with a similar remark on $2$-categories). Movements in conducting and dance are related because dancers hear the music, and they find conductor’s beats through the music. If the dancers are dancing on their own without any music, they have to think tempo pulse in their own, too;[^12] if there is music, dancers can rely on its pulse. Ideally, dancers, musicians, and of course the conductor, are thinking the same tempo pulse. For the sake of completeness, we should cite the Cunningham school, where dance is not connected with the pulse of music. In this case, we cannot define a functor from pulse to dance. We might describe diagram \[music-to-dance\_2\] as a simple functor from pulse to dance; however, it stresses the ‘generator’ role of pulse for both dance and conducting. We can imagine a ‘pulse’ that is first an abstract thought and then is embodied in conducting gestures and dance movements. In the practical reality, dancers extract, as in a filter operation, the information about tempo pulse from music. Thus: pulse *is the basis of* (is mapped into) conducting gestures, that *are the basis of* orchestral playing, that *is the basis of* listening to music, *that helps* dancers catch pulse from music, *allowing them* to move accordingly; see diagram \[conductor\_musician\_dancer\].
$$\label{music-to-dance_2}
\begin{tikzcd}[swap,bend angle=30] %45
\mbox{conductor's r.h. position 1}
\arrow[r,bend right=20, "hand's\,movement\,2"{name=D}]
\arrow[r,bend left=20, "hand's\,movement\,1"{name=U, above}]
&
\arrow[Rightarrow, from=U, to=D, " "]
\mbox{conductor's r.h. position 2}
\\
\\
\\ & & & &
\\
\arrow[mapsto]{uuuu}{pulse-cond.} % maps to
\mbox{pulse 1}
\arrow[mapsto]{dddd}{pulse-dance} % maps to
\arrow[r,bend right=20, "time\,interval\,2"{name=D}]
\arrow[r,bend left=20, "time\,interval\,1"{name=U, above}]
&
\arrow[Rightarrow, from=U, to=D, " "]
\arrow[mapsto]{uuuu}{pulse-cond.}
\mbox{pulse 2}
\arrow[mapsto]{dddd}{pulse-dance} % maps to
\\
\\
\\ & & & &
\\
\mbox{dance figure 1}
\arrow[r,bend right=20, "body's\,movement\,2"{name=Z}]
\arrow[r,bend left=20, "body's\,movement\,1"{name=V, above}]
&
\mbox{dance figure 2} % maps to
\arrow[Rightarrow, from=V, to=Z, " "]
\end{tikzcd}$$
The arrow from conductor to dancer is dashed because the conductor does not directly give tempo to the dancer: he or she catches it from music, thus, from musicians’ activity. $$\label{conductor_musician_dancer}
\begin{footnotesize}
\begin{diagram}
& &\mbox{conductor}
\\ & \ldTo^{gives\,tempo} & & \rdDashto^{gives\,tempo}
\\ \mbox{musician} & & \rTo^{suggests\,tempo} & & \mbox{dancer}
\end{diagram}
\end{footnotesize}$$
The mapping from tempo-pulse to dance can be formally described via a [*pulse-dance functor*]{}; see diagram \[pulse\_dance\_2\]. $$\label{conductor_musician_dancer_listener}
\begin{footnotesize}
\begin{diagram}
& &\mbox{conductor}
%\\ & \ldTo^{gives\,tempo} & \dTo^{\,\,reaches} & \rdDashto^{gives\,tempo}
\\ & \ldTo^{gives\,tempo} & & \rdDashto^{gives\,tempo}
\\ \mbox{musician} & & \lImplies & & \mbox{dancer}
\\ & \rdTo^{\,reaches\,via\,sound} & & \ldTo^{reaches\,via\,image\,}
\\ & & \mbox{audience}
\end{diagram}
\end{footnotesize}$$
Overall musical character and expressivity shape expressive dance movements. For example, a sudden orchestral [*forte*]{} should imply a corresponding choreography variation for ballet dancers.[^13] We can navigate through complexity of dance, by considering comparisons between gestures of orchestral musicians within time, and gestures of dancers. Such a complex mapping can be formalized as a [*Choreography functor*]{}, corresponding to the choices of a choreographer that instructs ballet dancers on how to move on stage while listening to the orchestra. In folk music, this functor connects traditional instruments’ playing with folk dancers’ movements. Intuitively, a choreography functor should map marked points of the musical score corresponding to groups of musical gestures into dance figures of a group of dancers, and morphisms between these points of the score (and the associate connecting groups of gestures) into ‘connecting’ group dance movements. In this framework, natural transformations between two choreography functors would describe differences between two different choreographies based on the same music.
We can schematize the action of conductor, orchestral musicians, and dancers, connecting them with listeners in the audience. In diagram \[conductor\_musician\_dancer\_listener\], both music played by musicians and choreography made by dancers reach the final target, the audience. Here, the conductor metaphorically plays the role of a limit, while the audience plays that of a colimit.[^14] Arrows appear thus inverted with respect to [@mannone_jmm], but the two representations are both valid. See details in [@mannone_jmm] about conductor and listener. As noted by one of the reviewers, diagram \[conductor\_musician\_dancer\_listener\] does not commute, but it does up to a 2-cell as indicated by the double arrow. The reason is that impressions of the public with respect to musicians and with respect to dancers can be similar, but they will never be equal. The commutativity is verified if we consider a single listener, or if we consider the equivalence class of all impressions of the public as a whole. More pictorially, we can imagine diagram \[conductor\_musician\_dancer\_listener\] as an ellipsoid (Figure \[ellipsoid\]), with the larger horizontal section representing the connections between musicians’ and dancers’ movements, and the vertical one, the separation between music and dance worlds.[^15] The conductor’s role appears as being more relevant for a ballet than for other dance styles. Diagrams \[dance\_functor\] and \[conductor\_musician\_dancer\_listener\] are given for completeness, and to connect our study with former studies in mathematical theory of musical gestures of orchestral musicians [@mannone_jmm].
![Ellipsoid where conductor and audience can be thought of as limit and colimit, respectively.[]{data-label="ellipsoid"}](ellipsoid){width="6.5cm"}
Discussion
==========
In this paper, we introduced the use of category theory, an abstract branch of mathematics, to the study of dance in relationship with music. The nested structures of categories help connecting different perspectives on dance, such as the relationship with musical pulse [@amiot] and the geometry in dance [@borkovitz]. Also, the ‘trajectory,’ that is, the change over space/time of the center of attention as a trajectory [@schaffer] can be schematized via an arrow, and the comparisons between centers of attention for different choreographies, via arrows between arrows. This would also allow for a comparison between the geometry of different choreographies, combining into a unitary vision the collection of examples proposed in literature [@borkovitz; @schaffer]. The geometry of choreographies can be investigated via connected categories.[^16] In the case of the tango ball, we can also think of connected musical sequences, the so-called [*tandas*]{}, with their different styles.
Also, one can establish a connection between dancers’ positions and movements and their center of attention mass [@wasilewska], and one can compare variations of this center within a choreography with variations of this center within another choreography, quantitatively characterizing choreographers’ styles and different editions/productions of the same ballet. More generally, a comparison between variations of the center of attention mass would well characterize differences and analogies of choreographies for different dance styles.
We can wonder if nested structures of categories may help connecting even more layers of understanding and aesthetic appreciation for dance. We can start from the symmetry of the single, idealized human body to reach the most complex choreography. The possible steps of formal analysis would be:
1. the beauty[^17] of the human body as part of nature;
2. the beauty of different poses of the human body in dance, as photographical shoots;
3. the beauty of dancing movements of the human body, that is, of the connections and transitions between static poses of dance;
4. the beauty of several people dancing together: ‘static poses’ as well as their connecting movements.
If, on the one hand, we tend towards abstraction and generalization, on the other hand we can also wonder about the applicability of our model in concrete setups and in pedagogical frameworks, suggesting categorical developments of recent research and maybe new software applications [@amiot]. Both points and arrows are part of dance learning: dance students learn the poses and the way to smoothly, and expressively, reach them. The movements of each dancer have to be well-coordinated with movements of the other dancer of the couple, and with all the other dancers involved, if any. Thus, the formalism of monoidal categories[^18] can catch the ‘simultaneity’ of transformations. We need monoidal categories and not just categories because, having more dancers dancing simultaneously, with monoidal categories we can easily model their different movements with different transformations.
In fact, next developments of this research can concern an abstract description of group dancing, from couples to larger groups. Diagram \[couple\_dance\_1\] refers to monoidal categories to represent the movements of a couple of dancers. Let $A,\,B$ be two dancers, and $P^A_i,\,P^B_i$ their $i$-th poses (or ‘figures’), respectively. If dancer $A$ changes figure while dancer $B$ remains steady, the overall movement can be schematized by the action of the morphism $t\otimes 1$, where $t$ indicates the movement, and $1$ is the identity. Conversely, if $A$ is steady and $B$ moves, the movement will be represented by $1\otimes t'$. In general, both dancers are moving, thus we have $t\otimes t'$ in the simplified case where $A$ is moving via $t$ and $B$ via $t'$.
$$\label{couple_dance_1}
\begin{diagram}
P_1^A\otimes P_1^B & \rTo^{t\otimes 1} & P_2^A\otimes P_1^B
\\ \dTo^{1\otimes t'} & & \dTo^{1\otimes t'}
\\ P_1^A\otimes P_2^B & \rTo^{t\otimes 1} & P_2^A\otimes P_2^B
\end{diagram}$$
Diagram \[couple\_dance\_2\] shows the action of a ‘smoothness’ operator $s$, that changes the movement of the first dancer — let us say, a beginner – from a [*not smooth*]{} movement to a [*smooth*]{} one. The complete operator, indicated in the diagram by the double arrow, is $s\otimes 1$, because it acts as an identity with respect to the second dancer, that is already moving smoothly. In diagram \[couple\_dance\_2\], the final poses are the same but the way to reach them are different. If dancers stop moving but music continues, we can describe dancers’ movements via identities leading to the same positions. Also, if the leader role permutes during the show, we can use braids in monoidal categories. Finally, the formalism we developed for dance can be extended to other movements that ‘have to be made in a specific way’: a reviewer thought of a worker building a wall, or a tourist walking while visiting a church. This might raise a question: when is a movement ‘dance’? We can think of formal characteristics, such as smoothness of movements and music-pulse following, but we can also think of aesthetics (and motivation) of movement in itself. According to [@charnavel], “Everyday movements are usually goal-oriented, which makes them fundamentally different from dance.”
$$\label{couple_dance_2}
\begin{tikzcd}
P_1^A\otimes P_1^B
\arrow[r,bend left=40, "not\,smooth\otimes smooth"{name=U, above}]
\arrow[r,bend right=40, "smooth\otimes smooth"{name=D, below}]
&
P_2^A\otimes P_2^B
\arrow[Rightarrow, from=U, to=D,"s\otimes 1"]
\end{tikzcd}$$
Conclusion
==========
Summarizing, we applied to dance categorical formalism formerly used for music [@mannone_knots; @mannone_jmm], comparing gestures in music and dance, and highlighting some of their connections. We started from conceptual considerations in Section \[categories\_dance\], we discussed metaphorical yet formal applications of functors and commutative diagrams to music and dance in Section \[diagrammatic\_details\], and we finally made some references to monoidal categories in Section \[discussion\]. Previous work concerned mathematical description of gestures [@mazzola_andreatta], connections of categorical and topological formalism with function spaces to investigate spaces of gestures [@arias], and application of $2$-categories and $n$-categories to orchestral playing and to describe recursive musical gestures [@mannone_knots; @mannone_jmm]. Former works about a formal and often mathematical description of dance topics [@borkovitz; @charnavel; @schaffer; @wasilewska] motivated us to include dance within the aforementioned diagrammatic and categorical formalism, trying to envisage general characteristics in different dance styles, and their general connection with music and musical gestures. This paper aimed to contribute towards a unified vision of gestures of conductor, musicians, and dancers. The mathematical reason behind that is to stress the importance of the unifying power of diagrammatic thinking to navigate within the complexity of the arts and to connect in a simple way things whose nature appears at first as deeply different. The artistic reason is to stress the importance of mutual knowledge between artistic roles, and the relevance of a well-working communication and continuous exchange via sounds, images, and gestures. We can try to approach expressivity in the arts in a simple and rational way. We can do the same thing while admiring beauty of nature via scientific investigation. This is an attempt to find keys and hidden rules of aesthetics[^19] and beauty, their hidden skeleta constituting the structure of a magnificent building. Future research may include a more detailed description of particular musical genres, such as tango, and its connection with tango music [@amiot]. We can investigate if advanced topics in category theory do have a meaningful application in dance, and, vice versa, how essential topics in dance can be categorically described. Also, generative theories used to modeling natural language and dance [@charnavel; @patel] and natural forms (with their growth processes) [@lindenmayer] can be compared within a categorical framework, looking for formal analogies and differences, hoping to find hidden connections between beauty in dance (and music) and beautiful natural forms, looking for some natural ‘roots’ of beauty in the arts. And yes, let’s (math and) dance!
Acknowledgments
===============
The authors are grateful to the mathematician, musician, and tango dancer Emmanuel Amiot for his helpful suggestions.
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Kubota, A., Hori, H., Naruse, M., and Akiba, F.: A New Kind of Aesthetics — The Mathematical Structure of the Aesthetic. Philosophies, **3** (14), 1–10 (2017)
Lawvere, W., and S. Schanuel. Conceptual Mathematics. A first introduction to categories. Cambridge, Cambridge University Press (2009)
Lerdahl, F., and R. Jackendoff. Generative Theory of Tonal Music. Cambridge: MIT Press (1983)
Mac Lane, Saunders. Categories for the Working Mathematician. New York, Springer (1971)
Mannone, M. cARTegory Theory: framing Aesthetics of Mathematics, Journal of Humanistic Mathematics, **9** (18), 277–294 (2019)
Mannone, M. Knots, Music and DNA. Journal of Creative Music Systems, **2** (2), 1–20, <https://www.jcms.org.uk/article/id/523/> (2018)
Mannone, M. Introduction to Gestural Similarity in Music. An application of Category Theory to the Orchestra. Journal of Mathematics and Music, **18** (2), 63–87 (2018)
Mazzola, G., et al. The Topos of Music: I-IV. Heidelberg, Springer (2018)
Mazzola, G., and Andreatta, M. Diagrams, gestures and formulae in music. Journal of Mathematics and Music, **1** (1), 23–46 (2010)
Patel-Grosz, P., P. G. Grosz, T. Kelkar, and A. R. Jensenius. Coreference and disjoint reference in the semantics of narrative dance. Proceedings of Sinn und Bedeutung, **22**, 199–216 (2018)
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[^1]: An Archimedean solid.
[^2]: Families of circles where every circle of a family intersects every circle of the other family orthogonally.
[^3]: We can define $2$-categories and $n$-categories.
[^4]: More precisely, an abstract (and oriented) diagram is mapped into points and paths in a topological space.
[^5]: Conversely, the violent bow’s movements of strings in the movie [*Psycho*]{} by Hitchcock well evoke the knife hitting.
[^6]: According to [@karin], “in the dance studio, conscious use of breathing patterns can enhance the phrasing and expressivity in movement.”
[^7]: The category of gestures has gestures as objects and gestures of gestures (hypergestures) as the morphisms between them. The composition of hypergestures (paths) is associative up to a path of paths [@mannone_knots].
[^8]: From [@charnavel]: “The physical signal of dance consists of a change in position of the dancer body in space with respect to time.” We can see positions as vertices and changes of positions as arrows. Also: “movement is physically created by an infinite sequence of continuous positions in space unfolding in time.” We can represent the transition from a position to another one via an arrow representing a morphing.
[^9]: In fact, a transformation between movements requires a homotopy, and homotopy is not associative (it requires a reparametrization). However, homotopy classes are associative.
[^10]: In the case of total improvisation (music and dance), the diagram would be commutative.
[^11]: As defined in [@maclane], a $2$-functor between two categories $A$ and $B$ is a triple of functions that map objects, arrows, and $2$-cells of $A$ into objects, arrows, and $2$-cells of $B$ respectively, preserving “all the categorical structures.”
[^12]: Dancers can move without any (external) music if they are able to think of their pulse and to communicate with each other via touch and non-verbal indications (and the leader should be particularly clear in such indications), especially for couple dancing; but this is an unstable situation. Permutations of roles should be clearly signaled via, for example, a variation in touch or visual communication.
[^13]: For tango music, this is more complicated than a simple ‘pulse extraction.’ We can instead think of categories enriched with maps from a beat to another beat containing inner maps, that is, inner pulses. In salsa, there often are recognizable patterns with clave rhythms between strong beats. The pulse can depend on the specific music style, and it can be the object of future research in itself.
[^14]: Limits and colimits are generalizations of products and coproducts, respectively; they are obtained the ones from the others via reversing arrows [@lawvere].
[^15]: An ellipse is used for the center of attention in [@schaffer]. The ellipsoid of Fig. \[ellipsoid\] is a diagram connecting different elements. If the listener/audience recovers the pulse contained in conducting gestures, the two extremities of the ellipsoid can be joined, transforming it into a torus with a section collapsed into a point.
[^16]: A connected category $J$ is a category where, for each couple of objects $j, k\in J$, there is a finite sequence of objects $j_0,j_1,...,j_{2n}$ connecting them, that is: $j=j_0\rightarrow j_1 \leftarrow j_2 \rightarrow... \rightarrow j_{2n-1} \leftarrow j_{2n}=k$, where both directions are allowed [@maclane]. Here, the morphisms are the arrows $f_i:j_i\rightarrow j_{i+1}$ or $f_i:j_{i+1}\rightarrow j_{i}$.
[^17]: The definition of [*beauty*]{} is beyond the scope of this paper, and it would start a philosophical debate. We can just say that a mixture of symmetry, balance, proportion, and smoothness of movements can be overall thought of and mathematically investigated as ‘beauty’ in dance.
[^18]: In a nutshell, a monoidal category, also called a tensor category, is a category $C$ having a bifunctor $\otimes:C\times C\rightarrow C$, that verifies pentagonal and triangular identities [@lawvere]. See [@popoff2; @jed; @mannone_knots] for examples of monoidal categories in music.
[^19]: Categorical approaches on aesthetics of the arts [@kubota] and of mathematics and the arts [@mannone_aesthetics] can be joined with studies of nature.
|
---
abstract: 'Quantum walks have been shown to have a wide range of applications, from artificial intelligence, to photosynthesis, and quantum transport. Quantum stochastic walks (QSWs) generalize this concept to additional non-unitary evolution. In this paper, we propose a trajectory-based quantum simulation protocol to effectively implement a family of discrete-time QSWs in a quantum device. After deriving the protocol for a 2-vertex graph with a single edge, we show how our protocol generalizes to a graph with arbitrary topology and connectivity. The straight-forward generalization leads to simple scaling of the protocol to complex graphs. Finally, we show how to simulate a restricted class of continuous-time QSWs by a discrete-time QSW, and how this is amenable to our simulation protocol for discrete-time QSWs.'
address:
- '$^1$ Theoretical Physics, Saarland University, Campus, 66123 Saarbrücken, Germany'
- '$^2$ Raytheon BBN Technologies, 10 Moulton Street, Cambridge, MA, 02138, USA'
- '$^3$ Departamento de Física, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, Brazil.'
author:
- 'Peter K. Schuhmacher$^1$, Luke C. G. Govia$^{1,2}$, Bruno G. Taketani$^{1,3}$ and Frank K. Wilhelm$^1$'
bibliography:
- 'BibQuantumWalks.bib'
title: 'Quantum Simulation of a Discrete-Time Quantum Stochastic Walk'
---
Introduction {#sec:intro}
============
The quantum mechanical analogue to the ubiquitous classical random walk on a graph is the so-called quantum walk [@Aharonov1993]. Quantum walks can be either continuous-time [@Farhi1998], or discrete-time [@Aharonov2001; @ambainis2001one], and as both versions have been shown to be universal for quantum computation [@Childs2003; @Lovett2010] they offer a powerful paradigm for both studying, and harnessing quantum mechanics for computational applications. Examples of this include machine learning [@Schuld2014c; @Rebentrost2014], search algorithms [@Shenvi2003] and photosynthetic excitation transfer [@Mohseni2008; @Walschaers:2013aa].
Quantum walks are completely coherent and hence, the walk is naturally reversible and undirected, as it follows Hamiltonian dynamics. In directed walks, time-reversal symmetry is broken as vertices can be connected by one-way edges. This condition implies that these walks are described by non-Hermitian dynamics and thus cannot be directly implemented in a quantum computer. Quantum stochastic walks (QSWs) are a simple way to represent such evolutions as they combine both quantum unitary walks and non-unitary, stochastic evolution [@Whitfield2010]. In the continuous-time case it was recently shown that the reservoir engineering required to implement such walks is a problem as hard as the problem the QSW is designed to solve [@Taketani2016], though in some cases a quantum simulation approach can be taken [@Govia2017]. For discrete-time QSWs the scenario is different as one can take advantage of measurement-based feed-forward schemes to implement the directionality [@Schuld2014c].
In this work we propose an algorithm to simulate discrete-time QSWs. The central concept behind our protocol is that if one performs randomly chosen unitary dynamics from a carefully designed set, this can implement a specific non-unitary evolution in the ensemble average [@CrispinGardiner2004]. Our protocol is based on ancilla systems and a feed-forward scheme to implement the required evolution. The simplicity of the implementation of a single edge lends to straight-forward scaling to more complex graphs, and is a key feature of the protocol.
This paper is organized as follows. In Section \[sec:Kraus\] we present the ensemble average formalism using the Kraus decomposition. Section \[sec:Protocol\] describes the algorithm, starting from a 2-vertex directed graph and generalizing it to complex graphs. In section \[sec:ApproximateTimeEvolution\], we show how to simulate a restricted class of continuous-time QSWs via a discrete-time QSW. Finally, Section \[sec:conc\] presents our concluding remarks.
Quantum Simulation of a Kraus Map {#sec:Kraus}
=================================
The algorithm proposed here will be formulated on the Kraus decomposition of the desired QSW. Any completely-positive and trace preserving quantum operation [@Nielsen2000], can be written in Kraus operator form as $$\begin{aligned}
\mathcal{B}[\rho] = \sum_j \hat{K}_j\rho\hat{K}_j^\dagger,\end{aligned}$$ where $\{\hat{K}_j\}$ are the Kraus operators, which must satisfy $\sum_j\hat{K}_j^\dagger\hat{K}_j = \hat{\mathbb{I}}$ to preserve the trace of the quantum state. This condition implies that $$\begin{aligned}
{\rm Tr}\left[\sum_j\hat{K}_j^\dagger\hat{K}_j\rho\right] = 1~~\Rightarrow&\sum_j{\rm Tr}\left[\hat{K}_j^\dagger\hat{K}_j\rho\right] = \sum_j\tilde{P}_j = 1,\end{aligned}$$ where we have defined the probabilities $\tilde{P}_j = {\rm Tr}[\hat{K}_j^\dagger\hat{K}_j\rho]$, which are guaranteed to be non-negative as $\hat{K}_j^\dagger\hat{K}_j$ is Hermitian. We can then rewrite our original quantum operation as $$\begin{aligned}
\mathcal{B}[\rho] = \sum_j \tilde{P}_j\tilde{K}_j\rho\tilde{K}_j^\dagger,
\label{eq:SingleStepWalk}\end{aligned}$$ where $\tilde{K}_j = \hat{K}_j/\sqrt{\tilde{P}_j}$. This definition will easily allow us to define our protocol through the ensemble average of quantum trajectories.
We now suppose that we have a protocol (which in our case uses ancilla systems and quantum measurement), labeled $\tilde{\mathcal{B}}$, that implements one of the $\tilde{K}_j$ sampled from the set $\{\tilde{K}_j\}$ with the correct probability $\tilde{P}_j$. Then, if we implement this protocol many times on identical copies of the same initial state $\rho$, we have that $$\begin{aligned}
\mathbb{E}\left(\tilde{\mathcal{B}}\left[\rho\right]\right) = \sum_j \tilde{P}_j\tilde{K}_j\rho\tilde{K}_j^\dagger = \mathcal{B}[\rho],\end{aligned}$$ where $\mathbb{E}\left(.\right)$ is the ensemble average. We will use the above description as a single time-step of the discrete-time QSW.
Let us now consider repeated action of $\tilde{\mathcal{B}}$, and in analogy to the “quantum trajectories on a quantum computer” scheme developed in Ref. [@Govia2017] we shall refer to each instance of such repeated action as a *trajectory*. By linearity of the quantum operations, we see that $$\begin{aligned}
\mathbb{E}\left(\tilde{\mathcal{B}}\left[\tilde{\mathcal{B}}\left[\rho\right]\right]\right) = \mathbb{E}\left(\tilde{\mathcal{B}}\left[\mathbb{E}\left(\tilde{\mathcal{B}}\left[\rho\right]\right)\right]\right) = \mathcal{B}\left[\mathcal{B}[\rho]\right],\end{aligned}$$ which can be trivially extended to any number of actions of $\tilde{\mathcal{B}}$. Thus, by averaging over the final outcome of many trajectories we can simulate the action of an arbitrary number of repetitions of the Kraus map $\mathcal{B}$. In the rest of this manuscript, we detail how to implement a map of the form of $\tilde{\mathcal{B}}$ for the case of a discrete-time QSW.
Quantum Simulation of a Discrete-Time Quantum Stochastic Walk {#sec:Protocol}
=============================================================
Let $\mathcal{G}=\left(V(\mathcal{G}),E(\mathcal{G})\right)$ be an arbitrarily connected (and possibly directed) graph with vertices $V(\mathcal{G})$ and edges $E(\mathcal{G})$, and $\{|n\rangle,1\leq n\leq |V(\mathcal{G}|\}$ a set of pairwise orthonormal quantum states which enumerate the location of a “walker” on the graph vertices. We will restrict our system to the single excitation subspace, so that $|n\rangle$ denotes a quantum state with a single excitation in vertex $n$ and all other vertices empty.
For any connected graph $\mathcal{G}$, we consider a quantum stochastic map $\mathcal{B}$ representing a single time-step of a QSW, which can be written in Kraus form as $$\begin{aligned}
\label{StochasticMap}
\mathcal{B}[\rho]:=\alpha\hat{U}_{\mathcal{G}}(\Delta t)\rho(t)\hat{U}_{\mathcal{G}}^{\dagger}(\Delta t)+\sum_{(m,n)\in E\left(\mathcal{G}\right)}\kappa_{nm}{|m\rangle\langle n|\rho|n\rangle\langle m|}.\end{aligned}$$ Here, $\hat{U}_{\mathcal{G}}(\Delta t):=e^{-i\hat{H}_{\mathcal{G}}\Delta t}$ is the propagator of the graph coherent evolution for a time-step of length $\Delta t$, generated by the Hamiltonian $\hat{H}_{\mathcal{G}}$ of the graph $\mathcal{G}$. The coefficients $\alpha,\kappa_{nm}\in[0,1]$ represent the weights for coherent or incoherent processes to happen and satisfy $\sum_{m}\kappa_{nm}=1-\alpha$ for all $n\in V\left(\mathcal{G}\right)$ due to trace-preservation.
We define a discrete-time quantum stochastic walk by the repeated application of the single time-step quantum stochastic map $\mathcal{B}$ to the initial state $\rho_0$ $$\begin{aligned}
\label{DiscreteQSW}
\rho_n=\mathcal{B}^n\left[\rho_0\right]:=\underbrace{\mathcal{B}[\mathcal{B}[\ldots\mathcal{B}\left[\rho_0\right]}_{n\textnormal{ times}}\ldots]].\end{aligned}$$ In the following, we show how to construct the quantum stochastic map $\tilde{\mathcal{B}}$ that, as described previously, can be used to simulate $\mathcal{B}$ via the ensemble average. To do this for any connected graph $\mathcal{G}$, we use its key building-block: the 2-vertex graph $\mathcal{G}_2$ with a single (possibly directed) edge. Notice that equation (\[StochasticMap\]) is of the form of the single time-step quantum operation, see equation (\[eq:SingleStepWalk\]). We thus need to define how to implement each of its Kraus operators.
A general 2-vertex graph {#sec:SingleEdge}
------------------------
Let us consider the most general 2-vertex graph $\mathcal{G}_2$ with coherent edge coupling and all possible directed[^1] edges, see figure \[fig:Edge\]. As we will argue, the procedure outlined below easily generalizes to larger graphs. For such 2-vertex graphs, equation (\[StochasticMap\]) becomes $$\begin{aligned}
\label{StochasticMap2}
\mathcal{B}[\rho]:=\alpha\hat{U}_{\mathcal{G}_2}(\Delta t)\rho(t)\hat{U}_{\mathcal{G}_2}^{\dagger}(\Delta t)+\sum_{m,n=1}^2\kappa_{nm}{|m\rangle\langle n|\rho|n\rangle\langle m|},\end{aligned}$$ where trace-preservation implies that $\kappa_{11}+\kappa_{12}=\kappa_{21}+\kappa_{22}=1-\alpha$. The full system will be comprised of the original graph vertices, represented by the basis states ${\left|n\right\rangle}$, and one ancillary quantum state coupled to graph vertex. We shall refer to the graph vertices simply as the [*system*]{}, and the ancillary states as the [*ancillae*]{}. The ancillae will be used to implement the stochastic processes. A single time-step of the QSW, given by equation (\[StochasticMap2\]), will be divided into three parts (see figure \[fig:protocolSingleEdge\]):
\(1) [*Initialization*]{}: At the start of each time-step, the system and ancillae are uncoupled with no excitations in the ancillae. The density matrix can then be written as $$\begin{aligned}
\label{eq:rho0}
\rho_0= \left(\begin{array}{rrrr} \rho_{11} & \rho_{12} & \rho_{1a_1} & \rho_{1a_2}\\
\rho_{21} & \rho_{22} & \rho_{2a_1} & \rho_{2a_2}\\
\rho_{a_11} & \rho_{a_12} & \rho_{a_1a_1} & \rho_{a_1a_2}\\
\rho_{a_21} & \rho_{a_22} & \rho_{a_2a_1} & \rho_{a_2a_2}\\
\end{array}\right)= \left(\begin{array}{rrrr} \rho_{11} & \rho_{12} & 0 & 0\\
\rho_{21} & \rho_{22} & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
\end{array}\right),\end{aligned}$$ where the subscript 1 (2) denotes vertex 1 (2), and subscripts $a_1,~a_2$ denote the corresponding ancillae. All vertices are then coupled to their corresponding ancillae via the interaction Hamiltonian $$\begin{aligned}
\hat{H}_{\rm init}:=g\left(|1\rangle\langle a_1|+|a_1\rangle\langle 1|+|2\rangle\langle a_2|+|a_2\rangle\langle 2|\right),\end{aligned}$$ for time $\Delta t_{\rm init}$. All couplings are equal and $\Delta t_{\rm init}$ is chosen such that $$\begin{aligned}
\label{Arccos}
g\Delta t_{\rm init}=\arccos\left(\sqrt{\alpha}\right).\end{aligned}$$ This choice is crucial, as will become clear in the second step discussed below. It results in a density matrix $\rho_{\rm init}$ right after the initialization step that is given by $$\begin{aligned}
\label{Initialization}
\rho_{\rm init}=\hat{U}_{\rm init}\rho_0\hat{U}^{\dagger}_{\rm init}=
\left(\begin{array}{cc}
\begin{array}{cc}
\alpha\rho_{11} & \alpha\rho_{12}\\
\alpha\rho_{21} & \alpha\rho_{22}
\end{array} & \Upsilon \\
\Upsilon & \Upsilon \\
\end{array}\right),\end{aligned}$$ where $\hat{U}_{\rm init}=e^{-i\hat{H}_{\rm init}\Delta t_{\rm init}}$ and the $\Upsilon$ symbols represent generic $2\times2$ matrices whose precise form is not relevant at this stage.
\(2) [*Coherent Evolution*]{}: We now decouple the ancillae from the system and implement the desired coherent evolution between the graph vertices within the system $$\begin{aligned}
\hat{H}_{\mathcal{G}_2}:=g_{\rm coh}\left(|1\rangle\langle 2|+|2\rangle\langle 1|\right)\end{aligned}$$ for the desired length of the time-step $\Delta t$. Note that $\hat{H}_{\mathcal{G}_2}$ is the Hamiltonian of the graph $\mathcal{G}_2$, which has one free element $g_{\rm coh} \geq 0$. The density matrix $\rho_{\rm coh}$ after the coherent evolution is $$\begin{aligned}
\rho_{\rm coh}=\hat{U}_{\mathcal{G}_2}(\Delta t)\rho_{\rm init}\hat{U}^{\dagger}_{\mathcal{G}_2}(\Delta t),\end{aligned}$$ where $\hat{U}_{\mathcal{G}_2}=e^{-i\hat{H}_{\mathcal{G}_2}\Delta t}$ is the propagator of the graph Hamiltonian. Explicitly, at the end of the coherent evolution step we obtain
$$\begin{aligned}
\rho_{\rm coh}=
\left(\begin{array}{cc}
\alpha\hat{U}_{\mathcal{G}_2}(\Delta t)\left(\begin{array}{cc}
\rho_{11} & \rho_{12}\\
\rho_{21} & \rho_{22}
\end{array}\right)\hat{U}^{\dagger}_{\mathcal{G}_2}(\Delta t) & \Upsilon \\
\Upsilon & \Upsilon \\
\end{array}\right).\end{aligned}$$
which is an implementation of the first term (coherent evolution) on the right-hand side of equation (\[StochasticMap2\]) in the subspace of the vertices.
\(3) [*Measurement and Feed-Forward*]{}: The final part of the protocol uses quantum measurement to randomly determine which term from equation (\[StochasticMap2\]) is implemented for each time-step in a given trajectory. The coherence between vertices and ancillae is also removed, guaranteeing that once each time-step is concluded the system state is of the form of equation (\[eq:rho0\]). To do this, we decouple the system vertices and measure all ancillae simultaneously. As the system is restricted to the single-excitation subspace, there are three possible results:
1. the excitation is measured in ancilla $a_1$,
2. the excitation is measured in ancilla $a_2$,
3. all ancillae are found empty.
For an $n$-vertex graph there are $n+1$ possible measurement outcomes. The last outcome guarantees that the walker is in one of the system vertices, and the second step implements the coherent evolution part of equation (\[StochasticMap2\]) in this case. No further action is required, and we can proceed to the next time-step in the trajectory.
The other measurement results are interpreted as one of the incoherent processes having taken place. To determine which, we use the incoherent rates $\kappa_{ij}$ of the intended QSW as follows. If the excitation is found in ancilla $a_i$ this fixes the index $i$ in $\kappa_{ij}$, i.e. the starting vertex of the incoherent process. To determine the index $j$ and implement the incoherent evolution, we randomly choose $j$ from a probability distribution given by the conditional probabilities $P(j|i)$, and then move the excitation to system vertex $j$. These conditional probabilities are given by $$\begin{aligned}
\label{CondProbs}
P(j|i)=\frac{\kappa_{ij}}{\sum_j\kappa_{ij}}.\end{aligned}$$ In this feed-forward operation, the outcome of the quantum measurement combined with the outcome of the classical random choice determines which of the incoherent terms in equation (\[StochasticMap2\]) is implemented in this time-step of the trajectory.
The complete set of operators which describe the measurement and feed-forward step for a two-vertex graph is given by $\big\{\hat{M}_0,\hat{M}^{a_1}_{1},\hat{M}^{a_1}_{2},\hat{M}^{a_2}_{1},\hat{M}^{a_2}_{2}\big\}$, with matrix representations given in \[app:Matrices\]. $\hat{M}_0$ describes the measurement outcome where both ancillae are found to be empty, and we write $$\begin{aligned}
\hat{M}^{a_1}_{1/2}=\hat{F}^{a_1}_{1/2}\hat{M}_{a_1}\textnormal{ and }\hat{M}^{a_2}_{1/2}=\hat{F}^{a_2}_{1/2}\hat{M}_{a_2},\end{aligned}$$ where $\hat{M}_{a_1/a_2}$ describes the measurement where the excitation is found in ancilla $a_1/a_2$, and $\hat{F}^{a_1/a_2}_{1/2}$ describes the conditional feed-forward according to the measurement result and classical random choice.
The three step procedure outlined above implements a single step of a single trajectory of the discrete-time QSW. Averaging over many trajectories we obtain the density matrix
$$\begin{aligned}
\label{Rhof4x4}
\rho_{\Delta t}&:=\hat{M}_0\rho_{\rm coh}\hat{M}^{\dagger}_0+\sum_{y\in\{1,2\}}\sum_{x\in\{a_1,a_2\}}\hat{M}^{x}_{y}\rho_{\rm coh}{\hat{M}^{x\dagger}_{y}}\\
&=\left(\begin{array}{rr}
\mathcal{B}\left[\left(\begin{array}{rr} \rho_{11} & \rho_{12}\\ \rho_{21} & \rho_{22}\end{array}\right)\right] &\begin{array}{rr} 0 & 0\\ 0 & 0\end{array}\\ \begin{array}{rr} 0 & 0\\ 0 & 0\end{array}\hspace{.8cm} & \begin{array}{rr} 0 & 0\\ 0 & 0\end{array}\\
\end{array}\right).\end{aligned}$$
A $k$-step trajectory is performed by $k$ successive implementations of the above protocol, with its ensemble average having the desired statistics to simulate the QSW.
Arbitrary Graphs {#sec:ArbitraryGraphs}
----------------
The protocol proposed in section \[sec:SingleEdge\] generalizes trivially to any larger graph $\mathcal{G}$, with each system vertex requiring an ancilla. As before, a single time-step is split into three parts:
\(1) [*Initialization*]{}: System states are coupled to their corresponding ancillae via $$\begin{aligned}
\label{Hinit}
\hat{H}_{\rm init}:=\sum_{m\in V\left(\mathcal{G}\right)}g\left(|m\rangle\langle a_m|+|a_m\rangle\langle m|\right),\end{aligned}$$ for a time $\Delta t_{\rm init}$. Here, the summation covers all the graph vertices $m\in V\left(\mathcal{G}\right)$ and the state $|a_m\rangle$ denotes the ancilla state which corresponds to vertex $m$. Again, $\Delta t_{\rm init}$ is chosen such that $$\begin{aligned}
g\Delta t_{\rm init}=\arccos\left(\sqrt{\alpha}\right).\end{aligned}$$
\(2) [*Coherent Evolution*]{}: The ancillae are now decoupled from the system and the system evolves coherently with $$\begin{aligned}
\label{JCH}
\hat{H}_{\mathcal{G}}:=\sum_{(n,m)\in E\left(\mathcal{G}\right)}g_{nm}\left(|m\rangle\langle n|+|n\rangle\langle m|\right),\end{aligned}$$ for a time $\Delta t$. Note that equation (\[JCH\]) is the full Hamiltonian of the graph as the summation covers all the edges of the graph.
\(3) [*Measurement and Feed-Forward*]{}: Finally, the ancillae are measured. As before, if the ancilla are all found to be empty the time-step is complete. If the excitation is found in an ancilla, then the excitation will be incoherently moved to a randomly chosen system vertex that is connected to the system vertex corresponding to the excited ancilla. This process is identical to that described previously for a two-vertex graph, but with the choice of final vertex expanded to include all vertices connected incoherently ($\kappa_{ij} > 0$) to the initial vertex.
Parts (1)-(3) implement a single time-step of equation (\[StochasticMap2\]). Again, the complete walk will be given by iterating this procedure. This simple generalization is possible for two main reasons: (i) each graph vertex is only coupled to a single ancilla, and (ii) the ancillae are never directly coupled to each other.
Simulating a Continuous-Time QSW by a Discrete-Time QSW {#sec:ApproximateTimeEvolution}
=======================================================
In the previous sections we showed how to simulate discrete-time quantum stochastic walks as defined by equations (\[StochasticMap\]) and (\[DiscreteQSW\]), using a trajectory approach. However, currently the majority of applications of QSWs use the continuous-time version, as is widely documented in literature [@Schuld2014c; @Cuevas; @Mohseni2008; @Zimboras:2013aa; @Caruso:2014aa; @Viciani:2015aa; @Caruso:2016aa; @Park:2016aa]. Therefore, we now show how to implement a restricted set of continuous-time QSW by a discrete-time QSW, such that our method for simulating discrete-time QSWs is also applicable.
The continuous-time quantum stochastic walk of Ref. [@Whitfield2010] is given by a Lindblad master equation of the form $$\begin{aligned}
\label{QSW}
\dot{\rho}= (\omega-1)i\left[\hat{H}_{\mathcal{G}},\rho\right] + \omega\sum_{k}\gamma_k{\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right)},\end{aligned}$$ where $\rho$ is the density operator of the system, $\hat{L}_k$ are the Lindblad operators with $\gamma_k$ their associated incoherent transition rates, $\hat{H}_{\mathcal{G}}$ is the Hamiltonian of the underlying graph $\mathcal{G}$ and $\omega\in[0,1]$. For $\omega=0$, we obtain the completely coherent quantum walk and for $\omega=1$, the classical random walk. Hence, for $\omega\in(0,1)$, equation (\[QSW\]) leads to dynamics we could not obtain in a purely coherent or incoherent framework.
We write the Liouvillian $\mathcal{L}_{\omega}$ of equation (\[QSW\]) as $$\begin{aligned}
\mathcal{L}_{\omega}\rho=(1-\omega)\mathcal{H}\rho+\omega\Lambda\rho
,\label{Liouvillian}\end{aligned}$$ where $\mathcal{H}\rho=-i\left[\hat{H}_{\mathcal{G}},\rho\right]$ and $$\begin{aligned}
\Lambda\rho=\sum_{k}\gamma_k{\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right)}.\end{aligned}$$ The Liouvillian (\[Liouvillian\]) is the generator of a quantum dynamical semigroup [@Breuer:2006uq]. Therefore, we can write $$\begin{aligned}
\rho(t+\Delta t)=\exp\left(\mathcal{L}_{\omega}\Delta t\right)\rho(t)=\left(\sum_{l=0}^{\infty}\frac{1}{l!}\mathcal{L}_{\omega}^l\Delta t^l\right)\rho(t).
\label{Semigroup}\end{aligned}$$ Inserting equation (\[Liouvillian\]) into equation (\[Semigroup\]) yields, up to first order in $\Delta t$, $$\begin{aligned}
\fl\qquad\qquad\rho(t+\Delta t)&=\Big(1+\Delta t\left((1-\omega)\mathcal{H}_{\mathcal{G}}+\omega\Lambda\right)\nonumber +\mathcal{O}\left(\Delta t^2\right)\Big)\rho(t)\nonumber \\&=\Big((1-\omega)\left(1+\Delta t\mathcal{H}_{\mathcal{G}}\right)+\omega\left(1+\Delta t\Lambda\right)+\mathcal{O}\left(\Delta t^2\right)\Big)\rho(t).
\label{FirstOrderApproximation}\end{aligned}$$ The first term on the right-hand side of equation (\[FirstOrderApproximation\]) can be interpreted as pure coherent evolution that occurs with probability $(1-\omega)$. Analogously, we interpret the second term as describing incoherent evolution occurring with probability $\omega$.
We now consider continuous-time QSWs where the incoherent evolution describes incoherent excitation transfer between system vertices [@Cuevas], such that $\hat{L}_k = \left|m\rangle\langle n\right|$. In this case, the incoherent evolution of equation (\[FirstOrderApproximation\]) becomes $$\begin{aligned}
\fl\qquad\omega\left(1+\Delta t\Lambda\right)\rho
=\omega\rho+\sum_{(m,n)\in E\left(\mathcal{G}\right)}\omega\Delta t\gamma_{nm}{\left(|m\rangle\langle n|\rho|n\rangle\langle m|-\frac{1}{2}\big\{|n\rangle\langle n|,\rho\big\}\right)}.\end{aligned}$$ Here, the $k$-th Lindblad operator $\hat{L}_k=|m\rangle\langle n|$ generates an incoherent jump from vertex $n$ to vertex $m$, and $\gamma_{nm}$ describes the transition rate for this process, with $E\left(\mathcal{G}\right)$ the set of connected edges of the graph.
To first order in $\Delta t$, we see that $p_{nm} =\Delta t\gamma_{nm}$ can be treated as the conditional probability to transition to vertex $m$ if the excitation is in vertex $n$ during time-step $\Delta t$, if the Lindblad rates satisfy $$\begin{aligned}
\label{CondProb}
\sum_{m\in V\left(\mathcal{G}\right)}p_{nm}= \sum_{m\in V\left(\mathcal{G}\right)}\Delta t\gamma_{nm} =1, \label{eqn:probcon}\end{aligned}$$ to ensure conservation of probability. This condition must be satisfied simultaneously for all $n$, which is possible if and only if $$\begin{aligned}
\sum_{m\in V\left(\mathcal{G}\right)}\gamma_{nm}=\gamma\hspace{1cm}\forall n\in V\left(\mathcal{G}\right),\end{aligned}$$ such that $\Delta t=\gamma^{-1}$ can be chosen uniquely to simultaneously guarantee equation (\[eqn:probcon\]) for all $n$. We note that this is the same restriction on the Lindblad rates as was necessary for protocols to simulate continuous time QSWs using quantum trajectories on a quantum computer [@Govia2017].
Under this restriction, we can write the incoherent evolution as $$\begin{aligned}
\nonumber\fl\omega\rho+\omega\sum_{(m,n)\in E\left(\mathcal{G}\right)}p_{nm} {\left(|m\rangle\langle n|\rho|n\rangle\langle m|-\frac{1}{2}\big\{|n\rangle\langle n|,\rho\big\}\right)}
\\ \fl\nonumber=\omega\rho+\omega\sum_{(m,n)\in E\left(\mathcal{G}\right)}p_{nm} |m\rangle\langle n|\rho|n\rangle\langle m| -\frac{\omega}{2} \underbrace{\sum_{n\in V\left(\mathcal{G}\right)}\big\{|n\rangle\langle n|,\rho\big\}}_{=2\rho}\underbrace{\sum_{m\in V\left(\mathcal{G}\right)}p_{nm} }_{=1}
\\ \fl \label{eqn:kappadef} =\sum_{(m,n)\in E\left(\mathcal{G}\right)}\kappa_{nm}|m\rangle\langle n|\rho|n\rangle\langle m|,\end{aligned}$$ where in the last line we have defined $\kappa_{nm} = \omega p_{nm}$. Thus, we see that the short-time incoherent evolution for this restricted class of continuous-time QSWs has the same form of the incoherent part of the discrete-time QSW we have used throughout this manuscript.
Similarly, we replace $\left(1+\Delta t\mathcal{H}_{\mathcal{G}}\right)$ with the unitary propagator $\hat{U}_{\mathcal{G}}(\Delta t) =e^{-i\hat{H}_{\mathcal{G}}\Delta t} $ for the coherent evolution. Combining these results, and defining $\alpha=1-\omega$, we see that we can write the continuous-time evolution for short $\Delta t$ as $$\begin{aligned}
\label{ApproximationFinal}
\fl\rho(t+\Delta t)=\alpha\hat{U}_{\mathcal{G}}(\Delta t)\rho(t)\hat{U}_{\mathcal{G}}^{\dagger}(\Delta t) + \sum_{(m,n)\in E\left(\mathcal{G}\right)}\kappa_{nm}{|m\rangle\langle n|\rho|n\rangle\langle m|}+\mathcal{O}\left(\Delta t^2\right).\end{aligned}$$ This has the form of a Kraus map for a discrete-time QSW, and as such, we have shown how to implement the short time evolution of a restricted class of continuous-time QSWs with a discrete-time QSW, broadening the applicability of our simulation method for discrete-time QSWs.
Conclusion {#sec:conc}
==========
In this work, we developed a trajectory-based protocol to simulate discrete-time QSWs on a coherent quantum computer. This ancilla-based protocol breaks down each time-step of the QSW into three parts that require only coherent couplings, measurements, and feed-forward operations, and thus are suitable to implementation on quantum hardware. Subsequent applications of this process create a single quantum trajectory, and we show that, as with the standard quantum trajectories approach, ensemble averages over many trajectories mimics the desired QSW dynamics.
The full time-step was carefully detailed for the most general graph of two vertices and we have shown that this serves as a building block to simulate arbitrary graphs. The simple procedure to generalize to complex graphs is one of the key features of our proposal, as no complicated design of system-ancillae interaction is needed. The protocol can also be employed for simulations of continuous-time QSWs satisfying certain conditions, which are also present in previously proposed simulation methods using quantum computers.
We note that as our protocol is designed on the single-excitation subspace, hardware implementations using qubits to represent vertices and ancillae are not resource efficient. Such qubit implementations use a $2^N$-dimensional Hilbert space to simulate a graph $\mathcal{G}$ with $|V\left(\mathcal{G}\right)|=N$, which only requires $2N$ degrees of freedom including ancillae. An alternative approach could use qutrits to represent each vertex, with the third energy level representing the ancillae.
As is usual for quantum trajectory based protocols, our proposal will be more suited for simulations for which convergence scales faster than $d^2$, where $d$ is the Hilbert space dimension of the system to be simulated. As this heavily depends on the underlying graph $\mathcal{G}$, no general statement is possible. However, we note that as the system evolution is reset to a specific state after any ancilla is measured to be occupied, the protocol requires coherence times much shorter than the total simulation time and could therefore be useful for near-term quantum hardware implementations.
B.G.T. acknowledges support from FAPESC and CNPq INCT-IQ (465469/2014- 0).
Matrix representations of the measurement operators for a single edge {#app:Matrices}
=====================================================================
The matrix representations of the measurement operators in Section \[sec:SingleEdge\] for a 2-vertex graph are: $$\begin{aligned}
\fl\hat{M}_0=\left(\begin{array}{rrrr} 1 & 0 & 0 & 0\\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}\right)\hspace{.8cm}\hat{M}_{a_1/a_2}=\left(\begin{array}{rrrr} 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 \\
0 & 0 & 1/0 & 0 \\
0 & 0 & 0 & 0/1 \\
\end{array}\right)\\
\fl\hat{F}^{a_1}_{1}=P(1|1)\left(\begin{array}{rrrr} 0 & 0 & 1 & 0\\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}\right)\hspace{.8cm}\hat{F}^{a_1}_{2}=P(2|1)\left(\begin{array}{rrrr} 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}\right)\\
\fl\hat{F}^{a_1}_{1}=P(1|2)\left(\begin{array}{rrrr} 0 & 0 & 0 & 1\\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}\right)\hspace{.8cm}\hat{F}^{a_2}_{2}=P(2|2)\left(\begin{array}{rrrr} 1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
\end{array}\right)\end{aligned}$$
References {#references .unnumbered}
==========
[^1]: We call an edge directed if any of the $\kappa_{nm} > 0$, since that is the defining difference between a QSW and a coherent QW.
|
---
abstract: 'Bars in galaxies are mainly supported by particles trapped around closed periodic orbits. These orbits respond to the bar’s forcing frequency only and lack free oscillations. We show that a similar situation takes place in double bars: particles get trapped around orbits which only respond to the forcing from the two bars and lack free oscillations. We find that writing the successive positions of a particle on such an orbit every time the bars align generates a closed curve, which we call a loop. Loops allow us to verify consistency of the potential. As maps of doubly periodic orbits, loops can be used to search the phase-space in double bars in order to determine the fraction occupied by ordered motions.'
author:
- Witold Maciejewski
title: 'Chaos or Order in Double Barred Galaxies?'
---
Introduction
============
Bars within bars appear to be a common phenomenon in galaxies. Recent surveys show that up to 30% of early-type barred galaxies contain nested bars [@erw+s02]. The relative orientation of the two bars is random, therefore it is likely that the bars rotate with different pattern speeds. Inner bars, like large bars, are made of relatively old stellar populations, since they remain distinct in near infrared [@wozn96]. Galaxies with two independently rotating bars do not conserve the Jacobi integral, and it is a complex dynamical task to explain how such systems are sustained. To account for their longevity, one has to find sets of particles that support the shape of the potential in which they move. Particle motion in a potential of double bars belongs to the general problem of motion in a pulsating potential [@l+g89] [@sridh88], of which the restricted elliptical 3-body problem is the best known example. Families of closed periodic orbits have been found in this last problem, where the test particle moves in the potential of a binary star with components on elliptical orbits [@bro69]. However, such families are parameterized by values that also characterize the potential (i.e. ellipticity of the stellar orbit and the mass ratio of the stars), and their orbital periods are commensurate with the pulsation period of the potential. For a given potential, these families are reduced to single orbits separated in phase-space. The solution for double bars is formally identical, and there an orbit can close only when the orbital period is commensurate with the relative period of the bars. Such orbits are separated in phase-space, and therefore families of closed periodic orbits are unlikely to provide orbital support for nested bars. Another difficulty in supporting nested bars is caused by the piling up of resonances created by each bar, which leads to considerable chaotic zones. In order to minimize the number of chaotic zones, resonant coupling between the bars has been proposed [@sygnet88], so that the resonance generated by one bar overlaps with that caused by the other bar.
Finding support for nested bars has been hampered by the fact that closed periodic orbits are scarce there. However, it is particles, not orbits, which create density distributions that support the potential. The concept of closed periodic orbit is too limiting in investigation of nested bars, and another description of particle motion, which does not have its limitations, is needed. Naturally, in systems with two forcing frequencies, double-periodic orbits play a fundamental role. Thus in double bars a large fraction of particle trajectories gets trapped around a class of double-periodic orbits. Although such orbits do not close in any reference frame, they can be conveniently mapped onto the loops [@m+s00], which are an efficient descriptor of orbital structure in a pulsating potential. The loop is a closed curve that is made of particles moving in the potential of a doubly barred galaxy, and which pulsates with the relative period of the bars. Orbital support for nested bars can be provided by placing particles on the loops.
Here I give a systematic description of the loop approach, which recovers families of stable double-periodic orbits, and which can be applied to any pulsating potential. In §2 I use the epicyclic approximation to introduce the basic concepts, and in §3 I outline the general method.
The epicyclic solution for any number of bars
=============================================
If a galaxy has a bar that rotates with a constant pattern speed, it is convenient to study particle orbits in the reference frame rotating with the bar. If two or more bars are present, and each rotates with its own pattern speed, there is no reference frame in which the potential remains unchanged. In order to point out formal similarities in solutions for one and many bars, I solve the linearized equations in the inertial frame. This is equivalent to the solution in any rotating frame, and the transformation is particularly simple: in the rotating frame the centrifugal and Coriolis terms are equivalent to the Doppler shift of the angular velocity. It is convenient to show it in cylindrical coordinates $(R,\varphi,z)$: if ${\bf e}_z$ is the rotation axis, then the $R$ and $\varphi$ components of the equation of motion for the rotating frame, $\ddot{\bf r} = -\nabla \Phi - 2 ({\bf \Omega_B} \times \dot{\bf r})
- {\bf \Omega_B} \times ( {\bf \Omega_B} \times {\bf r})$, can be written as $$\begin{aligned}
\ddot{R} - R ( \dot{\varphi} + \Omega_B )^2
& = & -\frac{\partial \Phi}{\partial R}, \\
R \ddot{\varphi} + 2 \dot{R} ( \dot{\varphi} + \Omega_B )
& = & -\frac{1}{R} \frac{\partial \Phi}{\partial \varphi}.\end{aligned}$$ These equations are identical with the components of the equation of motion in the inertial frame, $$\ddot{\bf r} = -\nabla \Phi ,
\label{eqmi}$$ where clearly the angular velocity $\dot{\varphi}$ in the rotating frame corresponds to $\dot{\varphi}+\Omega_B$ in the inertial frame. For the rest of this section I assume the inertial frame, in which the equation of motion (\[eqmi\]) has the following $R$ and $\varphi$ components in cylindrical coordinates $$\begin{aligned}
\label{eqmri}
\ddot{R} - R \dot{\varphi}^2
& = & -\frac{\partial \Phi}{\partial R}, \\
\label{eqmfi}
R \ddot{\varphi} + 2 \dot{R} \dot{\varphi}
& = & -\frac{1}{R} \frac{\partial \Phi}{\partial \varphi} .\end{aligned}$$ The $z$ component in any frame is $\ddot{z} = - \partial \Phi / \partial z$, but I consider here motions in the plane of the disc only, hence I neglect the dependence on $z$.
To linearize equations (\[eqmri\]) and (\[eqmfi\]), one needs expansions of $R$, $\varphi$ and $\Phi$ to first order terms. The epicyclic approximation is valid for particles whose trajectories oscillate around circular orbits. For such particles one can write $$\begin{aligned}
R(t) & = & R_0 + R_I(t) , \\
\varphi(t) & = & \varphi_{00} + \Omega_0 t + \varphi_I(t) , \\
\Phi(R,\varphi,t) & = & \Phi_0(R) + \Phi_I(R,\varphi,t) ,\end{aligned}$$ where terms with index $I$ are small to the first order, and second- and higher-order terms were neglected. The parameter $\varphi_{00}$ allows the particle to start from any position angle at time $t=0$, so that $\varphi_0 = \varphi_{00} + \Omega_0 t$. Asymmetry $\Phi_I$ in the potential is small and may be time-dependent. The angular velocity $\Omega_0$ on the circular orbit of radius $R_0$ relates to the potential $\Phi_0$ through the zeroth order of (\[eqmri\]): $\Omega_0^2 = (1/R_0) (\partial \Phi_0 / \partial R) \; |_{R_0}$. The zeroth order of (\[eqmfi\]) is identically equal to zero, and the first order corrections to (\[eqmri\]) and (\[eqmfi\]) take respectively forms $$\begin{aligned}
\label{rlin}
\ddot{R_I} - 4 A \Omega_0 R_I - 2 R_0 \Omega_0 \dot{\varphi_I} & = &
-\frac{\partial \Phi_I}{\partial R} \; |_{R_0,\varphi_0}, \\
\label{flin}
R_0 \ddot{\varphi_I} + 2 \Omega_0 \dot{R_I} & = &
-\frac{1}{R_0} \frac{\partial \Phi_I}{\partial \varphi} \; |_{R_0,\varphi_0},\end{aligned}$$ where $A$ is the Oort constant defined by $ 4 A \Omega_0 = \Omega_0^2 - \frac{\partial^2 \Phi_0}{\partial R^2} |_{R_0}$.
We assume that the bars are point-symmetric with respect to the galaxy centre. Thus to first order the departure of the barred potential from axial symmetry can be described by a term $\cos (2 \varphi)$. If multiple bars, indexed by $i$, rotate independently as solid bodies with angular velocities $\Omega_i$, the time-dependent first-order correction $\Phi_I$ to the potential can be written as $$\label{phi1}
\Phi_I(R,\varphi,t) = \sum_i \Psi_i(R) \cos[ 2 (\varphi - \Omega_i t) ] ,$$ where the radial dependence $\Psi_i(R)$ has been separated from the angle dependence. No phase in the trigonometric functions above means that we define $t=0$ when all the bars are aligned. Derivatives of (\[phi1\]) enter right-hand sides of (\[rlin\]) and (\[flin\]), which after introducing $\omega_i = 2 (\Omega_0 - \Omega_i)$ take the form $$\begin{aligned}
\label{rlin1}
\ddot{R_I} - 4 A \Omega_0 R_I - 2 R_0 \Omega_0 \dot{\varphi_I} & = &
- \sum_i \frac{\partial \Psi_i}{\partial R} \; |_{R_0} \cos(\omega_i t + 2 \varphi_{00}) , \\
\label{flin1}
R_0 \ddot{\varphi_I} + 2 \Omega_0 \dot{R_I} & = &
\frac{2}{R_0} \sum_i \Psi_i(R_0) \sin (\omega_i t + 2 \varphi_{00}).\end{aligned}$$
In order to solve the set of equations (\[rlin1\],\[flin1\]), one can integrate (\[flin1\]) and get an expression for $R_0 \dot{\varphi_I}$, which furthermore can be substituted to (\[rlin1\]). This substitution eliminates $\varphi_I$, and one gets a single second order equation for $R_I$, which can be written schematically as $$\label{ddotR}
\ddot{R_I} + \kappa_0^2 R_I = \sum_i A_i \cos (\omega_i t + 2 \varphi_{00})
+ C_{\varphi},$$ where $ A_i = - \frac{4 \Omega_0 \Psi_i}{\omega_i R_0}
- \frac{\partial \Psi_i}{\partial R}_{| R_0} $, $\kappa_0^2 = 4 \Omega_0 ( \Omega_0 - A )$, and $C_{\varphi}/2\Omega_0$ is the integration constant that appears after integrating (\[flin1\]). This is the equation of a harmonic oscillator with multiple forcing terms, whose solution is well known. It can be written as $$\label{r1}
R_I (t) = C_1 \cos (\kappa_0 t + \delta)
+ \sum_i M_i \cos(\omega_i t + 2 \varphi_{00})
+ C_{\varphi}/\kappa_0^2 .$$ The first term of this solution corresponds to a free oscillation at the local epicyclic frequency $\kappa_0$, and $C_1$ is unconstrained. The terms under the sum describe oscillations resulting from the forcing terms in (\[phi1\]), and $M_i$ are functions of $A_i$. Hereafter I focus on solutions without free oscillations, thus I assume that $C_1=0$. These solutions will lead to closed periodic orbits and to loops. The formula for $\varphi_I(t)$ can be obtained by substituting (\[r1\]) into the time-integrated (\[flin1\]). As a result, one gets $$\label{phi1dot}
\dot{\varphi_I} = \sum_i N_i \cos(\omega_i t + 2 \varphi_{00})
- \frac{2 A C_{\varphi}}{\kappa_0^2 R_0},$$ where again $N_i$ are determined by the coefficients of the equations above. Note that to the first order $\Omega_0 [R_0 + C_{\varphi}/\kappa_0^2] =
\Omega_0 [R_0] - 2 A C_{\varphi}/{\kappa_0^2 R_0}$, thus the integration constants entering (\[r1\]) and (\[phi1dot\]) correspond to a change in the guiding radius $R_0$, and to the appropriate change in the angular velocity $\Omega_0$. They all can be incorporated into $R_0$, and in effect the unique solutions for $R_I$ and $\varphi_I$ are $$\begin{aligned}
\label{r1f}
R_I (t) & = & \sum_i M_i \cos(\omega_i t + 2 \varphi_{00}) , \\
\label{f1f}
\varphi_I(t) & = & \sum_i N'_i \cos(\omega_i t + 2 \varphi_{00}) + const ,\end{aligned}$$ where free oscillations have been neglected. The integration constant in (\[f1f\]) is an unconstrained parameter of the order of $\varphi_I$.
Closed periodic orbits in a single bar
--------------------------------------
In a potential with a single bar there is only one term in the sums (\[r1f\]) and (\[f1f\]), hereafter indexed with $B$. Consider the change in values of $R_I$ and $\varphi_I$ for a given particle after half of its period in the frame corotating with the bar. This interval is taken because the bar is bisymmetric, so its forcing is periodic with the period $\pi$ in angle. After replacing $t$ by $t + \pi / (\Omega_0 - \Omega_B)$ one gets $$\begin{aligned}
R_I & = &
M_B \cos[ \omega_B (t + \frac{\pi}{\Omega_0 - \Omega_B}) + 2 \varphi_{00}] \\
& = &
M_B \cos( \omega_B t + 2 \pi + 2 \varphi_{00} ).\end{aligned}$$ Thus the solution for $R_I$ after time $\pi / (\Omega_0 - \Omega_B)$ returns its starting value, and the same holds true for $\varphi_I$. After twice that time, i.e. in a full period of this particle in the bar frame, the epicycle centre returns to its starting point and the orbit closes. Thus (\[r1f\]) and (\[f1f\]) describe closed periodic orbits in the linearized problem of a particle motion in a single bar.
Loops in double bars
--------------------
When two independently rotating bars coexist in a galaxy (hereafter indexed by $B$ and $S$), there is no reference frame in which the potential is constant. Thus when a term from one bar in (\[r1f\]) and (\[f1f\]) returns to its starting value, the term from the other bar does not (unless the frequencies of the bars are commensurate). Therefore the particle’s trajectory does not close in any reference frame. However, consider the change in value of $R_I$ and $\varphi_I$ after time $\pi / (\Omega_S - \Omega_B)$, which is the relative period of the bars. One gets $$\begin{aligned}
R_I & = &
M_B \cos[ \omega_B (t + \frac{\pi}{\Omega_S - \Omega_B}) + 2 \varphi_{00}] +
M_S \cos[ \omega_S (t + \frac{\pi}{\Omega_S - \Omega_B}) + 2 \varphi_{00}] \\
& = &
M_B \cos( \omega_B t + 2 \pi \frac{\Omega_0 - \Omega_B}{\Omega_S - \Omega_B} + 2 \varphi_{00} ) +
M_S \cos( \omega_S t + 2 \pi \frac{\Omega_0 - \Omega_S}{\Omega_S - \Omega_B} + 2 \varphi_{00} ) \\
& = &
M_B \cos( \omega_B t + 2 \pi + 2 \varphi_{01} ) +
M_S \cos( \omega_S t + 2 \varphi_{01}),\end{aligned}$$ where $\varphi_{01} = \varphi_{00} + \pi \frac{\Omega_0 - \Omega_S}{\Omega_S - \Omega_B}$. The same result can be obtained for $\varphi_I$. This means that the time transformation $t \rightarrow t + \pi / (\Omega_S - \Omega_B)$ is equivalent to the change in the starting position angle of a particle from $\varphi_{00}$ to $\varphi_{01}$. Consider motion of a set of particles that have the same guiding radius $R_0$, but start at various position angles $\varphi_{00}$. This is a one-parameter set, therefore in the disc plane it is represented by a curve, and because of continuity of (\[r1f\]) and (\[f1f\]) this curve is closed. After time $\pi / (\Omega_S - \Omega_B)$, a particle starting at angle $\varphi_{00}$ will take the place of the particle which started at $\varphi_{01}$, a particle starting at $\varphi_{01}$ will take the place of another particle from this curve and so on. The whole curve will regain its shape and position every $\pi / (\Omega_S - \Omega_B)$ time interval, although positions of particles on the curve will shift. This curve is the epicyclic approximation to the [*loop*]{}: a curve made of particles moving in a given potential, such that the curve returns to its original shape and position periodically. In the case of two bars, the period is the relative period of the bars, and the loop is made out of particles having the same guiding radius $R_0$. Particles on the loop respond to the forcing from the two bars, but they lack any free oscillation. An example of a set of loops in a doubly barred galaxy in the epicyclic approximation can be seen in [@m+s97]. Since they occupy a significant part of the disc, one should anticipate large zones of ordered motions also in the general, non-linear solution for double bars.
Full nonlinear solution for loops in nested bars
================================================
Tools and concepts useful in the search for ordered motions in double bars are best introduced through the inspection of particle trajectories in such systems. For this inspection I chose the potential of Model 1 defined in [@m+s00], where the small bar is 60% in size of the big bar, and pattern speeds of the bars are not commensurate. Consider a particle moving in this potential inside the corotation of the small bar. Simple experiments with various initial velocities show that if the initial velocity is small enough, the particle usually remains bound. A typical trajectory is shown in the left panels of Fig.1 – since it depends on the reference frame, it is written twice, for reference frame of each bar. Further experimenting with initial velocities shows that particle trajectories are often even tidier: they look like those in the right panels of Fig.1, as if the trajectories were trapped around some regular orbit.
![Two example trajectories (one in the two left panels, one in the two right ones) of a particle that moves in the potential of two independently rotating bars. The particle is followed for 10 relative periods of the bars, and its trajectory is displayed in the frame corotating with the big bar (top panels), and the small bar (bottom panels). Each bar is outlined in its own reference frame by the dotted line. Large dot marks the starting point of the particle.[]{data-label="f1"}](af1.ps){width="125mm"}
Fine adjustments of the initial velocity lead to a highly harmonious trajectory (Fig.2), which looks like a loop orbit in a potential of a single bar (see e.g. Fig.3.7a in [@bt87]). This is only a formal similarity, but understanding it will let us find out what kind of orbit we see in Fig.2. The loop orbit in a single bar forms when a particle oscillates around a closed periodic orbit. Therefore two frequencies are involved: the frequency of the free oscillation, and the forcing frequency of the bar. On the other hand, the Fourier transform of the trajectory from Fig.2 shows two sharp peaks at frequencies equal to the forcing frequencies of the two bars (Fig.3). Thus the trajectory from Fig.2 also has two frequencies: this time these are the forcing frequencies from the two bars, while the free oscillation is absent. This is how the solution in the linear approximation (§2.2) was constructed. We conclude that in both the linear (epicyclic approximation) case and the general case we are dealing with doubly periodic orbits in an oscillating potential of a double bar, with frequencies equal to the forcing frequencies of the bars. In the epicyclic approximation, these orbits have a nice feature that particles following them populate loops: closed curves that return to their original shape and position at every alignment of the bars. One may therefore expect that also in the general case these particles gather on loops.
![A doubly periodic orbit in the doubly barred potential, followed for 20 relative periods of the bars, and written in the frame corotating with the big bar (left), and the small bar (right). The long axis of each bar is marked by the dashed line. Dots mark positions of the particle at every alignment of the bars.[]{data-label="f2"}](af2.eps){width="12cm"}
![Fourier transforms of the trajectories from right panels of Fig.1 (dotted line) and from Fig.2 (solid line). The peaks in the solid line are related to the forcing frequencies of the bars, and the peaks in the dotted line are not.[]{data-label="f3"}](af3.eps){width="8cm"}
If in the general case particles on doubly periodic orbits form a loop, one can construct it by writing positions of a particle on such an orbit every time the bars align. These positions are the initial conditions for particles forming the loop, because after every alignment, the $n^{th}$ particle generated in this way takes the position of particle $n+1$. The first 20 positions of a particle on a doubly periodic orbit are overplotted in Fig.2. They indeed seem to be arranged on a closed ellipse-like curve; the shape of this curve varies in time, but it returns to where it started at every alignment of the bars (Fig.4). This construction shows that in the general case particles on doubly periodic orbits also form loops. Note that positions of particles on other orbits, which involve free oscillations, when written at every alignment of the bars, densely populate some two-dimensional section of the plane, and do not gather on any curve. It is extremely useful for the investigation of the orbital structure in double bars that the appearance of the loop is frame-independent. Loops provide an efficient way to classify doubly periodic orbits, which has been hampered so far by the dependence of the last ones on the reference frame.
![Evolution of the loop from Fig.2 during one relative period of the bars. The bars, outlined with solid lines, rotate counterclockwise. The loop is made out of points that represent separate particles on doubly periodic orbits.[]{data-label="f4"}](af4.eps){width="10cm"}
It turns out that doubly periodic orbits play crucial role in providing orbital support for the pulsating potential of double bars. No closed periodic orbits have been proposed as candidates for the backbone of such a potential. If in a given potential of two bars there are loops that follow the inner bar, and other loops that follow the outer bar, then one may expect that such a potential is dynamically possible. An example of such a potential has been constructed in [@m+s00]. The loop from Fig.4 does not follow either bar in its motion, and therefore it is unlikely that it supports the assumed potential. It can be shown that in that potential, there are no loops which could support the two bars. Thus that potential is not self-consistent. This example shows how efficient is the loop approach in rejecting hypothetical doubly barred systems that have no orbital support.
![The width of the ring formed by particles trapped around loops in Model 2 from [@m+s00] as a function of the particle’s position along the minor axis of the aligned bars, and of its velocity (perpendicular to this axis). Darker color means smaller width. In the insert, the same is shown for rings around closed periodic orbits in a single bar (same model, but inner bar axisymmetric). Regions related to the $x_1$ and $x_2$ orbits, and to the loops originating from them, are marked.[]{data-label="f5"}](af5.ps){width="14cm"}
Doubly periodic orbits in double bars are surrounded by regular orbits in the same way as are the closed periodic orbits in a single bar. In both cases, the trapped regular orbits oscillate around the parent orbit. The trajectory from the right panels of Fig.1 is an example of a regular orbit that is trapped around the doubly periodic orbit from Fig.2. How much of the phase space in double bars is occupied by orbits trapped around doubly periodic orbits? It can be examined by launching a particle from e.g. the minor axis of the bar, in the direction perpendicular to this axis, when the bars are aligned. If the particle is trapped, its positions at every alignment of the two bars lie within a ring containing the loop. The width of this ring depends on the particle’s position along the minor axis, and on its velocity. It is displayed in Fig.5 for the potential of Model 2 defined in [@m+s00]. Two stripes of low width appear on the diagram, which correspond to the $x_1$ and $x_2$ orbits in a single bar [@c+p80] (displayed in the insert). Thus in double bars there are doubly periodic orbits that correspond to closed periodic orbits in single bars. There are possible regions of chaos in double bars (white stripes in Fig.5), but overall loops in double bars and periodic orbits in single bars trap similar volumes of phase-space around them.
Conclusions
===========
In a potential of two independently rotating bars, a large fraction of phase space can be occupied by trajectories trapped around parent regular orbits. These orbits are doubly periodic, with the two periods corresponding to the forcing frequencies of the two bars, but they do not close in any reference frame. Like particle trajectories oscillating around closed periodic orbits in a single bar, particle trajectories in double bars oscillate around the doubly periodic parent orbits. The structure of the parent regular orbits can be mapped using the loop approach, which allows us to single out dynamically possible double bars.
[**Acknowledgments.**]{} The concept of the loop as the organized form of motion in double bars benefits from the insight of Linda Sparke. I thank Lia Athanassoula for our collaboration that lead to this paper, and Peter Erwin for comments on the manuscript.
[8.]{}
Binney, J. & Tremaine, S. 1987, Galactic Dynamics (Princeton: Princeton Univ. Press) Broucke, R. A. 1969, Periodic Orbits in the Elliptic Restricted Three-Body Problem, Jet Propulsion Laboratory Report 32-1360 Contopoulos, G. & Papayannopoulos, Th. 1980, A&A, 92, 33 Erwin, P. & Sparke, L. S. 2002, AJ, 124, 65 Friedli, D., Wozniak, H., Rieke, M., Martinet, L., & Bratschi, P. 1996, A&AS, 118, 461 Louis, P. D. & Gerhard, O. E. 1988, MNRAS, 233, 337 Maciejewski, W. & Sparke, L. S. 1997, ApJL, 484, L117 Maciejewski, W. & Sparke, L. S. 2000, MNRAS, 313, 745 Sridhar, S. 1989, MNRAS, 238, 1159 Sygnet, J. F., Tagger, M., Athanassoula, E., & Pellat, R. 1988, MNRAS, 232, 733
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---
abstract: 'Given a real $n \times m$ matrix $B$, its operator norm can be defined as $$|B|=\max_{|v|=1}|Bv|.$$ We consider a matrix “small” if it has non-negative integer entries and its operator norm is less than $2$. These matrices correspond to bipartite graphs with spectral radius less than $2$, which can be classified as disjoint unions of Coxeter graphs. This gives a direct route to an $ADE$-classification result in terms of very basic mathematical objects. Our goal here is to see these results as part of a general program of classification of small objects, relating quadratic forms, reflection groups, root systems, and Lie algebras.'
address:
- 'Department of Mathematics and Statistics, Canisius College, 2001 Main Street, Buffalo, NY 14208'
- 'Department of Mathematics and Statistics, Canisius College, 2001 Main Street, Buffalo, NY 14208'
author:
- Terrence Bisson
- Jonathan Lopez
bibliography:
- 'bibliography.bib'
date: 'August, 21, 2017'
title: A classification of small operators using graph theory
---
Classifications
===============
In mathematics, a classification result describes all possible structures of a given type, usually by showing that every structure is equivalent to one which decomposes into “components”, each equivalent to one from a set of basic types.
Good examples of classification in mathematics are rare and interesting. The description of the possible structures of semi-simple Lie algebras over the complex numbers is one of the most important examples. Every such Lie algebra is isomorphic to a direct sum of simple types, known by the alphabet $A$, $B$, $C$, $D$, $E$, $F$, $G$ (see Fulton and Harris [@FH], for instance). The $ADE$ part of this classification has surprising similarities with classification results in many other areas of mathematics (see, for example, Hazewinkel, et al. [@HHSV]). For instance, the $ADE$-series appears in Coxeter’s classification of the simply-laced crystallographic finite reflection groups (see [@Cox]); and Cameron, et al. [@cameron] showed that the classification of certain sets of lines in ${\mathbb R}^n$ at mutual angles of $60^{\circ}$ or $90^{\circ}$ also involves the $ADE$-series (see Theorem 3.5 in [@cameron]). These results have led to the development of large areas of ongoing research.
In [@symposium], Arnol’d asked if the appearance of the $ADE$-series in these classifications was merely coincidence, or if there was some profound underlying cause. Though we do not claim to provide an answer to Arnol’d’s question, we do exposit a direct route to an $ADE$-classification result in terms of very basic mathematical objects. Namely, the non-negative integer matrices with operator norm less than $2$ can be classified by the $ADE$-series of graphs. We hope the simplicity of development will appeal to a wide audience, since the usual paths one takes to arrive at an $ADE$-classification can be long and difficult.
More generally, we want to recommend some notions of smallness in various parts of mathematics, and show how such “small” objects can be classified. In our examples, the small objects satisfy a quantitative bound and are defined over the non-negative integers.
In Section 2, we define operator norm and record some results concerning the operator norm needed later in the paper. In particular, we explain how the operator norm of a rectangular matrix can be determined from an associated square symmetric matrix and we note that the norm of a square symmetric matrix is equal to its spectral radius. In Section 3, we prove some results concerning the Perron-Frobenius theory of non-negative square matrices. Since we are interested only in symmetric matrices, some of the proofs differ from the standard proofs, and are simpler (and we hope more intuitive). In Section 4, we explain the connection between small operators and small graphs, and show how small operators can be classified by the Coxeter graphs associated to the $ADE$ series. In particular, any small operator can be represented by a disjoint union of the Coxeter graphs for the $ADE$ series. In the final section, we sketch related ideas of smallness for quadratic forms, reflection groups, and root systems, using graphs as an organizing principle.
Our ideas here are inspired by a very interesting chapter in the monograph by Goodman, de la Harpe, and Jones [@GdlaHJ].
Operators
=========
By an operator we mean a linear transformation $B:{\mathbb R}^n\to {\mathbb R}^m$. So an operator $B$ can be represented by a rectangular matrix.
Let $B$ be a rectangular real matrix, i.e., an operator ${\mathbb R}^n\to{\mathbb R}^m$. The operator norm $|B|$ can be defined as $$|B|={\rm max}_{v\neq 0}{\frac{|Bv|}{|v|}}={\rm max}_{|v|=1}|Bv|.$$
Let’s consider matrices with non-negative integer entries; they are just the finite sums of the basic operators $E_{i,j}$ (the rank 1 operators, the matrices with all entries $0$ except for just one $1$ in the $i,j$ position).
An operator $B$ is considered small if it has non-negative integer entries and $|B|<2$.
In the final section we will give some indications of why $2$ is a natural and useful bound here.
Our goal in this paper is to describe the classification of all small operators. The appropriate notion of equivalence is given by the action of symmetric groups on the entries of vectors in ${\mathbb R}^m$ and ${\mathbb R}^n$. The appropriate notion of decomposition corresponds to direct sum of matrices (juxtaposition of blocks).
Each $|E_{i,j}|=1$, so these are small operators.
Note that a small operator can only have $0, 1$ entries.
Which $3\times 2$ matrices are small? Consider all the $3\times 2$ matrices with $0,1$ entries; there are $64$ of these and $54$ of them are small. Each non-small example is equivalent to one of the following: $$\begin{pmatrix}1&1\\1& 1\\ 0&0 \end{pmatrix}\quad\begin{pmatrix}1&1\\1& 1\\ 1&0 \end{pmatrix}
\quad \begin{pmatrix}1&1\\1& 1\\ 1&1 \end{pmatrix}$$
The definition of operator norm, maximizing a continuous function over a compact set, seems based on real analysis. For instance, we could use Lagrange multipliers to calculate the operator norm. But for our purposes it is convenient to work with an associated symmetric matrix.
For any operator with matrix $B$ we have symmetric matrices $BB^\top$ and $B^\top B$; they have real eigenvalues (since they are symmetric) and their eigenvalues are non-negative, since $(B^\top B)v=av$ with $v\neq0$ implies $$av^\top v=v^\top(B^\top B)v=(Bv)^\top(Bv),$$ so that $a$ times a positive number is non-negative; and similarly for $BB^\top$. This is related to the following construction. Given an operator with matrix $B\in{\rm Mat}_{m,n}({\mathbb R})$, we may form a symmetric matrix $A\in{\rm Mat}_{m+n}({\mathbb R})$ by $$A=\begin{pmatrix}0&B\\B^\top& 0\\ \end{pmatrix}.$$ We will refer to $A$ as the symmetric matrix associated to $B$. Note that $$A^2=\begin{pmatrix}BB^\top&0\\0&B^\top B\\ \end{pmatrix}$$ has square diagonal blocks.
\[norm.is.max.of.block.norms\] If $M\in{\rm Mat}_{m+n}({\mathbb R})$ is block diagonal on square matrices $M_1\in{\rm Mat}_m({\mathbb R})$ and $M_2\in{\rm Mat}_n({\mathbb R})$, then $|M|={\rm max}(|M_1|,|M_2|)$.
Let $m={\rm max}(|M_1|,|M_2|)$. So $|M_1|\leq m$ and $|M_2|\leq m$. So $|M_1v_1|\leq m|v_1|$ and $|M_2v_2|\leq m|v_2|$ for any $v_1\in{\mathbb R}^m$ and any $v_2\in{\mathbb R}^n$. Consider $$v=\begin{pmatrix}v_1\\v_2\\ \end{pmatrix}\in{\mathbb R}^{m+n}\quad {\rm and}\quad
Mv=\begin{pmatrix}M_1&0\\0&M_2\\ \end{pmatrix}\begin{pmatrix}v_1\\v_2\\ \end{pmatrix}=\begin{pmatrix}M_1v_1\\M_2v_2\\ \end{pmatrix}$$ So $|Mv|^2=|M_1v_1|^2+|M_2v_2|^2\leq m^2(|v_1|^2+|v_2|^2)=m^2|v|^2$. Thus $|M|\leq m$. Suppose $|M_1|\geq |M_2|$ so that $|M_1|=m$. There is a unit vector $v_1$ such that $|M_1v_1|=|M_1|$. Taking $v_2=0$ in the above gives unit vector $v$ with $|Mv|=|M_1v_1|=|M_1|=m$. So $|M|\geq m$, and thus $|M| = m$.
For any $B$ consider the symmetric matrix $A$ associated to it. Lemma \[norm.is.max.of.block.norms\] can be used to prove the following result.
For an operator with matrix $B$, its operator norm is equal to the operator norm of its associated symmetric matrix $A$: $|A|=|B|$.
For the square matrix $A$ associated to $B$ shown above, $A^2$ has square diagonal blocks $M_1=BB^\top$ and $M_2=B^\top B$. For any vector $v$ with $|v|=1$, $$|Bv|^2=\langle Bv,Bv\rangle=\langle B^\top Bv,v\rangle\leq|B^\top Bv|\leq|B^\top B|.$$ This gives $|Bv|\leq\sqrt{|B^\top B|}$, so that $|B|^2\leq|B^\top B|\leq|B^\top|\cdot|B|$. Thus, for $B\neq 0$, $|B|\leq|B^\top|$. Replacing $B$ with $B^\top$ in the above argument gives $|B^\top|\leq|B|$, so that $|B|=|B^\top|$. So $$|B^\top|\cdot|B|=|B|^2\leq|B^\top B|\leq|B^\top|\cdot|B|,$$ which gives $|B|^2=|B^\top B|$. Replacing $B$ with $B^\top$ gives $|B|^2=|BB^\top|$. Now $|M_1|=|BB^\top|=|B|^2$ and $|M_2|=|B^\top B|=|B|^2$, so $|A^2|=|B|^2$ by Lemma \[norm.is.max.of.block.norms\]. But $A^2=A^\top A$ and $|A^\top A|=|A|^2$. So $|A|^2=|B|^2$ and thus $|A|=|B|$.
So the study of small operators can be carried out in the setting of small symmetric matrices. The set of eigenvalues of a square matrix $A$ is called its spectrum; and the spectral radius $\rho(A)$ is the radius of the smallest disk centered at 0 in the complex plane and containing the spectrum of $A$. When $A$ is a symmetric matrix, all its eigenvalues are real numbers, and $\rho(A)$ is the largest of these in absolute value, leading to the following well-known result.
\[norm.equals.spectral.radius\] For a symmetric matrix $A$, the operator norm $|A|$ is equal to the spectral radius $\rho(A)$.
A symmetric $n\times n$ matrix $A$ determines an orthonormal basis of eigenvectors $v_1,\ldots, v_n$, with real eigenvalues $\lambda_1,\ldots,\lambda_n$, such that $|\lambda_i|\leq |\lambda_n|$ for all $i$. So $\rho(A)=|\lambda_n|$, and $|A|\geq |\lambda_n|$ since $|Av_n|=|\lambda_n|$. Let $v=\sum a_i v_i$ with $|v|^2=\sum a_i^2=1$; then $|A|\leq |\lambda_n|$, since $$|Av|^2=\left|\sum a_i \lambda_iv_i\right|^2=\sum |a_i |^2|\lambda_i|^2 \leq \left(\sum |a_i |^2\right)|\lambda_n|^2=|\lambda_n|^2.$$
Non-negative square matrices
============================
For a matrix $B$ with real entries, we write $B\geq 0$ when all the entries of $B$ are non-negative, and say that $B$ is non-negative. When $B\geq 0$ and some entry is non-zero, we write $B>0$; when $B\geq 0$ and all entries are non-zero (positive), we write $B\gg 0$. Let $A\geq B$ mean $A-B\geq 0$, and $A>B$ mean $A-B>0$, and $A\gg B$ mean $A-B\gg 0$. In particular, the above notations apply for vectors with real entries. We want to use some part of the Perron-Frobenius theory of non-negative square matrices. The proofs in this theory tend to be rather intricate; see Gantmacher [@Gant] or Sternberg [@Stern], for instance. But we only need to consider symmetric matrices in this paper. So we present proofs for the symmetric case; they seem simpler than the usual proofs, making efficient use of the Rayleigh quotient function for a symmetric matrix.
Consider ${\mathbb R}^n$ with its real inner product (dot product) $\langle x,y \rangle=x^\top y$. For a square symmetric matrix $A$, define the real-valued function $R_A$ by $$R_A(x)={\frac{\langle Ax,x\rangle}{\langle x,x\rangle}}$$ for $x\neq 0$. Note that $R_A(ax)=R_A(x)$ for any non-zero number $a$; so we can consider $R_A$ to be defined on the set of rays, or on the set of unit vectors. Let $\lambda$ denote the maximal value achieved by $R_A$ on the unit sphere. The fact that $\lambda$ is the largest of all the eigenvalues of $A$ is part of the “minmax principle” for the Rayleigh quotient of $A$ (see [@lax]). Recall that a symmetric matrix has all its eigenvalues real.
\[Rayleigh.max\] The maximum value of the Rayleigh quotient $R_A$ is the largest eigenvalue $\lambda$ of $A$, and the maximum is achieved only at eigenvectors for $A$ and $\lambda$.
A symmetric $n\times n$ matrix $A$ determines an orthonormal basis of eigenvectors $v_1,\ldots, v_n$, with real eigenvalues $\lambda_1,\ldots,\lambda_n$, such that $\lambda_i\leq\lambda_n$ for all $i$. Let $v=\sum a_i v_i$ with $|v|^2=\sum a_i^2=1$; then $$R_A(v)=\left\langle \sum a_i\lambda_i v_i,\sum a_i v_i\right\rangle= \sum a_i^2 \lambda_i \leq \left(\sum a_i^2\right) \lambda_n=\lambda_n.$$ Also, $R_A(v_n)=\lambda_n$, so the maximum value of $R_A(v)$ is $\lambda_n$. If $R_A(v)=\lambda_n$ then $\sum (\lambda_n-\lambda_i)a_i^2=0$, all non-negative, so that we must have $(\lambda_n-\lambda_i)a_i=0$ for all $i$; then $\sum a_i\lambda_i v_i=\sum a_i\lambda v_i$, and $Av=\lambda_nv$.
For any vector $x$, let $\operatorname{abs}(x)$ denote the vector whose entries are the absolute values of the entries of $x$. Note that $ \langle\operatorname{abs}(x), \operatorname{abs}(x)\rangle= \langle x, x\rangle$. Also, if $A\geq 0$ then $\operatorname{abs}(A\ x)\leq A\ \operatorname{abs}(x)$, by the triangle inequality.
\[eigenvector.thm\] If $A>0$ and $A$ is symmetric, then the maximum value $\lambda$ of $R_A$ is achieved at some $z>0$ with $Az=\lambda z$. Also, $|\lambda'|\leq\lambda$ for every eigenvalue $\lambda'$ of $A$.
Assume $A>0$ and $A$ symmetric. Let $\lambda$ be the maximum value of $R_A$ on the unit sphere, achieved at $x$. We have $\lambda>0$, since $R_A(e)>0$ where $e$ is the vector of $1$’s. Also, $A x=\lambda x$ by Theorem \[Rayleigh.max\].
Now let $\lambda'$ be any eigenvalue of $A$, with $Ay=\lambda'y$ and $|y|=1$. Apply the $\operatorname{abs}$ operator to $Ay=\lambda' y$, and use $\langle\operatorname{abs}(y), \operatorname{abs}(y)\rangle= \langle y, y\rangle=1$ to get: $$|\lambda'|\ \operatorname{abs}(y)=\operatorname{abs}(\lambda' y)=\operatorname{abs}(A y)\leq A\ \operatorname{abs}(y)\quad {\rm so}\quad$$ $$|\lambda'|=\langle|\lambda'| \ \operatorname{abs}(y), \operatorname{abs}(y)\rangle \leq \langle A \ \operatorname{abs}(y), \operatorname{abs}(y)\rangle=R_A(\operatorname{abs}(y))\leq \lambda,$$ since $\lambda$ is the maximum of $R_A$. Thus $|\lambda'|\leq \lambda$, and for the eigenvalue $\lambda>0$ with $Ax=\lambda x$, we have $\lambda\leq R_A(\operatorname{abs}(x))\leq\lambda$. By Theorem \[Rayleigh.max\], $z=\operatorname{abs}(x)$ is an eigenvector for $A$ with eigenvalue $\lambda$, and $z> 0$ since $x\neq 0$.
Note that for a symmetric matrix $A$ with $A>0$, we have $$\lambda=\max_{|x|=1}{R_A(x)}=\rho(A)=|A|$$ according to Theorems \[norm.equals.spectral.radius\], \[Rayleigh.max\], and \[eigenvector.thm\].
Any square matrix has an associated matrix of 1’s and 0’s, where 1 means non-zero; and we may interpret this matrix of 1’s and 0’s as the adjacency matrix of a [*directed*]{} graph. A directed graph is strongly connected if it contains a directed path from each vertex to every other vertex. Following Frobenius, let us say that a non-negative square matrix is irreducible when its underlying directed graph is strongly connected.
When $A$ is irreducible, there exists a square matrix $P$ with all its entries non-zero and with $AP=PA$. In fact, since the directed graph is strongly connected, we can choose an integer $N$ so large that there exists a path of length at most $N$ from each vertex to every other vertex. Then in the directed graph for matrix $I+A$, each vertex has a path of length $N$ to each other vertex. But the entries of $(I+A)^N$ count the paths of length $N$ in this directed graph-with-loops; thus $P=(I+A)^N$ for large $N$ has the desired properties. Since we consider symmetric matrices, we don’t need to consider the underlying graph as a directed graph; and it is strongly connected if and only if it is connected.
\[pos.eigenvector\] If $A$ is irreducible, $A>0$, and $A$ is symmetric, then there exists $y\gg 0$ with $Ay=\rho(A) y$.
Since $A$ is irreducible and $A>0$, $P=(I+A)^N$ for large $N$ gives a (symmetric square) matrix $P\gg 0$ with $AP=PA$. Moreover, there exists $z>0$ with $Az=\lambda z=\rho(A)z$, as in Theorem \[eigenvector.thm\]. Then $APz=PAz=\lambda Pz=\rho(A)Pz$, and $Pz\gg 0$. So let $y=Pz$.
\[comparison.theorem\] If $A$ is irreducible and symmetric and $A>B>0$, then $\rho(A)>\rho(B)$.
We start by applying Theorem \[eigenvector.thm\] to $B$. Let $R_B$ achieve its maximum value $\mu>0$ at unit vector $y>0$. Then by Theorems \[Rayleigh.max\] and \[eigenvector.thm\], $\mu=\rho(B)$. Since $A-B>0$ and $y>0$, $(A-B)y\geq0$. This gives $\langle(A-B)y,y\rangle\geq0$, so that $R_A(y)-R_B(y)\geq 0$.
Now apply Theorems \[Rayleigh.max\], \[eigenvector.thm\], and \[pos.eigenvector\] to $A$, with $R_A$ achieving its maximum value $\lambda=\rho(A)$ at unit vector $x\gg 0$. Then $\lambda=R_A(x)\geq R_A(y)\geq R_B(y)=\mu$. So $\lambda\geq \mu$.
We now show $\lambda\neq \mu$. Suppose $\lambda=\mu$; then $R_A$ achieves its maximum value at the unit vectors $x\gg 0$ and $y>0$. Suppose $x\neq y$; then $x$ and $y$ are linearly independent. Let $c=\max\left\lbrace y_i/x_i\right\rbrace$. Then $z=cx-y>0$, $Az=\lambda z$, and $z$ has some entry $z_i=0$ and some entry $z_j > 0$. But $A$ is irreducible, so the underlying graph of $A$ has a path from vertex $i$ to vertex $j$, say of length $m$; then $A^m$ has its $(i,j)$ entry non-zero. Then $z'=A^m z$ has $z'_i\geq(A^m)_{ij}z_j>0$; but this contradicts $z'_i=\lambda^m z_i=0$, which follows from $A^m z=\lambda^m z$. Thus, $x=y$ and $(A-B)x=(\lambda-\mu)x=0$, which is impossible since $(A-B)>0$ and $x\gg 0$ implies $(A-B)x>0$. Thus, $\lambda>\mu$, i.e., $\rho(A)>\rho(B)$.
Graphs
======
A graph is a finite set of vertices and edges. Let’s exclude loops and multiple edges. We say that vertices $x$ and $y$ are adjacent when $xy$ is an edge. Enumerating the vertices of a graph gives an adjacency matrix which completely describes the graph; it is a symmetric matrix of 0’s and 1’s, indicating which vertices are adjacent.
A graph is bicolored if we have assigned a color red or blue to each vertex, so that each edge connects a red and a blue vertex. A bicolored graph is completely described by an $m\times n$ matrix $B$ of 0’s and 1’s, once we enumerate its $m$ red vertices and its $n$ blue vertices.
A small operator corresponds to a small matrix of 0’s and 1’s, which in turn corresponds to a small bicolored graph. If $B$ is an $m\times n$ matrix corresponding to a bicolored graph, then the adjacency matrix of the underlying graph (forgetting the bicoloring) is the symmetric matrix $A$ associated to $B$.
We will classify the small bicolored graphs, up to isomorphism and disjoint union of bicolored graphs. The first step (which turns out to be the main step for the classification of small operators) is the classification of “small” graphs, in the following sense.
Let us say that a graph is small when its adjacency matrix has spectral radius less than 2. So we have that the matrix of a small operator corresponds exactly to a bicoloring of a small graph.
Equivalence of graphs is isomorphism of graphs. Decomposition of graphs is disjoint union of graphs. A graph is small if and only if all its connected components are small graphs.
Now we use our results about non-negative square matrices from the previous section. Recall that the undirected graphs that we work with have symmetric adjacency matrix, which is irreducible if and only if the graph is connected.
In particular, if $A'$ is the adjacency matrix of a proper subgraph of a connected graph with adjacency matrix $A$, then $A>A'\geq 0$ and $\rho(A)>\rho(A')$.
When $A$ is the $n\times n$ adjacency matrix for one of our graphs, we can interpret an $n\times 1$ vector $v$ as assigning a number $v(x)$ to each vertex $x$ of the graph. Then $v'=Av$ means that $v'(x)=\sum v(y)$, where we sum over the vertices $y$ which are adjacent to $x$.
This helps us verify that the connected graphs in Figure \[forbidden.subgraphs\] all have spectral radius $2$: just assign a number $v(x)$ to each vertex $x$ so that $2v(x)=\sum v(y)$, where we sum over the vertices $y$ which are adjacent to $x$. We refer to these graphs as “forbidden subgraphs”, since a [*small*]{} connected graph cannot contain any of these as a subgraph (and still have spectral radius less than $2$).
A small graph is a disjoint union of connected small graphs. Each connected small graph is isomorphic to $A_n$, $D_n$, or $E_n$, for some $n$.
Let $\Gamma$ be a connected small graph. Note that $\Gamma$ cannot contain a cycle, a vertex of degree $4$ or more, or more than one vertex of degree $3$ (since then $\Gamma$ would contain one of the forbidden subgraphs in Figure \[forbidden.subgraphs\], and would have spectral radius at least $2$ by Theorem \[comparison.theorem\]). Let $T_{p,q,r}$ denote the “tripod” graph, consisting of three legs with $p$, $q$, and $r$ vertices. If $\Gamma$ is a small tripod graph, there are limitations on how long its legs can be (since $\Gamma$ cannot contain any of the forbidden tripods in Figure \[forbidden.subgraphs\] as subgraphs). Thus, a small connected graph is isomorphic to one from the $ADE$ series, shown in Figure \[small.graphs\]. Note that each of the graphs in Figure \[small.graphs\] is a proper subgraph of a graph in Figure \[forbidden.subgraphs\], and so must have spectral radius less than $2$ by Theorem \[comparison.theorem\].
Note that if $\Gamma$ is small but not connected, the vertices can be enumerated so that its adjacency matrix is block diagonal on square matrices. Using Lemma \[norm.is.max.of.block.norms\], each connected component of $\Gamma$ must be small so that $\Gamma$ is a disjoint union of connected small graphs.
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[([.75cm]{}\*1,[.75cm]{}\*.5) – ([.75cm]{}\*2,[.75cm]{}\*0);]{}; [([.75cm]{}\*1,[.75cm]{}\*-.5) – ([.75cm]{}\*2,[.75cm]{}\*0);]{}; [([.75cm]{}\*2,[.75cm]{}\*0) – ([.75cm]{}\*3,[.75cm]{}\*0);]{}; [([.75cm]{}\*3,[.75cm]{}\*0) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*4,[.75cm]{}\*0) – ([.75cm]{}\*5,[.75cm]{}\*0);]{}; [([.75cm]{}\*5,[.75cm]{}\*0) – ([.75cm]{}\*6,[.75cm]{}\*.5);]{}; [([.75cm]{}\*5,[.75cm]{}\*0) – ([.75cm]{}\*6,[.75cm]{}\*-.5);]{}; [([.75cm]{}\*1,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*1,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; [([.75cm]{}\*6,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*6,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; in [2,...,5]{} [ =100 [[ (-[1.5]{},0\*[.75cm]{}-[1.5]{}) – (+[1.5]{},0\*[.75cm]{}+[1.5]{}); (-[1.5]{},0\*[.75cm]{}+[1.5]{}) – (+[1.5]{},0\*[.75cm]{}-[1.5]{}); ]{}]{} ]{}
&&&
[([.75cm]{}\*1,[.75cm]{}\*.5) – ([.75cm]{}\*2,[.75cm]{}\*.5);]{}; [([.75cm]{}\*2,[.75cm]{}\*.5) – ([.75cm]{}\*3,[.75cm]{}\*0);]{}; [([.75cm]{}\*1,[.75cm]{}\*-.5) – ([.75cm]{}\*2,[.75cm]{}\*-.5);]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) – ([.75cm]{}\*3,[.75cm]{}\*0);]{}; [([.75cm]{}\*3,[.75cm]{}\*0) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*4,[.75cm]{}\*0) – ([.75cm]{}\*5,[.75cm]{}\*0);]{}; [([.75cm]{}\*1,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*1,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; [([.75cm]{}\*2,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; in [3,...,5]{} [ =100 [[ (-[1.5]{},0\*[.75cm]{}-[1.5]{}) – (+[1.5]{},0\*[.75cm]{}+[1.5]{}); (-[1.5]{},0\*[.75cm]{}+[1.5]{}) – (+[1.5]{},0\*[.75cm]{}-[1.5]{}); ]{}]{} ]{}
\
&&&\
&&&\
[([.75cm]{}\*1,[.75cm]{}\*.5) – ([.75cm]{}\*2,[.75cm]{}\*.5);]{}; [([.75cm]{}\*2,[.75cm]{}\*.5) – ([.75cm]{}\*3,[.75cm]{}\*.5);]{}; [([.75cm]{}\*3,[.75cm]{}\*.5) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*4,[.75cm]{}\*0) – ([.75cm]{}\*5,[.75cm]{}\*0);]{}; [([.75cm]{}\*1,[.75cm]{}\*-.5) – ([.75cm]{}\*2,[.75cm]{}\*-.5);]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) – ([.75cm]{}\*3,[.75cm]{}\*-.5);]{}; [([.75cm]{}\*3,[.75cm]{}\*-.5) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*1,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*2,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*3,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*1,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; [([.75cm]{}\*3,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; in [4,...,5]{} [ =100 [[ (-[1.5]{},0\*[.75cm]{}-[1.5]{}) – (+[1.5]{},0\*[.75cm]{}+[1.5]{}); (-[1.5]{},0\*[.75cm]{}+[1.5]{}) – (+[1.5]{},0\*[.75cm]{}-[1.5]{}); ]{}]{} ]{}
&&&
[([.75cm]{}\*0,[.75cm]{}\*.5) – ([.75cm]{}\*1,[.75cm]{}\*.5);]{}; [([.75cm]{}\*1,[.75cm]{}\*.5) – ([.75cm]{}\*2,[.75cm]{}\*.5);]{}; [([.75cm]{}\*2,[.75cm]{}\*.5) – ([.75cm]{}\*3,[.75cm]{}\*.5);]{}; [([.75cm]{}\*3,[.75cm]{}\*.5) – ([.75cm]{}\*4,[.75cm]{}\*.5);]{}; [([.75cm]{}\*4,[.75cm]{}\*.5) – ([.75cm]{}\*5,[.75cm]{}\*0);]{}; [([.75cm]{}\*5,[.75cm]{}\*0) – ([.75cm]{}\*6,[.75cm]{}\*0);]{}; [([.75cm]{}\*3,[.75cm]{}\*-.5) – ([.75cm]{}\*4,[.75cm]{}\*-.5);]{}; [([.75cm]{}\*4,[.75cm]{}\*-.5) – ([.75cm]{}\*5,[.75cm]{}\*0);]{}; [([.75cm]{}\*5,[.75cm]{}\*0) circle ([.08cm]{});]{}; [([.75cm]{}\*6,[.75cm]{}\*0) circle ([.08cm]{});]{}; [([.75cm]{}\*3,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; [([.75cm]{}\*4,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; in [0,...,4]{} [ =100 [[ (-[1.5]{},0\*[.75cm]{}-[1.5]{}) – (+[1.5]{},0\*[.75cm]{}+[1.5]{}); (-[1.5]{},0\*[.75cm]{}+[1.5]{}) – (+[1.5]{},0\*[.75cm]{}-[1.5]{}); ]{}]{} ]{}
\
[cccc]{} $A_n$
[([.75cm]{}\*1,[.75cm]{}\*0) – ([.75cm]{}\*2,[.75cm]{}\*0);]{}; [([.75cm]{}\*2,[.75cm]{}\*0) – ([.75cm]{}\*3,[.75cm]{}\*0);]{}; [([.75cm]{}\*3,[.75cm]{}\*0) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; in [1,...,4]{} [ =100 [[ (-[1.5]{},0\*[.75cm]{}-[1.5]{}) – (+[1.5]{},0\*[.75cm]{}+[1.5]{}); (-[1.5]{},0\*[.75cm]{}+[1.5]{}) – (+[1.5]{},0\*[.75cm]{}-[1.5]{}); ]{}]{} ]{}
&&&
$D_n$
[([.75cm]{}\*1,[.75cm]{}\*0) – ([.75cm]{}\*2,[.75cm]{}\*0);]{}; [([.75cm]{}\*2,[.75cm]{}\*0) – ([.75cm]{}\*3,[.75cm]{}\*0);]{}; [([.75cm]{}\*3,[.75cm]{}\*0) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*4,[.75cm]{}\*0) – ([.75cm]{}\*5,[.75cm]{}\*.5);]{}; [([.75cm]{}\*4,[.75cm]{}\*0) – ([.75cm]{}\*5,[.75cm]{}\*-.5);]{}; [([.75cm]{}\*5,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*5,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; in [1,...,4]{} [ =100 [[ (-[1.5]{},0\*[.75cm]{}-[1.5]{}) – (+[1.5]{},0\*[.75cm]{}+[1.5]{}); (-[1.5]{},0\*[.75cm]{}+[1.5]{}) – (+[1.5]{},0\*[.75cm]{}-[1.5]{}); ]{}]{} ]{}
\
&&&\
&&&\
$E_6$
[([.75cm]{}\*1,[.75cm]{}\*.5) – ([.75cm]{}\*2,[.75cm]{}\*.5);]{}; [([.75cm]{}\*2,[.75cm]{}\*.5) – ([.75cm]{}\*3,[.75cm]{}\*0);]{}; [([.75cm]{}\*3,[.75cm]{}\*0) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*1,[.75cm]{}\*-.5) – ([.75cm]{}\*2,[.75cm]{}\*-.5);]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) – ([.75cm]{}\*3,[.75cm]{}\*0);]{}; [([.75cm]{}\*1,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*2,[.75cm]{}\*.5) circle ([.08cm]{});]{}; [([.75cm]{}\*1,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; in [3,...,4]{} [ =100 [[ (-[1.5]{},0\*[.75cm]{}-[1.5]{}) – (+[1.5]{},0\*[.75cm]{}+[1.5]{}); (-[1.5]{},0\*[.75cm]{}+[1.5]{}) – (+[1.5]{},0\*[.75cm]{}-[1.5]{}); ]{}]{} ]{}
&&&
$E_7$
[([.75cm]{}\*1,[.75cm]{}\*.5) – ([.75cm]{}\*2,[.75cm]{}\*.5);]{}; [([.75cm]{}\*2,[.75cm]{}\*.5) – ([.75cm]{}\*3,[.75cm]{}\*.5);]{}; [([.75cm]{}\*3,[.75cm]{}\*.5) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*4,[.75cm]{}\*0) – ([.75cm]{}\*5,[.75cm]{}\*0);]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) – ([.75cm]{}\*3,[.75cm]{}\*-.5);]{}; [([.75cm]{}\*3,[.75cm]{}\*-.5) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*4,[.75cm]{}\*0) circle ([.08cm]{});]{}; [([.75cm]{}\*5,[.75cm]{}\*0) circle ([.08cm]{});]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; [([.75cm]{}\*3,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; in [1,...,3]{} [ =100 [[ (-[1.5]{},0\*[.75cm]{}-[1.5]{}) – (+[1.5]{},0\*[.75cm]{}+[1.5]{}); (-[1.5]{},0\*[.75cm]{}+[1.5]{}) – (+[1.5]{},0\*[.75cm]{}-[1.5]{}); ]{}]{} ]{}
\
&&&\
&&&\
$E_8$
[([.75cm]{}\*0,[.75cm]{}\*.5) – ([.75cm]{}\*1,[.75cm]{}\*.5);]{}; [([.75cm]{}\*1,[.75cm]{}\*.5) – ([.75cm]{}\*2,[.75cm]{}\*.5);]{}; [([.75cm]{}\*2,[.75cm]{}\*.5) – ([.75cm]{}\*3,[.75cm]{}\*.5);]{}; [([.75cm]{}\*3,[.75cm]{}\*.5) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*4,[.75cm]{}\*0) – ([.75cm]{}\*5,[.75cm]{}\*0);]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) – ([.75cm]{}\*3,[.75cm]{}\*-.5);]{}; [([.75cm]{}\*3,[.75cm]{}\*-.5) – ([.75cm]{}\*4,[.75cm]{}\*0);]{}; [([.75cm]{}\*4,[.75cm]{}\*0) circle ([.08cm]{});]{}; [([.75cm]{}\*5,[.75cm]{}\*0) circle ([.08cm]{});]{}; [([.75cm]{}\*2,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; [([.75cm]{}\*3,[.75cm]{}\*-.5) circle ([.08cm]{});]{}; in [0,...,3]{} [ =100 [[ (-[1.5]{},0\*[.75cm]{}-[1.5]{}) – (+[1.5]{},0\*[.75cm]{}+[1.5]{}); (-[1.5]{},0\*[.75cm]{}+[1.5]{}) – (+[1.5]{},0\*[.75cm]{}-[1.5]{}); ]{}]{} ]{}
&&&\
A small operator corresponds to a small bicolored graph, which is isomorphic to a disjoint union of connected small graphs together with a bicoloration.
Remarks
=======
Here are some brief remarks on some famous classification results from different areas of mathematics Each of these areas has a natural notion of decomposition into indecomposables, and the classification is largely parallel to the classification of small operators.
Let’s start with a historical sketch of the classification story. In the 1880’s Wilhelm Killing worked on classifying possible types of geometries. He used recent developments in linear algebra to work out a classification of (what turned out to be) the semi-simple Lie algebras over the complex numbers. He was partly inspired by Sophus Lie’s ongoing work on “continuous groups”. In particular, Killing used sophisticated ideas about eigenvalues to record an isomorphism class of semi-simple Lie algebras in terms of a “root system”. The root systems and Lie algebras are then built up as direct sums of indecomposable ones; and Killing essentially classified the indecomposables into types $A$, $B$, $C$, $D$, $E$, $F$, and $G$.
Eli Cartan organized and completed this classification in his 1894 thesis. The data for a root system can be encoded in a matrix of integers, now called the Cartan matrix of the root system; it determines an integer-valued bilinear form on a maximal abelian subalgebra of the Lie algebra. See Coleman [@Coleman] and Hawkins [@Hawkins] for more on the history of these developments. Donald Coxeter made a separate contribution through his study of kaleidoscopes. By the early 1930’s he had classified those sets of mirrors in a real finite-dimensional inner product space which generate a finite group of reflections (see [@roberts]). In his 1934 paper “Discrete groups generated by reflections” [@Cox], Coxeter used graphs to describe his mirror systems. In particular, certain of his finite reflection groups were encoded by connected undirected graphs, without loops and multiple edges. We refer to these as the $ADE$ series of graphs. Hermann Weyl gave a series of lectures on Lie algebras at Princeton that year, and Coxeter observed that his crystallographic reflection groups (those preserving a lattice) correspond to certain permutation groups of roots in a root system, now called the Weyl groups. The connected $ADE$ graphs determine the Cartan matrices for the “simply-laced” simple Lie algebras. The mimeographed lecture notes, published 1934-1935, include an appendix by Coxeter in which these graphs appear [@Weyl]. These classification ideas continued to be developed in work by Eugene Dynkin (1947, 1952), Bourbaki (1968, with exposition attributed to Jacques Tits), and many others.
The book by Fulton and Harris [@FH] is one good reference for the theory. Let us close with a presentation of these $ADE$ classification results, organized around our notion of small graph. [**A graph determines a quadratic form:**]{} Let $X$ be a graph with vertex set $X_0$. Let ${{\mathbb Z}}X_0$ be the free abelian group with basis $X_0$, so that the elements of ${{\mathbb Z}}X_0$ are the integer linear combinations of the elements in $X_0$. Define an integer-valued symmetric bilinear form on ${{\mathbb Z}}X_0$ by describing its values on $X_0$: $$(x|x)=2, \quad{\rm and}\quad
(x|y)=\begin{cases} -1, &\text{if $xy$ is an edge;}\\
\ \ 0, &\text{if $xy$ is not an edge.}\\
\end{cases}$$ This bilinear form is “even”, in that $q_X(v)=(v|v)/2$ defines an integral-valued quadratic form on ${{\mathbb Z}}X_0$. This means $q_X(v+w)=q_X(v)+q_X(v|w)+q_X(w)$. Note that over the integers, it is more convenient to not include the usual factor of $2$ in the middle correction term.
A graph $X$ is small if and only if its quadratic form $q_X$ is positive-definite.
Let $A$ be the adjacency matrix of the graph; so $X$ is small if and only if $\rho(A)<2$, if and only if $C=2I-A$ is positive definite. But $C$ is the symmetric matrix recording the bilinear form corresponding to $q_X$.
From this perspective, an integral-valued quadratic form on ${{\mathbb Z}}X_0$ is [*small*]{} when it is positive definite, and we have classified the small quadratic forms.
[**A graph determines a group:**]{} For each vertex $x$, define an additive involution $s_x:{{\mathbb Z}}X_0\to {{\mathbb Z}}X_0$ by describing its values on $X_0$: $$s_x(x)=-x, \quad{\rm and}\quad
s_x(y)=\begin{cases} y+x, &\text{if $xy$ is an edge;}\\
y, &\text{if $xy$ is not an edge.}\\
\end{cases}$$ If $v=\sum_x v_x x$ in ${\mathbb Z}X_0$, we have $s_x(v)=v'$ where $v'_x=-v_x+\sum v_y$ where the sum is over those vertices $y$ which are adjacent to $x$, and $v'_z=v_z$ for $z\neq x$. So $s_x$ replaces the label at vertex $x$ by the sum of surrounding labels, minus the original label. In terms of the bilinear form for the graph $X$, the involution $s_x$ associated to vertex $x$ is given by $v\mapsto v-(v|x)x$. Let $G_X$ be the group of additive isomorphisms of ${{\mathbb Z}}X_0$ generated by the $s_x$. The group $G_X$ preserves the quadratic form $q_X$.
The graph $X$ is small if and only if the group $G_X$ is finite.
From the definition, $s_x s_x=1$ for every vertex $x$, $s_x s_y s_x=s_y s_x s_y$ if $xy$ is an edge, and $s_y s_x=s_x s_y$ if $xy$ not an edge. This means that $s_y s_x$ has order 2 if $xy$ not an edge, and $s_y s_x$ has order 3 if $xy$ is an edge. This establishes the connection between graphs and the presentation of the simply-laced crystallographic Coxeter groups. For more details, see Coxeter’s paper [@Cox].
The small graphs correspond to the simply-laced Weyl groups, so the above work completes the classification of the simply-laced Weyl groups.
Let us go on to explain the notion of “root system” associated to these ideas. [**Graphs and (simply-laced) root systems:**]{} A graph $X$ determines a group $W=G_X$ and a lattice $\Lambda={\mathbb Z}X_0$, together with a quadratic form $q=q_X$ with $q(x)=1$ for $x\in X_0$. Thus the set $\Gamma=\{v\in \Lambda: q(v)=1\}$ generates $\Lambda$ as a ${\mathbb Z}$-module.
The graph is small if and only if $q$ is positive definite. For such a triple $(\Lambda,q,\Gamma)$ the real vector space $V$ generated by $\Gamma$ (the set of roots) is an inner product space with norm $q$, and $\Gamma$ is finite, since $\Gamma$ is the the intersection of a lattice and the unit sphere in a Euclidean space. Then $\Lambda$ is called the root lattice in this Euclidean space, and $\Gamma$ is called the set of roots. Moreover, the involutions $s_x$, which generate $W$ are orthogonal reflections in this Euclidean space. These are the root systems of the simply-laced semi-simple Lie algebras over the complex numbers; see Lurie’s discussion in [@Lurie]. This is the $ADE$ classification result:
Simply-laced root systems are classified by small graphs.
|
---
abstract: '[ In this paper, we consider adversarial attacks against a system of monocular depth estimation (MDE) based on convolutional neural networks (CNNs). The motivation is two-fold. One is to study the security of MDE systems, which has not been actively considered in the community. The other is to improve our understanding of the computational mechanism of CNNs performing MDE. Toward this end, we apply the method recently proposed for visualization of MDE to defending attacks. It trains another CNN to predict a saliency map from an input image, such that the CNN for MDE continues to accurately estimate the depth map from the image with its non-salient part masked out. We report the following findings. First, unsurprisingly, attacks by IFGSM (or equivalently PGD) succeed in making the CNNs yield inaccurate depth estimates. Second, the attacks can be defended by masking out non-salient pixels, indicating that the attacks function by perturbing mostly non-salient pixels. However, the prediction of saliency maps is itself vulnerable to the attacks, even though it is not the direct target of the attacks. We show that the attacks can be defended by using a saliency map predicted by a CNN trained to be robust to the attacks. These results provide an effective defense method as well as a clue to understanding the computational mechanism of CNNs for MDE. ]{}'
author:
- |
Junjie Hu$^{1,2}$ Takayuki Okatani$^{1,2}$\
$^1$ Graduate School of Information Sciences, Tohoku University, Japan\
$^2$ Center for Advanced Intelligence Project, RIKEN, Japan\
[{junjie.hu, okatani}@vision.is.tohoku.ac.jp]{}
bibliography:
- 'egbib.bib'
title: |
Analysis of Deep Networks for Monocular Depth Estimation\
Through Adversarial Attacks with Proposal of a Defense Method
---
Introduction
============
Monocular depth estimation, i.e., estimating the depth of a three-dimensional scene from its single image, has a long history of research in the fields of computer vision and visual psychophysics. The recent employment of convolutional neural networks (CNNs) has gained significant improvement in the estimation accuracy [@Eigen2014depth; @Li2015DepthAS; @laina2016deeper; @Kendall2017WhatUD; @fu2018deep]. In this paper, we consider adversarial attacks to the CNNs performing this task and particularly defense methods against them.
The motivation of this study is two-fold. First, we study the security of CNNs designed and trained for this task, which are beginning to be deployed in real-world applications. Unsurprisingly, they are vulnerable to adversarial attacks, as shown in Fig. \[fig\_depth\_attack\], although this has not been reported in the literature to the authors’ knowledge.
[p[0.095]{}<p[0.095]{}<p[0.095]{}<p[0.095]{}<]{} & & &\
& & &\
& & &\
(a)&(b)&(c)&(d)
Second, we attempt to understand, through analyses of adversarial examples, in what computational mechanism these CNNs infer depth from a single image. Recently, a few studies have been published on this question, which indicates that these CNNs utilize similar cues to those that are believed to be used by the human vision [@Dijk_2019_ICCV; @Hu2019VisualizationOC]. However, we are still on the way to a full understanding of the mechanism. If there are adversarial examples that lead the CNNs to malfunction, they should invalidate (some of) the cues that the CNNs utilize. By analyzing these, we wish to gain a deeper understanding of the inference mechanism.
Motivated by these two goals, we consider a defense method against adversarial attacks that is based on the recent study of Hu et al. [@Hu2019VisualizationOC] on the visualization of CNNs for the monocular depth estimation task. They propose a method for obtaining a saliency map, i.e., a set of a small number of pixels from which a CNN can estimate depth accurately. They show that such a saliency map can be predicted from the input image by another CNN trained for the purpose. They then show that CNNs can estimate a depth map fairly accurately from only a sparse subset of image pixels specified by the predicted saliency map.
We utilize their framework for visualization to construct a defense method, as illustrated in Fig. \[fig\_defense\_method\]. Given an adversarial input that will force the CNN $N$ to yield an erroneous depth map, a saliency map is first predicted by an auxiliary CNN $G$, which is multiplied with the (adversarial) input in a pixel-wise fashion; the resulting image is then fed to $N$, thereby aiming at obtaining an accurate depth map.
![Illustration of the proposed approach. Upper row: An adversarial image forces a depth estimator $N$ yield erroneous output. Lower row: The same adversarial image is detoxified by masking it with a saliency map, which is estimated by a robust estimator $G_{\mathrm{adv}}$. []{data-label="fig_defense_method"}](figs/adv_basicidea.pdf){width="0.95\columnwidth"}
This idea is based on the following considerations. Observing the saliency maps predicted by the method of Hu et al. [@Hu2019VisualizationOC], we can say that the CNNs use the portions of images that are similar to those used by the human vision. Thus, it is natural to hypothesize that the CNNs use similar cues and perform similar computations based on them to the human vision. If so, why do the adversarial attacks still succeed? (Why is it possible to affect CNNs by adding slight perturbation to inputs that do not affect the human vision?) To reconcile these two, considering that the CNN receives all the pixels at their input, we conjecture that [*the attacks are made possible not by perturbing salient pixels containing important depth cues but by mostly perturbing non-salient pixels.*]{}
A recent study has made an intriguing argument about adversarial attacks on classification tasks [@Ilyas2019AdversarialEA]. There are a number of features in images that could be effective for classification [*within*]{} a given dataset. Dividing them into robust and non-robust features, the authors argue that the human vision uses only the former, whereas CNNs may use them both, and that adversarial attacks affect only the non-robust features, explaining why the attacks succeed without affecting recognition by the human vision. Our approach may be restated from their perspective that we define robust and non-robust ‘features’ for monocular depth estimation by spatially dividing an input image based on the pixel saliency.
In the rest of the paper, we will show the followings:
- Adversarial attacks by IFGSM for a depth estimation CNN can be defended by masking out non-salient pixels in the inputs; see Fig. \[fig:config\](e). This indicates that the attack functions mostly by perturbing non-salient pixels.
- Attacks tailored for a depth prediction CNN also lead the auxiliary CNN (for saliency map prediction) to malfunction; see Fig. \[fig:config\](d). Thus, the attacks cannot be defended using the saliency maps predicted by them.
- The attacks can be defended by immunizing the auxiliary CNN against the attacks, which is made possible by employing adversarial training for it; it is not necessary to make any change on the depth CNN. This will be a valid defense method; see Fig. \[fig:config\](f).
{width="90.00000%"}
Backgrounds
===========
Adversarial Attack and Defense
------------------------------
In this paper, we mainly consider adversarial attacks by IFGSM [@Kurakin2017AdversarialEI], the iterative version of the fast gradient sign method (FGSM) [@Goodfellow2015ExplainingAH], although there are a number of attack methods; we refer the reader to a survey [@Akhtar2018ThreatOA]. This is because IFGSM, which is also known as projected gradient descent (PGD), is considered to be the strongest attack utilizing the local first order information about the network [@Madry2017TowardsDL], and thus is primarily considered in the previous studies on defense methods.
IFGSM is briefly summarized as follows. Suppose a network that has been trained for predicting $y$ from an input $x$ by minimizing a loss $\ell(x,y)$. Assuming we have access to the gradient of the loss, IFGSM iteratively updates inputs starting from a given input $x(\rightarrow x_0^*)$ according to $$\label{eq_ifgsm}
\quad x^*_{t+1} = \text{Clip}_{\epsilon,x} \left\{ x^*_{t} + \alpha
\cdot \mathrm{sign}(\triangledown \ell(x_t{^*},y))\right\}.$$ where $\epsilon$ denotes $l_\infty$ bound of adversarial perturbation and $\alpha$ is the step size. As shown in Fig. \[fig\_depth\_attack\], this method is effective for monocular depth estimation. For the purpose of clarification, we will use the following notation to indicate that an adversarial input $x^*$ is generated from a clean input $x$ targeting at a network $N$: $$x^* = Adv(x; N),$$ as shown in Fig. \[fig:config\](a).
There are many other studies on defense methods. They are roughly categorized into two classes. The first is to improve the robustness of the target network against adversarial images in some ways. The method of [*adversarial training*]{} [@Tramr2018EnsembleAT; @Na2018CascadeAM] creates adversarial images and uses them for training along with normal images. There are a number of methods that incorporate some regularizers to training, such as the employment of network distillation [@Hinton2015DistillingTK], in which the prediction of another network is used as a soft label, the deep contrastive network, which employs a layer-wise contrastive penalty [@Gu2014TowardsDN], and non-standard training schemes, such as saturating networks [@Nayebi2017BiologicallyIP], parseval networks [@Ciss2017ParsevalNI] and adversarial-trained bayesian neural networks [@Liu2019AdvBNNIA]. The other class of methods aim to remove perturbations from adversarial images before they are fed into the target network. It is proposed to detect adversarial images and then use an auto-encoder to project them onto a learned manifold of clean images [@Meng2017MagNetAT]. A similar method is employed in Defense-GAN [@Samangouei2018DefenseGANPC]. It is also proposed to remove the perturbation from adversarial images by learning a denoiser network [@Liao2018DefenseAA]. The proposed method differs from any of the existing methods. It may be considered to lie in an intersection of the above two classes.
Visualizing CNNs on Depth Estimation {#Hu_visualization}
------------------------------------
It is a popular approach to visualize a saliency map, aiming at understanding inference conducted by a CNN for an input [@Smilkov2017SmoothGradRN; @Mahendran2016SalientDN; @Sundararajan2017AxiomaticAF]. In the study of Hu et al. [@Hu2019VisualizationOC], this is attempted for the monocular depth estimation task. Their method finds a set of a small number of pixels from which a CNN can estimate depth accurately. Specifically, they consider a mask $m$, or equivalently a saliency map, which is used to mask out the irrelevant pixels from $x$ by element-wise multiplication $x\otimes m$. Then, the goal is to find $m$ such that the original inference will be maintained as much as possible, that is, $N(x)\sim N(x\otimes m)$, and also $m$ is as sparse as possible, which reduces to the following optimization: $$\min\limits_m \; \ell_{\rm dif} (N(x),N(x \otimes m)) + \lambda
\frac{1}{n} \lVert m \rVert_1,
\label{eqn_M}$$ where $\ell_{\rm dif}$ is a difference measure; $\lambda$ is a control parameter for the sparseness of $m$; $n$ is the number of pixels; and $\lVert m\rVert_1$ is the $\ell_1$ norm (of a vectorized version) of $m$. To avoid artifacts generated by direct optimization of (\[eqn\_M\]), they propose to incorporate an auxiliary network $G$ and train it to predict $m$ from $x$ according to $$\min\limits_G \; \ell_{\rm dif} (y,N(x \otimes G(x)))+\lambda \frac{1}{n}
\lVert G(x) \rVert_1.
\label{eqn_G}$$ They show through experiments that the CNN $N$ continues to yield accurate depth from only a small number of pixels of $x$ that are delineated by the saliency map predicted from $x$ by $G$.
Method
======
Saliency Map and Adversarial Perturbation
-----------------------------------------
We conducted a preliminary experiment to understand how the method of Sec. \[Hu\_visualization\], which was originally developed for visualization, reacts to adversarial attacks. In the experiment, we used the same network $N$ and $G$ as well as the dataset as in Sec. \[white\_box\_experiments\]. The results are shown in Table \[attack\_validation\]. Here, adversarial inputs $x^*$’s are generated targeting at $N$, i.e., $x^*=Adv(x,N)$. The fist row, ‘$N(x^*)$’, shows that $N$ is indeed vulnerable to this attack; error (RMSE) increases from 0.555 for $\epsilon=0$ (no attack) to 1.055 and 1.139 for $\epsilon=0.05$ and $0.1$, respectively.
The second row, ‘$N(x^*\otimes G(x^*))$’, shows the cases where the perturbed $x^*$ is masked using the saliency map (predicted by $G$ from the same input $x^*$) and inputted to $N$. The results indicate that this configuration is similarly affected by the attack, although its effect is somewhat mitigated by the use of the mask. Note that the accuracy of this configuration $N(x^*\otimes G(x^*))$ is already a bit lower for clean images than $N(x^*)$, i.e., $0.683$ vs $0.555$. What comes as a surprise is that as shown in the third row, ‘$N(x^*\otimes G(x))$’, when the mask $G(x)$ computed from a clean $x$ is applied to $x^*$, this immunizes the whole system to the attack. The increase of error is small, i.e., from $0.683$ to $0.696$ and $0.712$ for $\epsilon=0.05$ and $0.1$, respectively.
These indicate that
- FGSM generates an adversarial input $x^*=Adv(x,N)$ by perturbing mainly the non-salient pixels of $x$.
- The adversarial inputs generated targeting at $N$ force the network $G$ to yield an inaccurate saliency map.
Therefore, if we can make $G$ predict an accurate saliency map from $x^*$, or equivalently, $G(x^*)\sim G(x)$, then we will be able to defend the attacks.
It should be noted here that if the attacker knows the configuration $C(x) \equiv N(x\otimes G(x))$ and can generate $x^{**}=Adv(x, C)$, this defense strategy may not work. In fact, our experiments showed that the error for $N(x^{**}\otimes G(x^{**})$ is 1.052 and 1.108 for $\epsilon=0.05$ and $0.1$, respectively. Thus, in what follows, we will consider the scenario where the attacker assumes that $N$ alone is used to predict depth.
[|p[.12]{}<|p[.08]{}<|p[.08]{}<|p[.08]{}<|]{} & $\epsilon=0$ & $\epsilon=0.05$ &$\epsilon=0.1$\
$N(x^*)$ & 0.555 &1.055 &1.139\
$N(x^*\otimes G(x^*))$ & 0.683 & 0.813&0.943\
$N(x^*\otimes G(x))$& 0.683 & 0.696 &0.712\
Saliency Prediction Robust to Attacks
-------------------------------------
Thus, we wish to obtain the saliency prediction network $G$ that is robust to attacks. Then, our objective is two-fold. One is that, given an input image of a scene, we wish to identify as few pixels in the image as possible from which the depth can be predicted as accurately as possible. This leads to prediction of saliency map with a sparseness constraint, which is proposed in [@Hu2019VisualizationOC]. The other is that we want to make this prediction robust to possible adversarial attacks.
To met the first requirement, we follow the method of $\cite{Hu2019VisualizationOC}$ for training with a slight modification. It is the use of ground truth depth $\bar{y}$ as the desired target; in the original method, the output $y=N(x)$ obtained from mask-free input is used as the target. We found in our experiments that this yields better performance.
To satisfy the second requirement, we employ adversarial training; that is, we use not only clean images $x$’s but adversarially perturbed images $x^*$’s for training $G$. Note that we do not touch the depth prediction network $N$ in this process.
Then, the training of $G$ is expressed as the following optimization: $$G_{\mathrm{adv}} = \mathop{\mathrm{argmin}}_G \; \ell_{\rm dif} (\bar{y},N(x' \otimes G(x')))+\lambda \frac{1}{n}
\lVert G(x') \rVert_1,
\label{eqn_G2}$$ where $x'$ is either an input clean image or an adversarial image generated from a clean image.
$N$: a target, fully-trained network for depth estimation; $\chi$: a training set, , pairs of an RGB image of a scene and its depth map; $\epsilon$: $\ell_\infty$ bound for IFGSM; $K$: training epochs; and $J$: iterations per epoch.
$G_{\mathrm{adv}}$: a network for predicting a saliency map. Select an RGBD pair $\{x,\bar{y}\}$ from $\chi$ $p=\mathrm{Uniform}(0,1)$ $\epsilon=\mathrm{Uniform}(0.01, 0.3)$ $T=\lfloor \mathrm{Uniform}(1, 10) \rfloor$ $x^*_0 = x$ $t=0$ $x^*_{t+1} = \mathrm{IFGSM}(x^*_{t},\epsilon)$ $x'=x^*_t$ $x'=x$ $L = \ell_{\rm dif} (\bar{y},N(x' \otimes G(x')))+\lambda \frac{1}{n} \lVert G(x') \rVert_1$ Backpropagate $L$ Update $G$ $G_{\mathrm{adv}}\leftarrow G$
The details of the procedure of training are given in Algorithm \[alg\_leaning\_G\]. The adversarial examples $x^*$’s are generated by IFGSM so that prediction error measured by $\ell_1$ norm of depth map will be maximized. The $\ell_\infty$ bound $\epsilon$ and the number of iterations for IFGSM are randomly chosen from uniform distributions in the range of $(0.01, 0.3)$ and $(1, 10)$, respectively. Following most of the previous studies, the step size $\alpha$ in Eq.(\[eq\_ifgsm\]) is set to 1. Thus, if the number of iterations is one, it reduces to FGSM. For $\ell_{\mathrm{dif}}$, we use the loss function proposed in [@hu2019revisiting], which calculates the errors of depth, gradient, and normals, as: $$\ell_{\rm dif} = l_{\rm depth} + l_{\rm grad} + l_{\rm normal},$$ where $l_{\rm depth}=\frac{1}{n}\sum_{i=1}^n F(e_i)$ with $F(e_i) = \ln(e_i+0.5)$, $e_i = \|\bar{y}_i - y_i\|_1$, and $\bar{y}_i$ and $y_i$ are true and estimated depths; $l_{\rm grad}=\frac{1}{n}\sum_{i=1}^n (F(\nabla_{x}(e_i))+F(\nabla_{y}(e_i)))$; and $ l_{\rm normal} = \frac{1}{n}\sum_{i=1}^n \left(1-\cos\theta_i\right)$ where $\theta_i$ is the angle between the surface normals computed from the true and estimated depth map.
Defense by Masking Least Salient Pixels
---------------------------------------
We expect that the above procedure will yield $G_{\mathrm{adv}}$ that can robustly predict a correct saliency map even from an adversarial input $x^*=Adv(x, N)$, i.e., $G_{\mathrm{adv}}(x^*)\sim G(x)$. Recall here that we do not touch the depth estimation network $N$ during the training of $G_{\mathrm{adv}}$. This is different from a standard defense strategy that tries to make $N$ itself robust to attacks, such as employment of adversarial training on $N$. This could be an advantage in some practical applications. In this sense, our method is more similar to the ‘denoising’ approach to cope with attacks, such as projection of adversarial images to the manifold of clean images [@Liao2018DefenseAA]. However, it also differs from them in that it identifies and masks irrelevant pixels for depth estimation, thereby minimizing the impact of attacks.
Experiments
===========
Experimental Setting
--------------------
#### Dataset
We use the NYU-v2 dataset [@Silberman2012IndoorSA] for all the experiments. The dataset consists of a variety of indoor scenes and is used in most of the previous studies. We use the standard procedure for preprocessing [@Eigen2014depth; @laina2016deeper; @ma2017sparse]. To be specific, the official splits of 464 scenes are used, i.e., 249 scenes for training and 215 scenes for testing. The official toolbox is used to extract RGB images and depth maps from the raw data, and then fill in missing pixels in the depth maps to generate ground truths. This results in approximately 50K unique pairs of an image and a depth map of 640$\times$480 pixel size. The images are then resized down to 320$\times$240 pixels using bilinear interpolation, and then crop their central parts of 304$\times$228 pixels, which are used as inputs to networks. The depth maps are resized to $152\times 114$ pixels. For testing, following the previous studies, we use the same small subset of 654 samples.
#### Networks
For $N$, we use the network built on ResNet-50 in [@hu2019revisiting]. It is trained on the data mentioned above. For $G_{\mathrm{adv}}$ (and $G$), we employ the same encoder-decoder network as [@Hu2019VisualizationOC]. It employs a dilated residual network (DRN) proposed in [@Yu2017] for the encoder part and three stacks of the up-projection blocks proposed in [@laina2016deeper] for the decoder part. It outputs a saliency map of the same size as the input image. We train $G_{\mathrm{adv}}$ according to Algorithm 1 for 60 epochs. The parameter $\lambda$ controlling the sparseness of the saliency map is set to 1. We use the Adam optimizer with a learning rate of 0.0001, $(\beta_{1},\beta_{2})=(0.9,0.999)$ and weight decay of 0.0001.
Defense to White-box Attacks {#white_box_experiments}
----------------------------
#### FGSM attacks with different $\epsilon$’s
We first evaluated the performance of the plain network $N(x^*)$ (Fig. \[fig:config\](a)) and the proposed method $N(x^*\otimes G_{\mathrm{adv}}(x^*))$ (Fig. \[fig:config\](f)) for FGSM attacks with different magnitude $\epsilon$ of perturbation. Each adversarial example $x^*$ is generated by FGSM with an $\epsilon$ to maximize the $\ell_1$ norm of the difference between the estimate $N(x^*)$ and its ground truth. We generated one $x^*$ for each of the NYU-v2 test images and calculate the average of depth estimation errors (measured by RMSE) by the two models. The results are shown in Fig. \[fig\_white\_attack\]. It is first confirmed from the plot that the attack is indeed effective for the plain network $N(x^*)$ without any defense; the error increases rapidly even with small $\epsilon$. It is also seen that our method $N( x^*\otimes G_{\mathrm{adv}}(x^*) )$ works fairly well; the error only increases slowly with increasing $\epsilon$.
\[quan\_res\]
Attack Prediction Method RMSE $\downarrow$ REL$\downarrow$ $\log10$ $\downarrow$ $\delta<1.25$ $\uparrow$ $\delta<1.25^{2}$ $\uparrow$ $\delta<1.25^{3}$ $\uparrow$
------------------------- --------------------------------------- ------------------- ----------------- ----------------------- -------------------------- ------------------------------ ------------------------------
No attack (clean) $N(x)$ **0.555** **0.126** **0.054** **0.843** **0.968** **0.991**
$N_{\mathrm{adv}}(x)$ 0.682 0.167 0.071 0.757 0.935 0.979
$N(x\otimes G(x))$ 0.683 0.154 0.068 0.773 0.939 0.982
$N(x\otimes G_{\mathrm{adv}}(x))$ [0.615]{} [0.148]{} [0.063]{} [0.792]{} [0.952]{} [0.987]{}
IFGSM ($\epsilon=0.05$) $N(x^*)$ 1.465 0.419 0.200 0.249 0.568 0.774
$N_{\mathrm{adv}}(x^*)$ 0.666 0.160 0.067 **0.774** 0.943 0.982
$N(x^*\otimes G(x))$ 0.692 **0.156** 0.069 0.768 0.937 0.981
$N(x^*\otimes G_{\mathrm{adv}}(x^*))$ **0.644** 0.158 **0.067** 0.771 **0.945** **0.984**
IFGSM ($\epsilon=0.1$) $N(x^*)$ 1.792 0.373 0.273 0.161 0.373 0.571
$N_{\mathrm{adv}}(x^*)$ 0.677 0.159 0.068 0.769 0.942 0.981
$N(x^*\otimes G(x))$ 0.706 **0.158** 0.070 0.763 0.934 0.980
$N(x^*\otimes G_{\mathrm{adv}}(x^*))$ **0.655** 0.160 **0.067** **0.770** **0.942** **0.983**
IFGSM ($\epsilon=0.15$) $N(x^*)$ 1.988 0.516 0.325 0.109 0.263 0.442
$N_{\mathrm{adv}}(x^*)$ 0.724 0.167 0.073 0.741 0.931 0.976
$N(x^*\otimes G(x))$ 0.720 **0.159** 0.071 0.759 0.931 0.978
$N(x^*\otimes G_{\mathrm{adv}}(x^*))$ **0.677** 0.162 **0.068** **0.767** **0.939** **0.981**
IFGSM ($\epsilon=0.2$) $N(x^*)$ 2.107 0.541 0.360 0.075 0.201 0.370
$N_{\mathrm{adv}}(x^*)$ 0.798 0.180 0.081 0.701 0.911 0.970
$N(x^*\otimes G(x))$ 0.743 **0.161** 0.074 0.751 0.927 0.977
$N(x^*\otimes G_{\mathrm{adv}}(x^*))$ **0.703** 0.165 **0.071** **0.754** **0.933** **0.979**
#### IFGSM attacks
We also evaluated their performance for IFGSM attacks. Figure \[fig\_rmse\_iterations\] shows the averaged RMSE over all the inputs $x^*$’s generated by IFGSM with various iterations and a constant $\epsilon=0.1$. It is seen that the plain network $N(x^*)$ suffers more from $x^*$ generated with larger iterations, whereas the proposed method $N( x^*\otimes G_{\mathrm{adv}}(x^*) )$ is able to stably defend the attack. Although it is not shown here, it continues to perform well for larger iterations ($>$ 10).
We then compared the above two methods and additionally two other methods on IFGSM attacks with different perturbation magnitude. One additional method is the network $N$ directly trained by adversarial training [@Kurakin2017AdversarialEI], denoted by $N_{\mathrm{adv}}$ (Fig. \[fig:config\](c)). Specifically, similarly to the training of $G_{\mathrm{adv}}$, it is trained using both clean and adversarial images, but its encoder part (Resnet-50) starts with a pretrained model on ImageNet, following [@hu2019revisiting]. The other is $N(x^*\otimes G(x))$ (Fig. \[fig:config\](e)), i.e., inputting to the plain network $N$ the multiplication of $x^*$ and a saliency map predicted from a clean input $x$ by $G$. Here, $G$ is trained using only clean images, for which the sparseness parameter is set to $\lambda=5$ as recommended in [@Hu2019VisualizationOC].
[p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<]{} & & & & & & &\
& & & & & & &\
& & & & & & &\
& & & & & & &\
& & & & & & &\
& & & & & & &\
[(a) $x$]{}& [(b) True depth]{} & [(c) $N(x)$]{}&[(d) $x^*$]{}&[(e) $N(x^*)$]{} &[(f) $N_{\mathrm{adv}}(x^*)$]{} &[(g) $N(\!x^*\!\otimes\! G(x)\!)$]{} &[(h) $N(\!x^*\!\otimes\! G_{\mathrm{adv}}(\!x^*\!)\!)$]{}
Table \[quan\_res\] shows the evaluation of the four methods using multiple accuracy metrics. Adversarial examples were generated by IFGSM with four different $\epsilon$’s and 10 iterations. It is observed from the table that the three methods are all effective in defending the attacks for all the $\epsilon$’s. It is also seen that the proposed method, $N(x^*\otimes G_{\mathrm{adv}}(x^*))$, achieves the best performance among the three; it is better than $N_{\mathrm{adv}}(x^*)$ by a good margin and is equivalent or slightly better than $N(x^*\otimes G(x))$. Note that $N(x^*\otimes G(x))$ cannot be used in practice, as it needs a clean $x$. It is an ideal implementation of our conjecture that an adversarial input $x^*$ can be [*detoxified*]{} by masking $x^*$ with an accurate saliency map. The results show that this ideal implementation works well as a defense method, which well validates our conjecture. The results also show that the proposed method achieves comparable to or even better performance than it, which confirms that $G_{\mathrm{adv}}(x^*)$ can yield accurate saliency map even from $x^*$ and help defense to attacks. Figure \[fig\_depth\_nyu\] shows the depth maps predicted by these methods from adversarial inputs for different scenes.
#### Effects of Losses Used for Adversarial Example Generation
Adversarial inputs are generated so that they will maximize errors. There are a number of choices in the measure of the errors, or losses, such as $\ell_1$ norm (used in the experiments so far), $\ell_2$, etc. To investigate if the defending performance will be affected by the choice of this loss, we conducted an experiment. Specifically, we use the same $G_{\mathrm{adv}}$ trained assuming $\ell_1$ and test its performance against adversarial images generated using $\ell_2$, REL, $\log10$ and $\ell_{\rm dif}$, respectively. Here we use IFGSM with 5 iterations and $\epsilon=0.1$. Table \[attack\_losses\] shows the results, indicating that all the losses are effective in making the plain network $N(x^*)$ yield erroneous depths, whereas the proposed method $N(x^*\otimes G_{\mathrm{adv}}(x^*))$ consistently performs well independently of the loss used for the adversarial example generation.
[|p[.08]{}<|p[.09]{}<|p[.15]{}<|]{} Loss & $N(x^*)$ & $N(x^*\otimes G_{\mathrm{adv}}(x^*))$\
$\ell_1$ &1.778 &0.664\
$\ell_2$ &1.842 &0.666\
REL &1.694 &0.664\
$\log10$ &1.783 & 0.659\
$\ell_{\rm dif}$ &1.718&0.662\
#### Effects of Number of Layers of $G_{\mathrm{adv}}$
We also investigate how the number of layers of the saliency estimation network $G_{\mathrm{adv}}$ affects defense performance. We tested $G_{\mathrm{adv}}$ having different numbers of layers for its encoder part: 22, 38, and 54 layers. Adversarial examples are generated by IFGSM with 5 iterations and $\epsilon=0.1$. Table \[layer\_depth\] shows the results. It indicates that the more layers $G_{\mathrm{adv}}$ has, the better performance we have, although the improvements are fairly small.
\[layer\_depth\]
[|p[.24]{}<|p[.12]{}<|]{} Layers of the encoder in $G_{\mathrm{adv}}$ & RMSE\
W/o $G_{\mathrm{adv}}$ (plain $N$) &1.778\
22&0.664\
38&0.659\
54&0.655\
\[saliency\_map\]
[p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<p[0.115]{}<]{} & & & & & & &\
& & & & & & &\
& & & & & & &\
& & & & & & &\
& & & & & & &\
& & & & & & &\
[(a) $x^*$]{} &[(b) $x-x^*$]{} & [(c) $G(x)$]{} & [(d) $G(x^{*})$]{} & [(e) $G_{\mathrm{adv}}(x)$]{} & [(f) $G_{\mathrm{adv}}(x^{*})$]{} & [(g) $|G(x)-G(x^{*})|$]{} &[(h) $|G_{\mathrm{adv}}(x)-G_{\mathrm{adv}}(x^{*})|$]{}
Qualitative Analysis
--------------------
Figure \[fig\_saliency\_nyu\] shows adversarial examples and the saliency maps from them predicted by $G$ and $G_{\mathrm{adv}}$, along with their differences showing the effects of attacks. Here, the same images as Fig. \[fig\_depth\_nyu\] are chosen, for which adversarial inputs are generated by IFGSM with $\epsilon=0.1$ and 10 iterations. It is first observed that the perturbations (Fig. \[fig\_saliency\_nyu\](b)) caused by the attack do not appear to be correlated with the saliency map and rather appear to be random. Although this agrees with the previously reported observation (mostly on classification tasks), it may not appear to be consistent with our hypothesis that adversarial examples function by perturbing non-salient pixels. This may be because that in the attack by FGSM/IFGSM (Eq.(\[eq\_ifgsm\])), a constraint is imposed only on the magnitude of perturbation but not on the number of perturbed pixels. On the other hand, it can be confirmed from $|G(x)-G(x^*)|$ (Fig. \[fig\_saliency\_nyu\](g)) that the attacks affect $G$ in its prediction, although $x^*$ is created for $N$ not for $G$. It is also seen from $|G_{\mathrm{adv}}(x)-G_{\mathrm{adv}}(x^*)|$ (Fig. \[fig\_saliency\_nyu\](h)) that $G_{\mathrm{adv}}$ is indeed robust to the attacks.
Conclusion
==========
In this paper, we have analyzed how CNNs for monocular depth estimation react against adversarial attacks. For the IFGSM attack, we have validated the three items listed at the last of Sec. 1 through a number of experiments. They can be summarized as follows. IFGSM attacks can be defended by masking out non-salient pixels in the inputs. The non-salient pixels can be identified either by predicting saliency maps from clean images or by predicting saliency maps from adversarial inputs using an estimator trained to be robust to the attacks. This study has two contributions. One is the proposal of an effective defense method against the IFGSM attack, provided that the attacker has the knowledge of the depth estimation CNN including its weights but does not recognize that we are using this defense method. The other is that our results indicate that the attacks function mostly by perturbing non-salient pixels, which will be a clue to understand how the CNNs yield an accurate depth map from a single (clean) image.
|
---
abstract: 'The Relativistic Hartree Bogoliubov model in coordinate space, with finite range pairing interaction, is applied to the description of $\Lambda$-hypernuclei with a large neutron excess. The addition of the $\Lambda$ hyperon to Ne isotopes with neutron halo can shift the neutron drip by stabilizing an otherwise unbound core nucleus. The additional binding of the halo neutrons to the core originates from the increase in magnitude of the spin-orbit term. Although the $\Lambda$ produces only a fractional change in the central mean-field potential, through a purely relativistic effect it increases the spin-orbit term which binds the outermost neutrons.'
author:
- |
D. Vretenar, W. Pöschl, G.A. Lalazissis and P. Ring\
Physik-Department, Technische Universität München,\
D-85748 Garching, Germany
title: 'Relativistic mean-field description of light $\Lambda$ hypernuclei with large neutron excess'
---
=16 true cm =23.5 true cm =cmbx10 scaled2
1.5cm
The production mechanisms, spectroscopy, and decay modes of hypernuclear states have been the subject of many theoretical studies. Extensive reviews of the experimental and theoretical status of strange- particle nuclear physics can be found in Refs. [@CD.89; @DMG.89; @BMZ.90]. The most studied hypernuclear system consists of a single $\Lambda$ particle coupled to the nuclear core. And although strangeness in principle can be used as a measure of quark deconfinement in nuclear matter, a single $\Lambda$ behaves essentially as a distinguishable particle in the nucleus. Theoretical models used in studies of hypernuclei extend from nonrelativistic approaches based on OBE models for $\Lambda$-N interaction, to the relativistic mean field approximation and quark-meson coupling models. However, our knowledge of the $\Lambda$-nucleus interaction, and of hypernuclear systems in general, is restricted to the valley of $\beta$-stability.
In recent years the study of the structure of exotic nuclei, produced by radioactive nuclear beams, has become one of the most active fields in nuclear physics. The structure of nuclei far from the stability line presents many interesting phenomena. In particular, in the present work we consider the extremely weak binding of the outermost nucleons, large spatial dimensions and the coupling between bound states and the particle continuum. By adding either more protons or neutrons, the particle drip lines are reached. Nuclei beyond the drip lines are unbound with respect to nuclear emission. Exotic nuclei on the neutron-rich side are especially important in nuclear astrophysics. They are expected to play an important role in nucleosynthesis by neutron capture (r-processes). Knowledge of their structure and properties would help the determination of astrophysical conditions for the formation of neutron-rich stable isotopes. On the neutron-rich side, the drip line has only been reached for very light nuclei.
It has been recently suggested [@Maj.95] that a study of $\Lambda$-hypernuclei with a large neutron excess could also display interesting phenomena. On one hand such hypernuclei, corresponding to core nuclei which are unbound or weakly bound, are of considerable theoretical interest. Some possibilities for unusual light hypernuclei were already analyzed in the early work of Dalitz and Levi Setti [@Dal.63]. On the other hand, one could speculate on the possible role of neutron-rich $\Lambda$-hypernuclei in the process of nucleosynthesis. The $\Lambda$ particle provides the nuclear core with additional binding. Even-core nuclei that are either weakly bound or unbound attain normal binding [@Gal.75].
In Refs. [@PVL.97; @LVR.97] we have investigated, in the framework of relativistic mean- field theory, light nuclear systems with large neutron excess. For such nuclei the separation energy of the last neutrons can become extremely small. The Fermi level is found close to the particle continuum, and the lowest particle-hole or particle-particle modes couple to the continuum. In Ref. [@PVL.97] the Relativistic Hartree Bogoliubov (RHB) model has been applied, in the self-consistent mean-field approximation, to the description of the neutron halo in the mass region above the s-d shell. As an extension of non-relativistic HFB-theory [@Doba.84], the RHB theory in coordinate space provides a unified description of mean-field and pairing correlations. Pairing correlations and the coupling to particle continuum states have been described by finite range two-body Gogny-type interaction. Finite element methods have been used in the coordinate space discretization of the coupled system of Dirac-Hartree-Bogoliubov and Klein-Gordon equations. Solutions in coordinate space are essential for the correct description of the coupling between bound and continuum states. Calculations have been performed for the isotopic chains of Ne and C nuclei. Using the NL3 [@LKR.96] parameter set for the mean-field Lagrangian, and D1S [@BGG.84] parameters for the Gogny interaction, we have found evidence for the occurrence of multi-neutron halo in heavier Ne isotopes. We have shown that the properties of the 1f-2p orbitals near the Fermi level and the neutron pairing interaction play a crucial role in the formation of the halo. In the present work we essentially repeat the calculations of Ref. [@PVL.97] for the Ne isotopes, but we add a $\Lambda$ particle to the system. We are interested in the effects that the $\Lambda$ particle in its ground state has on the core halo-nucleus.
Relativistic mean-field models have been successfully applied in calculations of nuclear matter and properties of finite nuclei throughout the periodic table [@Rin.96]. The model describes the nucleus as a system of Dirac nucleons which interact in a relativistic covariant manner the isoscalar scalar $\sigma$-meson, the isoscalar vector $\omega$-meson and the isovector vector $\rho$-meson. The photon field $(A)$ accounts for the electromagnetic interaction. For hypernuclear systems the original model has to be extended to the strange particle sector. In particular, the effective Lagrangian for $\Lambda$ hypernuclei reads $$\begin{aligned}
\label{equ.1}
{\cal L}&=&\bar\psi\left(\gamma(i\partial-g_\omega\omega
-g_\rho\vec\rho\vec\tau-eA)-m-g_\sigma\sigma\right)\psi
\nonumber\\
&&+\frac{1}{2}(\partial\sigma)^2-U(\sigma )
-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}
+\frac{1}{2}m^2_\omega\omega^2\nonumber\\
&&-\frac{1}{4}{\vec{\rm R}}_{\mu\nu}{\vec{\rm R}}^{\mu\nu}
+\frac{1}{2}m^2_\rho\vec\rho^{\,2}
-\frac{1}{4}{\rm F}_{\mu\nu}{\rm F}^{\mu\nu}
\nonumber\\
&&+\bar\psi_{\Lambda}\left(\gamma(i\partial-g_{\omega\Lambda}\omega)
- m_\Lambda - g_{\sigma\Lambda}\sigma \right)\psi_\Lambda.\end{aligned}$$
The Dirac spinors $(\psi )$ and $\psi_\Lambda$ denote the nucleon and the $\Lambda$ particle, respectively. Coupling constants $g_\sigma$, $g_\omega$, $g_\rho$, and unknown meson masses $m_\sigma$, $m_\omega$, $m_\rho$ are parameters, adjusted to fit data on nuclear matter and finite nuclei. The model includes the nonlinear self-coupling of the $\sigma$-field (coupling constants $g_2$, $g_3$) $$U(\sigma)~=~\frac{1}{2}m^2_\sigma\sigma^2+\frac{1}{3}g_2\sigma^3+
\frac{1}{4}g_3\sigma^4.$$ Since the $\Lambda$ particle is neutral and isoscalar, it only couples to the $\sigma$ and $\omega$ mesons (coupling constants $g_{\sigma\Lambda}$ and $g_{\omega\Lambda}$). We only consider the $\Lambda$ in the $1s$ state, and therefore do not include the tensor $\Lambda$ - $\omega$ interaction.. The generalized single-particle hamiltonian of HFB theory contains two average potentials: the self-consistent field $\hat\Gamma$ which encloses all the long range [*ph*]{} correlations, and a pairing field $\hat\Delta$ which sums up the [*pp*]{}-correlations. In the Hartree approximation for the self-consistent mean field, the Relativistic Hartree-Bogoliubov (RHB) equations read $$\begin{aligned}
\left( \matrix{ \hat h_D -m- \lambda & \hat\Delta \cr
-\hat\Delta^* & -\hat h_D + m +\lambda
} \right) \left( \matrix{ U_k \cr V_k } \right) =
E_k\left( \matrix{ U_k \cr V_k } \right).\end{aligned}$$ where $\hat h_D$ is the single-nucleon Dirac hamiltonian, and $m$ is the nucleon mass. $U_k$ and $V_k$ are quasi-particle Dirac spinors, and $E_k$ denote the quasi-particle energies. The Dirac equation for the $\Lambda$ particle $$\label{equ.5}
\bigl[-i{\bf\alpha\nabla}+\beta (m_\Lambda+g_{\sigma\Lambda}\sigma({\bf r})) +
g_{\omega\Lambda}\omega^0({\bf r})\bigr]\psi_\Lambda
= \epsilon_\Lambda\psi_\Lambda$$
The RHB equations for the nucleons and the Dirac equation for the $\Lambda$ are solved self-consistently, with potentials determined in the mean-field approximation from solutions of Klein-Gordon equations for mesons and Coulomb field: $$\begin{aligned}
\bigl[-\Delta + m_{\sigma}^2\bigr]\,\sigma({\bf r})&=&
-g_{\sigma}\,
\sum\limits_{E_k > 0} V_k^{\dagger}({\bf r})\gamma^0 V_k({\bf r})
-g_2\,\sigma^2({\bf r})-g_3\,\sigma^3({\bf r})
\nonumber \\
&\,&-g_{\sigma\Lambda}\psi_\Lambda^{\dagger}({\bf r})\gamma^0
\psi_\Lambda({\bf r})\, \\
\bigl[-\Delta + m_{\omega}^2\bigr]\,\omega^0({\bf r})&=&
\sum\limits_{E_k > 0} V_k^{\dagger}({\bf r}) V_k({\bf r})
+g_{\omega\Lambda}\psi_\Lambda^{\dagger}({\bf r})
\psi_\Lambda({\bf r})\, \\
\bigl[-\Delta + m_{\rho}^2\bigr]\,\rho^0({\bf r})&=&
\sum\limits_{E_k > 0} V_k^{\dagger}({\bf r})\tau_3 V_k({\bf r}), \\
-\Delta \, A^0({\bf r})&=&
\,\sum\limits_{E_k > 0} V_k^{\dagger}({\bf r}) {{1-\tau_3}\over 2}
V_k({\bf r}).\end{aligned}$$ The sums run over all positive energy states. The system of equations is solved self-consistently in coordinate space by discretization on the finite element mesh. In the coordinate space representation of the pairing field $\hat\Delta $, the kernel of the integral operator is $$\Delta_{ab} ({\bf r}, {\bf r}') = {1\over 2}\sum\limits_{c,d}
V_{abcd}({\bf r},{\bf r}') {\bf\kappa}_{cd}({\bf r},{\bf r}').$$ where $V_{abcd}({\bf r},{\bf r}')$ are matrix elements of a general two-body pairing interaction and ${\bf\kappa}_{cd}({\bf r},{\bf r}')$, is the pairing tensor, defined as $$\label{equ.4}
{\bf\kappa}_{cd}({\bf r},{\bf r}') :=
\sum_{E_k>0} U_{ck}^*({\bf r})V_{dk}({\bf r}').$$ The integral operator $\hat\Delta$ acts on the wave function $V_k({\bf r})$: $$\label{equ.2.4}
(\hat\Delta V_k)({\bf r})
= \sum_b \int d^3r' \Delta_{ab} ({\bf r},{\bf r}') V_{bk}({\bf r}').$$ In the particle-particle ($pp$) channel the pairing interaction is approximated by the finite range two-body Gogny interaction $$V^{pp}(1,2)~=~\sum_{i=1,2}
e^{-[ ({\bf r}_1- {\bf r}_2)
/ {\mu_i} ]^2}\,
(W_i~+~B_i P^\sigma
-H_i P^\tau -
M_i P^\sigma P^\tau),$$ with the parameters $\mu_i$, $W_i$, $B_i$, $H_i$ and $M_i$ $(i=1,2)$.
As in Ref. [@PVL.97], the even-even Ne isotopes have been calculated with the NL3 [@LKR.96] effective interaction for the mean-field Lagrangian, and the parameter set D1S [@BGG.84] has been used for the finite range pairing force. The coupling constants for the $\Lambda$ particle are from Ref. [@MJ.94], where the relativistic mean-field theory was used to study characteristics of $\Lambda$, $\Sigma$ and $\Xi$ hypernuclei. While the values for the $g_{\omega Y}$ coupling constants were determined from the naive quark model, that is $g_{\omega \Lambda} = \frac{2}{3} g_{\omega N}$; the values of $g_{\sigma Y}$ were deduced from the available experimental information of hyperon binding in the nuclear medium. For the $\Lambda$ hyperon $g_{\sigma \Lambda}$ was fitted to reproduce the binding energy of a $\Lambda$ in the $1s$ state of $^{17}_{\Lambda}$O: $g_{\sigma \Lambda} =
0.621 g_{\sigma N}$. The coupling constant determined from only this experimental quantity gives a reasonable description of binding energies in $\Lambda$ hypernuclei for a wide range of mass number.
In Ref. [@PVL.97] we have shown that the neutron $rms$ radii of the Ne isotopes follow the mean-field N$^{1/3}$ curve up to N $\approx$ 22. For larger values of N both neutron and matter $rms$ radii display a sharp increase, while the proton radii stay practically constant. The sudden increase in neutron $rms$ radii has been interpreted as evidence for the formation of a multi-particle halo. The phenomenon was also clearly seen in the plot of proton and neutron density distributions. The proton density profiles do not change with the number of neutrons, while the neutron density distributions display an abrupt change between $^{30}$Ne and $^{32}$Ne. The microscopic origin of the neutron halo has been found in a delicate balance of the self-consistent mean-field and the pairing field. This is shown in Fig. 1a, where we display the neutron single-particle states 1f$_{7/2}$, 2p$_{3/2}$ and 2p$_{1/2}$ in the canonical basis, and the Fermi energy as functions of the mass number A. For A$\leq$32 (N$\leq$ 22) the triplet of states is high in the continuum, and the Fermi level uniformly increases toward zero. The triplet approaches zero energy, and a gap is formed between these states and all other states in the continuum. The shell structure dramatically changes at N$\geq$ 22. Between A=32 (N = 22) and A=42 (N = 32) the Fermi level is practically constant and very close to the continuum. The addition of neutrons in this region of the drip does not increase the binding. Only the spatial extension of neutron distribution displays an increase. The formation of the neutron halo is related to the quasi-degeneracy of the triplet of states 1f$_{7/2}$, 2p$_{3/2}$ and 2p$_{1/2}$. The pairing interaction promotes neutrons from the 1f$_{7/2}$ orbital to the 2p levels. Since these levels are so close in energy, the total binding energy does not change significantly. Due to their small centrifugal barrier, the 2p$_{3/2}$ and 2p$_{1/2}$ orbitals form the halo. The last bound isotope is $^{40}$Ne. For N $\geq$ 32 (A=42) the neutron Fermi level becomes positive, and heavier isotopes are not bound any more.
In the present work we have repeated the calculation of Ne isotopes, but a $\Lambda$ hyperon has been added to the even-even cores. As one would expect, there are no excessive changes in the bulk properties. For example, the neutron $rms$ radii are reduced on the average by 2%. Therefore we do not display changes in macroscopic quantities, but going to the microscopic level, in Fig. 1b we illustrate the effect of the $\Lambda$ hyperon on the triplet of neutron states that form the halo: 1f$_{7/2}$, 2p$_{3/2}$ and 2p$_{1/2}$, and on the Fermi level. The energies are displayed as function of the core mass number A$_{c}$. Due to the extra binding provided by the $\Lambda$, the single-neutron energies and the Fermi level are lower. The most important effect that we observe, however, is that the Fermi level is negative for the isotope $^{42+\Lambda}$Ne. Without the $\Lambda$, the nucleus $^{42}$Ne was unbound. The presence of the strange baryon stabilizes the otherwise unbound core. This could have interesting consequences for the process of nucleosynthesis, especially r-processes. It should be noted, however, that heavier nuclei, and in particular those with atomic number around Z=40, are important in the rapid neutron-capture mechanisms. In a very recent study [@MR.97], it has been shown that RHB theory predicts the last bound Zr isotope to be $^{138}$Zr (N=98). Therefore we have calculated the ground state of the hypernucleus $^{140+\Lambda}$Zr. It turns out that although the presence of the $\Lambda$ lowers the Fermi level, the $\Lambda$-N interaction does not have enough strength to stabilize the core A$_{c}$=140. But if a single $\Lambda$ cannot stabilize such a large core, maybe two $\Lambda$ particles could. An interesting question is whether such objects could be found in the environment in which the processes of nucleosynthesis occur.
It is important to understand the microscopic mechanism through which the $\Lambda$ binds the additional pair of neutrons in $^{42+\Lambda}$Ne. In a recent study [@LVR.97] we have used the relativistic Hartree-Bogoliubov model to analyze the isospin dependence of the spin-orbit interaction in light neutron-rich nuclei. It has been shown that the magnitude of the spin-orbit potential is considerably reduced in drip line nuclei. With the increase of the neutron number, the effective one-body spin-orbit potential becomes weaker. This results in a reduction of the energy splittings between spin-orbit partners. The reduction of the spin-orbit potential is especially pronounced in the surface region. In the relativistic mean-field approximation, the spin-orbit potential originates from the addition of two large fields: the field of the vector mesons (short range repulsion), and the scalar field of the sigma meson (intermediate attraction). In the first order approximation, and assuming spherical symmetry, the spin orbit term can be written as $$\label{so1}
V_{s.o.} = {1 \over r} {\partial \over \partial r} V_{ls}(r),$$ where $V_{ls}$ is the spin-orbit potential [@Rin.96; @Koepf.91] $$\label{so2}
V_{ls} = {m \over m_{eff}} (V-S).$$ V and S denote the repulsive vector and the attractive scalar potentials, respectively. $m_{eff}$ is the effective mass $$\label{so3}
m_{eff} = m - {1 \over 2} (V-S).$$
Using the vector and scalar potentials from the self-consistent ground-state solutions, we have computed from (\[so1\]) - (\[so3\]) the spin-orbit terms for several Ne isotopes and corresponding $\Lambda$-hypernuclei. They are displayed in Fig. 2 as function of the radial distance from the center of the nucleus. The magnitude of the spin-orbit term $V_{s.o.}$ in Ne nuclei (dashed lines) decreases as we add more neutrons, i.e. more units of isospin. The reduction for nuclei close to the neutron drip is $\approx 40\%$ in the surface region, as compared to values which correspond to beta stable nuclei. For the corresponding $\Lambda$-hypernuclei (solid lines) the spin-orbit term displays an increase in magnitude of about 10% (smaller as we approach the drip line). This is a rather surprising result, as we have seen that the single-particle energies and bulk properties seem to be less affected by the presence of the $\Lambda$ particle. The effect is purely relativistic and it appears to be strong enough to bind an additional pair of neutrons at the drip line. The mean field potential, in which the nucleons move, results from the cancelation of two large meson potentials: the attractive scalar potential S and the repulsive vector potential V: V+S. The spin-orbit potential, on the other hand, arises from the very strong anti-nucleon potential V-S. Therefore, while in the presence of the $\Lambda$ the changes in V and S cancel out in the mean-field potential, they are amplified in $V_{ls}$. We illustrate this effect on the example of $^{30}$Ne and the corresponding hypernucleus $^{31}_{\Lambda}$Ne. For the core $^{30}$Ne the values of the scalar (S) and vector (V) potential in the center of the nucleus are -380 MeV and 308 MeV, respectively. For $^{31}_{\Lambda}$Ne the corresponding values are: -412 MeV and 336 MeV. The addition of the $\Lambda$ particle changes the value of the mean-field potential in the center of the nucleus by 4 MeV, but it changes the anti-nucleon potential by 60 MeV. This is reflected in the corresponding spin-orbit term (Fig. 2), which provides more binding for states close to the Fermi surface. The additional binding stabilizes the hypernuclear core.
In conclusion, we report results of the first application of the Relativistic Hartree Bogoliubov model in coordinate space, with finite range pairing interaction, to the description of $\Lambda$-hypernuclei with a large neutron excess. In particular, we have studied the effects of the $\Lambda$ hyperon in its ground state on Ne nuclei with neutron halo. Although the inclusion of the $\Lambda$ hyperon does not produce excessive changes in bulk properties, we find that it can shift the neutron drip by stabilizing an otherwise unbound core nucleus at drip line. The microscopic mechanism through which additional neutrons are bound to the core originates from the increase in magnitude of the spin-orbit term in presence of the $\Lambda$ particle. We find that the $\Lambda$ in its ground state produces only a fractional change in the central mean-field potential. On the other hand, through a purely relativistic effect, it notably changes the spin-orbit term in the surface region, providing additional binding for the outermost neutrons. Neutron-rich $\Lambda$-hypernuclei might have an important role in the process of nucleosynthesis by neutron capture.
This work has been supported by the Bundesministerium für Bildung und Forschung under project 06 TM 875. G. A. L. acknowledges support from DAAD.
[999]{} R. E. Chrien and C. B. Dover, Annu. Rev. Nucl. Part. Sci. [**39**]{}, 113 (1989). C. B. Dover, D. J. Millener, and A. Gal, Phys. Rep. [**184**]{}, 1 (1989). H. Bando, T. Motoba, and J. Žofka, Int. J. Mod. Phys. [**A5**]{}, 4021 (1990). L. Majling, Nucl. phys. [**A585**]{}, 211c (1995). R. H. Dalitz and R. Levi Setti, Nuovo Cim. [**30**]{}, 489 (1963). A. Gal, Adv. Nucl. Phys. [**8**]{}, 1 (1975). W. Pöschl, D. Vretenar, G.A. Lalazissis, and P. Ring, Phys. Rev. Lett. (1997). G.A. Lalazissis, D. Vretenar, W. Pöschl, and P. Ring, Phys. Lett. (1997). J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys. [**A 422**]{}, 103 (1984). G. A. Lalazissis, J. König and P. Ring; Phys. Rev. [**C55**]{}, 540 (1997). J. F. Berger, M. Girod and D. Gogny; Nucl. Phys. [**A428**]{}, 32 (1984). P. Ring, Progr. Part. Nucl. Phys. [**37**]{}, 193 (1996). J. Mareš and B. K. Jennings, Phys. Rev. [**C49**]{}, 2472 (1994). J. Meng and P. Ring, Phys. Lett. (1997). W. Koepf and P. Ring; Z. Phys. [**A339**]{} (1991) 81.
**Figure Captions**
- [**Fig.1**]{} 1f-2p single-particle neutron levels in the canonical basis for the $Ne$ (a), and $Ne + \Lambda$ (b) isotopes. The dotted line denotes the Fermi level.
- [**Fig.2**]{} Radial dependence of the spin-orbit potential in self-consistent solutions for the ground-states of $Ne$ (a), and $Ne + \Lambda$ (b) isotopes.
|
---
abstract: 'This article addresses the degree distribution of subnetworks, namely the number of links between the nodes in each subnetwork and the remainder of the structure (cond-mat/0408076). The transformation from a subnetwork-partitioned model to a standard weighted network, as well as its inverse, are formalized. Such concepts are then considered in order to obtain scale free subnetworks through design or through a dynamics of node exchange. While the former approach allows the immediate derivation of scale free subnetworks, in the latter nodes are sequentially selected with uniform probability among the subnetworks and moved into another subnetwork with probability proportional to the degree of the latter. Comparison of the designed scale-free subnetworks with random and Barabási-Albert counterparts are performed in terms of a set of hierarchical measurements.'
author:
- Luciano da Fontoura Costa
bibliography:
- 'inst.bib'
date: 27th January 2005
title: Scale Free Subnetworks by Design and Dynamics
---
Introduction
============
In a short period of time, complex network research progressed all the way from uniform random models [@Flory:1941; @Rapoport:1957; @Erdos:1959] to the scale free networks of Barabási [@Albert_Barab:2002]. A good deal of the motivation for such developments has been accounted for by the scale free distribution of node degrees observed in models such as that proposed by Barabási and Albert [@Albert_Barab:2002]. One of the principal consequences of such a type of distribution is that it promotes the appearance of *hubs*, namely nodes with particularly high degree. By concentrating connections, hubs play a critical role in defining the network connectivity as well as other topological properties such as minimal paths. Another concept which has been found to be particularly useful in understanding complex networks is that of *community*, which can be informally understood as a group of nodes which are intensely interconnected but loosely connected to the remainder of the network(e.g. [@Newman:2003; @Bollt:2004; @Latapy:2004; @Costa_hub:2004].
The relationship between hubs and communities has motivated some recent works [@Costa_hub:2004; @Bollt:2004] which considered hubs as references for obtaining communities. Another concept directly related, but not necessarily equivalent, to communities is that of a *subnetwork* [@Costa_gener:2004]. Given a network $\Gamma$, a subnetwork of $\Gamma$ is defined as a graph including a subset of nodes of $\Gamma$ plus their respective interconnections. Therefore, each community in a network can be understood as a densely linked subnetwork which is loosely connected with the remainder of the network. Every community is a subnetwork, but not every subnetwork is a community, i.e. communities are special cases of subnetworks. Because of their generality, subnetworks represent an interesting resource for theoretical and practical investigations of complex networks which has only scantly been explored [@Costa_gener:2004]. One particularly interesting situation is the *partition* of a network into several subnetworks, in the sense that every node belongs exactly to one and only subnetwork. The concept of *subnetwork degree* was recently formalized [@Costa_gener:2004] as the number of edges linking nodes inside the subnetwork to nodes in the remainder network.
The present work addresses subnetwork-partitioned models characterized by scale free subnetwork degrees. More specifically, we introduce a transformation from scale free subnetworks to traditional weighted networks, as well as its inverse. Two approaches to obtain scale free subnetworks from the random network $\Gamma$ are proposed: (i) *by design* and (ii) *by dynamics*. The former approach starts from the desired log-log curve and applies a direct, non-interactive method in order to obtain a subnetwork partition having similar node degree distribution. In the second methodology, nodes are sequentially selected from a subnetwork and reinserted into (possibly) another subnetwork with probability proportional to the degree of the latter. The comparison between the design scale free subnetworks and traditional random and Barabási-Albert models is also considered in terms of a set of recently introduced hierarchical features [@Costa_gener:2004].
Basic Concepts {#sec:basic}
==============
An undirected, *unweighted* network can be represented in terms of its *adjacency* matrix $K$, such that $K(i,j)=K(j,i)=1$ whenever there is a link between nodes $i$ and $j$, with $1 \leq i,j
\leq N$, and $K(i,j)=K(j,i)=0$ otherwise. Similarly, an undirected, *weighted* network can be represented in terms of its *weight* matrix, in the sense that $W(i,j)=W(j,i)\geq 0$ corresponds to the weight of the edge between nodes $i$ and $j$. The absence of edges between those nodes is represented by making $W(i,j)=W(j,i)=0$. Random networks, in the sense of Ërdos and Rényi [@Erdos:1959; @Albert_Barab:2002], can be obtained by selecting among the $N(N-1)/2$ possible edges with uniform probability $\gamma$, yielding average degree $\left< k \right> = \gamma (N-1)$.
![One of the possible subnetwork partitions of a simple network (a), and its respective subsumed network (b). \[fig\_1\]](fig_1a.eps "fig:") (a)\
![One of the possible subnetwork partitions of a simple network (a), and its respective subsumed network (b). \[fig\_1\]](fig_1b.eps "fig:") (b)
The network of interest $\Gamma$ can be *partitioned* into $n$ subnetworks, such that each subnetwork $c_i$ includes $N_i$ nodes from $\Gamma$ as well as the respective interconnections. Note that every node should belong to exactly one subnetwork. Figure \[fig\_1\](a) illustrates a simple random network with $N=14$ nodes and its partition into $n=4$ subnetworks. It is henceforth assumed that the original network $\Gamma$ to be partitioned into subnetworks follows the Ërdos and Rényi uniform model with Poisson rate $\gamma$. Now, given any two subnetworks $c_i$ and $c_j$, the mean expected number of edges inside each subnetwork are $e_i=\gamma N_i(N_i-1)/2$ and $e_j=\gamma N_j(N_j-1)/2$, respectively. Let the total number of edges in the network constituted by the two subnetworks $c_i$ and $c_j$ be $E_{i,j} = e_i + e_j + e_{i,j}$, where $e_{i,j}$ is the average number of edges extending between the two subnetworks. Because $E_{i,j}=N_{i,j}(N_{i,j}-1)/2$, where $N_{i,j}=N_i+N_j$, and $\left< k \right> = \gamma (N_{i,j}-1) \approx \gamma N_{i,j}$, it follows that
$$e_{i,j}=\gamma N_j N_i \approx \frac{N_i N_j}{N_{i,j}} \left< k \right > \label{eq:e}$$
The degree of a subnetwork $c_i$, hence $k(c_i)$, can now be calculated as suggested in [@Costa_gener:2004], i.e. as the number of edges between elements of $c_i$ and the remainder of the network $\Gamma$. The degree of subnetwork $c_i$ can be immediately obtained as $k(c_i) = e_{i,j}$. Now, considering the subnetwork $c_i$ with respect to all other $n-1$ subnetworks in the partition, i.e. $c_j=\bigcup_{j \neq i} c_j$, we have $N_j=N-N_i$. From Equation \[eq:e\] and the fact that $\left< k \right> \approx \gamma
N$, it follows that
$$k(c_i)=\gamma (N-N_i)N_i \approx \frac{N-N_i}{N} N_i \left< k \right> \label{eq:k}$$
In case $N \gg N_i$, we have $k(c_i) \approx N_i \left< k \right>$.
Given the original random network $\Gamma$, it is possible to construct a subnetwork-partioned version by assigning nodes of $\Gamma$ to each community $c_i$ according to some criterion. The opposite operation, namely the transformation of a partitioned network into a traditional weighted network, henceforth called the *subsumption* of $\Gamma$ is also possible through the following steps: (i) each community $c_i$ is subsumed into a single node $c_i$ and (ii) the weight of the edge linking two nodes $c_i$ and $c_j$ is defined as the number of edges between the respective subnetworks. Figure \[fig\_1\] illustrates the subsumption of the subnetwork partitioned structure in (a) into the weighted network in (b). The inverse transformation can be obtained by using the design approach described in the following.
Scale Free by Design
====================
In this section we present how scale free subnetwork partitions of a random network $\Gamma$ can be immediately obtained such that the subnetwork degree follows a pre-specified scale free distribution.
As described in the previous section, provided $N \gg N_j$, the average degree of a subjetwork $c_j$ can be approximated as $k_j \approx N_j
\left< k \right>$, i.e. this degree becomes independent of the overall size of the random network $\Gamma$. This fact allows the immediate design of subnetwork partitions following virtually any subnetwork degree distribution, including the particularly important case of scale free models. The generic scale free log-log distribution of the degrees of a network is illustrated in Figure \[fig\_2\]. In order to have the subnetwork degree histogram $h(k)$ such that $h(k)
\propto k^\xi$, we start by imposing that $ln(h(k_j))=(m-j) \Delta a$ for some pre-specified $da$, with $j=1, 2, \ldots, m$, so that the values of $ln(h(k))$ are uniformly distributed from $a$ down to 0 with step $\Delta a = a/(m-1)$ along the $y-$axis, as $k_j$ varies from $k_1$ to $k_m$. It follows that $h(k_j) = exp((m-j)\Delta a)$ and $\Delta k = - \Delta a /\xi$. Without loss of generality, we impose that $ln(k_1)=0$, which implies $ln(k_j)=(j-1)\Delta k$ and $ln(k_m)=(m-1) \Delta k =-a/\xi$. So, $ln(h(k_j))=\xi ln(k_j) +a$.
![The basic construction used in the scale free subnetwork design. \[fig\_2\]](fig_2.eps)
From the above developments, we have that $k_j=exp((j-1)\Delta k)$. In other words, it is desired that community $c_j$ has degree $k_(c_j)=k_j$. We have from Section \[sec:basic\] that $k(c_j)
\approx N_j \left< k \right>$. Therefore, in order to have $k(c_j) =
k_j$, we must have $N_j \approx k_j \left< k \right>$. The total required communities is $n=round(\sum_{j=1}^m h(k_j))$ and, because $h(k_j)$ communities with $N_j$ nodes each are needed, with $j=1, 2,
\ldots m$, the total number of nodes in the random network is given as $N=\sum_{j=1}^{m}h(k_j) N_j$.
Observe that, for a specified $h(k_j)$, the total number $N$ of nodes can be increased by reducing $\Delta a$.
Figure \[fig\_design\] illustrates the average $\pm$ standard deviation of log-log node degree distributions obtained for 50 realizations of a designed subnetwork assuming $\xi=-1.0$, $a=4$, $\Delta a = 0.5$ and $\left< k \right>=2$, implying $m=9$, $n=137$ and $N=275$. The obtained average curve falls reasonably close to the desired profile (dashed straight line). The average and standard deviation of the number of subnetworks with degree higher than zero were 122 and 3.93, respectively.
![The average $\pm$ standard deviation of 50 realizations of a design scale free subnetwork partition assuming $\xi=-1.0$, $a=4$, $\Delta a = 0.5$ and $\left< k \right>=2$. The dashed line corresponds to the originally desired distribution. \[fig\_design\]](fig_design.eps)
The 50 realizations of the scale free subnetwork partitioned models considered in the above example had their hierarchical topological features estimated as described in [@Costa_gener:2004; @Costa_hier_char:2004]. In order to do so, the subnetwork partitions were transformed into a traditional weighted complex network by applying the subsumption methodology described in Section \[sec:basic\].
Let $R_d(i)$ be the ring defined as the subnetwork including the nodes at minimal distance $d$, corresponding to the hierarchical level, from a reference node $i$ and the edges between such nodes. The considered measurements include the average (over all nodes) of: (i) the *hierarchical number of nodes*, i.e. the number of nodes in $R_d(i)$; (ii) the *hierarchical node degree*, defined as the number of edges between rings $R_d(i)$ and $R_{d}(i+1)$; (iii) the *intra ring degree*, i.e. the average degree among the elements of $R_d(i)$; (v) the *common degree*, namely the average of the traditional node degree considering the nodes in $R_d(i)$; and (vi) the *hierarchical clustering coefficient*, corresponding to the clustering coefficient of $R_d(i)$. Figure \[fig\_hier\] presents the average $\pm$ standard deviations of such measurements obtained for the above 50 simulations as well as for random and Barabási-Albert scale free models with the same number of nodes and average degree. It is clear from such results that the designed models have topological properties strikingly similar to those of the respective Barabási-Albert models, except for the hierarchical common degree, which resulted remarkably distinct, exhibiting a peak near at the higher hierarchical levels. Slightly higher values of clustering coefficient are also observed for the design models.
Scale Free by Dynamics
======================
The concepts and methods described in the previous sections can also be used to implement a dynamics of node exchange between the subnetworks in a partitioned system. Among the several possibilities, we investigate the scheme starting with a uniform subnetwork partition of a random network $\Gamma$ (i.e. each community $i$ initially has $N_i=N/n$ nodes) and involving sequential random selection of a subnetwork $c_i$, from which a node is randomly selected (uniform probability) and moved to (possibly) another subnetwork $c_j$ chosen with probability proportional to its respective degree $k(c_j)$. It is suggested that such a dynamical node exchange can be used to model several real-world phenomena such as the continuous exchange of individuals between institutions, e.g. music performers moving from an ensemble to another, animal species changing their environment, and so on. Figure \[fig\_dyn\] shows the log-log plot of the subnetwork degree distributions for three successive steps — i.e. $t=1$, $t=50$ and $t=185$ — along the node exchange iteractions. It is clearly perceived that the left-hand side of the log-log distribution tends to increase as the nodes are redistributed among the subnetworks.
![Three stages of the subnetwork degree evolution by using the suggested node exchange dynamics. \[fig\_dyn\]](fig_dyn.eps)
Concluding Remarks
==================
The concepts of subnetwork degree [@Costa_gener:2004] as well as the presently introduced notion of subnetwork partitions, have allowed interesting developments such as the design and evolution of scale free subnetworks. The hierarchical characterization of experimental results of a designed subnetwork partitioned model indicates that such networks present similar features to equivalent Barabási-Albert models, except for the hierarchical common degree, which tended to present a peak at higher hierarchical levels. Although we have concentrated attention on scale free degree distribution, the proposed concepts and methods can be immediately applied to many other situations including the design of *community* organized networks with generic degree distribution. Because for large values of $N$ the subnetwork degree can be well-approximated by the product between the number of nodes inside the subnetwork and the average degree of the underlying random network, the subnetwork degree distribution ultimately follows the distribution of the number of nodes in the subnetworks. As a consequence, geographical networks where nodes are uniformly distributed along the space and the subnetworks cover areas which follow a power law will result naturally scale free. Another issue deserving further attention is the dynamical redistribution of nodes among the subnetworks.
Luciano da F. Costa is grateful to FAPESP (proc. 99/12765-2), CNPq (proc. 308231/03-1) and the Human Frontier Science Program (RGP39/2002) for financial support.
|
---
abstract: 'In [@Ras-2-bridge], Rasmussen observed that the Khovanov-Rozansky homology of a link is a finitely generated module over the polynomial ring generated by the components of this link. In the current paper, we study the module structure of the middle HOMFLYPT homology, especially the Betti numbers of this module. For each link, these Betti numbers are supported on a finite subset of $\mathbb{Z}^4$. One can easily recover from these Betti numbers the Poincaré polynomial of the middle HOMFLYPT homology. We explain why the Betti numbers can be viewed as a generalization of the reduced HOMFLYPT homology of knots. As an application, we prove that the projective dimension of the middle HOMFLYPT homology is additive under split union of links and provides a new obstruction to split links.'
address: 'Department of Mathematics, The George Washington University, Phillips Hall, Room 739, 801 22nd Street NW, Washington DC 20052, USA. Telephone: 1-202-994-0653, Fax: 1-202-994-6760'
author:
- Hao Wu
title: Betti Numbers of the HOMFLYPT Homology
---
[^1]
Introduction {#sec-intro}
============
In [@Ras-2-bridge], Rasmussen observed that the ${\mathfrak{sl}}(N)$ homology of a link defined in [@KR1] is a finitely generated module over the polynomial ring generated by the components of this link. His observation applies to other versions of the Khovanov-Rozansky homology too. And it is not hard to see that the module structure of the Khovanov-Rozansky homology over this polynomial ring is a link invariant. In the current paper, we study the module structures of the middle HOMFLYPT homology $H$ defined in [@Ras2 Definition 2.9] and its reduction $H_r$ with respect to one component.[^2] Please see Section \[sec-HOMFLYPT-mod\] below for a brief review of $H$ and $H_r$, especially Lemmas \[lemma-HOMFLYPT-module\] and \[lemma-HOMFLYPT-module-inv\] for their module structures.
Let $B$ be a closed braid, and $K_1,\dots,K_m$ the components of $B$. To each $K_i$, we assign a homogeneous variable $X_i$ of degree $2$. Define graded rings $R_B:={\mathbb{Q}}[X_1,\dots,X_m]$ and $R_{B,r}:={\mathbb{Q}}[X_2-X_1,\dots,X_m-X_1]$.[^3] Let $H^{\star,j,k}(B) = \oplus_{i\in {\mathbb{Z}}} H^{i,j,k}(B)$ and $H_r^{\star,j,k}(B) = \oplus_{i\in {\mathbb{Z}}} H_r^{i,j,k}(B)$. According to Lemma \[lemma-HOMFLYPT-module-inv\], $H^{\star,j,k}(B)$ (resp. $H_r^{\star,j,k}(B)$) is a ${\mathbb{Z}}$-graded $R_B$-module (resp. $R_{B,r}$-module.) Let ${\mathfrak{m}}=(X_1,\dots,X_m)$ be the maximal homogeneous ideal of $R_B$, and ${\mathfrak{m}}_r=(X_2-X_1,\dots,X_m-X_1)$ be the maximal homogeneous ideal of $R_{B,r}$. Note that $R_B/{\mathfrak{m}}$ (resp. $R_{B,r}/{\mathfrak{m}}_r$) is also a ${\mathbb{Z}}$-graded $R_B$-module (resp. $R_{B,r}$-module.) So ${\mathrm{Tor}}^{R_B}_p(R_B/{\mathfrak{m}},H^{\star,j,k}(B))$ and ${\mathrm{Tor}}^{R_{B,r}}_p(R_{B,r}/{\mathfrak{m}}_r,H_r^{\star,j,k}(B))$ are both ${\mathbb{Z}}$-graded ${\mathbb{Q}}$-spaces.
\[def-Betti\] The Betti numbers of $H(B)$ and $H_r(B)$ are defined to be $$\begin{aligned}
\beta_B(p,q,j,k) & := & \dim_{\mathbb{Q}}{\mathrm{Tor}}^{R_B}_p(R_B/{\mathfrak{m}},H^{\star,j,k}(B))^q, \\
\beta_{B,r}(p,q,j,k) & := & \dim_{\mathbb{Q}}{\mathrm{Tor}}^{R_{B,r}}_p(R_{B,r}/{\mathfrak{m}}_r,H_r^{\star,j,k}(B))^q,\end{aligned}$$ where ${\mathrm{Tor}}^{R_B}_p(R_B/{\mathfrak{m}},H^{\star,j,k}(B))^q$ (resp. ${\mathrm{Tor}}^{R_{B,r}}_p(R_{B,r}/{\mathfrak{m}}_r,H_r^{\star,j,k}(B))^q$) is the homogeneous component of ${\mathrm{Tor}}^{R_B}_p(R_B/{\mathfrak{m}},H^{\star,j,k}(B))$ (resp. ${\mathrm{Tor}}^{R_{B,r}}_p(R_{B,r}/{\mathfrak{m}}_r,H_r^{\star,j,k}(B))$) of degree $q$.
Clearly, $\beta_B(p,q,j,k)$ and $\beta_{B,r}(p,q,j,k)$ are defined for $(p,q,j,k) \in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}^3$.
The main technical tool we use to study the Betti numbers are minimal free resolutions. We will review these in Section \[sec-minimal-res\] below. The following are some basic properties of the Betti numbers of the HOMFLYPT homology.
\[lemma-Betti-inv\]
1. For every $(p,q,j,k) \in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}^3$, $\beta_B(p,q,j,k)$ and $\beta_{B,r}(p,q,j,k)$ are invariant under Markov moves of $B$.
2. $\beta_B(p,q,j,k) = \beta_{B,r}(p,q,j,k) = 0$ for all but finitely many $(p,q,j,k) \in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}^3$.
It is a standard fact that one can recover from the Betti numbers the graded dimension of a graded module over a polynomial ring. Based on this, we can easily recover the Poincaré polynomials of $H(B)$ and $H_r(B)$ from the Betti numbers $\beta_B(p,q,j,k)$ and $\beta_{B,r}(p,q,j,k)$. Let us first normalize the binomial numbers by $$\label{eq-binomial}
{\genfrac{(}{)}{0pt}{}{n}{k}}= \begin{cases}
\frac{n!}{k!(n-k)!} &\text{if } 0\leq k \leq n, \\
1 & \text{if } n=-1 \text{ and } k=0, \\
0 &\text{otherwise.}
\end{cases}$$
\[def-Betti-polynomial\] $$\begin{aligned}
\mathcal{P}_B(x, y, a, b) & := & \sum_{(p,q,j,k) \in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}^3} \beta_B(p,q,j,k) \cdot x^p \cdot (\sum_{i\in {\mathbb{Z}}} y^{2i+q} \cdot {\genfrac{(}{)}{0pt}{}{i+m-1}{i}})\cdot a^j \cdot b^{\frac{k-j}{2}} \\
& = & \sum_{(p,q,j,k) \in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}^3} \beta_B(p,q,j,k) \cdot x^p \cdot \frac{y^q}{(1-y^2)^m}\cdot a^j \cdot b^{\frac{k-j}{2}}\end{aligned}$$ and $$\begin{aligned}
\mathcal{P}_{B,r}(x, y, a, b) & := & \sum_{(p,q,j,k) \in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}^3} \beta_{B,r}(p,q,j,k) \cdot x^p \cdot (\sum_{i\in {\mathbb{Z}}} y^{2i+q} \cdot {\genfrac{(}{)}{0pt}{}{i+m-2}{i}})\cdot a^j \cdot b^{\frac{k-j}{2}} \\
& = & \sum_{(p,q,j,k) \in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}^3} \beta_B(p,q,j,k) \cdot x^p \cdot \frac{y^q}{(1-y^2)^{m-1}}\cdot a^j \cdot b^{\frac{k-j}{2}},\end{aligned}$$ where $m$ is the number of components of the closed braid $B$.
\[lemma-Poincare-polynomials\] Polynomials $\mathcal{P}_B(x, y, a, b)$ and $\mathcal{P}_{B,r}(x, y, a, b)$ are invariant under Markov moves of $B$. Moreover, $$\begin{aligned}
\mathcal{P}_B(-1, y, a, b) & = & \sum_{(i,j,k) \in {\mathbb{Z}}^3} y^i \cdot a^j \cdot b^{\frac{k-j}{2}} \cdot \dim_{\mathbb{Q}}H^{i,j,k}(B), \\
\mathcal{P}_{B,r}(-1, y, a, b) & = & \sum_{(i,j,k) \in {\mathbb{Z}}^3} y^i \cdot a^j \cdot b^{\frac{k-j}{2}} \cdot \dim_{\mathbb{Q}}H_r^{i,j,k}(B).\end{aligned}$$ Consequently, $\mathcal{P}_B(-1, y, a, -1) = -\frac{P_B(a,y)}{y-y^{-1}}$ and $\mathcal{P}_{B,r}(-1, y, a, -1) = P_B(a,y)$, where $P_B$ is a normalization of the HOMFLYPT polynomial.
Note that, for any non-vanishing homogeneous element of the middle HOMFLYPT homology, its first and second ${\mathbb{Z}}$-gradings always have the opposite parity. By Lemma \[lemma-Poincare-polynomials\], for fixed $(j,k)\in {\mathbb{Z}}^2$, $$\label{eq-hilbert}
\dim_{\mathbb{Q}}H^{2T+1-j,j,k}(B)= \sum_{(p,q)\in {\mathbb{Z}}_{\geq0}\times {\mathbb{Z}}} (-1)^p\cdot\beta_B(p,q,j,k)\cdot{\genfrac{(}{)}{0pt}{}{T+m-\frac{j+q+1}{2}}{T-\frac{j+q-1}{2}}}.$$ Note that, when $T \geq \max\{\frac{j+q-1}{2}~|~\beta_B(p,q,j,k)\neq 0\}$, the right hand side of Equation is a polynomial of $T$. Thus, the Betti numbers determine when the Hilbert function of $H^{\star,j,k}(B)$ becomes its Hilbert polynomial. This was implicitly asked in [@Wu-hilbert Question 1.7].
It is another standard fact that one can recover from the Betti numbers the projective dimension of a graded module over a polynomial ring. For $H(B)$ and $H_r(B)$, we have the following lemma.
\[lemma-pd\]
1. ${\mathrm{pd}}_{R_B} H(B)=\deg_x \mathcal{P}_B(x, y, a, b)$, where ${\mathrm{pd}}_{R_B} H(B)$ is the projective dimension of $H(B)$ over $R_B$.
2. ${\mathrm{pd}}_{R_{B,r}} H_r(B)=\deg_x \mathcal{P}_{B,r}(x, y, a, b)$, where ${\mathrm{pd}}_{R_{B,r}} H_r(B)$ is the projective dimension of $H_r(B)$ over $R_{B,r}$.
Lemmas \[lemma-Betti-inv\], \[lemma-Poincare-polynomials\] and \[lemma-pd\] will be proved in Section \[sec-Betti\] below.
Our results start with the observation that the Betti numbers of the middle HOMFLYPT homology $H$ and its reduction $H_r$ are essentially the same.
\[thm-Betti\] $\beta_B(p,q,j,k) = \beta_{B,r}(p,q-1,j,k)$ for all $(p,q,j,k) \in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}^3$. In particular, if $B$ is a knot, then $\beta_B(0,q,j,k) = \dim_{\mathbb{Q}}H_r^{q-1,j,k}(B) =\dim_{\mathbb{Q}}\overline{H}^{q-1,j,k}(B)$ and $\beta_B(p,q,j,k)=0$ whenever $p>0$, where $\overline{H}$ is the reduced HOMFLYPT homology defined in [@Ras2].
Theorem \[thm-Betti\] will also be proved in Section \[sec-Betti\] below.
By Lemma \[lemma-Poincare-polynomials\], the Betti numbers of $H(B)$ determine the Poincaré polynomial of $H(B)$. By Theorem \[thm-Betti\], the Betti numbers of $H(B)$ further determine the Poincaré polynomial of $H_r(B)$. Comparing the definition of $H_r$ in Section \[sec-HOMFLYPT-mod\] below and that of the reduced HOMFLYPT homology $\overline{H}$ in [@Ras2], we know the Poincaré polynomial of $\overline{H}(B)$ is equal to that of $H_r(B)$ times $(1+a^{-2}b)^n$ for some $n \in \mathbb{N}_{\geq0}$. So, provided we know what $n$ is, the Betti numbers of $H(B)$ also determine the Poincaré polynomial of $\overline{H}(B)$.
Some researchers prefer to work with link homologies represented by a finite set of data. If $B$ is a knot, its reduced HOMFLYPT homology is finite dimensional, which is why these researchers prefer this version of the HOMFLYPT homology over others. Theorem \[thm-Betti\] shows that, for knots, the Betti numbers are the dimensions of homogeneous components of the reduced HOMFLYPT homology. If $B$ is a link with multiple components, then its reduced HOMFLYPT homology becomes infinite dimensional. However, its Betti numbers remain a finite set of data. In this sense, the Betti numbers may play the same role for links as that played by the reduced HOMFLYPT homology for knots.
It turns out that, up to a factor of $ab^{-1}$, the polynomial $\mathcal{P}_B(x, y, a, b)$ is multiplicative under split union of closed braids. So it follows from Lemma \[lemma-pd\] that the projective dimension of $H(B)$ is additive under split union of closed braids. This leads to a new obstruction to split links. First, let us recall the definition of the split union of braids.
\[def-split\] Denote by $\mathbf{B}_k$ the braid group on $k$ strands with standard generators $\sigma_1^{\pm1},\dots,\sigma_{k-1}^{\pm1}$. Let $B_1$ and $B_2$ be closed braids with braid words $w_1=\sigma_{i_1}^{\mu_1}\cdots \sigma_{i_{l_1}}^{\mu_{l_1}} \in \mathbf{B}_{k_1}$ and $w_2=\sigma_{j_1}^{\nu_1}\cdots \sigma_{j_{l_2}}^{\nu_{l_2}} \in \mathbf{B}_{k_2}$, respectively. The split union $B_1 \sqcup B_2$ of $B_1$ and $B_2$ is the closed braid with the braid word $\sigma_{i_1}^{\mu_1}\cdots \sigma_{i_{l_1}}^{\mu_{l_1}}\sigma_{j_1+k_1}^{\nu_1}\cdots \sigma_{j_{l_2}+k_1}^{\nu_{l_2}} \in \mathbf{B}_{k_1+k_2}$.
Clearly, the operation of split union is associative. And it is commutative up to Markov moves.
A closed braid $B$ is $n$-split if and only if there exist $n$ closed braids $B_1,\dots,B_n$ such that $B=B_1\sqcup\cdots\sqcup B_n$.
A link is $n$-split if and only if it is equivalent to an $n$-split closed braid.
One can see that every link is $1$-split. And a link is $2$-split if and only if it is split in the classical sense.
Now we can state our results on split links.
\[thm-split\]
1. For any two closed braids $B_1$ and $B_2$, $$\mathcal{P}_{B_1 \sqcup B_2}(x, y, a, b)= a\cdot b^{-1} \cdot \mathcal{P}_{B_1}(x, y, a, b) \cdot \mathcal{P}_{B_2}(x, y, a, b).$$ Consequently, ${\mathrm{pd}}_{R_{B_1 \sqcup B_2}} H(B_1 \sqcup B_2) = {\mathrm{pd}}_{R_{B_1}} H(B_1)+{\mathrm{pd}}_{R_{B_2}} H(B_2)$.
2. If $B$ is a closed braid diagram of an $m$-component $n$-split link, then ${\mathrm{pd}}_{R_B} H(B) \leq m-n$.
Theorem \[thm-split\] will be proved in Section \[sec-split\] below.
The distant from a link to being split is usually measure by the splitting number, that is, the minimal number of crossing changes needed to make the link split. Consider the $2$-strand closed braid $B_n$ with the braid word $\sigma_1^{2n}\in \mathbf{B}_2$. The splitting number of $B_n$ is $n$. But, by Theorem \[thm-split\], $0 \leq {\mathrm{pd}}_{R_{B_n}} H(B_n) \leq 1$. This example shows that ${\mathrm{pd}}_{R_B} H(B)$ is not a good indicator of how far a link is from being split. See [@Batson-Seed] for lower bounds of the splitting number from the Khovanov homology.
On the other hand, ${\mathrm{pd}}_{R_B} H(B)$ may turn out to be a good indicator of how many times we can split a link. Theorem \[thm-split\] seems to suggest that, the more times we can split a link, the smaller the projective dimension of its HOMFLYPT homology gets in comparison to the number of components of this link.
\[conj-split\] An $m$-component closed braid $B$ represents an $n$-split link if and only if ${\mathrm{pd}}_{R_B} H(B) \leq m-n$.
\[eg-hopf\] Consider the positive Hopf link $B_1$ with braid word $\sigma_1^{2}\in \mathbf{B}_2$. In Section \[sec-hopf\] below, we will prove that ${\mathrm{pd}}_{R_{B_1}} H(B_1) = 1$. By Theorem \[thm-split\], this confirms the well known fact that the Hopf link does not split.
For a closed braid $B$, its ${\mathfrak{sl}}(N)$ homology $H_N(B)$ is also a module over $R_B$. But the projective dimension of $H_N(B)$ over $R_B$ is far less interesting.
\[lemma-sl-N-proj-dim\] Let $B$ be a closed braid of $m$ components. Then, for any $N\geq 1$, ${\mathrm{pd}}_{R_B} H_N(B) =m$.
Lemma \[lemma-sl-N-proj-dim\] will be proved in Section \[sec-sl-N\] below.
Note that the ${\mathfrak{sl}}(N)$ homology $H_N(B)$ is not just a module over $R_B$. For each component $K_j$, the monomial $X_j^N$ acts on $H_N(B)$ as $0$. So $H_N(B)$ is actually a module over the quotient ring $R_{B,N}:= {\mathbb{Q}}[X_1,\dots,X_m]/(X_1^N,\dots,X_m^N)$. Since $R_{B,N}$ is a local ring, techniques based on minimal free resolutions should still work. It would be interesting to see what topological information the Betti numbers of $H_N(B)$ over $R_{B,N}$ contain.
Module Structure of the HOMFLYPT Homology {#sec-HOMFLYPT-mod}
=========================================
In this section, we briefly review the middle HOMFLYPT homology $H$ defined in [@Ras2] and its reduction $H_r$. For more details, see [@Ras2].
Base rings of chain complexes
-----------------------------
For a closed braid, an edge of it is a part of the closed braid that starts and ends at crossings, but contains no crossings in its interior. In the rest of this section, we fix a closed braid $B$ with $m$ components $K_1,\dots,K_m$. We order the edges of $B$ as $1^{st},2^{nd},\dots,M^{th}$ so that the $l^{th}$ edge is on the component $K_l$ for $1 \leq l \leq m$. For $1 \leq l \leq M$, we assign a variable $X_l$ of degree $2$ to the $l^{th}$ edge. For a crossing $c$ of $B$, assume the $k^{th}$ and $l^{th}$ edges are pointing out of $c$, and the $i^{th}$ and $j^{th}$ edges are pointing into $c$. Then $c$ defines a relation $\rho(c)=X_k+X_l-X_i-X_j$. The edge ring of $B$ is the ring $R(B):={\mathbb{Q}}[X_1,\dots,X_M]/(\rho(c_1),\dots,\rho(c_n))$, where $c_1,\dots,c_n$ are all the crossings of $B$. The reduced edge ring of $B$ is the ring $R_r(B):={\mathbb{Q}}[X_2-X_1,\dots,X_M-X_1]/(\rho(c_1),\dots,\rho(c_n))$. One can see that $R_r(B)$ is a subring of $R(B)$. Moreover, $$\label{eq-ring-tensor}
R(B) = R_r(B) \otimes_{\mathbb{Q}}{\mathbb{Q}}[X_1].$$ Note that $R(B)$ and $R_r(B)$ are not the rings $R_B$ and $R_{B,r}$ defined in the introduction.
HOMFLYPT homologies
-------------------
As defined in [@Ras2], the middle complex $(C_0(B), d_+, d_v)$ is a ${\mathbb{Z}}^3$-graded double cochain complex[^4] of finitely generated graded free $R(B)$-modules with homogeneous differential maps. Its first grading is the grading of the underlying $R(B)$-module. Its second and third gradings are the horizontal and vertical gradings of the double complex. These two gradings are both bounded. The reduced complex $(C_r(B), d_+, d_v)$ is defined by replacing each summand of $R(B)$ in $C_0(B)$ by a summand of $R_r(B)$. Clearly, $(C_r(B), d_+, d_v)$ is a ${\mathbb{Z}}^3$-graded double cochain complex of finitely generated graded free $R_r(B)$-module with homogeneous differential maps. By [@Ras2 Lemma 2.12], we know that $$\begin{aligned}
\label{eq-complex-quotient} C_r(B) & \cong & C_0(B) / X_1 C_0(B), \\
\label{eq-complex-tensor} C_0(B) & \cong & C_r(B) \otimes_{\mathbb{Q}}{\mathbb{Q}}[X_1].\end{aligned}$$ Note that $C_0(B)$ (resp. $C_r(B)$) is a finitely generated module over $R(B)$ (resp. $R_r(B)$.)
The middle HOMFLYPT homology $H$ and its reduction $H_r$ are defined by $$\begin{aligned}
\label{eq-H-def} H(B) & := & H(H(C_0(B), d_+), d_v)\{-w+b,w+b-1,w-b+1\},\\
\label{eq-Hbar-def} H_r(B) & := & H(H(C_r(B), d_+), d_v)\{-w+b-1,w+b-1,w-b+1\},\end{aligned}$$ where $w$ is the writhe of $B$, $b$ is the number of strands in $B$, and “$\{s, t, u\}$" means shifting the ${\mathbb{Z}}^3$-grading by the vector $(s, t, u)$. Note here that the definition of $H_r(B)$ is different from that of $\overline{H}(B)$ in [@Ras2]. The difference occurs when the closed braid $B$ splits. See [@Ras2 Section 2.10].
It is proved in [@KR2] that $H(B)$ and $H_r(B)$ are invariant as ${\mathbb{Z}}^3$-graded ${\mathbb{Q}}$-spaces under Markov moves of $B$.
One of the main advantages of the middle HOMFLYPT homology over the other normalizations of the HOMFLYPT homology is that, up to a grading shift, it is tensorial over ${\mathbb{Q}}$ under the split union. More precisely, let $B_1$ and $B_2$ be two closed braids. Then $$\begin{aligned}
\label{eq-C-tensor} C_0(B_1\sqcup B_2) & \cong & C_0(B_1) \otimes_{\mathbb{Q}}C_0(B_2), \\
\label{eq-H-tensor} H(B_1\sqcup B_2) & \cong & H(B_1) \otimes_{\mathbb{Q}}H(B_2)\{0,1,-1\},\end{aligned}$$ where, of course, the isomorphisms preserve the ${\mathbb{Z}}^3$-grading.
Module structures of the HOMFLYPT homologies
--------------------------------------------
Since $R(B)$ (resp. $R_r(B)$) is Noetherian, $H(B)$ (resp. $H_r(B)$) is a finitely generated module over $R(B)$ (resp. $R_r(B)$.) But there is no chance for these module structures to be invariant under Markov moves. This is simply because $R(B)$ and $R_r(B)$ change under Markov moves. But, in [@Ras-2-bridge Lemma 3.4], Rasmussen observed that, if $X_k$ and $X_l$ are assigned to edges on the same component of $B$, then their actions on $H(B)$ are the same.[^5] So $H(B)$ is a finitely generated module over the quotient ring $$R(B)/(\{X_k-X_l~|~\text{the } k^{th} \text{ and } l^{th} \text{ edges are on the same component of }B\}) \cong R_B = {\mathbb{Q}}[X_1,\dots,X_m].$$ Similar conclusion holds for $H_r(B)$. We have the following lemma.
[@Ras-2-bridge Lemma 3.4]\[lemma-HOMFLYPT-module\] Let $B$ be a closed braid diagram, and $K_1,\dots,K_m$ be the components of $B$. To each $K_i$, we assign a variable $X_i$ with degree $2$. Then:
- $H(B)$ is a finitely generated ${\mathbb{Z}}^3$-graded module over the ${\mathbb{Z}}$-graded ring $R_B:={\mathbb{Q}}[X_1,\dots,X_m]$, where the action of any homogeneous element of $R_B$ on $H(B)$ fixes the last two ${\mathbb{Z}}$-gradings of $H(B)$, but shifts the first by its own degree.
- $H_r(B)$ is a finitely generated ${\mathbb{Z}}^3$-graded module over the ${\mathbb{Z}}$-graded ring $R_{B,r}:={{\mathbb{Q}}[X_2-X_1,\dots,X_m-X_1]}$, where the action of any homogeneous element of $R_{B,r}$ on $H_r(B)$ fixes the last two ${\mathbb{Z}}$-gradings of $H_r(B)$, but shifts the first by its own degree.
The proof of this lemma is a straightforward adaptation of the proof of [@Ras-2-bridge Lemma 3.4]. We leave the details to the reader.
\[remark-quotient\] Applying the Universal Coefficient Theorem over ${\mathbb{Q}}$ to the right hand side of , we get that $H(B) \cong H_r(B) \otimes_{\mathbb{Q}}{\mathbb{Q}}[X_1]\{1,0,0\}$ as ${\mathbb{Z}}^3$-graded $R_B$-modules. Note that $R_{B,r}\cong R_B/(X_1)$ with the isomorphism given by $R_{B,r} \hookrightarrow R_B \twoheadrightarrow R_B/(X_1)$, where “$\hookrightarrow$" is the standard inclusion, and “$\twoheadrightarrow$" is the standard quotient map. Identify $R_{B,r}$ and $R_B/(X_1)$ via this isomorphism. Then $H_r(B) \cong H(B)/X_1H(B)\{-1,0,0\}$ as ${\mathbb{Z}}^3$-graded $R_{B,r}$-modules.
Next, we show that the module structures of $H(B)$ and $H_r(B)$ in Lemma \[lemma-HOMFLYPT-module\] are invariant under Markov moves.
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\[lemma-HOMFLYPT-module-inv\] Assume that $B'$ is another closed braid diagram of the same link. Fix a sequence of Markov moves that changes $B$ to $B'$. Denote by $K_i'$ the component of $B'$ that is identified to $K_i$ through this sequence of Markov moves. To each $K_i'$, we assign a variable $X_i'$ of degree $2$. Set $R_{B'}:={\mathbb{Q}}[X_1',\dots,X_m']$ and $R_{B',r}:={\mathbb{Q}}[X_2'-X_1',\dots,X_m'-X_1']$. We identify the rings $R_B$ with $R_{B'}$ (resp. $R_{B,r}$ with $R_{B',r}$) via the equations $X_i=X_i'$ (resp. $X_i-X_1=X_i'-X_1'$.) Then this sequence of Markov moves induces:
- an isomorphism $H(B)\cong H(B')$ of ${\mathbb{Z}}^3$-graded $R_B$-modules,
- an isomorphism $H_r(B)\cong H_r(B')$ of ${\mathbb{Z}}^3$-graded $R_{B,r}$-modules.
We only need to prove this lemma in the case when $B$ and $B'$ differ by a single braid-like Reidemeister move. For each braid-like Reidemeister move, Khovanov and Rozansky constructed in [@KR2] a ${\mathbb{Q}}$-linear isomorphism of $H(B)$ and $H(B')$ preserving the ${\mathbb{Z}}^3$-grading. This isomorphism commutes with the actions of the variables assigned to edges that are *not entirely with in* the part of $B$ and $B'$ changed by the braid-like Reidemeister move, that is, *not entirely with in* one of the dashed boxes in Figure \[fig-Reidemeister-boxes\]. Note that every component of $B$ and $B'$ contains an edge not entirely with in this dashed box. With out loss of generality, we choose the variables assigned to each pair of corresponding components of $B$ and $B'$ to be the variables assigned to a pair of corresponding edges on these components that are not entirely with in this dashed box. Then Khovanov and Rozansky’s isomorphism commutes with the variables assigned to all components of $B$ and $B'$. Thus, this isomorphism is an isomorphism of $R_B$-modules. This proves $H(B)\cong H(B')$ as ${\mathbb{Z}}^3$-graded $R_B$-modules. $H_r(B)\cong H_r(B')$ follows from $H(B)\cong H(B')$ and Remark \[remark-quotient\].
Note that we did not claim the naturality of the isomorphisms in Lemma \[lemma-HOMFLYPT-module-inv\]. We do not need the naturality for our results.
Minimal Free Resolutions {#sec-minimal-res}
========================
Betti numbers of a module are often understood through the minimal free resolution of this module. In this section, we review basics of minimal free resolutions of graded modules over a polynomial ring. For more details, see for example [@Eisenbud-book-2 Chapter 1].
Let $R={\mathbb{Q}}[X_1,\dots,X_m]$ be a polynomial ring graded by $\deg X_j =2$ for all $j=1,\dots,m$. The maximal homogeneous ideal of $R$ is ${\mathfrak{m}}=(X_1,\dots,X_m)$.
\[thm-syzygy\] Assume that $M$ is a finitely generated graded $R$-module. Then there is a graded free resolution $$0 \rightarrow F_l \rightarrow F_{l-1} \rightarrow \cdots \rightarrow F_1 \rightarrow F_0$$ of $M$ over $R$, in which each $F_j$ is finitely generated over $R$, each arrow preserves the module grading, and $l \leq m$.
For a detailed elementary proof of Hilbert’s Syzygy Theorem, see for example [@Arrondo-notes Theorem 4.3].
\[def-minimal-res\] A chain complex of graded $R$-modules $\cdots \rightarrow C_p \xrightarrow{d_p} C_{p-1} \rightarrow \cdots$ is called minimal if $d_P(C_p) \subset {\mathfrak{m}}C_{p-1}$ for each $p$.
A graded free resolution of a graded $R$-module is called a minimal free resolution if it is also a minimal chain complex of graded $R$-modules.
[@Eisenbud-book-2 Theorem 1.6]\[thm-minimal-res\] If $M$ is a finitely generated graded $R$-module, then any finitely generated graded free resolution of $M$ over $R$ contains a minimal free resolution of $M$ as a direct summand. Moreover, any two minimal free resolutions of $M$ over $R$ are isomorphic as chain complexes of graded $R$-modules via an isomorphism that induces the identity map on $M$.
Clearly, Theorem \[thm-syzygy\] and the first half of Theorem \[thm-minimal-res\] guarantee the existence of the minimal free resolution of any graded $R$-module. The second half of Theorem \[thm-minimal-res\] gives the uniqueness of the the minimal free resolution.
The proof of the existence part of Theorem \[thm-minimal-res\] is quite elementary. Say, $0 \rightarrow F_l \rightarrow F_{l-1} \rightarrow \cdots \rightarrow F_1 \rightarrow F_0$ is a finitely generated graded free resolution of $M$ over $R$. Fix a homogeneous $R$-basis for each $F_p$. Then each map $F_p \rightarrow F_{p-1}$ is given by a matrix whose entries are all homogeneous elements of $R$. Clearly, this resolution is minimal if and only if, for every $1\leq p \leq l$, all entries of this matrix are in ${\mathfrak{m}}$. If this is not true, then, for some $p$, the matrix contains non-zero scalar $c$. Using this entry $c$, one can perform a change of bases for $F_p$ and $F_{p-1}$ to show that the original resolution has a direct summand of the form $0\rightarrow R \xrightarrow{c} R \rightarrow 0$. Removing this direct summand, we get a new “smaller" graded free resolution of $M$. Repeat this process till there are no more non-zero scalars in the matrices representing the boundary maps. Then we get a minimal free resolution of $M$ that is a direct summand of the original graded free resolution of $M$.
The proof of the uniqueness part of Theorem \[thm-minimal-res\] requires some basic knowledge of homological algebra. It can be found in for example [@Eisenbud-book-1 Theorem 20.2]. Note that the proof in [@Eisenbud-book-1] is for modules over local rings. But, with minor modifications, this proof also works for graded modules over polynomial rings. We do not actually use the uniqueness of the minimal free resolution in this paper.
\[def-Betti-general\] For a finitely generated graded $R$-module $M$, its $(p,q)^{th}$ Betti number is $\beta_M(p,q) := \dim_{\mathbb{Q}}{\mathrm{Tor}}^R_p(R/{\mathfrak{m}},M)^q$, where ${\mathrm{Tor}}^R_p(R/{\mathfrak{m}},M)^q$ is the homogeneous component of ${\mathrm{Tor}}^R_p(R/{\mathfrak{m}},M)$ of degree $q$.
The following lemma describes the relations between the minimal free resolution, Betti numbers and the projective dimension.
[@Eisenbud-book-2 Proposition 1.7]\[lemma-mfs-Betti\] Let $M$ be a finitely generated graded $R$-module, and $0 \rightarrow F_l \rightarrow F_{l-1} \rightarrow \cdots \rightarrow F_1 \rightarrow F_0$ a minimal free resolution of $M$ over $R$. Then, for every $p\geq 0$, as graded ${\mathbb{Q}}$-spaces, $$\label{eq-iso-Tor}
(R/{\mathfrak{m}})\otimes_R F_p \cong {\mathrm{Tor}}^R_p(R/{\mathfrak{m}},M).$$
Consequently,
- any homogeneous $R$-basis for $F_p$ contains exactly $\beta_M(p,q)$ elements of degree $q$,
- the projective dimension of $M$ over $R$ is ${\mathrm{pd}}_R M = \max\{p~|~\beta_M(p,q)\neq 0 \text{ for some } q \in {\mathbb{Z}}.\}$
Recall that ${\mathrm{Tor}}^R_p(R/{\mathfrak{m}},M)$ is the $p^{th}$ homology of the chain complex $$\label{eq-minimal-0-arrows}
0 \rightarrow (R/{\mathfrak{m}})\otimes_R F_l \rightarrow (R/{\mathfrak{m}})\otimes_R F_{l-1} \rightarrow \cdots \rightarrow (R/{\mathfrak{m}})\otimes_R F_1 \rightarrow (R/{\mathfrak{m}})\otimes_R F_0\rightarrow 0.$$ The free resolution of $M$ being minimal implies that all arrows in the chain complex are zero maps. This proves isomorphism .
The number of elements of degree $q$ in any homogeneous $R$-basis of $F_p$ is equal to the dimension over ${\mathbb{Q}}$ of the homogeneous component of $(R/{\mathfrak{m}})\otimes_R F_p$ of degree $q$, which, according to isomorphism , is equal to $\dim_{\mathbb{Q}}{\mathrm{Tor}}^R_p(R/{\mathfrak{m}},M)^q = \beta_M(p,q)$.
The projective dimension of $M$ satisfies the inequality $$\max\{p~|~{\mathrm{Tor}}^R_p(R/{\mathfrak{m}},M)\neq 0\} \leq {\mathrm{pd}}_R M \leq \max\{p~|~F_p\neq 0.\}$$ But, by isomorphism , $$\max\{p~|~F_p\neq 0\} = \max\{p~|~{\mathrm{Tor}}^R_p(R/{\mathfrak{m}},M)\neq 0\} = \max\{p~|~\beta_M(p,q)\neq 0 \text{ for some } q \in {\mathbb{Z}}.\}$$ So ${\mathrm{pd}}_R M = \max\{p~|~\beta_M(p,q)\neq 0 \text{ for some } q \in {\mathbb{Z}}.\}$
The following lemma explains how to recover the graded dimension of a graded $R$-module using its Betti numbers.
\[lemma-Poincare-polynomial-general\] Let $M$ be a finitely generated graded $R$-module, and $0 \rightarrow F_l \rightarrow F_{l-1} \rightarrow \cdots \rightarrow F_1 \rightarrow F_0$ a minimal free resolution of $M$ over $R$. Denote by $M^i$ the homogeneous component of $M$ of degree $i$, and by $F_p^i$ the homogeneous component of $F_p$ of degree $i$. Then, for $p\geq 0$, $$\label{eq-gdim-Fp}
\sum_{i \in {\mathbb{Z}}} y^i \cdot \dim_{\mathbb{Q}}F_p^i = \sum_{q\in {\mathbb{Z}}}\beta_M(p,q)\cdot (\sum_{i \in {\mathbb{Z}}} y^{2i+q} {\genfrac{(}{)}{0pt}{}{i+m-1}{i}}).$$ Consequently, $$\label{eq-gdim-M}
\sum_{i \in {\mathbb{Z}}} y^i \cdot \dim_{\mathbb{Q}}M^i = \sum_{(p,q)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}}(-1)^p\cdot \beta_M(p,q)\cdot (\sum_{i \in {\mathbb{Z}}} y^{2i+q} {\genfrac{(}{)}{0pt}{}{i+m-1}{i}}).$$
By Lemma \[lemma-mfs-Betti\], $F_p \cong \bigoplus_{q\in {\mathbb{Z}}} R\{q\}^{\oplus \beta_M(p,q)},$ where $R\{q\}$ is $R$ with grading raised by $q$. That is, the scalar $1$ in $R\{q\}$ has grading $q$. Note that $R\{q\}$ has graded dimension $\sum_{i \in {\mathbb{Z}}} y^{2i+q} {\genfrac{(}{)}{0pt}{}{i+m-1}{i}}$. (Here, recall that each $X_j$ is of degree $2$.) This implies equation . But the graded dimension of $M$ is the alternating sum of the graded dimensions of $F_p$’s. Thus, we have equation .
Betti Numbers {#sec-Betti}
=============
In this section, we prove Lemmas \[lemma-Betti-inv\], \[lemma-Poincare-polynomials\], \[lemma-pd\] and Theorem \[thm-Betti\].
For Lemma \[lemma-Betti-inv\], the invariance of the Betti numbers follows from Lemma \[lemma-HOMFLYPT-module-inv\]. Since $H(B)$ (resp. $H_r(B)$) is finitely generated over $R_B$ (resp. $R_{B,r}$,) $\beta_B(p,q,j,k)$ (resp. $\beta_{B,r}(p,q,j,k)$) is non-zero for only finitely many $(p,q,j,k) \in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}^3$.
For Lemma \[lemma-Poincare-polynomials\], polynomials $\mathcal{P}_B(x, y, a, b)$ and $\mathcal{P}_{B,r}(x, y, a, b)$ are invariant under Markov moves because their coefficients are invariant under Markov moves. Equations in this lemma follows from Lemma \[lemma-Poincare-polynomial-general\].
Lemma \[lemma-pd\] follows from Lemma \[lemma-mfs-Betti\].
It remains to prove Theorem \[thm-Betti\]. To do this, we use the following graded version of [@Rotman-book Theorem 10.59], which is a special case of the Grothendieck Spectral Sequence [@Rotman-book Theorem 10.48].
\[thm-SS-base-ring\] Assume that:
- $R$ and $S$ are graded Noetherian ${\mathbb{Q}}$-algebras;
- $A$ is a finitely generated graded right $R$-module;
- - $B$ is a left $R$-module and a right $S$-module,
- $B$ has a grading that makes it a graded left $R$-module and a graded right $S$-module;
- ${\mathrm{Tor}}^R_i(A,B\otimes_S P) \cong 0$ for all $i\geq 1$ whenever $P$ is a projective left $S$-module.
Then, for every finitely generated graded left $S$-module $C$, there is a first quadrant spectral sequence $\{E^r_{p,q}\}$ of graded ${\mathbb{Q}}$-spaces with $E^2_{p,q} \cong {\mathrm{Tor}}^R_p(A, {\mathrm{Tor}}^S_q(B,C))$ that converges to ${\mathrm{Tor}}^S_{\ast}(A\otimes_R B, C)$.
For the convenience of the reader, we include a proof of Theorem \[thm-SS-base-ring\]. For this purpose, we need the following well known lemma.
\[lemma-universal-projective\] Let $R$ be a graded ${\mathbb{Q}}$-algebra, and $Q$ a graded flat right $R$-module. Given any chain complex $(C_\ast,d)$ of graded left $R$-modules, we have $Q \otimes_R H_n(C_\ast) \cong H_n(Q \otimes_R C_\ast)$ as graded ${\mathbb{Q}}$-spaces for each $n$, where the isomorphism is the ${\mathbb{Q}}$-linear map given by $x\otimes[c] \mapsto [x\otimes c]$ for $x \in Q$ and $c \in \ker d_{n}$.
First consider the short exact sequence $0 \rightarrow {\mathrm{Im}}d_{n+1} \xrightarrow{\jmath_{n}} \ker d_n \xrightarrow{\pi_n} H_n(C\ast) \rightarrow 0$ of graded left $R$-modules, where $\pi_n$ is the standard quotient map, and $\jmath_{n}$ is the standard inclusion. Since $Q$ is flat, we get a short exact sequence $$0 \rightarrow Q \otimes_R {\mathrm{Im}}d_{n+1} \xrightarrow{{\mathrm{id}}_Q\otimes \jmath_{n}} Q\otimes_R\ker d_n \xrightarrow{{\mathrm{id}}_Q\otimes\pi_n} Q \otimes_R H_n(C\ast) \rightarrow 0$$ of graded ${\mathbb{Q}}$-spaces. So we have $(Q\otimes_R\ker d_n)/({\mathrm{id}}_Q\otimes \jmath_{n})(Q \otimes_R {\mathrm{Im}}d_{n+1}) \cong Q \otimes_R H_n(C\ast)$ as graded ${\mathbb{Q}}$-spaces, where the isomorphism is given by $x \otimes c + ({\mathrm{id}}_Q\otimes \jmath_{n})(Q \otimes_R {\mathrm{Im}}d_{n+1}) \mapsto x\otimes [c]$ for $x \in Q$ and $c \in \ker d_{n}$.
Now consider the short exact sequence $0 \rightarrow \ker d_n \xrightarrow{\iota_n} C_n \xrightarrow{d_n} {\mathrm{Im}}d_n \rightarrow 0$ of graded left $R$-modules, where $\iota_n$ is the standard inclusion. Since $Q$ is flat, this gives us a short exact sequence $$0 \rightarrow Q\otimes_R\ker d_n \xrightarrow{{\mathrm{id}}_Q\otimes\iota_n} Q\otimes_R C_n \xrightarrow{{\mathrm{id}}_Q\otimes d_n} Q\otimes_R {\mathrm{Im}}d_n \rightarrow 0$$ of graded ${\mathbb{Q}}$-spaces, which induces a long exact sequence $$\cdots \rightarrow Q \otimes_R {\mathrm{Im}}d_{n+1} \xrightarrow{\Delta_{n+1}} Q\otimes_R\ker d_n \xrightarrow{({\mathrm{id}}_Q\otimes\iota)_\ast} H_n(Q\otimes_R C_\ast) \rightarrow Q \otimes_R {\mathrm{Im}}d_{n} \xrightarrow{\Delta_{n}} \cdots$$ of graded ${\mathbb{Q}}$-spaces. A simple diagram chase gives that the connecting homomorphism is $\Delta_n={\mathrm{id}}_Q \otimes \jmath_{n-1}$, which, as shown above, is injective. Thus, we get a short exact sequence $$0 \rightarrow Q \otimes_R {\mathrm{Im}}d_{n+1} \xrightarrow{{\mathrm{id}}_Q\otimes \jmath_{n}} Q\otimes_R\ker d_n \xrightarrow{({\mathrm{id}}_Q\otimes\iota)_\ast} H_n(Q\otimes_R C_\ast) \rightarrow 0$$ of graded ${\mathbb{Q}}$-spaces. So we have $(Q\otimes_R\ker d_n)/({\mathrm{id}}_Q\otimes \jmath_{n})(Q \otimes_R {\mathrm{Im}}d_{n+1}) \cong H_n(Q\otimes_R C_\ast)$ as graded ${\mathbb{Q}}$-spaces, where the isomorphism is given by $x \otimes c + ({\mathrm{id}}_Q\otimes \jmath_{n})(Q \otimes_R {\mathrm{Im}}d_{n+1}) \mapsto [x\otimes c]$ for $x \in Q$ and $c \in \ker d_{n}$.
Thus, $Q \otimes_R H_n(C_\ast) \cong H_n(Q \otimes_R C_\ast)$ as graded ${\mathbb{Q}}$-spaces, where the isomorphism is the ${\mathbb{Q}}$-linear map given by $x\otimes[c] \mapsto [x\otimes c]$ for $x \in Q$ and $c \in \ker d_{n}$.
Now we are ready to prove Theorem \[thm-SS-base-ring\].
Let $\cdots\rightarrow Q_1 \rightarrow Q_0$ be a graded projective resolution of the right $R$-module $A$, and $\cdots\rightarrow P_1 \rightarrow P_0$ be a graded projective resolution of the left $S$-module $C$. Consider the first quadrant double complex $$\label{eq-double-complex}
\xymatrix{
\cdots \ar[d]& \cdots \ar[d] & \cdots \ar[d] & \\
Q_0 \otimes_R B \otimes_S P_2 \ar[d] & Q_1 \otimes_R B \otimes_S P_2 \ar[d] \ar[l] & Q_2 \otimes_R B \otimes_S P_2 \ar[d]\ar[l] & \cdots \ar[l] \\
Q_0 \otimes_R B \otimes_S P_1 \ar[d] & Q_1 \otimes_R B \otimes_S P_1 \ar[d] \ar[l] & Q_2 \otimes_R B \otimes_S P_1 \ar[d] \ar[l]& \cdots \ar[l] \\
Q_0 \otimes_R B \otimes_S P_0 & Q_1 \otimes_R B \otimes_S P_0 \ar[l] & Q_2 \otimes_R B \otimes_S P_0 \ar[l] & \cdots \ar[l]
}$$ Denote by $\{\hat{E}^r\}$ the spectral sequence of double complex induced by its row filtration and by $\{E^r\}$ the spectral sequence of double complex induced by its column filtration. Both of these are spectral sequences of graded ${\mathbb{Q}}$-spaces converging to the homology of the total complex of double complex .
First we consider the spectral sequence $\{\hat{E}^r\}$. Note that $$\hat{E}^0=\xymatrix{
\cdots & \cdots & \cdots & \\
Q_0 \otimes_R (B \otimes_S P_2) & Q_1 \otimes_R (B \otimes_S P_2) \ar[l] & Q_2 \otimes_R (B \otimes_S P_2) \ar[l] & \cdots \ar[l] \\
Q_0 \otimes_R (B \otimes_S P_1) & Q_1 \otimes_R (B \otimes_S P_1) \ar[l] & Q_2 \otimes_R (B \otimes_S P_1) \ar[l]& \cdots \ar[l] \\
Q_0 \otimes_R (B \otimes_S P_0) & Q_1 \otimes_R (B \otimes_S P_0) \ar[l] & Q_2 \otimes_R (B \otimes_S P_0) \ar[l] & \cdots \ar[l]
}$$ So $$\hat{E}^1=\xymatrix{
\cdots \ar[d]& \cdots \ar[d] & \cdots \ar[d] & \\
A \otimes_R (B \otimes_S P_2) \ar[d] & {\mathrm{Tor}}^R_1(A, B \otimes_S P_2) \ar[d] & {\mathrm{Tor}}^R_2(A, B \otimes_S P_2) \ar[d] & \cdots \\
A \otimes_R (B \otimes_S P_1) \ar[d]& {\mathrm{Tor}}^R_1(A, B \otimes_S P_1) \ar[d] & {\mathrm{Tor}}^R_2(A, B \otimes_S P_1) \ar[d]& \cdots \\
A \otimes_R (B \otimes_S P_0) & {\mathrm{Tor}}^R_1(A, B \otimes_S P_0) & {\mathrm{Tor}}^R_2(A, B \otimes_S P_0) & \cdots
}$$ By assumption, all but the left most column in $\hat{E}_1$ vanish. Also note that $A \otimes_R (B \otimes_S P_j) \cong (A \otimes_R B) \otimes_S P_j$. So $$\hat{E}^2_{p,q} \cong \begin{cases}
{\mathrm{Tor}}^S_q (A \otimes_R B, C) & \text{if } p=0,\\
0 & \text{if } p>0.
\end{cases}$$ Thus, $\{\hat{E}^r\}$ collapses at its $E^2$-page. This implies that, as graded ${\mathbb{Q}}$-space, the $n^{th}$ homology of the total complex of double complex is isomorphic to ${\mathrm{Tor}}^S_n (A \otimes_R B, C)$.
Now consider the spectral sequence $\{E^r\}$. Note that $$E^0=\xymatrix{
\cdots \ar[d]& \cdots \ar[d]& \cdots \ar[d]& \\
Q_0 \otimes_R (B \otimes_S P_2) \ar[d]& Q_1 \otimes_R (B \otimes_S P_2) \ar[d]& Q_2 \otimes_R (B \otimes_S P_2) \ar[d]& \cdots \\
Q_0 \otimes_R (B \otimes_S P_1) \ar[d]& Q_1 \otimes_R (B \otimes_S P_1) \ar[d]& Q_2 \otimes_R (B \otimes_S P_1) \ar[d]& \cdots \\
Q_0 \otimes_R (B \otimes_S P_0) & Q_1 \otimes_R (B \otimes_S P_0) & Q_2 \otimes_R (B \otimes_S P_0) & \cdots
}$$ Recall that projective module are flat. By Corollary \[lemma-universal-projective\], $$E^1=\xymatrix{
\cdots & \cdots & \cdots & \\
Q_0 \otimes_R {\mathrm{Tor}}^S_2(B,C) & Q_1 \otimes_R {\mathrm{Tor}}^S_2(B,C) \ar[l]& Q_2 \otimes_R {\mathrm{Tor}}^S_2(B,C) \ar[l]& \cdots \ar[l] \\
Q_0 \otimes_R {\mathrm{Tor}}^S_1(B,C) & Q_1 \otimes_R {\mathrm{Tor}}^S_1(B,C) \ar[l]& Q_2 \otimes_R {\mathrm{Tor}}^S_1(B,C) \ar[l]& \cdots \ar[l] \\
Q_0 \otimes_R (B \otimes_S C) & Q_1 \otimes_R (B \otimes_S C) \ar[l]& Q_2 \otimes_R (B \otimes_S C) \ar[l]& \cdots \ar[l]
}$$ Therefore, $E^2_{p,q} \cong {\mathrm{Tor}}^R_p(A, {\mathrm{Tor}}^S_q(B,C))$. Moreover, recall that $\{E^r\}$ converges to the homology of the total complex of , which is ${\mathrm{Tor}}^S_\ast (A \otimes_R B, C)$
It is not too hard to prove Theorem \[thm-Betti\] using Theorem \[thm-SS-base-ring\].
$R_{B,r}:={\mathbb{Q}}[X_2-X_1,\dots,X_m-X_1]$ is a subring of $R_B:={\mathbb{Q}}[X_1,\dots,X_m]$. We have the isomorphism $R_B/{\mathfrak{m}}\cong R_{B,r}/{\mathfrak{m}}_r \otimes_{R_{B,r}} R_B/(X_1).$ Note that $R_B/(X_1)$ is a free module over $R_{B,r}$. So ${(R_B/(X_1))\otimes_{R_B}P}$ is projective over $R_{B,r}$ whenever $P$ is projective over $R_B$. Thus, $${{\mathrm{Tor}}^{R_{B,r}}_i(R_{B,r}/{\mathfrak{m}}_r,(R_B/(X_1))\otimes_{R_B}P)} =0$$ for all $i\geq 1$. This means that $R=R_{B,r}$, $S=R_B$, $A=R_{B,r}/{\mathfrak{m}}_r$ and $B= R_B/(X_1)$ satisfy the conditions required in Theorem \[thm-SS-base-ring\]. Now apply Theorem \[thm-SS-base-ring\] to the $R_B$-module $C=H^{\star,j,k}(B)$. From Remark \[remark-quotient\], we know that $H(B) \cong H_r(B)\otimes_{\mathbb{Q}}{\mathbb{Q}}[X_1]\{1,0,0\}$. In particular, $H^{\star,j,k}(B)$ is a free ${\mathbb{Q}}[X_1]$-module, and $H^{\star,j,k}(B)/X_1H^{\star,j,k}(B) \cong H_r^{\star,j,k}(B)\{1\}$, where “$\{1\}$" means shifting the $R_{B,r}$-module grading up by $1$. Note that $R_B/(X_1)$ has the simple minimal free resolution $0 \rightarrow R_B\{2\} \xrightarrow{X_1} R_B$. So ${\mathrm{Tor}}^{R_B}_\ast(R_B/(X_1), H^{\star,j,k}(B))$ is isomorphic to the homology of the chain complex $0 \rightarrow H^{\star,j,k}(B)\{2\} \xrightarrow{X_1} H^{\star,j,k}(B)$. But $H^{\star,j,k}(B)$ is a free ${\mathbb{Q}}[X_1]$-module. So ${\mathrm{Tor}}^{R_B}_q(R_B/(X_1), H^{\star,j,k}(B))\cong 0$ for all $q\geq1$. This shows that, in our case, the $E^2$-page of the spectral sequence in Theorem \[thm-SS-base-ring\] is supported on the degree $q=0$. Thus, this spectral sequence collapses at its $E^2$-page. Consequently, as graded ${\mathbb{Q}}$-spaces, $$\begin{aligned}
{\mathrm{Tor}}^{R_B}_{p}(R_B/{\mathfrak{m}}, H^{\star,j,k}(B)) & \cong & {\mathrm{Tor}}^{R_B}_{p}(R_{B,r}/{\mathfrak{m}}_r \otimes_{R_{B,r}} R_B/(X_1), H^{\star,j,k}(B)) \\
& \cong & {\mathrm{Tor}}^{R_{B,r}}_p (R_{B,r}/{\mathfrak{m}}_r, R_B/(X_1) \otimes_{R_B}H^{\star,j,k}(B)) \\
& \cong & {\mathrm{Tor}}^{R_{B,r}}_p (R_{B,r}/{\mathfrak{m}}_r, H^{\star,j,k}(B)/X_1H^{\star,j,k}(B)) \\
& \cong & {\mathrm{Tor}}^{R_{B,r}}_p (R_{B,r}/{\mathfrak{m}}_r, H_r^{\star,j,k}(B))\{1\}\end{aligned}$$ By the definition of the Betti numbers, this proves that $\beta_B(p,q,j,k) = \beta_{B,r}(p,q-1,j,k)$.
In the case when $B$ is a knot, we have $R_{B,r}={\mathbb{Q}}$ and ${\mathfrak{m}}_r=\{0\}$. So $\beta_{B,r}(p,q,j,k)=0$ if $p\geq1$, and $\beta_{B,r}(0,q,j,k) = \dim_{\mathbb{Q}}H_r^{q,j,k}(B)=\dim_{\mathbb{Q}}\overline{H}^{q,j,k}(B)$, since $H_r(B)=\overline{H}(B)$ when $B$ is a knot.
Split Union {#sec-split}
===========
To understand the behavior of the Betti numbers under split union, we need the following lemma.
\[lemma-mfs-tensor\] Let $X_1,\dots,X_m,Y_1,\dots,Y_n$ be pairwise distinct variables of degree $2$, $R_X={\mathbb{Q}}[X_1,\dots,X_m]$ and $R_Y={\mathbb{Q}}[Y_1,\dots,Y_n]$. Assume that $M_X$ is a finitely generated graded $R_X$-module with the minimal free resolution $0 \rightarrow F_l \rightarrow F_{l-1} \rightarrow \cdots \rightarrow F_1 \rightarrow F_0$ over $R_X$, and $M_Y$ is a finitely generated graded $R_Y$-module with the minimal free resolution $0 \rightarrow G_k \rightarrow G_{k-1} \rightarrow \cdots \rightarrow G_1 \rightarrow G_0$ over $R_Y$. Then the finitely generated $R_X\otimes_{\mathbb{Q}}R_Y$-module $M_X \otimes_{\mathbb{Q}}M_Y$ has the minimal free resolution $F_\ast \otimes_{\mathbb{Q}}G_\ast$ over $R_X\otimes_{\mathbb{Q}}R_Y$, which is of the form $$\label{eq-mfs-tensor}
0 \rightarrow F_l \otimes_{\mathbb{Q}}G_k \rightarrow \cdots \rightarrow \bigoplus_{p_1+p_2=p} F_{p_1}\otimes_{\mathbb{Q}}G_{p_2} \rightarrow \cdots \rightarrow F_0 \otimes_{\mathbb{Q}}G_0.$$
Denote by $\beta_{M_X}(p,q)$ the Betti number of $M_X$ over $R_X$, by $\beta_{M_Y}(p,q)$ the Betti number of $M_Y$ over $R_Y$ and by $\beta_{M_X\otimes_{\mathbb{Q}}M_Y}(p,q)$ the Betti number of $M_X\otimes_{\mathbb{Q}}M_Y$ over $R_X\otimes_{\mathbb{Q}}R_Y$. Let $$\begin{aligned}
\mathcal{P}_{M_X}(x,y) & = & \sum_{(p,q)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}} \beta_{M_X}(p,q) \cdot x^p \cdot \left(\sum_{i\in {\mathbb{Z}}} y^{2i+q} {\genfrac{(}{)}{0pt}{}{i+m-1}{i}}\right), \\
\mathcal{P}_{M_Y}(x,y) & = & \sum_{(p,q)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}} \beta_{M_Y}(p,q) \cdot x^p \cdot \left(\sum_{i\in {\mathbb{Z}}} y^{2i+q} {\genfrac{(}{)}{0pt}{}{i+n-1}{i}}\right), \\
\mathcal{P}_{M_X \otimes_{\mathbb{Q}}M_Y}(x,y) & = & \sum_{(p,q)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}} \beta_{M_{M_X\otimes_{\mathbb{Q}}M_Y}}(p,q) \cdot x^p \cdot \left(\sum_{i\in {\mathbb{Z}}} y^{2i+q} {\genfrac{(}{)}{0pt}{}{i+m+n-1}{i}}\right).\end{aligned}$$ Then $\mathcal{P}_{M_X \otimes_{\mathbb{Q}}M_Y}(x,y) = \mathcal{P}_{M_X}(x,y) \cdot \mathcal{P}_{M_Y}(x,y).$
It is clear that $F_\ast \otimes_{\mathbb{Q}}G_\ast$ is a minimal chain complex of graded free $R_X\otimes_{\mathbb{Q}}R_Y$-modules. So we only need to verify that it is a resolution of $M_X \otimes_{\mathbb{Q}}M_Y$. Since ${\mathbb{Q}}$ is a field, the Künneth Formula gives that $$H_p(F_\ast \otimes_{\mathbb{Q}}G_\ast) \cong \bigoplus_{p_1+p_2=p} H_{p_1}(F_\ast) \otimes_{\mathbb{Q}}H_{p_2}(G_\ast) \cong \begin{cases}
M_X \otimes_{\mathbb{Q}}M_Y & \text{if } p=0,\\
0 & \text{otherwise.}
\end{cases}$$ It shows that $F_\ast \otimes_{\mathbb{Q}}G_\ast$ is a minimal free resolution of $M_X \otimes_{\mathbb{Q}}M_Y$ over $R_X\otimes_{\mathbb{Q}}R_Y$.
By Lemma \[lemma-Poincare-polynomial-general\], $$\begin{aligned}
\mathcal{P}_{M_X}(x,y) & = & \sum_{(p,i)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}} x^p \cdot y^i \cdot \dim_{\mathbb{Q}}F_p^i, \\
\mathcal{P}_{M_Y}(x,y) & = & \sum_{(p,i)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}} x^p \cdot y^i \cdot \dim_{\mathbb{Q}}G_p^i,\end{aligned}$$ where $F_p^i$ (resp. $G_p^i$) is the homogeneous component of $F_p$ (resp. $G_p$) of degree $i$. Clearly, $$\left(\sum_{(p,i)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}} x^p \cdot y^i \cdot \dim_{\mathbb{Q}}F_p^i\right) \cdot \left(\sum_{(p,i)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}} x^p \cdot y^i \cdot \dim_{\mathbb{Q}}G_p^i\right) = \sum_{(p,i)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}} x^p \cdot y^i \cdot \dim_{\mathbb{Q}}\left( \bigoplus_{p_1+p_2=p} F_{p_1}\otimes_{\mathbb{Q}}G_{p_2}\right)^i,$$ where $\left(\bigoplus_{p_1+p_2=p} F_{p_1}\otimes_{\mathbb{Q}}G_{p_2}\right)^i$ is the homogeneous component of $\bigoplus_{p_1+p_2=p} F_{p_1}\otimes_{\mathbb{Q}}G_{p_2}$ of degree $i$. But $M_X\otimes_{\mathbb{Q}}M_Y$ has the minimal free resolution $F_\ast \otimes_{\mathbb{Q}}G_\ast$ over $R_X \otimes_{\mathbb{Q}}R_Y$, which is of form . So, by Lemma \[lemma-Poincare-polynomial-general\], $$\mathcal{P}_{M_X \otimes_{\mathbb{Q}}M_Y}(x,y) = \sum_{(p,i)\in {\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}} x^p \cdot y^i \cdot \dim_{\mathbb{Q}}\left( \bigoplus_{p_1+p_2=p} F_{p_1}\otimes_{\mathbb{Q}}G_{p_2}\right)^i.$$ Thus, $\mathcal{P}_{M_X \otimes_{\mathbb{Q}}M_Y}(x,y) = \mathcal{P}_{M_X}(x,y) \cdot \mathcal{P}_{M_Y}(x,y).$
The next lemma is a simple observation.
\[lemma-pd-m-1\] Let $B$ be a closed braid with $m$ components. Then ${\mathrm{pd}}_{R_B} H(B) \leq m-1$.
By Lemma \[lemma-pd\] and Theorem \[thm-Betti\], $$\begin{aligned}
{\mathrm{pd}}_{R_B} H(B) & = & \max\{p~|~ \beta_B(p,q,j,k)\neq 0 \text{ for some } (q,j,k)\in {\mathbb{Z}}^3\} \\
& = & \max\{p~|~ \beta_{B,r}(p,q,j,k)\neq 0 \text{ for some } (q,j,k)\in {\mathbb{Z}}^3\} = {\mathrm{pd}}_{R_{B,r}} H_r(B).\end{aligned}$$ But $R_{B,r}$ is a polynomial ring of $m-1$ variables. So its global dimension is $m-1$. Thus, ${\mathrm{pd}}_{R_B} H(B) = {\mathrm{pd}}_{R_{B,r}} H_r(B) \leq m-1$.
Now we are ready to prove Theorem \[thm-split\].
We prove Part (1) first. Using the polynomial notation in Lemma \[lemma-mfs-tensor\], we have $$\begin{aligned}
\mathcal{P}_{B_1} (x,y,a,b) & = & \sum_{(j,k)\in {\mathbb{Z}}^2} a^j \cdot b^{\frac{k-j}{2}} \cdot \mathcal{P}_{H^{\star,j,k}(B_1)}(x,y), \\
\mathcal{P}_{B_2} (x,y,a,b) & = & \sum_{(j,k)\in {\mathbb{Z}}^2} a^j \cdot b^{\frac{k-j}{2}} \cdot \mathcal{P}_{H^{\star,j,k}(B_2)}(x,y), \\
\mathcal{P}_{B_1\sqcup B_2} (x,y,a,b) & = & \sum_{(j,k)\in {\mathbb{Z}}^2} a^j \cdot b^{\frac{k-j}{2}} \cdot \mathcal{P}_{H^{\star,j,k}(B_1\sqcup B_2)}(x,y).\end{aligned}$$ By isomorphism , $$H^{\star,j,k}(B_1\sqcup B_2) \cong \bigoplus_{j_1+j_2=j-1,~k_1+k_2 =k+1} H^{\star,j_1,k_1}(B_1) \otimes_{\mathbb{Q}}H^{\star,j_2,k_2}(B_2).$$ Thus, by Lemma \[lemma-mfs-tensor\], $$\mathcal{P}_{H^{\star,j,k}(B_1\sqcup B_2)}(x,y) = \sum_{j_1+j_2=j-1,~k_1+k_2 =k+1} \mathcal{P}_{H^{\star,j_1,k_1}(B_1)}(x,y) \cdot \mathcal{P}_{H^{\star,j_2,k_2}(B_2)}(x,y).$$ Therefore, $$\begin{aligned}
&& \mathcal{P}_{B_1\sqcup B_2} (x,y,a,b) \\
& = & \sum_{(j,k)\in {\mathbb{Z}}^2} a^j \cdot b^{\frac{k-j}{2}} \cdot \mathcal{P}_{H^{\star,j,k}(B_1\sqcup B_2)}(x,y) \\
& = & ab^{-1}\sum_{(j,k)\in {\mathbb{Z}}^2} a^{j-1} \cdot b^{\frac{(k+1)-(j-1)}{2}} \cdot \left(\sum_{j_1+j_2=j-1,~k_1+k_2 =k+1} \mathcal{P}_{H^{\star,j_1,k_1}(B_1)}(x,y) \cdot \mathcal{P}_{H^{\star,j_2,k_2}(B_2)}(x,y)\right) \\
& = & ab^{-1}\left(\sum_{(j_1,k_1)\in {\mathbb{Z}}^2} a^{j_1} \cdot b^{\frac{k_1-j_1}{2}} \cdot \mathcal{P}_{H^{\star,j_1,k_1}(B_1)}(x,y)\right) \cdot \left(\sum_{(j_2,k_2)\in {\mathbb{Z}}^2} a^{j_2} \cdot b^{\frac{k_2-j_2}{2}} \cdot \mathcal{P}_{H^{\star,j_2,k_2}(B_1)}(x,y)\right) \\
& = & ab^{-1}\mathcal{P}_{B_1} (x,y,a,b) \cdot \mathcal{P}_{B_2} (x,y,a,b).\end{aligned}$$ Combining this and Lemma \[lemma-pd\], we get $$\begin{aligned}
{\mathrm{pd}}_{R_{B_1 \sqcup B_2}} H(B_1 \sqcup B_2) & = & \deg_x \mathcal{P}_{B_1\sqcup B_2} (x,y,a,b) \\
& = & \deg_x \mathcal{P}_{B_1} (x,y,a,b) + \deg_x \mathcal{P}_{B_2} (x,y,a,b) = {\mathrm{pd}}_{R_{B_1}} H(B_1)+{\mathrm{pd}}_{R_{B_2}} H(B_2).\end{aligned}$$ This completes the proof of Part (1).
For Part (2), without loss of generality, assume that $B$ is an $m$-component closed braid that is $n$-split. Then there are $n$ closed braid $B_1,\dots,B_n$ such that $B = B_1 \sqcup \cdots \sqcup B_n$. Denote by $m_l$ the number of components of $B_l$. Note that $\sum_{l=1}^n m_l =m$. By Part (1) and Lemma \[lemma-pd-m-1\], we have $${\mathrm{pd}}_{R_B} H(B) = \sum_{l=1}^n {\mathrm{pd}}_{R_{B_l}} H(B_l) \leq \sum_{l=1}^n (m_l-1) =m-n.$$ This completes the proof of Theorem \[thm-split\].
The Hopf Link {#sec-hopf}
=============
In this section, we compute the Betti numbers of the middle HOMFLYPT homology of the positive Hopf link $B_1$ and verify Example \[eg-hopf\].
$$\PandocStartInclude{hopf.tex}\PandocEndInclude{input}{664}{13}$$
Figure \[fig-hopf\] is a standard diagram of $B_1$. The variables assigned to the edges of this diagram are $X_1,X_2,X_3,X_4$ as shown in Figure \[fig-hopf\]. The base ring $R=R(B_1)$ of the double chain complex $C_0(B_1)$ is $R=R(B_1)={\mathbb{Q}}[X_1,X_2,X_3,X_4]/(X_1+X_2-X_3 -X_4) \cong {\mathbb{Q}}[X_1,X_2,X_3]$. The double chain complexes associated to the two crossings $c_1$ and $c_2$ of $B_1$ are $$C_0(c_1) = \xymatrix{
R\{0,-2,0\} \ar[rrr]^{X_2-X_3} &&& R\{0,0,0\} \\
R\{2,-2,-2\} \ar[rrr]^{(X_2-X_3)(X_1-X_3)} \ar[u]^{X_1-X_3} &&& R\{0,0,-2\} \ar[u]^{1}
}$$ and $$C_0(c_2) = \xymatrix{
R\{0,-2,0\} \ar[rrr]^{X_3-X_2} &&& R\{0,0,0\} \\
R\{2,-2,-2\} \ar[rrr]^{-(X_2-X_3)(X_1-X_3)} \ar[u]^{X_1-X_3} &&& R\{0,0,-2\}.\ar[u]^{1}
}$$ So the double chain complex $C_0(B_1)$ is $$C_0(B_1) = \xymatrix{
R\{0,-4,0\} \ar[rr]^{d_+^{-4,0}} &&
{\left.\begin{array}{l}
R\{0,-2,0\}\\
\oplus \\
R\{0,-2,0\}
\end{array}\right.} \ar[rr]^{d_+^{-2,0}} && R\{0,0,0\} \\
{\left.\begin{array}{l}
R\{2,-4,-2\}\\
\oplus \\
R\{2,-4,-2\}
\end{array}\right.} \ar[rr]^{d_+^{-4,-2}} \ar[u]^{d_v^{-4,-2}} &&
{\left.\begin{array}{l}
R\{0,-2,-2\} \\
\oplus \\
R\{2,-2,-2\}\\
\oplus \\
R\{2,-2,-2\} \\
\oplus \\
R\{0,-2,-2\}
\end{array}\right.} \ar[rr]^{d_+^{-2,-2}} \ar[u]^>>>>{d_v^{-2,-2}} &&
{\left.\begin{array}{l}
R\{0,0,-2\}\\
\oplus \\
R\{0,0,-2\}
\end{array}\right.} \ar[u]^{d_v^{0,-2}} \\
R\{4,-4,-4\} \ar[rr]^{d_+^{-4,-4}} \ar[u]^{d_v^{-4,-4}}
&&
{\left.\begin{array}{l}
R\{2,-2,-4\}\\
\oplus \\
R\{2,-2,-4\}
\end{array}\right.} \ar[rr]^{d_+^{-2,-4}} \ar[u]^>>>>{d_v^{-2,-4}} &&
R\{0,0,-4\}, \ar[u]^{d_v^{0,-4}}
}$$ where the horizontal chain maps are $$\begin{aligned}
d_+^{-2,0} & = & (X_2-X_3,X_3-X_2), \\
d_+^{-4,0} & = & {\left(\begin{array}{c}
X_2-X_3\\
X_2-X_3
\end{array}\right)}, \\
d_+^{-2,-2} & = & {\left(\begin{array}{cccc}
X_2-X_3 & -(X_2-X_3)(X_1-X_3) & 0 & 0 \\
0& 0& (X_2-X_3)(X_1-X_3) & X_3-X_2
\end{array}\right)}, \\
d_+^{-4,-2} & = & {\left(\begin{array}{cc}
(X_2-X_3)(X_1-X_3) & 0 \\
X_2-X_3 & 0\\
0 & X_2-X_3\\
0 & (X_2-X_3)(X_1-X_3)
\end{array}\right)}, \\
d_+^{-2,-4} & = & ((X_2-X_3)(X_1-X_3), -(X_2-X_3)(X_1-X_3)), \\
d_+^{-4,-4} & = & {\left(\begin{array}{c}
(X_2-X_3)(X_1-X_3) \\
(X_2-X_3)(X_1-X_3)
\end{array}\right)},\end{aligned}$$ and the vertical chain maps are $$\begin{aligned}
d_v^{0,-2} & = & (1,1) \\
d_v^{-2,-2}& = & {\left(\begin{array}{cccc}
1 & 0 & X_1-X_3 & 0 \\
0 & X_1-X_3 & 0 & 1
\end{array}\right)}, \\
d_v^{-4,-2} & = & (X_1-X_3,X_1-X_3), \\
d_v^{0,-4} & = & {\left(\begin{array}{c}
1\\
-1
\end{array}\right)}, \\
d_v^{-2,-4} & = & {\left(\begin{array}{cc}
X_1-X_3 & 0 \\
0 & 1 \\
-1 & 0 \\
0 & X_3-X_1
\end{array}\right)}, \\
d_v^{-4,-4} & = & {\left(\begin{array}{c}
X_1-X_3\\
X_3-X_1
\end{array}\right)}.\end{aligned}$$
Thus, the homology of $C_0(B_1)$ with respect to $d_+$ is $$\begin{aligned}
&& H(C_0(B_1),d_+) \cong\\
&& \xymatrix{
0 &
{\left(\begin{array}{c}
1\\
1
\end{array}\right)} \cdot R/(X_2-X_3) & R/(X_2-X_3) \\
0 \ar[u]^{d_v^{-4,-2}} &
{\left(\begin{array}{c}
0 \\
0 \\
1\\
X_1-X_3
\end{array}\right)\cdot R/(X_2-X_3)} \oplus {\left(\begin{array}{c}
X_1-X_3 \\
1 \\
0\\
0
\end{array}\right)\cdot R/(X_2-X_3)} \ar[u]^>>>>{d_v^{-2,-2}} &
{\left.\begin{array}{l}
R/(X_2-X_3)\\
\oplus \\
R/(X_2-X_3)
\end{array}\right.} \ar[u]^{d_v^{0,-2}} \\
0 \ar[u]^{d_v^{-4,-4}}
&
{\left(\begin{array}{c}
1\\
1
\end{array}\right)} \cdot R/((X_2-X_3)(X_1-X_3)) \{2\} \ar[u]^>>>>{d_v^{-2,-4}} &
R/((X_2-X_3)(X_1-X_3)), \ar[u]^{d_v^{0,-4}}
}\end{aligned}$$ where we omit the second and third ${\mathbb{Z}}$-gradings of the homology since these are represented by the position of the term in the diagram. Also, note that the first ${\mathbb{Z}}$-grading of the middle term of the bottom row is shifted up by $2$.
Now we take homology with respect to $d_v$. This gives $$H(H(C_0(B_1),d_+),d_v) \cong \xymatrix{
0 & \textcolor{red}{R/(X_2-X_3,X_1-X_3)} & 0 \\
0 & 0 & 0 \\
0 & R/(X_1-X_3)\{2\} & R/(X_1-X_3).
}$$ Note that $R/(X_1-X_3) \cong {\mathbb{Q}}[X_1,X_2]=R_{B_1}$. So $$H^{\star,j,k}(H(C_0(B_1),d_+),d_v) \cong \begin{cases}
\textcolor{red}{R_{B_1}/(X_1-X_2)} & \text{if } (j,k)=(-2,0), \\
R_{B_1} & \text{if } (j,k)=(0,-4), \\
R_{B_1}\{2\} & \text{if } (j,k)=(-2,-4), \\
0 & \text{otherwise.}
\end{cases}$$
Note that the writhe of $B_1$ is $w=2$ and $B_1$ has $2$ strands. So, by equation , $$H^{\star,j,k}(B_1) = H^{\star,j-3,k-1}(H(C_0(B_1),d_+),d_v) \cong \begin{cases}
\textcolor{red}{R_{B_1}/(X_1-X_2)} & \text{if } (j,k)=(1,1), \\
R_{B_1} & \text{if } (j,k)=(3,-3), \\
R_{B_1}\{2\} & \text{if } (j,k)=(-1,-3), \\
0 & \text{otherwise.}
\end{cases}$$ Using Lemma \[lemma-mfs-Betti\], one gets $$\beta_{B_1}(p,q,j,k) = \begin{cases}
1 & \text{if } (p,q,j,k)=\textcolor{red}{(1,2,1,1)}, (0,0,1,1),(0,0,3,-3),(0,2,-1,-3), \\
0 & \text{otherwise.}
\end{cases}$$ In particular, the projective dimension of $H(B_1)$ over $R_{B_1}$ is $1$ by Lemma \[lemma-pd\]. Therefore, by Theorem \[thm-split\], the positive Hopf link $B_1$ does not split.
Projective Dimension of the $\mathfrak{sl}(N)$ Homology {#sec-sl-N}
=======================================================
The proof of Lemma \[lemma-sl-N-proj-dim\] is quite straightforward. But, to state it, we need to recall the relation between the projective dimension and regular sequences via the depth. First, we recall the well-known Auslander-Buchsbaum Formula, which can be found in for example [@Peeva-graded-syzygies Section 15].
\[thm-Auslander-Buchsbaum\] Let $R={\mathbb{Q}}[X_1,\dots,X_m]$, graded so that which $X_j$ is homogeneous of degree $2$. Assume that $M$ is a finitely generated graded $R$-module. Then ${\mathrm{pd}}_R M = m-\mathrm{depth}(M)$, where $\mathrm{depth}(M)$ is the depth of $M$ over $R$ with respect to the maximal homogeneous ideal $\mathfrak{m}=(X_1,\dots,X_m)$ of $R$.
Next we recall the definition of regular sequences in [@Peeva-graded-syzygies Section 14].
\[def-regular\] Let $R={\mathbb{Q}}[X_1,\dots,X_m]$, graded so that which $X_j$ is homogeneous of degree $2$. Assume that $M$ is a finitely generated graded $R$-module.
An element $f \in R$ is a non-zero divisor on $M$ if $fu\neq 0$ for every non-zero element $u$ of $M$. Otherwise, $f$ is called a zero divisor on $M$.
A sequence $f_1,\dots,f_q\in R$ is an $M$-regular sequence if
- $(f_1,\dots,f_q)M \neq M$,
- for every $1\leq i \leq q$, $f_i$ is a non-zero divisor on the module $M/(f_1,\dots,f_{i-1})M$.
The following relation between the depth and regular sequences is stated in [@Peeva-graded-syzygies Proposition 20.1] and proved in [@Bruns-Herzog Propositions 1.5.11 and 1.5.12].
\[prop-depth-regular-sequences\] Let $R={\mathbb{Q}}[X_1,\dots,X_m]$, graded so that which $X_j$ is homogeneous of degree $2$. Assume that $M$ is a finitely generated graded $R$-module. If the depth of $M$ over $R$ with respect to its maximal homogeneous ideal is $s$, then there exists an $M$-regular sequence $f_1,f_2,\dots,f_s$ of homogeneous elements of $R$.
Now we are ready to prove Lemma \[lemma-sl-N-proj-dim\].
Recall that $H_N(B)$ is a graded $R_B$-module that is finite dimensional over ${\mathbb{Q}}$. This implies that $M$ is finitely generated over $R_B$. Moreover, this also implies that any homogeneous element of $R_B$ of positive degree is a zero divisor on $H_N(B)$. Thus, there are no $H_N(B)$-regular sequences of homogeneous elements of $R_B$ of any positive length. By Proposition \[prop-depth-regular-sequences\], this implies that $\mathrm{depth}(H_N(B))=0$, where the depth is over $R_B$ and with respect to its maximal homogeneous ideal. Now the Auslander-Buchsbaum Formula gives that ${\mathrm{pd}}_{R_B} H_N(B) = m-\mathrm{depth}(H_N(B))=m$.
[99]{}
V. Arrondo *Introduction to projective varieties,* http://www.mat.ucm.es/$\sim$arrondo/projvar.pdf J. Batson, C. Seed, *A link-splitting spectral sequence in Khovanov homology,* Duke Math. J. **164** (2015), no. 5, 801–841. W. Bruns, J. Herzog, *Cohen-Macaulay rings,* Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN: 0-521-41068-1 D. Eisenbud, *Commutative algebra. With a view toward algebraic geometry,* Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN: 0-387-94268-8; 0-387-94269-6 D. Eisenbud, *The geometry of syzygies. A second course in commutative algebra and algebraic geometry.* Graduate Texts in Mathematics, 229. Springer-Verlag, New York, 2005. xvi+243 pp. ISBN: 0-387-22215-4 M. Khovanov, L. Rozansky, *Matrix factorizations and link homology,* Fund. Math. **199** (2008), no. 1, 1–91. M. Khovanov, L. Rozansky, *Matrix factorizations and link homology II,* Geom. Topol. **12** (2008), no. 3, 1387–1425. I. Peeva, *Graded syzygies,* Algebra and Applications, 14. Springer-Verlag London, Ltd., London, 2011. xii+302 pp. ISBN: 978-0-85729-176-9. J. Rasmussen, *Khovanov-Rozansky homology of two-bridge knots and links,* Duke Math. Journal **136** (2007), 551–583. J. Rasmussen, *Some differentials on Khovanov-Rozansky homology,* Geom. Topol. **19** (2015) 3031–3104, DOI: 10.2140/gt.2015.19.3031 J. Rotman, *An introduction to homological algebra,* second edition, Universitext, Springer, New York, 2009. xiv+709 pp. ISBN: 978-0-387-24527-0 H. Wu, *On the Hilbert polynomial of the HOMFLYPT homology,* 14pp, arXiv:1604.05222.
[^1]: The author was partially supported by NSF grant DMS-1205879.
[^2]: The middle HOMFLYPT homology $H$ in the current paper is defined exactly as in [@Ras2]. However, its reduction $H_r$ defined in Section \[sec-HOMFLYPT-mod\] below is different from the reduced HOMFLYPT homology $\overline{H}$ in [@Ras2]. For non-split links, both $H_r$ and $\overline{H}$ are the same quotient of $H$. But, for links with split diagrams, $H_r$ remains a quotient of $H$, while $\overline{H}$ follows a more complex definition in [@Ras2 Section 2.10] and is no longer a quotient of $H$.
[^3]: Note that $R_{B,r}$ does note depend on the ordering of the components. In fact, for any $1\leq j \leq m$, ${R_{B,r}={\mathbb{Q}}[X_1-X_j,\dots,X_{j-1}-X_j,X_{j+1}-X_j,\dots,X_m-X_j]}$.
[^4]: Strictly speaking, $C_0(B)$ is not a double complex since squares in it commute, instead of anti-commute. But this does not affect any of our computations.
[^5]: [@Ras-2-bridge Lemma 3.4] is about the ${\mathfrak{sl}}(N)$ homology. But its conclusion and proof remain true for the HOMFLYPT homology.
|
---
abstract: 'Bennett *et al*. showed that allowing shared entanglement between a sender and receiver before communication begins dramatically simplifies the theory of quantum channels, and these results suggest that it would be worthwhile to study other scenarios for entanglement-assisted classical communication. In this vein, the present paper makes several contributions to the theory of entanglement-assisted classical communication. First, we rephrase the Giovannetti-Lloyd-Maccone sequential decoding argument as a more general packing lemma and show that it gives an alternate way of achieving the entanglement-assisted classical capacity. Next, we show that a similar sequential decoder can achieve the Hsieh-Devetak-Winter region for entanglement-assisted classical communication over a multiple access channel. Third, we prove the existence of a quantum simultaneous decoder for entanglement-assisted classical communication over a multiple access channel with two senders. This result implies a solution of the quantum simultaneous decoding conjecture for unassisted classical communication over quantum multiple access channels with two senders, but the three-sender case still remains open (Sen recently and independently solved this unassisted two-sender case with a different technique). We then leverage this result to recover the known regions for unassisted and assisted quantum communication over a quantum multiple access channel, though our proof exploits a coherent quantum simultaneous decoder. Finally, we determine an achievable rate region for communication over an entanglement-assisted bosonic multiple access channel and compare it with the Yen-Shapiro outer bound for unassisted communication over the same channel.'
author:
- |
Shen Chen Xu and Mark M. Wilde\
*School of Computer Science, McGill University,*\
*Montreal, Quebec, H3A 2A7 Canada*
bibliography:
- 'Ref.bib'
title: 'Sequential, successive, and simultaneous decoders for entanglement-assisted classical communication'
---
Shared entanglement between a sender and receiver leads to surprises such as super-dense coding [@PhysRevLett.69.2881] and teleportation [@PhysRevLett.70.1895], and these protocols were the first to demonstrate that entanglement, classical bits, and quantum bits can interact in interesting ways. For this reason, one could argue that these protocols and their noisy generalizations [@DHW05RI; @HW08GFP; @HW09] make quantum information theory [@book2000mikeandike; @W11] richer than its classical counterpart [@book1991cover]. A good way to think of the super-dense coding protocol is that it is a statement of resource conversion [@DHW05RI]: one noiseless qubit channel and one noiseless ebit are sufficient to generate two noiseless bit channels between a sender and receiver.
Bennett *et al*. explored a generalization of the super-dense coding protocol in which a sender and receiver are given noiseless entanglement in whatever form they wish and access to many independent uses of a noisy quantum channel, and the goal is to determine how many asymptotically perfect noiseless bit channels that the sender and receiver can simulate with the aforementioned resources [@BSST99; @BSST02; @H02]. The entanglement-assisted classical capacity theorem provides a beautiful answer to this question. The optimal rate at which they can communicate classical bits in the presence of free entanglement is equal to the mutual information of the channel [@BSST02; @H02], defined as$$I\left( \mathcal{N}\right) \equiv\max_{\phi^{AA^{\prime}}}I\left(
A;B\right) _{\rho},$$ where $\rho^{AB}\equiv\mathcal{N}^{A^{\prime}\rightarrow B}(\phi^{AA^{\prime}})$, $\mathcal{N}^{A^{\prime}\rightarrow B}$ is the noisy channel connecting the sender to the receiver, and $\phi^{AA^{\prime}}$ is a pure, bipartite state prepared at the sender’s end of the channel. This result is the strongest statement that quantum information theorists have been able to make in the theory of quantum channels, because the above channel mutual information is additive as a function of any two channels $\mathcal{N}$ and $\mathcal{M}$ [@PhysRevA.56.3470]: $$I\left( \mathcal{N}\otimes\mathcal{M}\right) =I\left( \mathcal{N}\right)
+I\left( \mathcal{M}\right) ,$$ and the mutual information $I\left( A;B\right) $ is concave in the input state when the channel is fixed [@PhysRevA.56.3470] (these two properties imply that we can actually calculate the entanglement-assisted classical capacity of *any* quantum channel). Furthermore, this information measure is particularly robust in the sense that a quantum feedback channel from receiver to sender does not increase it—Bowen showed that the classical capacity of a quantum channel in the presence of unlimited quantum feedback communication is equal to the entanglement-assisted classical capacity [@B04]. For these reasons, the entanglement-assisted classical capacity of a quantum channel is the best formal analogy of Shannon’s classical capacity of a classical channel [@bell1948shannon].
The simplification that shared entanglement brings to the theory of quantum channels suggests that it might be fruitful to explore other scenarios in which communicating parties share entanglement, and this is precisely the goal of the present paper. Indeed, we explore five different scenarios for entanglement-assisted classical communication:
1. Sequential decoding for entanglement-assisted classical communication over a single-sender, single-receiver quantum channel.
2. Sequential and successive decoding for entanglement-assisted classical communication over a quantum multiple access channel (a two-sender, single-receiver channel).
3. Simultaneous decoding for classical communication over an entanglement-assisted quantum multiple access channel.
4. Coherent simultaneous decoding for assisted and unassisted quantum communication over a quantum multiple access channel.
5. Entanglement-assisted classical communication over a bosonic multiple access channel.
We briefly overview each of these scenarios in what follows.
Our first contribution is a sequential decoder for entanglement-assisted classical communication, meaning that the receiver performs a sequence of measurements with yes/no outcomes in order to determine the message that the sender transmits (the receiver performs these measurements on the channel outputs and his share of the entanglement). The idea of this approach is the same as the recent Giovannetti-Lloyd-Maccone (GLM) sequential decoder for unassisted classical communication [@GLM10] (which in turn bears similarities to the Feinstein approach [@F54; @ON07; @itit1999winter]). In fact, our approach for proving that the sequential method works for the entanglement-assisted case is to rephrase their argument as a more general packing lemma [@HDW08; @W11] and exploit the entanglement-assisted coding scheme of Hsieh *et al*. [@HDW08; @W11].
Our next contribution is to extend this sequential decoding argument to a quantum multiple access channel. Winter [@W01] and Hsieh *et al*. [@HDW08] have already shown that successive decoding works well for unassisted and assisted transmission of classical information over a quantum multiple access channel, respectively. (Here, successive decoding means that the receiver first decodes one sender’s message and follows by decoding the other sender’s message). We show that a receiver can exploit a sequence of measurements with yes/no outcomes to determine the first sender’s message, followed by a different sequence of yes/no measurements to determine the second sender’s message. Thus, our decoder here is both sequential and successive and generalizes the GLM sequential decoding scheme.
Our third contribution is to prove that the receiver of an entanglement-assisted quantum multiple access channel can exploit a quantum simultaneous decoder to detect two messages sent by two respective senders. A simultaneous decoder is different from a successive decoder—it can detect the two senders’ messages asymptotically faithfully as long as their transmission rates are within the pentagonal rate region of the multiple access channel [@el2010lecture; @W01; @HDW08]. A simultaneous decoder is more powerful than a successive decoder for two reasons:
1. A simultaneous decoder does not require the use of time-sharing in order to achieve the rate region of the multiple access channel (whereas a successive decoder requires the use of time-sharing). Thus, the technique should generalize well to the setting of one-shot information theory [@DR09], where time-sharing does not apply because that theory is concerned with what is possible with a *single* use of a quantum channel.
2. Nearly every proof in classical network information theory exploits a simultaneous decoder [@el2010lecture]. Thus, a *quantum* simultaneous decoder would be of broad interest for a network theory of quantum information. In particular, the strategy for achieving the best known achievable rate region of the classical interference channel exploits a simultaneous decoder [@HK81; @el2010lecture]. (An interference channel has two senders and two receivers, and each sender is interested in communicating with one particular receiver.)
We should mention that Fawzi *et al*. could prove the existence of a quantum simultaneous decoder for certain quantum channels [@FHSSW11], but a proof for the general case remained missing and they did not address the entanglement-assisted case. Though, the results of this paper and recent work of Sen [@S11a] give a quantum simultaneous decoder for unassisted communication over a two-sender multiple access channel and solve the conjecture from Ref. [@FHSSW11] for the two-sender case. It remains unclear how to prove the conjecture for the case of three senders. The results of this work might be useful for establishing an achievable rate region for a quantum interference channel setting in which sender-receiver pairs share entanglement before communication begins, but this remains the topic of future work.
We then leverage the above result to recover the known regions for assisted and unassisted quantum communication over a quantum multiple access channel [@nature2005horodecki; @YHD05MQAC; @HDW08]. We call the decoder a *coherent quantum simultaneous decoder* because we construct an isometry from the above simultaneous decoding POVM, and the isometry is what enables quantum communication between both senders and the receiver.
Our final contribution is to determine an achievable rate region for entanglement-assisted classical communication over the multiple access bosonic channel studied in Ref. [@YS05]. This channel is simply a beamsplitter with two input ports, where the receiver obtains one output port and the environment of the channel obtains the other output port. The beamsplitter is a simplified model for light-based free-space communication in a multiple-access setting. In order to calculate the rate region for this setting, we apply the theorem of Hsieh *et al*. in Ref. [@HDW08] with both senders sharing a two-mode squeezed vacuum state [@GK04] with the receiver. Since this state achieves the entanglement-assisted capacity of the single-mode lossy bosonic channel [@GLMS03; @GLMS03a; @HW01], we might suspect that it should do well in the multiple access setting. Though, it still remains open to determine whether this strategy is optimal.
Packing Argument for a Sequential Decoder {#sec:sequential-packing}
=========================================
Giovannetti, Lloyd, and Maccone (GLM) offered a scheme for transmitting classical information over a quantum channel that exploits a sequential decoder [@GLM10]. In their sequential decoding scheme, the receiver tries to distinguish the transmitted message from a list of all possible messages one by one until the correct one is identified, by performing a sequence of projective measurements. We recast this procedure as a general packing argument in this section, and the next section demonstrates that the sequential decoding scheme works well for entanglement-assisted classical communication.
\[Sequential Packing\]\[thm:sequential-packing\]Let $\{p_{X}\left(
x\right) ,\rho_{x}\}_{x\in\mathcal{X}}$ be an ensemble of states indexed by letters in an alphabet $\mathcal{X}$. Each state $\rho_{x}$ has the following spectral decomposition:$$\rho_{x}=\sum_{y}\lambda_{x,y}\left\vert \psi_{x,y}\right\rangle \left\langle
\psi_{x,y}\right\vert ,$$ and the expected density operator of the ensemble is as follows:$$\rho\equiv\sum_{x\in\mathcal{X}}p_{X}\left( x\right) \rho_{x}.$$ Suppose there exists a code subspace projector $\Pi$ and codeword subspace projectors $\left\{ \Pi_{x}\right\} _{x\in\mathcal{X}}$ such that the following properties hold for some $D,d\geq0$, $1/2\geq\epsilon>0$, and for all $x\in\mathcal{X}$:$$\begin{aligned}
\mathrm{Tr}\left\{ \Pi\rho_{x}\right\} & \geq1-\epsilon
,\label{eq:unit-prob-1}\\
\mathrm{Tr}\left\{ \Pi_{x}\rho_{x}\right\} & \geq1-\epsilon
,\label{eq:unit-prob-2}\\
\Pi_{x}\rho_{x}\Pi_{x} & \geq\frac{1}{d}\Pi_{x},\label{eq:equi-part-1}\\
\Pi\rho\Pi & \leq\frac{1}{D}\Pi,\label{eq:equi-part-2}\\
\lbrack\Pi_{x},\rho_{x}] & =0.\end{aligned}$$ Then corresponding to a message set $\mathcal{M}$, we can construct a random code $\mathcal{C}=\left\{ c_{m}\right\} _{m\in\mathcal{M}}$ with $c_{m}\in\mathcal{X}$ such that the receiver can reliably distinguish between the states $\left\{ \rho_{c_{m}}\right\} _{m\in\mathcal{M}}$ by performing a sequence of projective measurements using the projectors $\Pi$ and $\Pi_{x}$. More precisely, suppose that our performance measure is the expectation of the average success probability where the expectation is with respect to all possible random choices of codes. Then we can bound this performance measure from below (as long as $2-\exp\left\{ d\left\vert \mathcal{M}\right\vert
/D\right\} $ is positive):$$\mathbb{E}_{\mathcal{C}}\left\{ \bar{p}_{\text{succ}}\left( \mathcal{C}\right) \right\} \geq\left\vert \left( 1-2\epsilon\right) \left(
2-e^{\frac{d}{D}\left\vert \mathcal{M}\right\vert }\right) \right\vert ^{2},$$ implying that the performance measure becomes arbitrarily close to one if $D/d$ is large, $\left\vert \mathcal{M}\right\vert \ll D/d$, and $\epsilon$ is arbitrarily small.
The proof of this lemma is similar to the GLM proof, and we thus place it in Appendix \[sec:sequential-packing-proof\].
Sequential Decoding for Entanglement-Assisted Communication
===========================================================
In this section, we show an application of the GLM sequential decoding scheme to entanglement-assisted classical communication by exploiting the coding approach of Hsieh *et al*. [@HDW08]. The approach thus gives another way of achieving the entanglement-assisted classical capacity of a quantum channel.
\[Entanglement-Assisted Sequential Decoding\]\[thm:ea-sequential\]The sequential decoding scheme can achieve the entanglement-assisted classical capacity of a quantum channel.
Suppose that a quantum channel $\mathcal{N}^{A^{\prime}\rightarrow A}$ connects Alice to Bob and that they share many copies of an arbitrary entangled pure state $\left\vert \phi\right\rangle ^{A^{\prime}A}$:$$\left\vert \phi\right\rangle ^{A^{\prime n}A^{n}}\equiv\left( \left\vert
\phi\right\rangle ^{A^{\prime}A}\right) ^{\otimes n}=\left\vert
\phi\right\rangle ^{A^{\prime}A}\otimes\left\vert \phi\right\rangle
^{A^{\prime}A}\otimes\dots\otimes\left\vert \phi\right\rangle ^{A^{\prime}A},$$ where Alice has access to the system $A^{\prime}$ and Bob has access to the system $A$. Alice chooses a message from her message set $\mathcal{M}$ uniformly at random, applies a corresponding encoder to her shares $A^{\prime
n}$ of the entanglement, and sends the systems $A^{\prime n}$ to Bob. Later in the analysis, we would like to be able to pull these encoding operations through the channel so that they are equivalent to some other operator acting at Bob’s end. In order to do this, we can write the many copies of the shared entanglement as a direct sum of maximally entangled states [@HDW08; @W11]. Starting from the Schmidt decomposition for one copy of the state $\left\vert \phi\right\rangle $$$\left\vert \phi\right\rangle ^{A^{\prime}A}=\sum_{z}\sqrt{p_{Z}\left(
z\right) }\left\vert z\right\rangle ^{A^{\prime}}\left\vert z\right\rangle
^{A},$$ we can derive the following using the method of types [@book1991cover; @W11]:$$\begin{aligned}
\left\vert \phi\right\rangle ^{A^{\prime n}A^{n}} & =\sum_{z^{n}}\sqrt{p_{Z^{n}}\left( z^{n}\right) }\left\vert z^{n}\right\rangle
^{A^{\prime n}}\left\vert z^{n}\right\rangle ^{A^{n}}\\
& =\sum_{t}\sum_{z^{n}\in T_{t}}\sqrt{p_{Z^{n}}\left( z^{n}\right)
}\left\vert z^{n}\right\rangle ^{A^{\prime n}}\left\vert z^{n}\right\rangle
^{A^{n}}\\
& =\sum_{t}\sqrt{p_{Z^{n}}\left( z_{t}^{n}\right) d_{t}}\frac{1}{d_{t}}\sum_{z^{n}\in T_{t}}\left\vert z^{n}\right\rangle ^{A^{\prime n}}\left\vert
z^{n}\right\rangle ^{A^{n}}\\
& =\sum_{t}\sqrt{p\left( t\right) }\left\vert \Phi_{t}\right\rangle
^{A^{\prime n}A^{n}},\end{aligned}$$ where$$p\left( t\right) \equiv p_{Z^{n}}\left( z_{t}^{n}\right) d_{t},$$ $T_{t}$ is a type class, $d_{t}$ is the dimension of a type class subspace $t$, $z_{t}^{n}$ is a representative sequence for the type class $t$, and each $\left\vert \Phi_{t}\right\rangle ^{A^{\prime n}A^{n}}$ is maximally entangled on the type class subspace specified by $t$ (see Refs. [@HDW08; @W11] for more details on this approach). Thus, applying an operator acting on type class subspaces at Alice’s end is equivalent to applying the transpose of the same operator at Bob’s end. As in Refs. [@HDW08; @W11], Alice constructs her encoders using the Heisenberg-Weyl set of operators $\left\{ X\left(
x_{t}\right) Z\left( z_{t}\right) \right\} _{x_{t},z_{t}}$ that act on each of the type class subspaces$$U\left( s\right) \equiv\bigoplus_{t}\left( -1\right) ^{b_{t}}X\left(
x_{t}\right) Z\left( z_{t}\right) ,$$ where $b_{t}$ determines a phase that is applied to the operators in each subspace. We denote this unitary by $U\left( s\right) $ where $s$ is some vector that contains all the needed indices $x_{t}$, $z_{t}$ and $b_{t}$. Let $\mathcal{S}$ denote the set of all such possible vectors. We construct a random code $\left\{ s_{m}\right\} _{m\in\mathcal{M}}$ where $s_{m}$ is a vector chosen uniformly at random from $\mathcal{S}$ and the corresponding set of encoders is then $\left\{ U\left( s_{m}\right) \right\} _{m\in
\mathcal{M}}$. Since the transpose trick holds for each of these unitaries, we have that $$U\left( s\right) ^{A^{\prime n}}\left\vert \phi\right\rangle ^{A^{\prime
n}A^{n}}=U^{T}\left( s\right) ^{A^{n}}\left\vert \phi\right\rangle
^{A^{\prime n}A^{n}}.$$ The induced ensemble at Bob’s end is then $$\left\{ \frac{1}{\left\vert \mathcal{S}\right\vert },\sigma_{s}\right\}
_{s\in\mathcal{S}},$$ where$$\begin{aligned}
\sigma_{s} & \equiv U^{T}\left( s\right) ^{A^{n}}\rho^{A^{n}B^{n}}U^{\ast
}\left( s\right) ^{A^{n}},\\
\rho^{A^{n}B^{n}} & \equiv\mathcal{N}^{A^{\prime n}\rightarrow B^{n}}\left(
\left\vert \phi\right\rangle \left\langle \phi\right\vert ^{A^{\prime n}A^{n}}\right) .\end{aligned}$$ Let $\overline{\sigma}$ denote the expected state of the ensemble:$$\overline{\sigma}\equiv\frac{1}{\left\vert \mathcal{S}\right\vert }\sum
_{s\in\mathcal{S}}\sigma_{s}.$$ We give Bob the following code subspace projector:$$\Pi\equiv\Pi_{\delta}^{A^{n}}\otimes\Pi_{\delta}^{B^{n}},
\label{eq:EA-code-projector}$$ and the codeword subspace projectors:$$\Pi_{s}\equiv U^{T}\left( s\right) ^{A^{n}}\Pi_{\delta}^{A^{n}B^{n}}U^{\ast
}\left( s\right) ^{A^{n}}, \label{eq:EA-message-proj}$$ where $\Pi_{\delta}^{A^{n}B^{n}}$, $\Pi_{\delta}^{A^{n}}$, and $\Pi_{\delta
}^{B^{n}}$ are the $\delta$-typical projectors for many copies of the states $\rho^{A^{n}B^{n}}$, $\rho^{A^{n}}=\mathrm{Tr}_{B}\left\{ \rho^{A^{n}B^{n}}\right\} $ and $\rho^{B^{n}}=\mathrm{Tr}_{A}\left\{ \rho^{A^{n}B^{n}}\right\} $, respectively.
At this point we would like to apply our packing argument from Theorem \[thm:sequential-packing\] and we would like to have the following conditions hold:$$\begin{aligned}
\mathrm{Tr}\left\{ \Pi\sigma_{s}\right\} & \geq1-\epsilon,\\
\mathrm{Tr}\left\{ \Pi_{s}\sigma_{s}\right\} & \geq1-\epsilon,\\
\Pi\overline{\sigma}\Pi & \leq2^{-n\left( H\left( A\right) _{\rho
}+H\left( B\right) _{\rho}-\eta\left( n,\delta\right) -\delta\right) }\Pi\\
\Pi_{s}\sigma_{s}\Pi_{s} & \geq2^{-n\left( H\left( AB\right) _{\rho
}+\delta\right) }\Pi_{s},\\
\left[ \Pi_{s},\sigma_{s}\right] & =0,\end{aligned}$$ where the function $\eta\left( n,\delta\right) $ goes to zero as $n\rightarrow\infty$ and $\delta\rightarrow0$. The first three conditions are shown in Refs. [@HDW08; @W11]. The fourth condition follows from the equipartition property of typical subspaces [@W11] and the fact that $U^{T}U^{\ast}=I$ for any unitary operator $U$. The fifth condition follows from the fact that the projector $\Pi_{s}$ commutes with the density operator $\sigma_{s}$. By our packing argument in Theorem \[thm:sequential-packing\] that gives a bound on the expectation of the average success probability, there exists a particular code, with which Alice can transmit messages from her set $\mathcal{M}$ and Bob can detect the transmitted state by performing a series of projective measurements, with its average success probability being greater than$$\begin{aligned}
\bar{p}_{\text{succ}} & \geq\left\vert \left( 1-2\epsilon\right) \left(
2-\exp\left\{ 2^{-n\left( H\left( A\right) _{\rho}+H\left( B\right)
_{\rho}-H\left( AB\right) _{\rho}-\eta\left( n,\delta\right)
-2\delta\right) }\left\vert \mathcal{M}\right\vert \right\} \right)
\right\vert ^{2}\\
& =\left\vert \left( 1-2\epsilon\right) \left( 2-\exp\left\{ 2^{-n\left(
I\left( A\,;\,B\right) _{\rho}-\eta\left( n,\delta\right) -2\delta\right)
}\left\vert \mathcal{M}\right\vert \right\} \right) \right\vert ^{2}$$ Therefore, Alice can pick the size of $\mathcal{M}$ to be $2^{n\left(
I\left( A\,;\,B\right) _{\rho}-\eta\left( n,\delta\right) -3\delta\right)
}$, and the rate of communication is then $$C=\frac{1}{n}\log_{2}\left\vert \mathcal{M}\right\vert =I\left( A;B\right)
_{\rho}-\eta\left( n,\delta\right) -3\delta,$$ with the average success probability becoming greater than$$\bar{p}_{\text{succ}}\geq\left\vert \left( 1-2\epsilon\right) \left(
2-\exp\left\{ 2^{-n\delta}\right\} \right) \right\vert ^{2}.$$ Thus, for sufficiently large $n$, the sequential decoding scheme achieves the entanglement-assisted classical capacity with arbitrarily high success probability.
As a final note, we should clarify a bit further: there is a codebook $\left\{ U\left( s_{m}\right) \right\} _{m\in\mathcal{M}}$ for Alice with entanglement-assisted quantum codewords of the following form:$$U^{A^{\prime n}}\left( s_{m}\right) \left\vert \phi\right\rangle ^{A^{\prime
n}A^{n}}.$$ If Alice sends message $m$, Bob performs a sequence of measurements in the following order (assuming a correct sequence of events):$$\Pi\rightarrow I-\Pi_{s_{1}}\rightarrow\Pi\rightarrow I-\Pi_{s_{2}}\rightarrow\Pi\rightarrow\cdots\rightarrow\Pi\rightarrow\Pi_{s_{m}},$$ with $\Pi$ and $\Pi_{s_{i}}$ of the form in (\[eq:EA-code-projector\]) and (\[eq:EA-message-proj\]), respectively.
Packing Argument for Sequential and Successive Decoding over a Multiple Access Channel
======================================================================================
We now extend the packing argument from Section \[sec:sequential-packing\] to a multiple-access setting, in which there are two senders and one receiver. The resulting scheme is both sequential and successive—sequential in the above sense where the receiver linearly tests one codeword at a time and successive in the sense that the receiver first decodes one sender’s message and follows by decoding the other sender’s message. After doing so, we then briefly remark how this argument achieves the known strategies for both unassisted [@W01] and assisted classical communication [@HDW08].
\[Sequential and Successive Decoding\]Suppose there exists a doubly-indexed ensemble of quantum states, where two independent distributions generate the different indices $x$ and $y$:$$\left\{ p_{X}\left( x\right) p_{Y}\left( y\right) ,\rho_{x,y}\right\} .$$ Averaging with the distributions $p_{X}\left( x\right) $ and $p_{Y}\left(
y\right) $ leads to the following states:$$\rho_{x}\equiv\sum_{y}p_{Y}\left( y\right) \rho_{x,y},\ \ \ \ \ \ \ \ \ \ \rho_{y}\equiv\sum_{x}p_{X}\left( x\right) \rho
_{x,y},\ \ \ \ \ \ \ \ \ \ \rho\equiv\sum_{x,y}p_{X}\left( x\right)
p_{Y}\left( y\right) \rho_{x,y}.$$ Suppose that there exist projectors $\Pi_{x}$, $\Pi_{y}$, $\Pi_{x,y}$, and $\Pi$ such that$$\begin{aligned}
\mathrm{Tr}\left\{ \Pi\rho_{x}\right\} & \geq1-\epsilon,\\
\mathrm{Tr}\left\{ \Pi_{x}\rho_{x}\right\} & \geq1-\epsilon,\\
\Pi_{x}\rho_{x}\Pi_{x} & \geq\frac{1}{d_{1}^{\left( -\right) }}\Pi_{x},\\
\Pi\rho\Pi & \leq\frac{1}{D_{1}}\Pi,\\
\lbrack\Pi_{x},\rho_{x}] & =0.\end{aligned}$$ and$$\begin{aligned}
\mathrm{Tr}\left\{ \Pi_{x}\rho_{x,y}\right\} & \geq1-\epsilon,\\
\mathrm{Tr}\left\{ \Pi_{x,y}\rho_{x,y}\right\} & \geq1-\epsilon,\\
\Pi_{x,y}\rho_{x,y}\Pi_{x,y} & \geq\frac{1}{d_{2}}\Pi_{x,y},\\
\Pi_{x}\rho_{x}\Pi_{x} & \leq\frac{1}{d_{1}^{\left( +\right) }}\Pi_{x},\\
\lbrack\Pi_{x,y},\rho_{x,y}] & =0.\end{aligned}$$ Suppose that $D_{1}/d_{1}^{\left( -\right) }$ is large, $\left\vert
\mathcal{L}\right\vert \ll D_{1}/d_{1}^{\left( -\right) }$, $d_{1}^{\left(
+\right) }/d_{2}$ is large, $\left\vert \mathcal{M}\right\vert \ll
d_{1}^{\left( +\right) }/d_{2}$, and $\epsilon$ is arbitrarily small. Then there exists a sequential and successive decoding scheme for the receiver that succeeds with high probability, in the sense that the expectation of the average success probability is arbitrarily high:$$\mathbb{E}_{\mathcal{C}}\left\{ \bar{p}_{\text{succ}}\left( \mathcal{C}\right) \right\} \geq\left\vert \left( 1-2\epsilon\right) \left(
2-e^{d_{2}\left\vert \mathcal{M}\right\vert /d_{1}^{\left( +\right) }}\right) \right\vert ^{2}-2\sqrt{2\left( \epsilon+\epsilon^{\prime}\right)
},$$ with $\epsilon^{\prime}$ chosen so that$$2-e^{d_{1}^{\left( -\right) }\left\vert \mathcal{L}\right\vert /D_{1}}\geq1-\epsilon^{\prime}.$$
The random construction of the code is similar to that in the proof of Theorem \[thm:sequential-packing\]. Given a message set $\mathcal{L}=\left\{ 1,2,\dots,\left\vert \mathcal{L}\right\vert \right\} $, we construct a code $\mathcal{C}_{1}\equiv\left\{ x\left( l\right) \right\}
_{l\in\mathcal{L}}$ for Alice randomly such that each $x\left( l\right) $ takes a value $x\in\mathcal{X}$ with probability $p_{X}\left( x\right) $. Similarly, given a message set $\mathcal{M}=\left\{ 1,2,\dots,\left\vert
\mathcal{M}\right\vert \right\} $, we construct a code $\mathcal{C}_{2}\equiv\left\{ y\left( m\right) \right\} _{m\in\mathcal{M}}$ for Bob randomly such that each $y\left( m\right) $ takes a value $y\in\mathcal{Y}$ with probability $p_{Y}\left( y\right) $. Using this code, Alice chooses a message $l$ from the message set $\mathcal{L}$, Bob chooses a message $m$ from the message set $\mathcal{M}$, and they encode their messages in the quantum codeword $\rho_{x\left( l\right) ,y\left( m\right) }$.
Suppose that the first sender Alice transmits message $l$ and the second sender Bob transmits message $m$. Without loss of generality, the receiver first tries to recover the message that Alice transmits. In order to do so, he measures $\Pi$ followed by $\Pi_{x\left( 1\right) }$ to determine if the transmitted message corresponds to the first codeword of Alice, with $\Pi_{x\left( 1\right) }$ corresponding to the outcome YES and $Q_{x\left(
1\right) }\equiv I-\Pi_{x\left( 1\right) }$ corresponding to the outcome NO. Suppose that the outcome is NO. He then measures $\Pi$ to project the state back into the large subspace. Assuming a correct sequence of events, the receiver continues and measures $Q_{x\left( i\right) }$ and $\Pi$ for $i\in\left\{ 2,\ldots,l-1\right\} $ until getting to the correct outcome $\Pi_{x\left( l\right) }$. Thus, the sequence of projectors measured is as follows, under the assumption of a correct sequence of events:$$\Pi\rightarrow Q_{x\left( 1\right) }\rightarrow\Pi\rightarrow Q_{x\left(
2\right) }\rightarrow\Pi\rightarrow\cdots\rightarrow Q_{x\left( i\right)
}\rightarrow\Pi\rightarrow\cdots\rightarrow\Pi\rightarrow\Pi_{x\left(
l\right) }.$$ After receiving a YES outcome from $\Pi_{x\left( l\right) }$, the receiver assumes that the first sender transmitted message $l$. The receiver then tries to determine the codeword that Bob transmitted by exploiting the projectors $\Pi_{x\left( l\right) }$ and $\Pi_{x\left( l\right) ,y\left( j\right)
}$. He does this in a similar fashion as above, proceeding in the following order (again under the assumption of a correct sequence of events):$$Q_{x\left( l\right) ,y\left( 1\right) }\rightarrow\Pi_{x\left( l\right)
}\rightarrow Q_{x\left( l\right) ,y\left( 2\right) }\rightarrow
\Pi_{x\left( l\right) }\rightarrow\cdots\rightarrow Q_{x\left( l\right)
,y\left( i\right) }\rightarrow\Pi_{x\left( l\right) }\rightarrow
\cdots\rightarrow\Pi_{x\left( l\right) }\rightarrow\Pi_{x\left( l\right)
,y\left( m\right) }.$$
The POVM corresponding to the above measurement strategy is as follows:$$\Lambda_{l,m}\equiv M_{l,m}^{\dag}M_{l,m},$$ where$$\begin{aligned}
M_{l,m} & \equiv\Pi_{x\left( l\right) ,y\left( m\right) }\overline
{\overline{Q}}_{x\left( l\right) ,y\left( m-1\right) }\cdots
\overline{\overline{Q}}_{x\left( l\right) ,y\left( 1\right) }\Pi_{x\left(
l\right) }\overline{Q}_{x\left( l-1\right) }\cdots\overline{Q}_{x\left(
1\right) },\\
\overline{\Theta} & \equiv\Pi\Theta\Pi,\\
\overline{\overline{\Theta}} & \equiv\Pi_{x\left( l\right) }\Theta
\Pi_{x\left( l\right) }.\end{aligned}$$ The average success probability of any particular code $c$ is$$\bar{p}_{\text{succ}}\left( c\right) \equiv\frac{1}{\left\vert
\mathcal{L}\right\vert \left\vert \mathcal{M}\right\vert }\sum_{l,m}\text{Tr}\left\{ \Lambda_{l,m}\rho_{x\left( l\right) ,y\left( m\right)
}\right\} ,$$ and the expectation of the average success probability is$$\begin{gathered}
\mathbb{E}_{X,Y}\left\{ \bar{p}_{\text{succ}}\left( C\right) \right\}
=\sum_{\substack{x\left( 1\right) ,\ldots,x\left( \left\vert \mathcal{L}\right\vert \right) ,\\y\left( 1\right) ,\ldots,y\left( \left\vert
\mathcal{M}\right\vert \right) }}p_{X}\left( x\left( 1\right) \right)
\cdots p_{X}\left( x\left( \left\vert \mathcal{L}\right\vert \right)
\right) p_{Y}\left( y\left( 1\right) \right) \cdots\label{eq:succ-prob-1}\\
\cdots p_{Y}\left( y\left( \left\vert \mathcal{M}\right\vert \right)
\right) \frac{1}{\left\vert \mathcal{L}\right\vert \left\vert \mathcal{M}\right\vert }\sum_{l,m}\text{Tr}\left\{ \Lambda_{l,m}\rho_{x\left( l\right)
,y\left( m\right) }\right\} .\end{gathered}$$$$=\frac{1}{\left\vert \mathcal{L}\right\vert \left\vert \mathcal{M}\right\vert
}\sum_{l,m}\sum_{x,y}p_{X}\left( x\right) p_{Y}\left( y\right)
\text{Tr}\left\{ \Psi_{x}^{m-1}\left( \Pi_{x}\Phi^{l-1}\left( \rho
_{x,y}\right) \Pi_{x}\right) \Pi_{x,y}\right\} ,$$ where$$\begin{aligned}
\Phi\left( \cdot\right) & \equiv\sum_{x}p_{X}\left( x\right)
\overline{Q}_{x}\left( \cdot\right) \overline{Q}_{x},\\
\Psi_{x}\left( \cdot\right) & \equiv\sum_{y}p_{Y}\left( y\right)
\overline{\overline{Q}}_{x,y}\left( \cdot\right) \overline{\overline{Q}}_{x,y}.\end{aligned}$$ Observe that we can rewrite the success probability in (\[eq:succ-prob-1\]) as follows:$$\sum_{\substack{x\left( 1\right) ,\ldots,x\left( \left\vert \mathcal{L}\right\vert \right) ,\\y\left( 1\right) ,\ldots,y\left( \left\vert
\mathcal{M}\right\vert \right) }}p_{X}\left( x\left( 1\right) \right)
\cdots p_{X}\left( x\left( \left\vert \mathcal{L}\right\vert \right)
\right) p_{Y}\left( y\left( 1\right) \right) \cdots p_{Y}\left( y\left(
\left\vert \mathcal{M}\right\vert \right) \right) \frac{1}{\left\vert
\mathcal{L}\right\vert \left\vert \mathcal{M}\right\vert }\sum_{l,m}\text{Tr}\left\{ \Gamma_{x,y,l,m}\omega_{x,y,l,m}\right\} ,
\label{eq:succ-prob-2}$$ where$$\begin{aligned}
\omega_{x,y,l,m} & \equiv\Pi_{x\left( l\right) }\overline{Q}_{x\left(
l-1\right) }\cdots\overline{Q}_{x\left( 1\right) }\rho_{x\left( l\right)
,y\left( m\right) }\overline{Q}_{x\left( 1\right) }\cdots\overline
{Q}_{x\left( l-1\right) }\Pi_{x\left( l\right) },\\
\Gamma_{x,y,l,m} & \equiv\overline{\overline{Q}}_{x\left( l\right)
,y\left( 1\right) }\cdots\overline{\overline{Q}}_{x\left( l\right)
,y\left( m-1\right) }\Pi_{x\left( l\right) ,y\left( m\right) }\overline{\overline{Q}}_{x\left( l\right) ,y\left( m-1\right) }\cdots\overline{\overline{Q}}_{x\left( l\right) ,y\left( 1\right) }.\end{aligned}$$ We can then obtain the following lower bound on (\[eq:succ-prob-2\]):$$\begin{gathered}
\sum_{\substack{x\left( 1\right) ,\ldots,x\left( \left\vert \mathcal{L}\right\vert \right) ,\\y\left( 1\right) ,\ldots,y\left( \left\vert
\mathcal{M}\right\vert \right) }}p_{X}\left( x\left( 1\right) \right)
\cdots p_{X}\left( x\left( \left\vert \mathcal{L}\right\vert \right)
\right) p_{Y}\left( y\left( 1\right) \right) \cdots
\label{eq:lower-bound-seq-mac}\\
\cdots p_{Y}\left( y\left( \left\vert \mathcal{M}\right\vert \right)
\right) \frac{1}{\left\vert \mathcal{L}\right\vert \left\vert \mathcal{M}\right\vert }\sum_{l,m}\left[ \text{Tr}\left\{ \Gamma_{x,y,l,m}\rho_{x\left( l\right) ,y\left( m\right) }\right\} -\left\Vert
\rho_{x\left( l\right) ,y\left( m\right) }-\omega_{x,y,l,m}\right\Vert
_{1}\right] ,\end{gathered}$$ by exploiting the following inequality:$$\text{Tr}\left\{ \Gamma_{x,y,l,m}\omega_{x,y,l,m}\right\} \geq
\text{Tr}\left\{ \Gamma_{x,y,l,m}\rho_{x\left( l\right) ,y\left( m\right)
}\right\} -\left\Vert \rho_{x\left( l\right) ,y\left( m\right) }-\omega_{x,y,l,m}\right\Vert _{1},$$ which holds for all positive operators $\Gamma_{x,y,l,m}$, $\omega_{x,y,l,m}$, and $\rho_{x\left( l\right) ,y\left( m\right) }$ that have spectrum less than one. So it remains to show that both Tr$\left\{ \Gamma_{x,y,l,m}\rho_{x\left( l\right) ,y\left( m\right) }\right\} $ is arbitrarily close to one and $\left\Vert \rho_{x\left( l\right) ,y\left( m\right) }-\omega_{x,y,l,m}\right\Vert _{1}$ is arbitrarily small when averaging over all codewords and taking the expectation over random codes. We can apply Theorem \[thm:sequential-packing\] to obtain the following inequality:$$\begin{aligned}
& \sum_{\substack{x\left( 1\right) ,\ldots,x\left( \left\vert
\mathcal{L}\right\vert \right) ,\\y\left( 1\right) ,\ldots,y\left(
\left\vert \mathcal{M}\right\vert \right) }}p_{X}\left( x\left( 1\right)
\right) \cdots p_{X}\left( x\left( \left\vert \mathcal{L}\right\vert
\right) \right) p_{Y}\left( y\left( 1\right) \right) \cdots p_{Y}\left(
y\left( \left\vert \mathcal{M}\right\vert \right) \right) \text{Tr}\left\{
\omega_{x,y,l,m}\right\} \\
& \geq\left\vert \left( 1-\epsilon\right) \left( 2-e^{d_{1}^{\left(
-\right) }\left\vert \mathcal{L}\right\vert /D_{1}}\right) \right\vert
^{2}\\
& \geq\left\vert \left( 1-\epsilon\right) \left( 1-\epsilon^{\prime
}\right) \right\vert ^{2}\\
& \geq1-2\left( \epsilon+\epsilon^{\prime}\right) ,\end{aligned}$$ with $\epsilon^{\prime}$ chosen as given in the statement of the theorem. We can then apply the Gentle Operator Lemma for ensembles (Lemma 9.4.3 in Ref. [@W11]) to prove the following inequality:$$\sum_{\substack{x\left( 1\right) ,\ldots,x\left( \left\vert \mathcal{L}\right\vert \right) ,\\y\left( 1\right) ,\ldots,y\left( \left\vert
\mathcal{M}\right\vert \right) }}p_{X}\left( x\left( 1\right) \right)
\cdots p_{X}\left( x\left( \left\vert \mathcal{L}\right\vert \right)
\right) p_{Y}\left( y\left( 1\right) \right) \cdots p_{Y}\left( y\left(
\left\vert \mathcal{M}\right\vert \right) \right) \frac{1}{\left\vert
\mathcal{L}\right\vert \left\vert \mathcal{M}\right\vert }\left\Vert
\rho_{x\left( l\right) ,y\left( m\right) }-\omega_{x,y,l,m}\right\Vert
_{1}\leq2\sqrt{2\left( \epsilon+\epsilon^{\prime}\right) }.$$ Invoking Theorem \[thm:sequential-packing\] one more time gives us the following lower bound:$$\begin{gathered}
\sum_{\substack{x\left( 1\right) ,\ldots,x\left( \left\vert \mathcal{L}\right\vert \right) ,\\y\left( 1\right) ,\ldots,y\left( \left\vert
\mathcal{M}\right\vert \right) }}p_{X}\left( x\left( 1\right) \right)
\cdots p_{X}\left( x\left( \left\vert \mathcal{L}\right\vert \right)
\right) p_{Y}\left( y\left( 1\right) \right) \cdots p_{Y}\left( y\left(
\left\vert \mathcal{M}\right\vert \right) \right) \frac{1}{\left\vert
\mathcal{L}\right\vert \left\vert \mathcal{M}\right\vert }\sum_{l,m}\text{Tr}\left\{ \Gamma_{x,y,l,m}\rho_{x\left( l\right) ,y\left( m\right)
}\right\} \\
\geq\left\vert \left( 1-2\epsilon\right) \left( 2-e^{d_{2}\left\vert
\mathcal{M}\right\vert /d_{1}^{\left( +\right) }}\right) \right\vert ^{2},\end{gathered}$$ and this completes the proof of the theorem, by combining the above two inequalities with the lower bound in (\[eq:lower-bound-seq-mac\]).
It is straightforward to apply this packing argument to either unassisted or assisted transmission of classical information over a quantum multiple access channel. For the unassisted case, one could exploit Winter’s coding scheme with conditionally typical projectors [@W01], and we would pick the parameters as$$\begin{aligned}
D_{1} & =2^{n\left[ H\left( B\right) -\delta\right] },\\
d_{1}^{\left( +\right) } & =2^{n\left[ H\left( B|X\right)
-\delta\right] },\\
d_{1}^{\left( -\right) } & =2^{n\left[ H\left( B|X\right)
+\delta\right] },\\
d_{2} & =2^{n\left[ H\left( B|XY\right) +\delta\right] },\end{aligned}$$ so that we would have$$\begin{aligned}
D_{1}/d_{1}^{\left( -\right) } & =2^{n\left[ I\left( X;B\right)
-2\delta\right] },\\
d_{1}^{\left( +\right) }/d_{2} & =2^{n\left[ I\left( Y;B|X\right)
-2\delta\right] }.\end{aligned}$$ For the entanglement-assisted case, one could exploit the coding structure of Hsieh *et al*. [@HDW08] that we have discussed throughout this article, and we would pick the parameters as$$\begin{aligned}
D_{1} & =2^{n\left[ H\left( A\right) +H\left( B\right) +H\left(
C\right) -\delta\right] },\\
d_{1}^{\left( +\right) } & =2^{n\left[ H\left( B\right) +H\left(
AC\right) -\delta\right] },\\
d_{1}^{\left( -\right) } & =2^{n\left[ H\left( B\right) +H\left(
AC\right) +\delta\right] },\\
d_{2} & =2^{n\left[ H\left( ABC\right) +\delta\right] },\end{aligned}$$ so that we would have$$\begin{aligned}
D_{1}/d_{1}^{\left( -\right) } & =2^{n\left[ I\left( A;C\right)
-2\delta\right] },\\
d_{1}^{\left( +\right) }/d_{2} & =2^{n\left[ I\left( B;AC\right)
-2\delta\right] }.\end{aligned}$$
Entanglement-Assisted Quantum Simultaneous Decoding {#sec:ea-simul}
===================================================
In this section, we prove the existence of a simultaneous decoder for entanglement-assisted classical communication over a quantum multiple access channel with two senders. A simultaneous decoder differs from a successive decoder in the sense that such a decoder allows for the receiver to reliably detect the messages of both senders with a single measurement as long as the rates are within the pentagonal rate region specified by Theorem 6 of Ref. [@HDW08] and Theorem \[thm:simul-decoder\] below (it might also be helpful to consult Ref. [@el2010lecture] to see the difference between classical successive and simultaneous decoders). The advantage of a simultaneous decoder over a successive decoder is that there is no need to invoke time-sharing in order to achieve the Hsieh-Devetak-Winter rate region of the entanglement-assisted multiple access channel in Ref. [@HDW08]. Also, an analogous classical decoder is required in order to achieve the Han-Kobayashi rate region for the classical interference channel [@HK81] (though it requires a simultaneous decoder for three senders).
Concerning the quantum interference channel, Fawzi *et al*. made progress towards demonstrating that a quantized version of the classical Han-Kobayashi rate region is achievable for classical communication over a quantum interference channel [@FHSSW11], though they were only able to prove this result up to a conjecture regarding the existence of a quantum simultaneous decoder for general channels. The importance of this conjecture stems not only from the fact that it would allow for a quantization of the Han-Kobayashi rate region, but also more broadly from the fact that many coding theorems in classical network information theory exploit the simultaneous decoding technique [@el2010lecture]. Thus, having a general quantum simultaneous decoder for an arbitrary number of senders should allow for the wholesale import of much of classical network information theory into quantum network information theory.
Our result below applies only to channels with two senders, and the technique unfortunately does not generally extend to channels with three senders. Thus, this important case still remains open as a conjecture. Sen independently arrived at the results here by exploiting both the proof structure outlined below and a different technique as well [@S11a].
\[Entanglement-Assisted Simultaneous Decoding\]\[thm:simul-decoder\]Suppose that Alice and Charlie share many copies of an entangled pure state $\left\vert \phi\right\rangle ^{A^{\prime}A}$ where Alice has access to the system $A^{\prime}$ and Charlie has access to the system $A$. Similarly, let Bob and Charlie share many copies of an entangled pure state $\left\vert
\psi\right\rangle ^{B^{\prime}B}$. Let $\mathcal{N}^{A^{\prime}B^{\prime
}\rightarrow C}$ be a multiple access channel that connects Alice and Bob to Charlie, and let$$\rho^{ABC}\equiv\mathcal{N}^{A^{\prime}B^{\prime}\rightarrow C}\left(
\left\vert \phi\right\rangle \left\langle \phi\right\vert ^{A^{\prime}A}\otimes\left\vert \psi\right\rangle \left\langle \psi\right\vert
^{B^{\prime}B}\right) . \label{eq:code-state}$$ Then there exists an entanglement-assisted classical communication code with a corresponding quantum simultaneous decoder, such that the following rate region is achievable for $R_{1},R_{2}\geq0$:$$\begin{aligned}
R_{1} & \leq I\left( A;C|B\right) _{\rho},\\
R_{2} & \leq I\left( B;C|A\right) _{\rho},\\
R_{1}+R_{2} & \leq I\left( AB;C\right) _{\rho},\end{aligned}$$ where the entropies are with respect to the state in (\[eq:code-state\]).
Suppose that Alice has a message set $\mathcal{L}$ and Bob has a message set $\mathcal{M}$ from which they will each choose a message $l\in\mathcal{L}$ and $m\in\mathcal{M}$ uniformly at random to send to Charlie. They construct random codes $C_{1}\equiv\left\{ s_{1}\left( l\right) \right\}
_{l\in\mathcal{L}}$ and $C_{2}\equiv\left\{ s_{2}\left( m\right) \right\}
_{m\in\mathcal{M}}$ in the same way as explained in the proof of Theorem \[thm:ea-sequential\]. Both of them encode their messages by applying unitary encoders to their respective shares of the entanglement, giving rise to the following states after applying the transpose trick to each type class [@HDW08; @W11]:$$\begin{aligned}
\left( U\left( s_{1}\left( l\right) \right) ^{A^{\prime n}}\otimes
I^{A^{n}}\right) \left\vert \phi\right\rangle ^{A^{\prime n}A^{n}} &
=\left( I^{A^{\prime n}}\otimes U^{T}\left( s_{1}\left( l\right) \right)
^{A^{n}}\right) \left\vert \phi\right\rangle ^{A^{\prime n}A^{n}},\\
\left( U\left( s_{2}\left( m\right) \right) ^{B^{\prime n}}\otimes
I^{B^{n}}\right) \left\vert \psi\right\rangle ^{B^{\prime n}B^{n}} &
=\left( I^{B^{\prime n}}\otimes U^{T}\left( s_{2}\left( m\right) \right)
^{B^{n}}\right) \left\vert \psi\right\rangle ^{B^{\prime n}B^{n}}.\end{aligned}$$ Then they both send their share of the state to Charlie over the multiple access channel $\mathcal{N}^{A^{\prime}B^{\prime}\rightarrow C}$, giving rise to a state $\sigma_{l,m}$ at Charlie’s receiving end:$$\sigma_{l,m}\equiv\left( U^{T}\left( s_{1}\left( l\right) \right)
^{A^{n}}\otimes U^{T}\left( s_{2}\left( m\right) \right) ^{B^{n}}\right)
\rho^{A^{n}B^{n}C^{n}}\left( U^{\ast}\left( s_{1}\left( l\right) \right)
^{A^{n}}\otimes U^{\ast}\left( s_{2}\left( m\right) \right) ^{B^{n}}\right) .$$
Charlie decodes with a simultaneous decoding POVM $\left\{ \Lambda
_{l,m}\right\} _{l\in\mathcal{L},m\in\mathcal{M}}$, defined as follows:$$\Lambda_{l,m}\equiv\left( \sum_{l^{\prime},m^{\prime}}\Upsilon_{l^{\prime
},m^{\prime}}\right) ^{-\frac{1}{2}}\Upsilon_{l,m}\left( \sum_{l^{\prime
},m^{\prime}}\Upsilon_{l^{\prime},m^{\prime}}\right) ^{-\frac{1}{2}},$$ where$$\Upsilon_{l,m}\equiv U^{T}\left( s_{1}\left( l\right) \right) ^{A^{n}}\hat{\Pi}_{3}\hat{\Pi}_{2}U^{T}\left( s_{2}\left( m\right) \right)
^{B^{n}}\Pi^{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{2}\left( m\right) \right)
^{B^{n}}\hat{\Pi}_{2}\hat{\Pi}_{3}U^{\ast}\left( s_{1}\left( l\right)
\right) ^{A^{n}},$$ and$$\begin{aligned}
\hat{\Pi}_{1} & \equiv\left( \Pi^{A^{n}}\otimes\Pi^{B^{n}C^{n}}\right) ,\\
\hat{\Pi}_{2} & \equiv\left( \Pi^{B^{n}}\otimes\Pi^{A^{n}C^{n}}\right) ,\\
\hat{\Pi}_{3} & \equiv\left( \Pi^{C^{n}}\otimes\Pi^{A^{n}B^{n}}\right) .\end{aligned}$$ The projectors $\Pi^{A^{n}}$, $\Pi^{B^{n}}$, $\Pi^{C^{n}}$, $\Pi^{A^{n}B^{n}}$, $\Pi^{A^{n}C^{n}}$, $\Pi^{B^{n}C^{n}}$ and $\Pi^{A^{n}B^{n}C^{n}}$ are $\delta$-typical projectors for the state $\rho^{A^{n}B^{n}C^{n}}$ onto the specified systems after tracing out all other systems.
The average error probability when Alice and Bob choose their messages independently and uniformly at random is$$\bar{p}_{e}\equiv\frac{1}{\left\vert \mathcal{L}\right\vert \cdot\left\vert
\mathcal{M}\right\vert }\sum_{l,m}\mathrm{Tr}\left\{ \left( I-\Lambda
_{l,m}\right) \sigma_{l,m}\right\} .\label{eq:avg-error-crit}$$ We can upper bound this error probability from above[^1] as$$\begin{aligned}
\bar{p}_{e} & \leq\frac{1}{\left\vert \mathcal{L}\right\vert \cdot\left\vert
\mathcal{M}\right\vert }\sum_{l,m}\mathrm{Tr}\left\{ \left( I-\Lambda
_{l,m}\right) U^{T}\left( s_{2}\left( m\right) \right) \hat{\Pi}_{1}U^{\ast}\left( s_{2}\left( m\right) \right) \sigma_{l,m}U^{T}\left(
s_{2}\left( m\right) \right) \hat{\Pi}_{1}U^{\ast}\left( s_{2}\left(
m\right) \right) \right\} +\nonumber\\
& \qquad\qquad\left\Vert U^{T}\left( s_{2}\left( m\right) \right)
\hat{\Pi}_{1}U^{\ast}\left( s_{2}\left( m\right) \right) \sigma_{l,m}U^{T}\left( s_{2}\left( m\right) \right) \hat{\Pi}_{1}U^{\ast}\left(
s_{2}\left( m\right) \right) -\sigma_{l,m}\right\Vert _{1}\\
& \leq\frac{1}{\left\vert \mathcal{L}\right\vert \cdot\left\vert
\mathcal{M}\right\vert }\sum_{l,m}\mathrm{Tr}\left\{ \left( I-\Lambda
_{l,m}\right) \theta_{l,m}\right\} +2\sqrt{\epsilon^{\prime}},\end{aligned}$$ where we define$$\begin{aligned}
\theta_{l,m} & \equiv U^{T}\left( s_{2}\left( m\right) \right) ^{B^{n}}\hat{\Pi}_{1}U^{\ast}\left( s_{2}\left( m\right) \right) ^{B^{n}}\sigma_{l,m}U^{T}\left( s_{2}\left( m\right) \right) ^{B^{n}}\hat{\Pi}_{1}U^{\ast}\left( s_{2}\left( m\right) \right) ^{B^{n}}\\
& =U^{T}\left( s_{2}\left( m\right) \right) ^{B^{n}}\hat{\Pi}_{1}U^{T}\left( s_{1}\left( l\right) \right) ^{A^{n}}\rho^{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{1}\left( l\right) \right) ^{A^{n}}\hat{\Pi}_{1}U^{\ast
}\left( s_{2}\left( m\right) \right) ^{B^{n}}.\end{aligned}$$ The first inequality follows from the inequality$$\text{Tr}\left\{ \Gamma\rho\right\} \leq\text{Tr}\left\{ \Gamma
\sigma\right\} +\left\Vert \rho-\sigma\right\Vert _{1},$$ for any operators $0\leq\Gamma,\rho,\sigma\leq I$ (Corollary 9.1.1 of Ref. [@W11]). The second inequality follows from the properties of quantum typicality, the Gentle Operator Lemma (Lemma 9.4.2 of Ref. [@W11]), and the inequality $\mathrm{Tr}\hat{\{\Pi_{1}}U^{T}\left( s_{1}\left( l\right)
\right) ^{A^{n}}\rho^{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{1}\left( l\right)
\right) ^{A^{n}}\}\geq1-\epsilon^{\prime}$ proved in Ref. [@HDW08]. We now recall the Hayashi-Nagaoka operator inequality [@hayashi2003general] which holds for any positive operator $S$ and $T$ such that $0\leq S\leq I$ and $T\geq0$:$$I-\left( S+T\right) ^{-\frac{1}{2}}S\left( S+T\right) ^{-\frac{1}{2}}\leq2\left( I-S\right) +4T.$$ Setting$$\begin{aligned}
S & =\Upsilon_{l,m},\\
T & =\sum_{\left( l^{\prime},m^{\prime}\right) \neq\left( l,m\right)
}\Upsilon_{l^{\prime},m^{\prime}},\end{aligned}$$ and applying the Hayashi-Nagaoka operator inequality, we obtain the following upper bound on the error probability:$$\bar{p}_{e}\leq\frac{1}{\left\vert \mathcal{L}\right\vert \cdot\left\vert
\mathcal{M}\right\vert }\sum_{l,m}\left( 2\mathrm{Tr}\left\{ \left(
I-\Upsilon_{l,m}\right) \theta_{l,m}\right\} +4\sum_{\left( l^{\prime
},m^{\prime}\right) \neq\left( l,m\right) }\mathrm{Tr}\left\{
\Upsilon_{l^{\prime},m^{\prime}}\theta_{l,m}\right\} \right) +2\sqrt
{\epsilon}.\label{eq:after-HN}$$ Considering the first term $\mathrm{Tr}\left\{ \left( I-\Upsilon
_{l,m}\right) \theta_{l,m}\right\} $, we can prove that $$\mathrm{Tr}\left\{ \left( I-\Upsilon_{l,m}\right) \theta_{l,m}\right\}
\leq\epsilon^{\prime\prime},$$ where $\epsilon^{\prime\prime}$ approaches zero when $n$ becomes large. This inequality follows from the following inequalities$$\begin{aligned}
\text{Tr}\left\{ \hat{\Pi}_{1}U^{T}\left( s_{1}\left( l\right) \right)
^{A^{n}}\rho^{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{1}\left( l\right) \right)
^{A^{n}}\right\} & \geq1-2\epsilon,\\
\text{Tr}\left\{ \hat{\Pi}_{3}U^{T}\left( s_{2}\left( m\right) \right)
^{B^{n}}\rho^{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{2}\left( m\right) \right)
^{B^{n}}\right\} & \geq1-2\epsilon,\\
\text{Tr}\left\{ \hat{\Pi}_{2}U^{T}\left( s_{2}\left( m\right) \right)
^{B^{n}}\rho^{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{2}\left( m\right) \right)
^{B^{n}}\right\} & \geq1-2\epsilon,\\
\text{Tr}\left\{ \Pi^{A^{n}B^{n}C^{n}}\rho^{A^{n}B^{n}C^{n}}\right\} &
\geq1-\epsilon,\end{aligned}$$ (which can be proved with the methods of Ref. [@HDW08]) and by applying measurement on approximately close states (Corollary 9.1.1 of Ref. [@W11]) and the Gentle Operator Lemma (Lemma 9.4.2 of Ref. [@W11]) several times.
In order to analyze the second term $\sum_{\left( l^{\prime},m^{\prime
}\right) \neq\left( l,m\right) }\mathrm{Tr}\left\{ \Upsilon_{l^{\prime
},m^{\prime}}\theta_{l,m}\right\} $, we need to take the expectation over all random codes and make several observations about the behavior of the codeword states under the expectation. Note that the encoding unitaries after the transpose trick and the channel commute because they act on different systems, so we can apply the encoding unitaries first. To simplify the calculation, we first consider only applying a random encoding unitary to the system $A^{n}$:$$\begin{aligned}
& \mathbb{E}_{\mathcal{C}_{1}}\left\{ U^{T}\left( s\right) ^{A^{n}}\left\vert \phi\right\rangle \left\langle \phi\right\vert ^{A^{\prime n}A^{n}}U^{\ast}\left( s\right) ^{A^{n}}\right\} \nonumber\\
= & \frac{1}{\left\vert \mathcal{S}_{1}\right\vert }\sum_{s\in
\mathcal{S}_{1}}U^{T}\left( s\right) ^{A^{n}}\left( \sum_{t}\sqrt{p\left(
t\right) }\left\vert \Phi_{t}\right\rangle ^{A^{\prime n}A^{n}}\right)
\left( \sum_{t^{\prime}}\sqrt{p\left( t^{\prime}\right) }\left\langle
\Phi_{t^{\prime}}\right\vert ^{A^{\prime n}A^{n}}\right) U^{\ast}\left(
s\right) ^{A^{n}}\\
= & \sum_{t}p\left( t\right) \pi_{t}^{A^{\prime n}}\otimes\pi_{t}^{A^{n}}$$ where $\pi_{t}$ is the maximally mixed state on the type subspace $t$. To see why the last equality holds, we note that when $t=t^{\prime}$, averaging over all elements in $\mathcal{S}_{1}$ gives rise to the state $\mathrm{Tr}_{A^{n}}\left\{ \left\vert \Phi_{t}\right\rangle \left\langle \Phi_{t}\right\vert
^{A^{\prime n}A^{n}}\right\} \otimes\pi_{t}^{A^{n}}=\pi_{t}^{A^{\prime n}}\otimes\pi_{t}^{A^{n}}$; when $t\neq t^{\prime}$, it can be shown that the whole expression sums up to zero [@HDW08; @W11]. Now we can append the other state at Bob’s side and send the overall state through the channel. Therefore, we have that $$\begin{aligned}
\mathbb{E}_{\mathcal{C}_{1}}\left\{ U^{T}\left( s\right) ^{A^{n}}\rho^{A^{n}B^{n}C^{n}}U^{\ast}\left( s\right) ^{A^{n}}\right\} &
=\mathcal{N}^{A^{\prime n}B^{\prime n}\rightarrow C^{n}}\left( \sum
_{t}p\left( t\right) \pi_{t}^{A^{\prime n}}\otimes\pi_{t}^{A^{n}}\otimes
\psi^{B^{\prime n}B^{n}}\right) \\
& =\sum_{t}p\left( t\right) \pi_{t}^{A^{n}}\otimes\mathcal{N}^{A^{\prime
n}B^{\prime n}\rightarrow C^{n}}\left( \pi_{t}^{A^{n}}\otimes\psi^{B^{\prime
n}B^{n}}\right) .\end{aligned}$$ Now consider the above state sandwiched between the projectors $\hat{\Pi}_{1}$:$$\begin{aligned}
& \hat{\Pi}_{1}\mathbb{E}_{\mathcal{C}_{1}}\left\{ U^{T}\left( s\right)
^{A^{n}}\rho^{A^{n}B^{n}C^{n}}U^{\ast}\left( s\right) ^{A^{n}}\right\}
\hat{\Pi}_{1}\nonumber\\
& =\left( \Pi^{A^{n}}\otimes\Pi^{B^{n}C^{n}}\right) \left( \sum
_{t}p\left( t\right) \pi_{t}^{A^{n}}\otimes\mathcal{N}^{A^{\prime
n}B^{\prime n}\rightarrow C^{n}}\left( \pi_{t}^{A^{n}}\otimes\psi^{B^{\prime
n}B^{n}}\right) \right) \left( \Pi^{A^{n}}\otimes\Pi^{B^{n}C^{n}}\right) \\
& =\sum_{t}p\left( t\right) \left( \Pi^{A^{n}}\pi_{t}^{A^{n}}\Pi^{A^{n}}\right) \otimes\left( \Pi^{B^{n}C^{n}}\mathcal{N}^{A^{\prime n}B^{\prime
n}\rightarrow C^{n}}\left( \pi_{t}^{A^{n}}\otimes\psi^{B^{\prime n}B^{n}}\right) \Pi^{B^{n}C^{n}}\right)\end{aligned}$$ At this point, we note that $\pi_{t}^{A^{n}}=\Pi_{t}^{A^{n}}/\mathrm{Tr}\left\{ \Pi_{t}^{A^{n}}\right\} $, $\mathrm{Tr}\left\{ \Pi_{t}^{A^{n}}\right\} \geq2^{n\left( H\left( A\right) -\eta\left( n,\delta\right)
\right) }$ for a typical type $t$, and $\Pi^{A^{n}}\Pi_{t}^{A^{n}}\Pi^{A^{n}}\leq\Pi^{A^{n}}$, where $\Pi_{t}^{A^{n}}$ is a projector onto the $t^{\mathrm{th}}$ type class subspace. Therefore, the above expression is bounded from above by the following one:$$\begin{aligned}
& \leq2^{-n\left( H\left( A\right) _{\rho}-\eta\left( n,\delta\right)
\right) }\Pi^{A^{n}}\otimes\left( \Pi^{B^{n}C^{n}}\mathcal{N}^{A^{\prime
n}B^{\prime n}\rightarrow C^{n}}\left( \sum_{t}p\left( t\right) \left(
\pi_{t}^{A^{n}}\right) \otimes\psi^{B^{\prime n}B^{n}}\right) \Pi
^{B^{n}C^{n}}\right) \nonumber\\
& =2^{-n\left( H\left( A\right) _{\rho}-\eta\left( n,\delta\right)
\right) }\Pi^{A^{n}}\otimes\left( \Pi^{B^{n}C^{n}}\mathcal{N}^{A^{\prime
n}B^{\prime n}\rightarrow C^{n}}\left( \phi^{A^{\prime n}}\otimes
\psi^{B^{\prime n}B^{n}}\right) \Pi^{B^{n}C^{n}}\right) \\
& \leq2^{-n\left( H\left( A\right) _{\rho}+H\left( BC\right) _{\rho
}-\eta\left( n,\delta\right) -c\delta\right) }\hat{\Pi}_{1}.
\label{eq:pi1sandwich}$$ We also note that similar observations can be made when applying random encoding unitaries to the system $B^{\prime n}$ alone or to both the systems $A^{\prime n}$ and $B^{\prime n}$.
Now we proceed to bound the second term in the RHS of (\[eq:after-HN\]) from above by taking the expectation over the random codes $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$:$$\begin{gathered}
\mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \sum_{\left( l^{\prime
},m^{\prime}\right) \neq\left( l,m\right) }\mathrm{Tr}\left\{
\Upsilon_{l^{\prime},m^{\prime}}\theta_{l,m}\right\} \right\} \\
=\mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \sum_{l^{\prime}\neq
l}\mathrm{Tr}\left\{ \Upsilon_{l^{\prime},m}\theta_{l,m}\right\}
+\sum_{m^{\prime}\neq m}\mathrm{Tr}\left\{ \Upsilon_{l,m^{\prime}}\theta_{l,m}\right\} +\sum_{l^{\prime}\neq l,\ m^{\prime}\neq m}\mathrm{Tr}\left\{ \Upsilon_{l^{\prime},m^{\prime}}\theta_{l,m}\right\}
\right\} .\end{gathered}$$ We bound the first error on the RHS above, which corresponds to Charlie correctly identifying the message from Bob only:$$\begin{aligned}
& \mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \sum_{l^{\prime}\neq
l}\mathrm{Tr}\left\{ \Upsilon_{l^{\prime},m}\theta_{l,m}\right\} \right\}
\nonumber\\
& =\sum_{l^{\prime}\neq l}\mathbb{E}_{\mathcal{C}_{2}}\left\{ \mathrm{Tr}\left\{ \mathbb{E}_{\mathcal{C}_{1}}\left\{ \Upsilon_{l^{\prime},m}\right\}
\mathbb{E}_{\mathcal{C}_{1}}\left\{ \theta_{l,m}\right\} \right\} \right\}
\\
& =\sum_{l^{\prime}\neq l}\mathbb{E}_{\mathcal{C}_{2}}\left\{ \mathrm{Tr}\left\{ \mathbb{E}_{\mathcal{C}_{1}}\left\{ \Upsilon_{l^{\prime},m}\right\}
U^{T}\left( s_{2}\left( m\right) \right) ^{B^{n}}\hat{\Pi}_{1}\mathbb{E}_{\mathcal{C}_{1}}\left\{ U^{T}\left( s_{1}\left( l\right)
\right) ^{A^{n}}\rho^{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{1}\left( l\right)
\right) ^{A^{n}}\right\} \hat{\Pi}_{1}U^{\ast}\left( s_{2}\left( m\right)
\right) ^{B^{n}}\right\} \right\} \\
& \leq2^{-n\left( H\left( A\right) _{\rho}+H\left( BC\right) _{\rho
}-\eta\left( n,\delta\right) -c\delta\right) }\sum_{l^{\prime}\neq
l}\mathbb{E}_{\mathcal{C}_{2}}\left\{ \mathrm{Tr}\left\{ \mathbb{E}_{\mathcal{C}_{1}}\left\{ \Upsilon_{l^{\prime},m}\right\} U^{T}\left(
s_{2}\left( m\right) \right) ^{B^{n}}\hat{\Pi}_{1}U^{\ast}\left(
s_{2}\left( m\right) \right) ^{B^{n}}\right\} \right\} \\
& \leq2^{-n\left( H\left( A\right) _{\rho}+H\left( BC\right) _{\rho
}-\eta\left( n,\delta\right) -c\delta\right) }\sum_{l^{\prime}\neq
l}\mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \mathrm{Tr}\left\{
\Pi^{A^{n}B^{n}C^{n}}\right\} \right\} \\
& \leq2^{-n\left( H\left( A\right) _{\rho}+H\left( BC\right) _{\rho
}-H\left( ABC\right) _{\rho}-\eta\left( n,\delta\right) -2c\delta\right)
}\left\vert \mathcal{L}\right\vert \\
& =2^{-n\left( I\left( A;C\mid B\right) \right) _{\rho}-\eta\left(
n,\delta\right) -2c\delta}\left\vert \mathcal{L}\right\vert .\end{aligned}$$ The first equality follows from the fact that the codewords for messages $l$ and $l^{\prime}$ are different and therefore independent (because of the way that we randomly selected the code). The first inequality follows from our observation in (\[eq:pi1sandwich\]).
We now bound the second error term, which corresponds to Charlie correctly identifying the message from Alice only:$$\begin{aligned}
& \mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \sum_{m^{\prime}\neq
m}\mathrm{Tr}\left\{ \Upsilon_{l,m^{\prime}}\theta_{l,m}\right\} \right\}
\nonumber\\
& =\sum_{m^{\prime}\neq m}\mathbb{E}_{\mathcal{C}_{1}}\left\{ \mathrm{Tr}\left\{ \mathbb{E}_{\mathcal{C}_{2}}\left\{ \Upsilon_{l,m^{\prime}}\right\}
\mathbb{E}_{\mathcal{C}_{2}}\left\{ \theta_{l,m}\right\} \right\} \right\}
\\
& =\sum_{m^{\prime}\neq m}\mathbb{E}_{\mathcal{C}_{1}}\left\{ \mathrm{Tr}\left\{ U^{T}\left( s_{1}\left( l\right) \right) ^{A^{n}}\hat{\Pi}_{3}\hat{\Pi}_{2}\mathbb{E}_{\mathcal{C}_{2}}\left\{ U^{T}\left(
s_{2}\left( m^{\prime}\right) \right) ^{B^{n}}\Pi_{A^{n}B^{n}C^{n}}U^{\ast
}\left( s_{2}\left( m^{\prime}\right) \right) ^{B^{n}}\right\} \hat{\Pi
}_{2}\hat{\Pi}_{3}U^{\ast}\left( s_{1}\left( l\right) \right) ^{A^{n}}\mathbb{E}_{\mathcal{C}_{2}}\left\{ \theta_{l,m}\right\} \right\}
\right\}\end{aligned}$$$$\begin{aligned}
& \leq2^{n\left( H\left( ABC\right) _{\rho}+c\delta\right) }\sum_{m^{\prime}\neq m}\mathbb{E}_{\mathcal{C}_{1}}\left\{ \mathrm{Tr}\left\{
\begin{array}
[c]{c}U^{T}\left( s_{1}\left( l\right) \right) ^{A^{n}}\hat{\Pi}_{3}\hat{\Pi
}_{2}\mathbb{E}_{\mathcal{C}_{2}}\left\{ U^{T}\left( s_{2}\left( m^{\prime
}\right) \right) ^{B^{n}}\rho_{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{2}\left(
m^{\prime}\right) \right) ^{B^{n}}\right\} \cdot\\
\hat{\Pi}_{2}\hat{\Pi}_{3}U^{\ast}\left( s_{1}\left( l\right) \right)
^{A^{n}}\mathbb{E}_{\mathcal{C}_{2}}\left\{ \theta_{l,m}\right\}
\end{array}
\right\} \right\} \\
& \leq2^{-n\left( H\left( B\right) _{\rho}+H\left( AC\right) _{\rho
}-H\left( ABC\right) _{\rho}-\eta\left( n,\delta\right) -2c\delta\right)
}\sum_{m^{\prime}\neq m}\mathbb{E}_{\mathcal{C}_{1}}\left\{ \mathrm{Tr}\left\{ U^{T}\left( s_{1}\left( l\right) \right) ^{A^{n}}\hat{\Pi}_{3}\hat{\Pi}_{2}\hat{\Pi}_{3}U^{\ast}\left( s_{1}\left( l\right) \right)
^{A^{n}}\mathbb{E}_{\mathcal{C}_{2}}\left\{ \theta_{l,m}\right\} \right\}
\right\} \\
& \leq2^{-n\left( H\left( B\right) _{\rho}+H\left( AC\right) _{\rho
}-H\left( ABC\right) _{\rho}-\eta\left( n,\delta\right) -2c\delta\right)
}\sum_{m^{\prime}\neq m}\mathrm{Tr}\left\{ \mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \theta_{l,m}\right\} \right\} \\
& =2^{-n\left( I\left( B;C|A\right) _{\rho}-\eta\left( n,\delta\right)
-2c\delta\right) }\left\vert \mathcal{M}\right\vert .\end{aligned}$$ The first equality follows because the codewords for messages $m$ and $m^{\prime}$ are different and thus independent. The first inequality follows from$$\Pi^{A^{n}B^{n}C^{n}}\leq2^{n[H\left( ABC\right) +c\delta]}\Pi^{A^{n}B^{n}C^{n}}\rho^{A^{n}B^{n}C^{n}}\Pi^{A^{n}B^{n}C^{n}}\leq2^{n[H\left(
ABC\right) +c\delta]}\rho^{A^{n}B^{n}C^{n}},$$ and we applied a similar observation as in (\[eq:pi1sandwich\]) to obtain the second inequality.
We now bound the third error term:$$\begin{aligned}
& \mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \sum_{l^{\prime}\neq
l,\ m^{\prime}\neq m}\mathrm{Tr}\left\{ \Upsilon_{l^{\prime},m^{\prime}}\theta_{l,m}\right\} \right\} \nonumber\\
& =\sum_{l^{\prime}\neq l,\ m^{\prime}\neq m}\mathrm{Tr}\left\{
\mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \Upsilon_{l^{\prime
},m^{\prime}}\right\} \mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{
\theta_{l,m}\right\} \right\} \\
& =\sum_{\substack{l^{\prime}\neq l\\m^{\prime}\neq m}}\mathrm{Tr}\left\{
\mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \Upsilon_{l^{\prime
},m^{\prime}}\right\} \mathbb{E}_{\mathcal{C}_{2}}\left\{ U^{T}\left(
s_{2}\left( m\right) \right) ^{B^{n}}\hat{\Pi}_{1}\mathbb{E}_{\mathcal{C}_{1}}\left\{ U^{T}\left( s_{1}\left( l\right) \right) ^{A^{n}}\rho
^{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{1}\left( l\right) \right) ^{A^{n}}\right\} \hat{\Pi}_{1}U^{\ast}\left( s_{2}\left( m\right) \right)
^{B^{n}}\right\} \right\} \label{eq:third-term-error}$$ Consider the following operator inequalities:$$\begin{gathered}
\hat{\Pi}_{1}\mathbb{E}_{\mathcal{C}_{1}}\left\{ U^{T}\left( s_{1}\left(
l\right) \right) ^{A^{n}}\rho^{A^{n}B^{n}C^{n}}U^{\ast}\left( s_{1}\left(
l\right) \right) ^{A^{n}}\right\} \hat{\Pi}_{1}\\
\leq2^{-n\left( H\left( A\right) _{\rho}-\eta\left( n,\delta\right)
\right) }\Pi^{A^{n}}\otimes\Pi^{B^{n}C^{n}}\mathcal{N}\left( \phi^{A^{\prime
n}}\otimes\psi^{B^{\prime n}B^{n}}\right) \Pi^{B^{n}C^{n}}\\
\leq2^{-n\left( H\left( A\right) _{\rho}-\eta\left( n,\delta\right)
\right) }\Pi^{A^{n}}\otimes\mathcal{N}\left( \phi^{A^{\prime n}}\otimes
\psi^{B^{\prime n}B^{n}}\right)\end{gathered}$$ The second inequality follows from the fact that the typical projector commutes with the state $\rho$ and therefore $\Pi\rho\Pi=\sqrt{\rho}\Pi
\sqrt{\rho}\leq\rho$. Thus the quantity in (\[eq:third-term-error\]) is upper bounded by the following one:$$\begin{aligned}
& \leq2^{-n\left( H\left( A\right) _{\rho}-\eta\left( n,\delta\right)
\right) }\sum_{l^{\prime}\neq l,\ m^{\prime}\neq m}\mathrm{Tr}\left\{
\mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \Upsilon_{l^{\prime
},m^{\prime}}\right\} \Pi^{A^{n}}\otimes\mathbb{E}_{\mathcal{C}_{2}}\left\{
U^{T}\left( s_{2}\left( m\right) \right) ^{B^{n}}\mathcal{N}\left(
\phi^{A^{\prime n}}\otimes\psi^{B^{\prime n}B^{n}}\right) U^{\ast}\left(
s_{2}\left( m\right) \right) ^{B^{n}}\right\} \right\} \\
& =2^{-n\left( H\left( A\right) _{\rho}-\eta\left( n,\delta\right)
\right) }\sum_{l^{\prime}\neq l,\ m^{\prime}\neq m}\mathrm{Tr}\left\{
\mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \Upsilon_{l^{\prime
},m^{\prime}}\right\} \Pi^{A^{n}}\otimes\left( \sum_{t}\pi_{t}^{B^{n}}\otimes\mathcal{N}\left( \phi^{A^{\prime n}}\otimes\pi_{t}^{B^{\prime n}}\right) \right) \right\} \\
& =2^{-n\left( H\left( A\right) _{\rho}-\eta\left( n,\delta\right)
\right) }\sum_{\substack{l^{\prime}\neq l\\m^{\prime}\neq m}}\mathrm{Tr}\left\{ \mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \Upsilon
_{l^{\prime},m^{\prime}}\right\} \hat{\Pi}_{3}\left( \Pi^{A^{n}}\otimes\left( \sum_{t}\pi_{t}^{B^{n}}\otimes\mathcal{N}\left( \phi
^{A^{\prime n}}\otimes\pi_{t}^{B^{\prime n}}\right) \right) \right)
\hat{\Pi}_{3}\right\}\end{aligned}$$$$\begin{aligned}
& \leq2^{-n\left( H\left( A\right) _{\rho}+H\left( B\right) _{\rho
}-2\eta\left( n,\delta\right) \right) }\sum_{\substack{l^{\prime}\neq
l\\m^{\prime}\neq m}}\mathrm{Tr}\left\{ \mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \Upsilon_{l^{\prime},m^{\prime}}\right\}
\Pi^{A^{n}}\otimes\Pi^{B^{n}}\otimes\Pi^{C^{n}}\mathcal{N}\left(
\phi^{A^{\prime n}}\otimes\sum_{t}\pi_{t}^{B^{\prime n}}\right) \Pi^{C^{n}}\right\} \\
& \leq2^{-n\left( H\left( A\right) _{\rho}+H\left( B\right) _{\rho
}+H\left( C\right) _{\rho}-2\eta\left( n,\delta\right) -c\delta\right)
}\sum_{l^{\prime}\neq l,\ m^{\prime}\neq m}\mathrm{Tr}\left\{ \mathbb{E}_{\mathcal{C}_{1},\mathcal{C}_{2}}\left\{ \Upsilon_{l^{\prime},m^{\prime}}\right\} \right\} \\
& \leq2^{-n\left( H\left( A\right) _{\rho}+H\left( B\right) _{\rho
}+H\left( C\right) _{\rho}-2\eta\left( n,\delta\right) -c\delta\right)
}\sum_{l^{\prime}\neq l,\ m^{\prime}\neq m}\mathrm{Tr}\left\{ \Pi^{A^{n}B^{n}C^{n}}\right\} \\
& \leq2^{-n\left( H\left( A\right) _{\rho}+H\left( B\right) _{\rho
}+H\left( C\right) _{\rho}-H\left( ABC\right) _{\rho}-2\eta\left(
n,\delta\right) -2c\delta\right) }\left\vert \mathcal{L}\right\vert
\cdot\left\vert \mathcal{M}\right\vert \\
& =2^{-n\left( I\left( AB;C\right) _{\rho}-2\eta\left( n,\delta\right)
-2c\delta\right) }\left\vert \mathcal{L}\right\vert \cdot\left\vert
\mathcal{M}\right\vert\end{aligned}$$ Thus, as long as we choose the message set sizes such that the corresponding rates obey the inequalities in the statement of the theorem, then this ensures the existence of a code with vanishing average error probability in the asymptotic limit of large blocklength $n$.
From Average to Maximal Error {#sec:avg-to-max}
-----------------------------
The above scheme for entanglement-assisted classical communication satisfies an average error criterion (as specified in (\[eq:avg-error-crit\])), but we would like it to satisfy a stronger maximal error criterion, where we can guarantee that every message pair has a low error probability. In the single-sender single-receiver case, the standard argument is just to invoke Markov’s inequality to demonstrate that throwing away half of the codewords ensures that the error for all codewords is less than $2\epsilon$ if the original average error probability is less than $\epsilon~$[@book1991cover]. This expurgation then only has a negligible impact on the rate of the code. We cannot employ such an argument for the multiple access case because the expurgation does not guarantee that the resulting expurgated codebook of message pairs decomposes as a product of two expurgated codebooks. Thus, the argument for average-to-maximal error needs to be a bit more clever.
Yard *et al*. introduced a straightforward scheme for constructing a code with low maximal error from one with low average error [@YHD05MQAC], based on some ideas in Ref. [@CK11] and some further ideas of their own. The first idea from Ref. [@CK11] is to suppose that the senders and receiver have access to uniform common randomness. That is, Alice and Charlie share some common randomness and so do Bob and Charlie. Let $S$ denote the Alice-Charlie common randomness and let $T$ denote the Bob-Charlie common randomness. Based on this common randomness, Alice and Bob each compute $l+S$ and $m+T$, where $l$ is Alice’s message and $m$ is Bob’s message and the addition is understood to be modulo the size of the respective message sets. Alice and Bob then encode according to $l+S$ and $m+T$ and Charlie decodes these messages. Using his share of the common randomness, he subtracts off $S$ and $T$ to obtain the intended messages $l$ and $m$. Now, the expected error probability for when Alice and Bob transmit the message pair $\left(
l,m\right) $, where the expectation is with respect to the common randomness, becomes as follows:$$\begin{aligned}
\mathbb{E}_{S,T}\left\{ \mathrm{Tr}\left\{ \left( I-\Lambda_{l+S,m+T}\right) \sigma_{l+S,m+T}\right\} \right\} & =\frac{1}{LM}\sum
_{s,t}\mathrm{Tr}\left\{ \left( I-\Lambda_{l+s,m+t}\right) \sigma
_{l+s,m+T}\right\} \\
& =\frac{1}{LM}\sum_{l,m}\mathrm{Tr}\left\{ \left( I-\Lambda_{l,m}\right)
\sigma_{l,m}\right\} .\end{aligned}$$ Thus, it becomes clear that the maximal error criterion for each message pair $\left( l,m\right) $ is equivalent to the average error criterion if the senders and receiver have access to common randomness.
Yard *et al*. then take this argument further to show that preshared common randomness is not actually necessary. The main idea is to divide the overall number of channel uses into $N+1$ blocks each of length $n$. For the first round, Alice and Bob use the channel $n$ times to establish common randomness of respective sizes $2^{nR_{1}}$ and $2^{nR_{2}}$ with Charlie. Since the common randomness is uniformly distributed and our protocol works well for the uniform distribution, this round fails with probability no larger than $\epsilon$. Alice, Bob, and Charlie then use this established common randomness and the randomized protocol given above for the next $N$ rounds (the key point is that they can use the same common randomness from the first round for all of the subsequent $N$ rounds). By choosing $N=1/\sqrt{\epsilon}$, the first round establishes common randomness at the negligible rates$$\frac{1}{nN}\log2^{nR_{i}}=\sqrt{\epsilon}R_{i},$$ while ensuring that the subsequent rounds have an error probability no larger than $N\epsilon=\sqrt{\epsilon}$. Now, the actual distribution resulting from the first round is $\epsilon$-close to perfect common randomness, but this only results in an error probability of $2\sqrt{\epsilon}$ for the $N$-blocked protocol. The resulting achievable rates for classical communication become $\left( 1-\sqrt{\epsilon}\right) R_{i}$ for $i\in\left\{ 1,2\right\} $. Thus, this blocked scheme shows how to convert a protocol with low average error probability to one with low maximal error probability.
Unassisted Simultaneous Decoding
--------------------------------
A simple corollary of Theorem \[thm:simul-decoder\] is a simultaneous decoder for unassisted classical communication. The proof is virtually identical to the above proof, but it takes advantage of the proof technique in Section III-B of Ref. [@HDW08] (thus we omit the details of the proof). The below result implies a complete solution of the strong interference case for transmitting classical data over a quantum interference channel [@el2010lecture], if the encoders are restricted to product-state inputs. Sen independently obtained a proof of the below corollary with a different technique [@S11a].
\[Unassisted Simultaneous Decoding\]\[thm:unassisted-simul-decoder\]Let $\mathcal{N}^{A^{\prime}B^{\prime}\rightarrow C}$ be a multiple access channel that connects Alice and Bob to Charlie, and let$$\rho^{XYC}\equiv\sum_{x,y}p_{X}\left( x\right) p_{Y}\left( y\right)
\left\vert x\right\rangle \left\langle x\right\vert ^{X}\otimes\left\vert
y\right\rangle \left\langle y\right\vert ^{Y}\otimes\mathcal{N}^{A^{\prime
}B^{\prime}\rightarrow C}\left( \rho_{x}^{A^{\prime}}\otimes\sigma
_{y}^{B^{\prime}}\right) . \label{eq-unassisted-code-state}$$ Then there exists a classical communication code with a corresponding quantum simultaneous decoder, such that the following rate region is achievable for $R_{1},R_{2}\geq0$:$$\begin{aligned}
R_{1} & \leq I\left( X;C|Y\right) _{\rho},\\
R_{2} & \leq I\left( Y;C|X\right) _{\rho},\\
R_{1}+R_{2} & \leq I\left( XY;C\right) _{\rho},\end{aligned}$$ where the entropies are with respect to the state in (\[eq-unassisted-code-state\]).
Quantum communication over a quantum multiple access channel
============================================================
In this section, we recover the previously known achievable rate regions for assisted and unassisted quantum communication over a multiple access channel [@HDW08; @nature2005horodecki; @YHD05MQAC]. We do so by employing a coherent version of the protocol from Section \[sec:ea-simul\] (many researchers have often employed this approach in quantum Shannon theory [@Har03; @DHW05RI; @HDW08; @W11]). Different from prior work, we show that this region can be achieved without the need for time sharing—the simultaneous nature of our decoding scheme guarantees this. Our scheme below achieves quantum communication by employing the blocked protocol from Section \[sec:avg-to-max\] (this is again related to the average versus maximal error issue).
We first recall the resource inequality formalism of Devetak *et al*. [@DHW05RI]. We denote one noiseless classical bit channel from Alice to Bob as $\left[ c\rightarrow c\right] _{AB}$, one noiseless qubit channel from Alice to Bob as $\left[ q\rightarrow q\right] _{AB}$, and one ebit of entanglement shared between Alice and Bob as $\left[ qq\right] _{AB}$. We will also be using a coherent channel from Alice to Bob, which is defined to implement the map [@Har03]:$$\left\vert i\right\rangle ^{A}\rightarrow\left\vert i\right\rangle
^{A}\left\vert i\right\rangle ^{B}.$$ We denote this communication resource as $\left[ q\rightarrow qq\right]
_{AB}$. Note that a coherent channel is a stronger resource than a classical channel because it can simulate a classical channel if Alice only sends computational basis states through it. Furthermore, let $\left\langle
\mathcal{N}\right\rangle $ denote one use of a multiple access channel $\mathcal{N}^{A^{\prime}B^{\prime}\rightarrow C}$. Resource inequalities describe ways of consuming some communication resources in order to create others. For example, the protocol described in Section 4 implements the following resource inequality:$$\left\langle \mathcal{N}\right\rangle +H\left( A\right) _{\rho}\left[
qq\right] _{AC}+H\left( B\right) _{\rho}\left[ qq\right] _{BC}\geq
R_{1}\left[ c\rightarrow c\right] _{AC}+R_{2}\left[ c\rightarrow c\right]
_{BC}.$$
We upgrade our protocol for entanglement-assisted classical communication from the previous section to one for entanglement-assisted coherent communication with two senders and one receiver.
The following resource inequality corresponds to an achievable coherent simultaneous decoding protocol for entanglement-assisted coherent communication over a noisy multiple access quantum channel$~\mathcal{N}$:$$\left\langle \mathcal{N}\right\rangle +H\left( A\right) _{\rho}\left[
qq\right] _{AC}+H\left( B\right) _{\rho}\left[ qq\right] _{BC}\geq
R_{1}\left[ q\rightarrow qq\right] _{AC}+R_{2}\left[ q\rightarrow
qq\right] _{BC},$$ where$$\rho^{ABC}\equiv\mathcal{N}^{A^{\prime}B^{\prime}\rightarrow C}\left(
\phi^{A^{\prime}A}\otimes\psi^{B^{\prime}B}\right) ,\label{eq:sec5rho}$$ as long as$$\begin{aligned}
R_{1} & \leq I\left( A;C|B\right) _{\rho},\label{eq:ea-rate-region-cond1}\\
R_{2} & \leq I\left( B;C|A\right) _{\rho},\label{eq:ea-rate-region-cond2}\\
R_{1}+R_{2} & \leq I\left( AB;C\right) _{\rho}.\label{eq:ea-rate-region-cond3}$$ The entropies are with respect to the state in (\[eq:sec5rho\]).
We again exploit the blocked protocol from Section \[sec:avg-to-max\]. We assume that Alice, Bob, and Charlie have already established their common randomness, and we describe how the protocol operates for the first of the $N$ rounds (the round after the one that establishes common randomness). Let $S$ denote the Alice-Charlie common randomness, and let $T$ denote the Bob-Charlie common randomness.
Suppose that Alice shares a state with a reference system $R_{A}$: $$\sum_{j,l=1}^{L}\alpha_{j,l}\left\vert j\right\rangle ^{R_{A}}\left\vert
l\right\rangle ^{A_{1}},$$ where $\left\{ \left\vert j\right\rangle \right\} $ and $\left\{ \left\vert
l\right\rangle \right\} $ are some orthonormal bases for $R_{A}$ and $A_{1}$ respectively. Similarly, Bob also shares a state with a reference system $R_{B}$:$$\sum_{k,m=1}^{M}\beta_{k,m}\left\vert k\right\rangle ^{R_{B}}\left\vert
m\right\rangle ^{B_{1}},$$ where $\left\{ \left\vert k\right\rangle \right\} $ and $\left\{ \left\vert
m\right\rangle \right\} $ are some orthonormal bases for $R_{B}$ and $B_{1}$ respectively. The parameters $L$ and $M$ for these states are chosen such that$$\begin{aligned}
R_{1} & \approx\frac{1}{n}\log L\leq I\left( A;C|B\right) _{\rho
},\label{eq:Schmidt-region-1}\\
R_{2} & \approx\frac{1}{n}\log M\leq I\left( B;C|A\right) _{\rho},\\
R_{1}+R_{2} & \approx\frac{1}{n}\log\left( LM\right) \leq I\left(
AB;C\right) _{\rho}.\label{eq:Schmidt-region-3}$$ Alice would like to simulate the action of a coherent channel on her system $A_{1}$ to a system $A_{2}$ for Charlie:$$\sum_{j,l}\alpha_{j,l}\left\vert j\right\rangle ^{R_{A}}\left\vert
l\right\rangle ^{A_{1}}\rightarrow\sum_{j,l}\alpha_{j,l}\left\vert
j\right\rangle ^{R_{A}}\left\vert l\right\rangle ^{A_{1}}\left\vert
l\right\rangle ^{A_{2}},$$ and Bob would like to do the same. We demand that they simulate these resources with vanishing error in the limit of many channel uses. As before, Alice and Charlie share many copies of a pure entangled state $\left\vert
\phi\right\rangle ^{AA^{\prime}}$, Bob and Charlie share many copies of $\left\vert \psi\right\rangle ^{BB^{\prime}}$, and they all have access to many uses of a noisy multiple access channel $\mathcal{N}^{A^{\prime}B^{\prime}\rightarrow C}$.
They have their encoding unitaries $\{U\left( s_{1}\left( l\right) \right)
^{A^{\prime n}}\}_{l}$ and $\{U\left( s_{2}\left( m\right) \right)
^{B^{\prime n}}\}_{m}$ as described in Section \[sec:ea-simul\], and they employ them now as the following controlled unitaries that act on the systems $A_{1}$ and $B_{1}$ and their shares of the entanglement:$$\begin{aligned}
& \sum_{l}\left\vert l\right\rangle \left\langle l\right\vert ^{A_{1}}\otimes
U\left( s_{1}\left( l\right) \right) ^{A^{\prime n}},\\
& \sum_{m}\left\vert m\right\rangle \left\langle m\right\vert ^{B_{1}}\otimes
U\left( s_{2}\left( m\right) \right) ^{B^{\prime n}}.\end{aligned}$$ The resulting global state after applying these unitaries and the transpose trick is as follows:$$\left( \sum_{j,l}\alpha_{j,l}\left\vert j\right\rangle ^{R_{A}}\left\vert
l\right\rangle ^{A_{1}}U^{T}\left( s_{1}\left( l\right) \right) ^{A^{n}}\left\vert \phi\right\rangle ^{A^{n}A^{\prime n}}\right) \otimes\left(
\sum_{k,m}\beta_{k,m}\left\vert k\right\rangle ^{R_{B}}\left\vert
m\right\rangle ^{B_{1}}U^{T}\left( s_{2}\left( m\right) \right) ^{B^{n}}\left\vert \psi\right\rangle ^{B^{n}B^{\prime n}}\right) .$$ Alice and Bob both then apply the respective unitaries $U\left( s_{1}\left(
S\right) \right) ^{A^{\prime n}}$ and $U\left( s_{2}\left( T\right)
\right) ^{B^{\prime n}}$, conditional on their common randomness shared with Charlie. The resulting state is$$\left( \sum_{j,l}\alpha_{j,l}\left\vert j\right\rangle ^{R_{A}}\left\vert
l\right\rangle ^{A_{1}}U^{T}\left( s_{1}\left( l+S\right) \right) ^{A^{n}}\left\vert \phi\right\rangle ^{A^{n}A^{\prime n}}\right) \otimes\left(
\sum_{k,m}\beta_{k,m}\left\vert k\right\rangle ^{R_{B}}\left\vert
m\right\rangle ^{B_{1}}U^{T}\left( s_{2}\left( m+T\right) \right) ^{B^{n}}\left\vert \psi\right\rangle ^{B^{n}B^{\prime n}}\right) ,$$ where the addition $l+S$ and $m+T$ is modulo $L$ and $M$, respectively. They both then send their shares of the states over the multiple access channel $\mathcal{N}^{A^{\prime}B^{\prime}\rightarrow C}$, whose isometric extension is $U_{\mathcal{N}}^{A^{\prime}B^{\prime}\rightarrow CE}$ and acts on $\left\vert \phi\right\rangle ^{A^{n}A^{\prime n}}\otimes\left\vert
\psi\right\rangle ^{B^{n}B^{\prime n}}$ as follows:$$\left\vert \varphi\right\rangle ^{A^{n}B^{n}C^{n}E^{n}}\equiv U_{\mathcal{N}}^{A^{\prime n}B^{\prime n}\rightarrow C^{n}E^{n}}\left( \left\vert
\phi\right\rangle ^{A^{n}A^{\prime n}}\otimes\left\vert \psi\right\rangle
^{B^{n}B^{\prime n}}\right) .$$ After the transmission, the overall state becomes$$\sum_{j,k,l,m}\alpha_{j,l}\beta_{k,m}\left\vert j\right\rangle ^{R_{A}}\left\vert l\right\rangle ^{A_{1}}\left\vert k\right\rangle ^{R_{B}}\left\vert m\right\rangle ^{B_{1}}\left( U^{T}\left( s_{1}\left(
l+S\right) \right) ^{A^{n}}\otimes U^{T}\left( s_{2}\left( m+T\right)
\right) ^{B^{n}}\right) \left\vert \varphi\right\rangle ^{A^{n}B^{n}C^{n}E^{n}}.\label{eq:state-after-mac}$$ Charlie performs the following coherent measurement constructed from the POVM $\left\{ \Lambda_{p,q}\right\} $ of Section \[sec:ea-simul\]:$$\Upsilon=\sum_{p,q}\left( \sqrt{\Lambda_{p,q}}\right) ^{A^{n}B^{n}C^{n}}\otimes\left\vert p\right\rangle ^{A_{2}}\otimes\left\vert q\right\rangle
^{B_{2}}.$$ Given that the original POVM is good on average, in the sense that$$\frac{1}{LM}\sum_{l,m}\mathrm{Tr}\left\{ \Lambda_{l,m}\sigma_{l,m}\right\}
\geq1-\epsilon,\label{eq:good-code-avg}$$ for all $\epsilon>0$ and sufficiently large $n$ (where each $\sigma_{l,m}$ is an entanglement-assisted quantum codeword as before), this coherent measurement also has little effect on the received state while coherently copying the basis states in registers $A_{1}$ and $B_{1}$. That is, the expected fidelity overlap between the states$$\Upsilon^{A^{n}B^{n}C^{n}A_{2}B_{2}}\left\vert \omega\right\rangle ^{F},$$ and$$\sum_{j,k,l,m}\alpha_{j,l}\beta_{k,m}\left\vert j\right\rangle ^{R_{A}}\left\vert l\right\rangle ^{A_{1}}\left\vert k\right\rangle ^{R_{B}}\left\vert m\right\rangle ^{B_{1}}\left( U^{T}\left( s_{1}\left(
l+S\right) \right) ^{A^{n}}\otimes U^{T}\left( s_{2}\left( m+T\right)
\right) ^{B^{n}}\right) \left\vert \varphi\right\rangle ^{A^{n}B^{n}C^{n}E^{n}}\left\vert l+S\right\rangle ^{A_{2}}\left\vert m+T\right\rangle
^{B_{2}}$$ is larger than $1-\epsilon$, where $\left\vert \omega\right\rangle $ denotes the state in (\[eq:state-after-mac\]), the system $F$ denotes all the systems $A_{1}B_{1}A^{n}B^{n}C^{n}E^{n}$, and the expectation is with respect to the common randomness $S$ and $T$. To see why this is true, consider the following chain of inequalities:$$\begin{aligned}
& \frac{1}{LM}\sum_{s,t}\sum_{j,k,l,m}\alpha_{j,l}^{\ast}\beta_{k,m}^{\ast
}\left\langle j\right\vert ^{R_{A}}\left\langle l\right\vert ^{A_{1}}\left\langle k\right\vert ^{R_{B}}\left\langle m\right\vert ^{B_{1}}\left\langle \varphi\right\vert \left( U^{\ast}\left( s_{1}\left(
l+s\right) \right) ^{A^{n}}\otimes U^{\ast}\left( s_{2}\left( m+t\right)
\right) ^{B^{n}}\right) \left\langle l+s\right\vert ^{A_{2}}\left\langle
m+t\right\vert ^{B_{2}}\nonumber\\
& \left( \sum_{p,q}\left( \sqrt{\Lambda_{p,q}}\right) ^{A^{n}B^{n}C^{n}}\left\vert p\right\rangle ^{A_{2}}\left\vert q\right\rangle ^{B_{2}}\right)
\nonumber\\
& \sum_{j^{\prime},k^{\prime},l^{\prime},m^{\prime}}\alpha_{j^{\prime
},l^{\prime}}\beta_{k^{\prime},m^{\prime}}\left\vert j^{\prime}\right\rangle
^{R_{A}}\left\vert l^{\prime}\right\rangle ^{A_{1}}\left\vert k^{\prime
}\right\rangle ^{R_{B}}\left\vert m^{\prime}\right\rangle ^{B_{1}}\left(
U^{T}\left( s_{1}\left( l^{\prime}+s\right) \right) ^{A^{n}}\otimes
U^{T}\left( s_{2}\left( m^{\prime}+t\right) \right) ^{B^{n}}\right)
\left\vert \varphi\right\rangle \nonumber\\
& =\frac{1}{LM}\sum_{s,t}\sum_{j,k,l,m}\left\vert \alpha_{j,l}\right\vert
^{2}\left\vert \beta_{k,m}\right\vert ^{2}\left\langle \varphi\right\vert
\left( U^{\ast}\left( s_{1}\left( l+s\right) \right) ^{A^{n}}\otimes
U^{\ast}\left( s_{2}\left( m+t\right) \right) ^{B^{n}}\right)
\sqrt{\Lambda_{l+s,m+t}}\ \ \times\nonumber\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( U^{T}\left( s_{1}\left( l+s\right)
\right) ^{A^{n}}\otimes U^{T}\left( s_{2}\left( m+t\right) \right)
^{B^{n}}\right) \left\vert \varphi\right\rangle \\
& =\sum_{j,k,l,m}\left\vert \alpha_{j,l}\right\vert ^{2}\left\vert
\beta_{k,m}\right\vert ^{2}\frac{1}{LM}\sum_{s,t}\left\langle \varphi
\right\vert \left( U^{\ast}\left( s_{1}\left( l+s\right) \right) ^{A^{n}}\otimes U^{\ast}\left( s_{2}\left( m+t\right) \right) ^{B^{n}}\right)
\sqrt{\Lambda_{l+s,m+t}}\ \ \times\nonumber\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( U^{T}\left( s_{1}\left( l+s\right)
\right) ^{A^{n}}\otimes U^{T}\left( s_{2}\left( m+t\right) \right)
^{B^{n}}\right) \left\vert \varphi\right\rangle \\
& \geq\sum_{j,k,l,m}\left\vert \alpha_{j,l}\right\vert ^{2}\left\vert
\beta_{k,m}\right\vert ^{2}\times\nonumber\\
& \frac{1}{LM}\sum_{s,t}\text{Tr}\left\{ \left( U^{T}\left( s_{1}\left(
l+s\right) \right) ^{A^{n}}\otimes U^{T}\left( s_{2}\left( m+t\right)
\right) ^{B^{n}}\right) \left\vert \varphi\right\rangle \left\langle
\varphi\right\vert \left( U^{\ast}\left( s_{1}\left( l+s\right) \right)
^{A^{n}}\otimes U^{\ast}\left( s_{2}\left( m+t\right) \right) ^{B^{n}}\right) \Lambda_{l+s,m+t}\right\} \\
& \geq\sum_{j,k,l,m}\left\vert \alpha_{j,l}\right\vert ^{2}\left\vert
\beta_{k,m}\right\vert ^{2}\left( 1-\epsilon\right) \\
& =1-\epsilon\end{aligned}$$ where the last inequality follows from (\[eq:good-code-avg\]). Thus, the resulting state is $2\sqrt{\epsilon}$-close in expected trace distance to the following state:$$\sum_{j,k,l,m}\alpha_{j,l}\beta_{k,m}\left\vert j\right\rangle ^{R_{A}}\left\vert l\right\rangle ^{A_{1}}\left\vert k\right\rangle ^{R_{B}}\left\vert m\right\rangle ^{B_{1}}\left( U^{T}\left( s_{1}\left(
l+S\right) \right) ^{A^{n}}\otimes U^{T}\left( s_{2}\left( m+T\right)
\right) ^{B^{n}}\right) \left\vert \varphi\right\rangle ^{A^{n}B^{n}C^{n}E^{n}}\left\vert l+S\right\rangle ^{A_{2}}\left\vert m+T\right\rangle
^{B_{2}}.$$ Now Charlie performs the following controlled unitary:$$\sum_{l,m}\left\vert l\right\rangle \left\langle l\right\vert ^{A_{2}}\otimes\left\vert m\right\rangle \left\langle m\right\vert ^{B_{2}}\otimes\left( U^{\ast}\left( s_{1}\left( l\right) \right) ^{A^{n}}\otimes
U^{\ast}\left( s_{2}\left( m\right) \right) ^{B^{n}}\right) ,$$ and the resulting state is as follows:$$\sum_{j,k,l,m}\alpha_{j,l}\beta_{k,m}\left\vert j\right\rangle ^{R_{A}}\left\vert l\right\rangle ^{A_{1}}\left\vert k\right\rangle ^{R_{B}}\left\vert m\right\rangle ^{B_{1}}\left\vert \varphi\right\rangle
^{A^{n}B^{n}C^{n}E^{n}}\left\vert l+S\right\rangle ^{A_{2}}\left\vert
m+T\right\rangle ^{B_{2}}$$ Charlie then performs the generalized Pauli shifts $X^{A_{2}}\left(
-S\right) $ and $X^{B_{2}}\left( -T\right) $ (based on his common randomness) to produce the state$$\left( \sum_{j,l}\alpha_{j,l}\left\vert j\right\rangle ^{R_{A}}\left\vert
l\right\rangle ^{A_{1}}\left\vert l\right\rangle ^{A_{2}}\right)
\otimes\left( \sum_{k,m}\beta_{k,m}\left\vert k\right\rangle ^{R_{B}}\left\vert m\right\rangle ^{B_{1}}\left\vert m\right\rangle ^{B_{2}}\right)
\otimes\left\vert \varphi\right\rangle ^{A^{n}B^{n}C^{n}E^{n}},$$ so that Alice and Bob have successfully generated coherent channels with the receiver Charlie for this round.
The above scheme constitutes the first of the $N$ blocks after establishing the common randomness. Alice, Bob, and Charlie perform the same scheme for the next $N-1$ blocks, and they use the same common randomness for each round. For similar reasons as given at the end of Section \[sec:avg-to-max\], this scheme works well if we set the number $N$ of rounds equal to $\epsilon
^{-1/4}$ (we require $\epsilon^{-1/4}$ this time because each round disturbs the state by $2\sqrt{\epsilon}$ so that the overall disturbance for all $N$ rounds is no larger than $N\left( 2\sqrt{\epsilon}\right) =2\epsilon^{1/4}$).
The coherent communication identity is a helpful tool in quantum Shannon theory, and it results from the protocols coherent teleportation and coherent super-dense coding [@Har03; @W11]. It states that two coherent channels are equivalent to a noiseless quantum channel and noiseless entanglement:$$2\log d\left[ q\rightarrow qq\right] =\log d\left[ q\rightarrow q\right]
+\log d\left[ qq\right] ,$$ where $d$ is the dimension of the underlying systems. Employing this identity gives us the following achievable rate region for entanglement-assisted quantum communication:
There exists an entanglement-assisted quantum communication protocol with a coherent quantum simultaneous decoder if the rates $\widetilde{R}_{1}$ and $\widetilde{R}_{2}$ of quantum communication satisfy the following inequalities:$$\begin{aligned}
\widetilde{R}_{1} & \leq\frac{1}{2}I\left( A;C|B\right) _{\rho},\\
\widetilde{R}_{2} & \leq\frac{1}{2}I\left( B;C|A\right) _{\rho},\\
\widetilde{R}_{1}+\widetilde{R}_{2} & \leq\frac{1}{2}I\left( AB;C\right)
_{\rho}.\end{aligned}$$
We simply recall the resource inequality from the previous theorem and apply the coherent communication identity: $$\begin{aligned}
\left\langle \mathcal{N}\right\rangle +H\left( A\right) _{\rho}\left[
qq\right] _{AC}+H\left( B\right) _{\rho}\left[ qq\right] _{BC} & \geq
R_{1}\left[ q\rightarrow qq\right] _{AC}+R_{2}\left[ q\rightarrow
qq\right] _{BC}\\
& \geq\frac{1}{2}R_{1}\left[ qq\right] _{AC}+\frac{1}{2}R_{1}\left[
q\rightarrow q\right] _{AC}+\frac{1}{2}R_{2}\left[ qq\right] _{BC}+\frac
{1}{2}R_{2}\left[ q\rightarrow q\right] _{BC}.\end{aligned}$$ Throughout out the rest of this section, we will assume that $R_{1}$ and $R_{2}$ satisfy the conditions (\[eq:ea-rate-region-cond1\])-(\[eq:ea-rate-region-cond3\]). If we allow catalytic protocols, that is we allow the use of some resources for free, provided that they are returned at the end of the protocol, then we obtain a protocol for entanglement-assisted quantum communication over a multiple access channel that implements the following resource inequality: $$\left\langle \mathcal{N}\right\rangle +\left( H\left( A\right) _{\rho
}-\frac{1}{2}R_{1}\right) \left[ qq\right] _{AC}+\left( H\left( B\right)
_{\rho}-\frac{1}{2}R_{2}\right) \left[ qq\right] _{BC}\geq\frac{1}{2}R_{1}\left[ q\rightarrow q\right] _{AC}+\frac{1}{2}R_{2}\left[ q\rightarrow
q\right] _{BC}.$$
Combining the above protocol further with entanglement distribution $\left[
q\rightarrow q\right] \geq\left[ qq\right] $ gives the following corollary:
There exists a catalytic quantum communication protocol (that consumes no net entanglement) with a coherent quantum simultaneous decoder if the rates $S_{1}$ and $S_{2}$ of quantum communication satisfy the following inequalities:$$\begin{aligned}
S_{1} & \leq I\left( A\rangle C|B\right) _{\rho},\\
S_{2} & \leq I\left( B\rangle C|A\right) _{\rho},\\
S_{1}+S_{2} & \leq I\left( AB\rangle C\right) _{\rho}.\end{aligned}$$
The protocol from the above corollary in turn leads to a proof of an achievable rate region for unassisted quantum communication over a multiple access channel:$$\begin{aligned}
& \left\langle \mathcal{N}\right\rangle +\left( H\left( A\right) _{\rho
}-\frac{1}{2}R_{1}\right) \left[ qq\right] _{AC}+\left( H\left( B\right)
_{\rho}-\frac{1}{2}R_{2}\right) \left[ qq\right] _{BC}\nonumber\\
\geq & \frac{1}{2}R_{1}\left[ q\rightarrow q\right] _{AC}+\frac{1}{2}R_{2}\left[ q\rightarrow q\right] _{BC}\\
\geq & \left( R_{1}-H\left( A\right) _{\rho}\right) \left[ q\rightarrow
q\right] _{AC}+\left( R_{2}-H\left( B\right) _{\rho}\right) \left[
q\rightarrow q\right] _{BC}+\left( H\left( A\right) _{\rho}-\frac{1}{2}R_{1}\right) \left[ qq\right] _{AC}+\left( H\left( B\right) _{\rho
}-\frac{1}{2}R_{2}\right) \left[ qq\right] _{BC}.\end{aligned}$$ The second inequality follows from the fact that we can perform entanglement distribution using noiseless quantum channels. After resource cancellation, this leads to $$\begin{aligned}
\left\langle \mathcal{N}\right\rangle & \geq\left( R_{1}-H\left( A\right)
_{\rho}\right) \left[ q\rightarrow q\right] _{AC}+\left( R_{2}-H\left(
B\right) _{\rho}\right) \left[ q\rightarrow q\right] _{BC}\\
& =S_{1}\left[ q\rightarrow q\right] _{AC}+S_{1}\left[ q\rightarrow
q\right] _{BC},\end{aligned}$$ where$$\begin{aligned}
S_{1} & \leq I\left( A\rangle B|C\right) _{\rho},\\
S_{2} & \leq I\left( B\rangle A|C\right) _{\rho},\\
S_{1}+S_{2} & \leq I\left( AB\rangle C\right) _{\rho}.\end{aligned}$$ Again, both of these two capacity regions can be achieved without time sharing, thanks to our simultaneous decoder.
Entanglement-Assisted Bosonic Multiple Access Channel
=====================================================
This final section details our last contribution—an achievable rate region for entanglement-assisted classical communication over a bosonic multiple access channel (see Refs. [@WPGCRSL11; @EW07] for a nice review of bosonic channels). Perhaps the simplest model for this channel is the following beamsplitter transformation (Yen and Shapiro [@YS05] considered unassisted communication over such a channel):$$\begin{aligned}
\hat{c} & =\sqrt{\eta}\hat{a}+\sqrt{1-\eta}\hat{b},\\
\hat{e} & =-\sqrt{1-\eta}\hat{a}+\sqrt{\eta}\hat{b},\end{aligned}$$ where $\hat{a}$ is the annihilation operator representing the first sender Alice’s input signal, $\hat{b}$ is the annihilation operator representing the second sender Bob’s input signal, $\hat{c}$ is the annihilation operator for the receiver’s output, and $\hat{e}$ is the annihilation operator for an inaccessible environment output of the channel. We prove the following theorem:
\[thm:ea-bosonic\] Suppose that Alice is allowed a mean photon number $N_{S_{a}}$ at her transmitter and Bob is allowed a mean photon number $N_{S_{b}}$ at his transmitter. Then the following rate region is achievable for entanglement-assisted transmission of classical information over the beamsplitter quantum multiple access channel:$$\begin{aligned}
R_{1} & \leq g\left( N_{S_{a}}\right) +g\left( \left( \lambda_{BC}^{+}+1\right) /2-1\right) +g\left( \left( \lambda_{BC}^{-}+1\right)
/2-1\right) -g\left( \eta N_{S_{b}}+\left( 1-\eta\right) N_{S_{a}}\right)
,\label{eq:EA-boson-region-1}\\
R_{2} & \leq g\left( N_{S_{b}}\right) +g\left( \left( \lambda_{AC}^{+}+1\right) /2-1\right) +g\left( \left( \lambda_{AC}^{-}+1\right)
/2-1\right) -g\left( \eta N_{S_{b}}+\left( 1-\eta\right) N_{S_{a}}\right)
,\\
R_{1}+R_{2} & \leq g\left( N_{S_{a}}\right) +g\left( N_{S_{b}}\right)
+g\left( \eta N_{S_{a}}+\left( 1-\eta\right) N_{S_{b}}\right) -g\left(
\eta N_{S_{b}}+\left( 1-\eta\right) N_{S_{a}}\right) ,
\label{eq:EA-boson-region-3}$$ where$$\begin{aligned}
g\left( N\right) & \equiv\left( N+1\right) \log\left( N+1\right)
-N\log N,\\
\lambda_{AC}^{\left( \pm\right) } & =\left( 1-\eta\right) \left\vert
N_{S_{a}}-N_{S_{b}}\right\vert \pm\sqrt{\left( 1-\eta\right) ^{2}\left(
N_{S_{a}}-N_{S_{b}}\right) ^{2}+2\left( 1-\eta\right) \left( 2N_{S_{a}}N_{S_{b}}+N_{S_{a}}+N_{S_{b}}\right) +1},\\
\lambda_{BC}^{\left( \pm\right) } & =\eta\left\vert N_{S_{a}}-N_{S_{b}}\right\vert \pm\sqrt{\eta^{2}\left( N_{S_{a}}-N_{S_{b}}\right) ^{2}+2\eta\left( 2N_{S_{a}}N_{S_{b}}+N_{S_{a}}+N_{S_{b}}\right) +1}.\end{aligned}$$ (Observe that $\lambda_{AC}^{\left( \pm\right) }$ and $\lambda_{BC}^{\left(
\pm\right) }$ are related by the substitution $\eta\leftrightarrow1-\eta$.)
We assume the most natural entangled states that Alice and Charlie and Bob and Charlie can share: a two-mode squeezed vacuum [@GK04; @WPGCRSL11]. This state has the following form:$$\sum_{n=0}^{\infty}\sqrt{\frac{N_{S}^{n}}{\left( N_{S}+1\right) ^{n+1}}}\left\vert n\right\rangle \left\vert n\right\rangle ,$$ where $N_{S}$ is the average number of photons in one mode (after tracing over the other), Alice or Bob has the first mode, and Charlie has the second mode. The covariance matrix for such a state is as follows [@WPGCRSL11]:$$V_{\text{TMS}}\left( N_{S}\right) \equiv\begin{bmatrix}
2N_{S}+1 & 0 & 2\sqrt{N_{S}\left( N_{S}+1\right) } & 0\\
0 & 2N_{S}+1 & 0 & -2\sqrt{N_{S}\left( N_{S}+1\right) }\\
2\sqrt{N_{S}\left( N_{S}+1\right) } & 0 & 2N_{S}+1 & 0\\
0 & -2\sqrt{N_{S}\left( N_{S}+1\right) } & 0 & 2N_{S}+1
\end{bmatrix}
.$$ The covariance matrix for the overall state before the channel acts is as follows:$$V^{AA^{\prime}BB^{\prime}}\equiv V_{\text{TMS}}\left( N_{S_{a}}\right)
\oplus V_{\text{TMS}}\left( N_{S_{b}}\right) ,$$ where $N_{S_{a}}$ is the average number of photons in one share of the state that Alice shares with Charlie and $N_{S_{b}}$ is the average number of photons in one share of the state that Bob shares with Charlie.
The symplectic operator for a beamsplitter unitary is as follows [@WPGCRSL11]:$$S_{\text{BS}}^{A^{\prime}B^{\prime}}\equiv\begin{bmatrix}
\sqrt{\eta}I & \sqrt{1-\eta}I\\
-\sqrt{1-\eta}I & \sqrt{\eta}I
\end{bmatrix}
,$$ and the covariance matrix of the state resulting from the beamsplitter interaction is$$V^{ACBE}\equiv\left( S_{\text{BS}}^{A^{\prime}B^{\prime}}\oplus
I^{AB}\right) V^{AA^{\prime}BB^{\prime}}\left( (S_{\text{BS}}^{A^{\prime
}B^{\prime}})^{T}\oplus I^{AB}\right) ,$$ where modes $C$ and $E$ emerge from the output ports of the beamsplitter (with input ports $A^{\prime}$ and $B^{\prime}$).
Hsieh *et al*. proved that the following rate region is achievable for entanglement-assisted communication over a quantum multiple access channel $\mathcal{M}$:$$\begin{aligned}
R_{1} & \leq I\left( A;BC\right) _{\rho},\\
R_{2} & \leq I\left( B;AC\right) _{\rho},\\
R_{1}+R_{2} & \leq I\left( AB;C\right) _{\rho},\end{aligned}$$ where $\rho^{ABC}$ is a state of the following form:$$\rho^{ABC}\equiv\mathcal{M}^{A^{\prime}B^{\prime}\rightarrow C}(\phi
^{AA^{\prime}}\otimes\psi^{BB^{\prime}}),$$ and $\phi^{AA^{\prime}}$ and $\psi^{BB^{\prime}}$ are pure, bipartite states [@HDW08]. Their theorem applies to finite-dimensional systems, but nevertheless, we apply their theorem to the infinite-dimensional setting by means of a limiting argument.[^2] By inspecting the above theorem, it becomes clear that it is necessary to compute just seven entropies in order to determine the achievable rate region: $H\left( A\right) _{\rho}$, $H\left(
B\right) _{\rho}$, $H\left( C\right) _{\rho}$, $H\left( AB\right) _{\rho
}$, $H\left( AC\right) _{\rho}$, $H\left( BC\right) _{\rho}$, and $H\left( ABC\right) _{\rho}$. Observe that $H\left( ABC\right) _{\rho
}=H\left( E\right) _{\rho}$ if we define $E$ as the environment of the channel. In order to determine these entropies, we just need to figure out the covariance matrices for each of the seven different systems corresponding to these entropies because the entropies are a function of the symplectic eigenvalues of these covariance matrices. These seven different covariance matrices are as follows:$$\begin{gathered}
V^{E}=\begin{bmatrix}
2\left( \eta N_{S_{b}}+\left( 1-\eta\right) N_{S_{a}}\right) +1 & 0\\
0 & 2\left( \eta N_{S_{b}}+\left( 1-\eta\right) N_{S_{a}}\right) +1
\end{bmatrix}
,\\
V^{A}=\begin{bmatrix}
2N_{S_{a}}+1 & 0\\
0 & 2N_{S_{a}}+1
\end{bmatrix}
,\\
V^{B}=\begin{bmatrix}
2N_{S_{b}}+1 & 0\\
0 & 2N_{S_{b}}+1
\end{bmatrix}
,\\
V^{C}=\begin{bmatrix}
2\left( \eta N_{S_{a}}+\left( 1-\eta\right) N_{S_{b}}\right) +1 & 0\\
0 & 2\left( \eta N_{S_{a}}+\left( 1-\eta\right) N_{S_{b}}\right) +1
\end{bmatrix}
,\end{gathered}$$$$V^{AB}=\begin{bmatrix}
2N_{S_{a}}+1 & 0 & 0 & 0\\
0 & 2N_{S_{a}}+1 & 0 & 0\\
0 & 0 & 2N_{S_{b}}+1 & 0\\
0 & 0 & 0 & 2N_{S_{b}}+1
\end{bmatrix}
,$$$$V^{AC}=\begin{bmatrix}
2N_{S_{a}}+1 & 0 & 2\sqrt{\eta}\sqrt{N_{S_{a}}\left( N_{S_{a}}+1\right) } &
0\\
0 & 2N_{S_{a}}+1 & 0 & -2\sqrt{\eta}\sqrt{N_{S_{a}}\left( N_{S_{a}}+1\right)
}\\
2\sqrt{\eta}\sqrt{N_{S_{a}}\left( N_{S_{a}}+1\right) } & 0 & 2\left( \eta
N_{S_{a}}+\overline{\eta}N_{S_{b}}\right) +1 & 0\\
0 & -2\sqrt{\eta}\sqrt{N_{S_{a}}\left( N_{S_{a}}+1\right) } & 0 & 2\left(
\eta N_{S_{a}}+\overline{\eta}N_{S_{b}}\right) +1
\end{bmatrix}
, \label{eq:CM-AC}$$$$V^{BC}=\begin{bmatrix}
2\left( \eta N_{S_{a}}+\overline{\eta}N_{S_{b}}\right) +1 & 0 &
2\sqrt{\overline{\eta}}\sqrt{N_{S_{b}}\left( N_{S_{b}}+1\right) } & 0\\
0 & 2\left( \eta N_{S_{a}}+\overline{\eta}N_{S_{b}}\right) +1 & 0 &
-2\sqrt{\overline{\eta}}\sqrt{N_{S_{b}}\left( N_{S_{b}}+1\right) }\\
2\sqrt{\overline{\eta}}\sqrt{N_{S_{b}}\left( N_{S_{b}}+1\right) } & 0 &
2N_{S_{b}}+1 & 0\\
0 & -2\sqrt{\overline{\eta}}\sqrt{N_{S_{b}}\left( N_{S_{b}}+1\right) } & 0 &
2N_{S_{b}}+1
\end{bmatrix}
, \label{eq:CM-BC}$$ where $\overline{\eta}\equiv1-\eta$. The five entropies $H\left( A\right)
_{\rho}$, $H\left( B\right) _{\rho}$, $H\left( C\right) _{\rho}$, $H\left( E\right) _{\rho}$, and $H\left( AB\right) _{\rho}$ are straightforward to compute because their covariance matrices all correspond to those for thermal states:$$\begin{aligned}
H\left( A\right) & =g\left( N_{S_{a}}\right) ,\\
H\left( B\right) & =g\left( N_{S_{b}}\right) ,\\
H\left( C\right) & =g\left( \eta N_{S_{a}}+\left( 1-\eta\right)
N_{S_{b}}\right) ,\\
H\left( ABC\right) & =H\left( E\right) =g\left( \eta N_{S_{b}}+\left(
1-\eta\right) N_{S_{a}}\right) ,\\
H\left( AB\right) & =g\left( N_{S_{a}}\right) +g\left( N_{S_{b}}\right) .\end{aligned}$$ We can calculate the other entropies $H\left( AC\right) $ and $H\left(
BC\right) $ by computing the symplectic eigenvalues of the covariance matrices in (\[eq:CM-AC\]) and (\[eq:CM-BC\]), respectively:$$\begin{aligned}
\lambda_{AC}^{\left( \pm\right) } & =\left( 1-\eta\right) \left\vert
N_{S_{a}}-N_{S_{b}}\right\vert \pm\sqrt{\left( 1-\eta\right) ^{2}\left(
N_{S_{a}}-N_{S_{b}}\right) ^{2}+2\left( 1-\eta\right) \left( 2N_{S_{a}}N_{S_{b}}+N_{S_{a}}+N_{S_{b}}\right) +1},\\
\lambda_{BC}^{\left( \pm\right) } & =\eta\left\vert N_{S_{a}}-N_{S_{b}}\right\vert \pm\sqrt{\eta^{2}\left( N_{S_{a}}-N_{S_{b}}\right) ^{2}+2\eta\left( 2N_{S_{a}}N_{S_{b}}+N_{S_{a}}+N_{S_{b}}\right) +1}.\end{aligned}$$ Recall that we find the symplectic eigenvalues of a matrix $V$ by computing the eigenvalues of the matrix $\left\vert iJV\right\vert $ [@WPGCRSL11] where$$J\equiv\bigoplus_{i=1}^{n}\begin{bmatrix}
0 & 1\\
-1 & 0
\end{bmatrix}
,$$ and $n$ is the number of modes. These symplectic eigenvalues lead to the following values for the entropies:$$\begin{aligned}
H\left( AC\right) & =g\left( \left( \lambda_{AC}^{+}+1\right)
/2-1\right) +g\left( \left( \lambda_{AC}^{-}+1\right) /2-1\right) ,\\
H\left( BC\right) & =g\left( \left( \lambda_{BC}^{+}+1\right)
/2-1\right) +g\left( \left( \lambda_{BC}^{-}+1\right) /2-1\right) ,\end{aligned}$$ by exploiting the fact that the entropy of a Gaussian state $\rho$ is the following function of its symplectic eigenvalues $\left\{ \nu_{k}\right\} $ [@WPGCRSL11]:$$H\left( \rho\right) =\sum_{k}g\left( \left( \nu_{k}+1\right) /2-1\right)
.$$ Thus, an achievable rate region for the entanglement-assisted bosonic multiple access channel is as stated in the theorem.
Figure \[fig:EA-bosonic\] plots several achievable rate regions given by Theorem \[thm:ea-bosonic\] as the transmissivity parameter $\eta$ varies from 0 to 1. The first plot has Alice’s mean photon number much higher than Bob’s, while the second plot sets them equal.
Comparison with the Unassisted Bosonic Multiple Access Rate Region
------------------------------------------------------------------
We would also like to compare the achievable rate region given by Theorem \[thm:ea-bosonic\] to the Yen-Shapiro outer bound for unassisted classical communication over the beamsplitter bosonic multiple access channel [@YS05]. Consider that the Yen-Shapiro outer bound is as follows:$$\begin{aligned}
R_{1} & \leq g\left( N_{S_{a}}\right) ,\\
R_{2} & \leq g\left( N_{S_{b}}\right) ,\\
R_{1}+R_{2} & \leq g\left( \eta N_{S_{a}}+\left( 1-\eta\right) N_{S_{b}}\right) . \label{eq:unassisted-sum-rate}$$ They derived this outer bound with two straightforward arguments. First, if $N_{S_{a}}$ and $N_{S_{b}}$ are the respective mean photon numbers at the channel input, then the mean photon number at the output is $\eta N_{S_{a}}+\left( 1-\eta\right) N_{S_{b}}$, and the Holevo quantity can never exceed $g\left( \eta N_{S_{a}}+\left( 1-\eta\right) N_{S_{b}}\right)
$ [@GGLMSY04]. The individual rate bounds follow by assuming that the receiver gets access to both output ports of the channel. The best strategy would then be simply to invert the beamsplitter, and the rate bounds follow from a similar argument (that the Holevo quantity for mean photon number constraints $N_{S_{a}}$ and $N_{S_{b}}$ cannot exceed $g\left( N_{S_{a}}\right) $ and $g\left( N_{S_{b}}\right) $, respectively).
It is straightforward to demonstrate that the sum rate bound in Theorem \[thm:ea-bosonic\] always exceeds the sum rate bound in (\[eq:unassisted-sum-rate\]). Consider that the difference between these two sum rate bounds is$$g\left( N_{S_{a}}\right) +g\left( N_{S_{b}}\right) -g\left( \eta
N_{S_{b}}+\left( 1-\eta\right) N_{S_{a}}\right) ,$$ and this quantity is always positive because $g\left( x\right) $ is positive and monotone increasing for $x\geq0$ (i.e., supposing WLOG that $N_{S_{a}}\geq N_{S_{b}}$, it follows that $N_{S_{a}}\geq\eta N_{S_{b}}+\left(
1-\eta\right) N_{S_{a}}$ and thus $g\left( N_{S_{a}}\right) \geq g\left(
\eta N_{S_{b}}+\left( 1-\eta\right) N_{S_{a}}\right) $). The individual rate bounds are incomparable as Figure \[fig:unassist-vs-assist\] demonstrates—there are examples of channels and photon number constraints for which the assisted region contains or does not contain the Yen-Shapiro unassisted outer bound.
Conclusion
==========
We have discussed five different scenarios for entanglement-assisted classical communication: sequential decoding for a single-sender, single-receiver channel, sequential and successive decoding for a multiple access channel, simultaneous decoding, coherent simultaneous decoding, and communication over a bosonic channel. Our third contribution gives further progress toward proving the quantum simultaneous decoding conjecture from Ref. [@FHSSW11] (see Appendix A in the thesis of Dutil for a different manifestation of this conjecture in distributed compression [@D11]).
Several open questions remain. It would of course be good to prove that the quantum simultaneous decoding conjecture holds in the general case for entanglement-assisted classical communication or even to broaden the classes of channels or the conditions for which it holds. It would be worthwhile to determine whether our strategy for entanglement-assisted classical communication over a bosonic multiple access channel is optimal.
We are grateful to Vittorio Giovannetti for suggesting the idea of extending the GLM sequential decoder to the entanglement-assisted case, and we thank Pranab Sen for sharing his results in Ref. [@S11a] and for pointing out that a slight modification of our proof technique from the first version of this article solves the quantum simultaneous decoding conjecture for two senders. MMW acknowledges useful discussions with Omar Fawzi, Patrick Hayden, Ivan Savov, and Pranab Sen during the development of Ref. [@FHSSW11]. MMW acknowledges financial support from the MDEIE (Québec) PSR-SIIRI international collaboration grant.
Appendix {#sec:sequential-packing-proof}
========
\[Proof of the Sequential Packing Lemma\]Our proof below is essentially identical to the proof given in Ref. [@GLM10], with the exception that it extracts only the most basic conditions needed (these conditions are given in the statement of the theorem). Given a message set $\mathcal{M}=\left\{
1,2,\dots,\left\vert \mathcal{M}\right\vert \right\} $, we construct a code $\mathcal{C}\equiv\left\{ c_{m}\right\} _{m\in\mathcal{M}}$ randomly such that each $c_{m}$ takes a value in $\mathcal{X}$ with probability $p_{X}\left( c_{m}\right) $. Using this code, Alice chooses a message $m$ from the message set $\mathcal{M}$ and encodes it in the quantum codeword $\rho_{c_{m}}$. To decode the message $m$, Bob performs the following steps:
1. Starting from $k=1$, Bob tries to determine if he received the $k$th message.
2. Bob first makes a projective measurement with the code subspace projector $\Pi$ to determine if the received state is in the code subspace.
3. If the answer is NO, then an error has occurred and Bob aborts the protocol.
4. If the answer is YES, Bob performs another projective measurement on the post-measurement state using the codeword subspace projector $\Pi_{c_{k}}$.
5. If the answer is YES, then Bob declares to have received the $k$th message and stops the protocol.
6. If the answer is NO, then Bob increments $k$ and goes back to Step 2 if $k<\left\vert \mathcal{M}\right\vert $. If $k=\left\vert \mathcal{M}\right\vert $, Bob declares that an error has occurred and aborts the protocol.
As derived in Ref. [@GLM10], the following POVM $\left\{ \Lambda
_{m}\right\} _{m\in\mathcal{M}}$ corresponds to the above sequential decoding scheme:$$\Lambda_{m}\equiv\bar{Q}_{c_{1}}\cdots\bar{Q}_{c_{m-1}}\bar{\Pi}_{c_{m}}\bar{Q}_{c_{m-1}}\cdots\bar{Q}_{c_{1}},$$ where for any operator $\Theta$, we define $\bar{\Theta}$ as$$\bar{\Theta}\equiv\Pi\Theta\Pi,$$ and$$Q_{x}\equiv I-\Pi_{x}.$$
We analyze the performance of this sequential decoding scheme by computing a lower bound on the expectation of the average success probability, where the expectation is with respect to all possible codes:$$\begin{aligned}
\mathbb{E}_{\mathcal{C}}\left\{ \bar{p}_{\text{succ}}\left( \mathcal{C}\right) \right\} & =\sum_{c_{1},\dots,c_{\left\vert \mathcal{M}\right\vert
}}p_{X}\left( c_{1}\right) \cdots p_{X}\left( c_{\left\vert \mathcal{M}\right\vert }\right) \frac{1}{\left\vert \mathcal{M}\right\vert }\sum
_{m=1}^{\left\vert \mathcal{M}\right\vert }\mathrm{Tr}\left\{ \Pi_{c_{m}}\bar{Q}_{c_{m-1}}\cdots\bar{Q}_{c_{1}}\rho_{c_{m}}\bar{Q}_{c_{1}}\cdots\bar
{Q}_{c_{m-1}}\right\} \\
& =\frac{1}{\left\vert \mathcal{M}\right\vert }\sum_{l=0}^{\left\vert
\mathcal{M}\right\vert -1}\sum_{x,c_{1},\dots,c_{l}}p_{X}\left( x\right)
p_{X}\left( c_{1}\right) \cdots p_{X}\left( c_{l}\right) \mathrm{Tr}\left\{ \Pi_{x}\bar{Q}_{c_{l}}\cdots\bar{Q}_{c_{1}}\rho_{x}\bar{Q}_{c_{1}}\cdots\bar{Q}_{c_{l}}\right\} \\
& =\frac{1}{\left\vert \mathcal{M}\right\vert }\sum_{l=0}^{\left\vert
\mathcal{M}\right\vert -1}\sum_{x,c_{1},\dots,c_{l}}p_{X}\left( x\right)
p_{X}\left( c_{1}\right) \cdots p_{X}\left( c_{l}\right) \sum_{y}\sum_{y^{\prime}\in\mathcal{T}_{x}}\lambda_{x,y}\left\vert \left\langle
\psi_{x,y^{\prime}}\right\vert \bar{Q}_{c_{1}}\cdots\bar{Q}_{c_{l}}\left\vert
\psi_{x,y}\right\rangle \right\vert ^{2}. \label{eq:succ-lower-bound-1}$$ We obtain the last equality by writing out the spectral decomposition of $\Pi_{x}$ and $\rho_{x}$:$$\begin{aligned}
\rho_{x} & =\sum_{y}\lambda_{x,y}\left\vert \psi_{x,y}\right\rangle
\left\langle \psi_{x,y}\right\vert ,\\
\Pi_{x} & =\sum_{y\in\mathcal{T}_{x}}\left\vert \psi_{x,y}\right\rangle
\left\langle \psi_{x,y}\right\vert .\end{aligned}$$ We note that $\rho_{x}$ and $\Pi_{x}$ commute by assumption and therefore share common eigenstates. We use $\mathcal{T}_{x}$ to index a subset of the eigenstates of $\rho_{x}$.
The following lower bound applies to the rightmost term in (\[eq:succ-lower-bound-1\]):$$\begin{aligned}
\sum_{y}\sum_{y^{\prime}\in\mathcal{T}_{x}}\lambda_{x,y}\left\vert
\left\langle \psi_{x,y^{\prime}}\right\vert \bar{Q}_{c_{1}}\cdots\bar
{Q}_{c_{l}}\left\vert \psi_{x,y}\right\rangle \right\vert ^{2} & \geq
\sum_{y\in\mathcal{T}_{x}}\lambda_{x,y}\left\vert \left\langle \psi
_{x,y}\right\vert \bar{Q}_{c_{1}}\cdots\bar{Q}_{c_{l}}\left\vert \psi
_{x,y}\right\rangle \right\vert ^{2}\\
& =\sum_{y\in\mathcal{T}_{x}}\lambda_{x,y}\left\vert \left\langle \psi
_{x,y}\right\vert \bar{Q}_{c_{1}}\cdots\bar{Q}_{c_{l}}\left\vert \psi
_{x,y}\right\rangle \right\vert ^{2}\sum_{y}\lambda_{x,y}\label{eq:beforeCS}\\
& \geq\left\vert \sum_{y\in\mathcal{T}_{x}}\lambda_{x,y}\left\langle
\psi_{x,y}\right\vert \bar{Q}_{c_{1}}\cdots\bar{Q}_{c_{l}}\left\vert
\psi_{x,y}\right\rangle \right\vert ^{2}\label{eq:afterCS}\\
& =\left\vert \mathrm{Tr}\left\{ \Pi_{x}\rho_{x}\Pi_{x}\bar{Q}_{c_{1}}\cdots\bar{Q}_{c_{l}}\right\} \right\vert ^{2}.\end{aligned}$$ The first inequality follows by eliminating some positive terms from the summation. The second inequality follows by applying the Cauchy-Schwarz inequality. The last equality follows by exploiting the assumed commutative relation between $\Pi_{x}$ and $\rho_{x}$. Therefore, the following lower bound applies to the expectation of the average success probability:$$\mathbb{E}_{\mathcal{C}}\left\{ \bar{p}_{\text{succ}}\left( \mathcal{C}\right) \right\} \geq\frac{1}{\left\vert \mathcal{M}\right\vert }\sum
_{l=0}^{\left\vert \mathcal{M}\right\vert -1}\sum_{x,c_{1},\dots,c_{l}}p_{X}\left( x\right) p_{X}\left( c_{1}\right) \cdots p_{X}\left(
c_{l}\right) \left\vert \mathrm{Tr}\left\{ \Pi_{x}\rho_{x}\Pi_{x}\bar
{Q}_{c_{1}}\cdots\bar{Q}_{c_{l}}\right\} \right\vert ^{2}.$$ We again apply the Cauchy-Schwarz inequality to the inner summation:$$\begin{aligned}
& \sum_{x,c_{1},\dots,c_{l}}p_{X}\left( x\right) p_{X}\left( c_{1}\right)
\cdots p_{X}\left( c_{l}\right) \left\vert \mathrm{Tr}\left\{ \Pi_{x}\rho_{x}\Pi_{x}\bar{Q}_{c_{1}}\cdots\bar{Q}_{c_{l}}\right\} \right\vert
^{2}\\
\geq & \left\vert \sum_{x,c_{1},\dots,c_{l}}p_{X}\left( x\right)
p_{X}\left( c_{1}\right) \cdots p_{X}\left( c_{l}\right) \mathrm{Tr}\left\{ \Pi_{x}\rho_{x}\Pi_{x}\bar{Q}_{c_{1}}\cdots\bar{Q}_{c_{l}}\right\}
\right\vert ^{2}\\
= & \left\vert \mathrm{Tr}\left\{ \left( \sum_{x}p_{X}\left( x\right)
\Pi_{x}\rho_{x}\Pi_{x}\right) \left( \sum_{c_{1}}p_{X}\left( c_{1}\right)
\bar{Q}_{c_{1}}\right) \cdots\left( \sum_{c_{l}}p_{X}\left( c_{l}\right)
\bar{Q}_{c_{l}}\right) \right\} \right\vert ^{2}\\
= & \left\vert \mathrm{Tr}\left\{ W_{1}\mathcal{Q}^{l}\right\} \right\vert
^{2},\end{aligned}$$ where we define$$\begin{aligned}
W_{q} & \equiv\sum_{x}p_{X}\left( x\right) \Pi_{x}\rho_{x}^{q}\Pi_{x},\\
\mathcal{Q} & \equiv\sum_{x}p_{X}\left( x\right) \bar{Q}_{x},\end{aligned}$$ and it is understood that $\mathcal{Q}^{0}=\Pi$ (an abuse of notation explained further below). Therefore, we obtain the following lower bound on the expectation of the average success probability:$$\mathbb{E}_{\mathcal{C}}\left\{ \bar{p}_{\text{succ}}\left( \mathcal{C}\right) \right\} \geq\frac{1}{\left\vert \mathcal{M}\right\vert }\sum
_{l=0}^{\left\vert \mathcal{M}\right\vert -1}\left\vert \mathrm{Tr}\left\{
W_{1}\mathcal{Q}^{l}\right\} \right\vert ^{2}. \label{eq:succ-prob-1-1}$$
In order to proceed, we note that $$\begin{aligned}
\mathcal{Q} & =\sum_{x}p_{X}\left( x\right) \bar{Q}_{x}\\
& =\Pi\left( \sum_{x}p_{X}\left( x\right) \left( I-\Pi_{x}\right)
\right) \Pi\\
& \leq I,\end{aligned}$$ and therefore$$\mathrm{Tr}\left\{ W_{1}\mathcal{Q}^{l}\right\} =\mathrm{Tr}\left\{
W_{1}\mathcal{Q}^{\frac{l-1}{2}}\mathcal{QQ}^{\frac{l-1}{2}}\right\}
\leq\mathrm{Tr}\left\{ W_{1}\mathcal{Q}^{l-1}\right\} .$$
Given this observation, we can further lower bound the success probability by taking the smallest term of the summation from (\[eq:succ-prob-1-1\]):$$\begin{aligned}
\mathbb{E}_{\mathcal{C}}\left\{ \bar{p}_{\text{succ}}\left( \mathcal{C}\right) \right\} & \geq\left\vert \mathrm{Tr}\left\{ W_{1}\mathcal{Q}^{\left\vert \mathcal{M}\right\vert -1}\right\} \right\vert ^{2}\\
& =\left\vert \mathrm{Tr}\left\{ W_{1}\left( \bar{I}-\bar{W}_{0}\right)
^{\left\vert \mathcal{M}\right\vert -1}\right\} \right\vert ^{2}\\
& =\left\vert \mathrm{Tr}\left\{ \sum_{z=0}^{\left\vert \mathcal{M}\right\vert -1}\binom{\left\vert \mathcal{M}\right\vert -1}{z}\left(
-1\right) ^{z}W_{1}\bar{I}^{\left\vert \mathcal{M}\right\vert -z}\bar{W}_{0}^{z}\right\} \right\vert ^{2}\\
& =\left\vert \sum_{z=0}^{\left\vert \mathcal{M}\right\vert -1}\binom{\left\vert \mathcal{M}\right\vert -1}{z}\left( -1\right)
^{z}\mathrm{Tr}\left\{ W_{1}\Pi\bar{W}_{0}^{z}\right\} \right\vert ^{2}$$ Here, we define a function $f_{z}$ as$$f_{z}\equiv\mathrm{Tr}\left\{ W_{1}\Pi\bar{W}_{0}^{z}\right\} ,$$ where $z$ is a nonnegative integer. We abused the notation of $\bar{W}_{0}^{z}$ here, which does *not* mean to raise the eigenvalues of $\bar
{W}_{0}$ to the power $z$ in its spectral decomposition, but rather$$\bar{W}_{0}^{z}=\prod_{i=1}^{z}\bar{W_{0}},$$ as it arises from the binomial expansion. We note that $f_{0}=\mathrm{Tr}\left\{ W_{1}\Pi\right\} $ and the function $f_{z}$ is always positive. Thus, the above expression is equal to the following one:$$\begin{aligned}
& \left\vert \sum_{z=0}^{\left\vert \mathcal{M}\right\vert -1}\binom
{\left\vert \mathcal{M}\right\vert -1}{z}\left( -1\right) ^{z}f_{z}\right\vert ^{2}\\
& =\left\vert f_{0}+\sum_{z=1}^{\left\vert \mathcal{M}\right\vert -1}\binom{\left\vert \mathcal{M}\right\vert -1}{z}\left( -1\right) ^{z}f_{z}\right\vert ^{2}\\
& =\left\vert A\right\vert ^{2},\end{aligned}$$ with$$A\equiv f_{0}+\sum_{z=1}^{\left\vert \mathcal{M}\right\vert -1}\binom
{\left\vert \mathcal{M}\right\vert -1}{z}\left( -1\right) ^{z}f_{z}.$$ We then have$$A\geq2f_{0}-\sum_{z=0}^{\left\vert \mathcal{M}\right\vert -1}\binom{\left\vert
\mathcal{M}\right\vert -1}{z}f_{z}.$$ The function $f_{z}$ satisfies the following two properties:$$\begin{aligned}
f_{0} & \geq1-2\epsilon,\\
f_{z} & \leq\left( \frac{d}{D}\right) ^{z}f_{0}.\end{aligned}$$ We now prove this. First we show that $f_{0}$ is $\epsilon$-close to one: $$\begin{aligned}
f_{0} & =\mathrm{Tr}\left\{ W_{1}\Pi\right\} \\
& =\sum_{x}p_{X}\left( x\right) \mathrm{Tr}\left\{ \Pi_{x}\rho_{x}\Pi
_{x}\Pi\right\} \\
& =\sum_{x}p_{X}\left( x\right) \mathrm{Tr}\left\{ \Pi_{x}\rho_{x}\Pi\right\} \\
& =\sum_{x}p_{X}\left( x\right) \mathrm{Tr}\left\{ \left( I-\left(
I-\Pi_{x}\right) \right) \rho_{x}\Pi\right\} \\
& =\sum_{x}p_{X}\left( x\right) \mathrm{Tr}\left\{ \rho_{x}\Pi\right\}
-\sum_{x}p_{X}\left( x\right) \mathrm{Tr}\left\{ (I-\Pi_{x})\rho_{x}\Pi\right\} \\
& \geq\sum_{x}p_{X}\left( x\right) \mathrm{Tr}\left\{ \rho_{x}\Pi\right\}
-\sum_{x}p_{X}\left( x\right) \mathrm{Tr}\left\{ (I-\Pi_{x})\rho
_{x}\right\} \\
& \geq\sum_{x}p_{X}\left( x\right) \mathrm{Tr}\left\{ \rho_{x}\Pi\right\}
-\epsilon\nonumber\\
& \geq1-2\epsilon\end{aligned}$$ The third equality follows by the commutative relation between $\Pi_{x}$ and $\rho_{x}$. The second inequality follows from the condition (\[eq:unit-prob-2\]). Now we will upper bound the function $f_{z}$ in terms of $f_{z-1}$, and we show that we can upper bound $f_{z}$ in terms of $f_{z-1}$, and as a result, in terms of $f_{0}$. $$\begin{aligned}
f_{z} & =\mathrm{Tr}\left\{ W_{1}\bar{W}_{0}^{z}\right\} \\
& =\mathrm{Tr}\left\{ \sqrt{W_{1}}\bar{W}_{0}^{\frac{z-1}{2}}\bar{W}_{0}\bar{W}_{0}^{\frac{z-1}{2}}\sqrt{W_{1}}\right\} \\
& =\mathrm{Tr}\left\{ \sqrt{W_{1}}\bar{W}_{0}^{\frac{z-1}{2}}\Pi\left(
\sum_{x}p_{X}\left( x\right) \Pi_{x}\right) \Pi\bar{W}_{0}^{\frac{z-1}{2}}\sqrt{W_{1}}\right\} \\
& \leq d\cdot\mathrm{Tr}\left\{ \sqrt{W_{1}}\bar{W}_{0}^{\frac{z-1}{2}}\Pi\left( \sum_{x}p_{X}\left( x\right) \Pi_{x}\rho_{x}\Pi_{x}\right)
\Pi\bar{W}_{0}^{\frac{z-1}{2}}\sqrt{W_{1}}\right\} \\
& \leq d\cdot\mathrm{Tr}\left\{ \sqrt{W_{1}}\bar{W}_{0}^{\frac{z-1}{2}}\Pi\rho\Pi\bar{W}_{0}^{\frac{z-1}{2}}\sqrt{W_{1}}\right\} \\
& \leq\frac{d}{D}\mathrm{Tr}\left\{ \sqrt{W_{1}}\bar{W}_{0}^{\frac{z-1}{2}}\Pi\bar{W}_{0}^{\frac{z-1}{2}}\sqrt{W_{1}}\right\} \\
& \leq\frac{d}{D}\mathrm{Tr}\left\{ W_{1}\bar{W}_{0}^{z-1}\right\} \\
& =\frac{d}{D}f_{z-1}\\
\Rightarrow f_{z} & \geq\left( \frac{d}{D}\right) ^{z}f_{0}$$ In this derivation, we used the conditions (\[eq:equi-part-1\]) and (\[eq:equi-part-2\]) and the fact that $W_{1}$ and $\bar{W}_{0}$ are positive.
Therefore, using the above two inequalities, we get that$$\begin{aligned}
A & \geq2f_{0}-f_{0}\sum_{z=0}^{\left\vert \mathcal{M}\right\vert -1}\binom{\left\vert \mathcal{M}\right\vert -1}{z}\left( \frac{d}{D}\right)
^{z}\\
& =f_{0}\left( 2-\left( 1+\frac{d}{D}\right) ^{\left\vert \mathcal{M}\right\vert -1}\right) \\
& \geq\left( 1-2\epsilon\right) \left( 2-e^{\frac{d}{D}\left\vert
\mathcal{M}\right\vert }\right) .\end{aligned}$$ The last inequality follows from the fact that $1+x\leq e^{x}$ for all $x$, and our analysis completes with the observation that$$\mathbb{E}_{\mathcal{C}}\left\{ \bar{p}_{\text{succ}}\left( \mathcal{C}\right) \right\} \geq\left\vert A\right\vert ^{2}\geq\left\vert \left(
1-2\epsilon\right) \left( 2-e^{\frac{d}{D}\left\vert \mathcal{M}\right\vert
}\right) \right\vert ^{2},$$ as long as $2-\exp\left\{ d\left\vert \mathcal{M}\right\vert /D\right\} $ is positive.
[^1]: We are indebted to Pranab Sen for this observation [@S11] (c.f., versions 1 and 2 of this paper).
[^2]: The argument is similar to those appearing Refs. [@YS05; @G08], for example, and is simply that an infinite-dimensional Hilbert space with a mean photon-number constraint is effectively identical to a finite-dimensional Hilbert space. Suppose that we truncate the Hilbert space at the channel input so that it is spanned by the Fock number states $\left\{ \left\vert 0\right\rangle ,\left\vert
1\right\rangle ,\ldots,\left\vert K\right\rangle \right\} $ where $K\gg
N_{S}$. Thus, all coherent states, squeezed states, and thermal states become truncated to this finite-dimensional Hilbert space. Applying the Hsieh-Devetak-Winter theorem to squeezed states in this truncated Hilbert space gives a capacity region which is strictly an inner bound to the region in (\[eq:EA-boson-region-1\]-\[eq:EA-boson-region-3\]). As we let $K$ grow without bound, the entropies given by the Hsieh-Devetak-Winter theorem converge to the entropies in (\[eq:EA-boson-region-1\]-\[eq:EA-boson-region-3\]).
|
---
abstract: 'Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to quantum mechanics. We will concentrate our attention into quantum theories and we will show how to use in a systematic way the transition from algebraic to geometrical structures to explore their geometry, mainly its Jordan-Lie structure.'
author:
- 'J.F. Cariñena$^{a)}$, A. Ibort$^{b)}$, G. Marmo$^{c)}$ and G. Morandi$^{d)}$'
title: 'Geometrical description of algebraic structures: Applications to Quantum Mechanics'
---
$^{a)}$ Departamento de Física Teórica, Universidad de Zaragoza,
50009 Zaragoza, Spain
jfc@unizar.es
$^{b)}$ Departamento de Matemáticas, Universidad Carlos III de Madrid,
Avda. de la Universidad 30, 28911 Leganés, Spain
albertoi@math.uc3m.es
$^{c)}$ Dipartimento di Scienze Fisiche, Universitá Federico II di Napoli e Sezione INFN di Napoli,
Via Cintia, 80125, Napoli, Italy
marmo@na.infn.it
$^{d)}$ Dipartimento di Fisica, Universitá di Bologna e Sezione INFN di Bologna
6/2 viale B. Pichat, I-40127 Bologna, Italy
Giuseppe.Morandi@bo.infn.it
[*Keywords:*]{} Symplectic, Poisson, connection, Hermitean
PACS codes: 02.40.Yy; 03.65.Ca; 45.20.Jj
“[*... I realized that the foundations of geometry have physical relevance*]{}.”
“[*Tensor calculus knows physics better than most physicists*]{}”
Introduction
=============
The two quotations above should make clear why we would like to privilege a geometrical description of physical systems. The geometrical description of physical systems uses more general objects than the traditional Euclidean spaces: differentiable manifolds, sometimes endowed with particular structures. After the formulation of General Relativity by means of the (pseudo-) Riemannian geometry, it is accepted without any doubt that the equations used to describe physical systems should be written in tensorial form. For instance, we may indeed consider classical Gauge Theories as the Theory of Connections.
Quantum theories, due to the superposition rule, are always formulated as theories on complex vector spaces or algebras (the Schrödinger equation on a Hilbert space and the Heisenberg equation on a $\mathbb{C}^*$-algebra). It is however convenient to analyse the problem from a more general perspective, which is manifestly necessary when the character of rays rather than vector of pures states is taken into account. We hope that a geometrization of quantum mechanics may be used for a more sound theory of Quantum Gravity. It follows that to ‘geometrize’ quantum theories we should first describe some fundamental algebraic structures in tensorial terms and then apply this procedure to describe quantum theories by means of tensorial entities. The general ideology of this presentation is being elaborated in a book provisionally entitled: *Geometrical Theory of Classical and Quantum Dynamical Systems*.
We shall proceed by explaining first what we mean by a geometrical description of physical systems by considering Newton’s equations and Maxwell’s equations, then we consider the usual formulation of quantum mechanics and we finally introduce a tensorial description of the algebraic structures emerging in the usual formulation.
Newton’s equations
------------------
We shall start by considering [*Newton’s equations*]{}, this is a second-order differential equation on some connected and simply connected configuration space $Q$: $$\frac{d^2x}{dt^2}= F\left(x,\frac {dx}{dt}\right) .$$ With this equation we associate a vector field $\Gamma$ on the tangent bundle $TQ$, or velocity phase space, of $Q$, say [@MSSV], $$\Gamma= v\,\pd{}{x}+F(x,v)\pd{}{v}\,.$$ Having a vector field on a manifold we can use all transformations on $TQ$ to transform it and find an easier way to integrate it for instance. Usually the existence of additional geometrical structures compatible with the given dynamics will uncover properties of it and will help with its integration. Thus, given a dynamics $\Gamma$ one usually looks for compatible structures, among them and most noticingly, Poisson brackets. In other words, one tries to find out if the given dynamics is Hamiltonian with respect to some (in principle unknown) Poisson brackets. Poisson brackets are encoded into a Poisson bivector field, i.e. a $\Gamma$-invariant, contravariant skew-symmetric 2-tensor field $\Lambda$ such that $\mathcal{L}_{\Gamma}\Lambda=0$. This equation has a clear tensorial meaning. This bivector field is required to satisfy $[\Lambda,\Lambda]_{S}=0$, where $[\cdot ,\cdot]_{S}$ is the Schouten bracket or, equivalently, the associated Poisson bracket: $$\left\{ f_{1},f_{2}\right\} =\Lambda\left( df_{1},df_{2}\right)$$ should satisfy the Jacobi identity, which is a quadratic relation. This condition for $\Lambda$ is a nonlinear partial differential equation. It may admit no solution, one solution or many solutions. When it has more than one solution the dynamical system we are describing is called a bi-Hamiltonian system an exhibits some integrability properties [@magri]. As a matter of fact we have to distinguish the case of degenerate and non-degenerate tensors. Whenever the Poisson tensor is non-degenerate one can define (‘modulo’ an arbitrary and irrelevant additive constant) a Hamiltonian function $H$ via $$\Lambda(\cdot,dH)=\Gamma\,,$$ and the dynamics can be written in Hamiltonian form as: $${\mathcal{L}}_\Gamma
f=\{f,H\}\,.$$ In this case we say that $\Gamma$ defines an inner derivation of the Poisson algebra defined by the Poisson bivector field on the space $\mathcal{F}(M)$ of smooth functions on the manifold (the tangent bundle in the case of second-order differential equations). If, instead, $\Lambda$ is degenerate, then $\Gamma$ is still a derivation of the Poisson algebra but it need not to be inner and it may define what is called an outer derivation, i.e. it will not be the image under $\Lambda$ of a 1-form. When it is the image of a 1-form, the 1-form needs to be closed only on vector fields which define inner derivations. We shall not insist on these aspects and refer the reader to the literature [@MFLMR].
When $\Lambda$ is non-degenerate, the condition $$\Lambda(df,dh)=0\,, \forall f\in \mathcal{F}(M)$$ implies that the function $h$ is a constant function and we may associate a (symplectic) structure $\omega_{\Lambda}$ to $\Lambda$ and the quadratic condition coming from the Schouten bracket becomes $d\omega_\Lambda=0$.
In the case of a second-order differential equation, by using $\tau_Q:TQ\to Q$ we may further require the localization property (i.e., the possibility of measuring simultaneously observables depending only on configuration variables): $$\Lambda(\tau^*_Qdg_1,\tau^*_Qdg_2)=0\,, \qquad \forall g_1,g_2\in {\mathcal{F}}(Q)\,,$$ and then we find (see [@CIMS]) that there exists a function $L\in {\mathcal{F}}(TQ)$ such that $$\omega_{\Lambda}=-d\theta_L\,,$$ where $\theta_L = S^*(dL)$ and $S$ denotes the soldering (1,1)-tensor field (or vertical endomorphism) [@MFLMR]: $$S=dx^j\otimes \pd{}{v^j} .$$ In such a case the dynamics $\Gamma$ may be described in terms of a Lagrangian function $L\in
{\mathcal{F}}(TQ)$ and if we introduce the Liouville vector field [@MFLMR] $$\Delta=v^i\pd{}{v^i}\,,$$ which is the infinitesimal generator of dilations along the fibres, and the energy function $E_L=\Delta L-L$, then $(TQ,\omega_\Lambda,E_L)$ is a Hamiltonian system such that $i(\Gamma)\omega_\Lambda=dE_L$.
Then starting from Newton’s equations on $Q$ we have defined a second-order vector field on $TQ$ and if there exists a localizable compatible non-degenerate Poisson tensor $\Lambda$ we have defined a symplectic structure and a Lagrangian function such that the original dynamics is both a Hamiltonian and Lagrangian system, completing a geometrization of the original equations of motion.
[**Remark:**]{} This approach shows very clearly how we reduce ${\rm
Diff\,}(TQ)$ to ${\rm Diff\,}(TQ,\Lambda)$ and further to tangent bundle diffeomorphisms according to Klein’s Erlangen programme, i.e. we may start with the diffeomorphism group and ‘break it’ to appropriate subgroups by means of additional structures. These subgroups in general are enough to identify the manifold along with the additional structures.
Maxwell’s equations
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[*Maxwell’s equations*]{} for the electric and magnetic fields, ${\bf E}$ and ${\bf B}$, in empty space and without sources can be written as: $$\frac d{dt}\matriz{c}{{\bf B}\\{\bf E}}=\matriz{cc}{0&-{\rm rot}\\{\rm
rot}&0}\matriz{c}{{\bf B}\\{\bf E}}\,,$$ which are evolution or dynamical equations on the space of electric and magnetic fields, and: $$\label{constraints}
\nabla\cdot\mathbf{B}=0,\qquad \nabla\cdot\mathbf{E}=0,$$ which are [*constraint*]{} equations. Here again we may rewrite this system of equations in Hamiltonian form, but the constraints Eqs. (\[constraints\]) will restrict the possible Cauchy data we may evolve with the evolutionary equations. It is possible to argue, and it is often done, that a Lagrangian description for these equations is possible by means of a degenerate (or gauge invariant) Lagrangian written on a bigger carrier space described by vector potentials $A = ({\bf A}, \phi)$, such that ${\rm rot\,} {\bf A} = {\bf B}$, and ${\bf E} = \dot{\bf A} - \nabla \phi$, The introduction of the vector potential is a way to take into account holonomic constraints given by div $\mathbf{B}=0$, the constraint on $\mathbf{E}$ being on ‘velocities’ is a non-holonomic constraint. Thus we can achieve a geometrization of Maxwell’s equations in empty space without sources as a non-holonomic degenerate Lagrangian system on the space of vector potentials. The geometrization of Maxwell’s equations in empty space is completed in covariant form when considered as a theory of connections on a $U(1)$-principal bundle over space-time.
For completeness we comment on the covariant geometrical formulation of Maxwell’s equations when we consider also sources. A covariant geometrical formulation [@MPT; @MT] requires the introduction of the Faraday 2-form $$D=E\wedge dt -B\,,$$ and the Ampère’s odd 2-form $$G=H\wedge dt +D\,,$$ with the odd 3-form $$J=\rho+j\wedge dt\,.$$
The equations $$\left\{\begin{array}{rcl} dF&=&0\,,\\
dG&=&J\,,\end{array}\right.$$ must be supplemented with phenomenological constitutive equations $${\mathcal{C}}(F,G,J)=0\,.$$ The constitutive equations are ‘phenomenological’ relations between the Faraday tensor, the Ampere’s tensor and the sources. They need not be local, i.e. $${\mathcal{C}}(F,G,J)(x,t)\neq{\mathcal{C}}(F(x,t),G(x,t),J(x,t))\,.$$ When they are local, additional geometrical structures may be associated with them.
More general Gauge Theories [@BMSS] are also geometrical theories, indeed they can be considered, as Maxwell’s equations, theories of connections in principal bundles, for instance with structural group $SU(3)\times SU(2)\times
U(1)$. In this respect also General Relativity may be considered a theory of (pseudo-) Riemannian connections. Both theories have been considered jointly in the framework of Kaluza–Klein theories [@CLM1; @CLM2]. Once again we have achieved a purely tensorial description of our physical system.
We will turn now our attention to quantum theories.
Geometrical description of Quantum Mechanics
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The geometry of quantum mechanics, contraryly to what has happened with other physical theories, has played a minor rôle after its beginning. In fact, von Neumann’s formulation of quantum mechanics in terms of the theory of Hilbert spaces constituted already a formidable geometrization of quantum mechanics, however, further geometrical analysis of the theory was not pursued.
Quantum Mechanics
-----------------
Quantum mechanics [@EM] is usually described in the realm of Hilbert space theory in different ways called ‘pictures’. We will concentrate here only on the dynamics of quantum systems, i.e. on the equation describing the evolution of quantum states. Thus we will associate a complex separable Hilbert space $\Hil$ (the set of pure states) with our physical system and the observables of the system are the self-adjoint operators (not necessarily bounded) on $\Hil$. We will not analyze here other physical aspects of quantum systems like the measure process, etc.
- Schrödinger’s picture: The equation of motion, Schrödinger equation, is written as: $$\label{schrodinger}
\ii\hbar \pd{\ket{\psi}}t =H\ket{\psi}\,,\qquad \ket{\psi}\in \Hil,\,$$ where $H$ is a self-adjoint operator, typically unbounded, on the Hilbert space $\Hil$.
- Heisenberg’s picture: The equations of motion are written as: $$\ii\hbar \frac{dA}{dt} =[H,A]\,,$$ where $A$ and $H$ are self-adjoint operators on $\Hil$.
- Dirac’s interaction picture: The evolution equation is written now as: $$\left(\ii\hbar \frac d{dt}U \right)U^{-1}=H\,,$$ where $U$ is a unitary operator on $\Hil$. This is a generalized Dirac picture written on the group of unitary transformations, in this sense it is a quantum theory written on a group manifold.
All these images can be seen as different realizations in associated vector bundles of a principal connection (see e.g. [@ACP]).
Geometrical description of the Schrödinger picture
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Previous descriptions of quantum mechanics are given, except for the generalized Dirac picture, on carrier spaces which are complex linear spaces. To describe the equations in tensorial terms [@CJM] we should replace linear spaces with real manifolds and linear operators with tensor fields. We first consider the complex linear space as a real linear space. Then, to this end we may use our experience in going from special relativity to general relativity which replaces the affine Minkowski space with a general Lorentzian manifold. Let us recall what is done to go from special relativity described on some affine space modelled on a Minkowski vector space $V$ with Minkowskian metric (inner product) $\eta_{\mu\nu}\, x^\mu\, x^\nu$ to a description on a pseudo-Riemannian manifold $M$ by replacing the Minkowskian inner product with the metric tensor $g=\eta_{\mu\nu}\, dx^\mu\otimes d x^\nu$.
We may perform a similar trick by formally replacing the inner Hermitean product $\braket\psi\psi$ on the complex Hilbert space $\Hil$ with the Hermitean tensor $\braket{d\psi}{d\psi}.$
Let us consider an orthonormal Hilbert basis in $\Hil$, say $\{\ket{e_j}\mid
j=1,2,\ldots\}$, $\braket{e_j}{e_k}=\delta_{jk}$, and define complex coordinate functions: $$\label{complex_coordinates}
\braket{e_j}{\psi}=z^j=x^j+\ii y^j,\quad \ket{d\psi}=(dz^j)\ket{e_j}.$$ Our Hermitean tensor will give rise to: $$\braket{d\psi}{d\psi}=(d\bar z^k\otimes dz^j)\,\braket{e_k}{e_j},$$ that written in real coordinates looks like: $$\braket{d\psi}{d\psi}=(dx^k\otimes dx^j+dy^k\otimes dy^j)\delta_{kj}+\ii
(dx^k\otimes dy^j-dy^j\otimes dx^k)\delta_{kj}\,.$$ In this way we obtain a symmetric, Riemannian tensor, and a skew-symmetric symplectic tensor, i.e. $$g=dx^k\otimes dx^k+dy^k\otimes dy^k\,,\qquad \omega=dx^k\wedge dy^k\,.\label{thetwostr}$$
In a more rigorous way, the Hilbert space $\Hil$ can be seen as a real space and then both, a Riemann and a symplectic structure, are defined by: $$g(v,w)={\rm Re\,}\langle v,w\rangle\,,\qquad \omega(v,w)={\rm Im\,}\langle v,w\rangle\,.$$ The 2-form $\omega$ can be shown to be an exact 1-form, i.e. there exists a 1-form $\theta$ such that $\omega =-d\theta$. Moreover, there is a complex structure $J$ in $\mathcal{H}$ when considered as a real space: the $\mathbb{R}$-linear map corresponding to multiply by the complex number $\ii$, $Jv=\ii v$, and therefore such that $$J^2=- I .$$ This complex structure relates the Riemann and the symplectic structures: $$g(v_1,v_2)=-\omega(Jv_1,v_2),\qquad \omega
(v_1,v_2)=g(Jv_1,v_2) ,$$ together with: $$g(Jv_1,Jv_2)=g(v_1,v_2)\,.$$
By passing to a contravariant form the structures (\[thetwostr\]) can be substituted by the corresponding contravariant tensors: $$\label{GLambda}
G=\pd{}{x^k}\otimes \pd{}{x^k}+\pd{}{y^k}\otimes \pd{}{y^k}\,,\qquad
\Lambda=\pd{}{x^k}\wedge \pd{}{y^k}\,.$$ We may associate with the first tensor a bi-differential operator: $$\label{symm}
(f_1,f_2)\equiv G(df_1,df_2)=[[\Delta,f_1],f_2]\,,$$ where the Laplacian $\Delta$ is the second-order differential operator: $$\Delta f=\pd{^2f}{{x^k}^2}+\pd{^2f}{{y^k}^2}\,.$$ With the skew-symmetric tensor we may associate a Poisson bracket defined as: $$\label{poiss}
\{ f , g \} = \Lambda(df,dg)=\pd{f}{x^k}\, \pd{g}{y^k}-\pd{f}{y^k}\, \pd{g}{x^k}\,.$$ To the Schrödinger equation, Eq. (\[schrodinger\]), we associate the linear equation $$\label{linear}
\frac {dz^k}{dt}=A^k\,_j\, z^j\,.$$ in the coordinate system $z^k$ introduced above, Eq. (\[complex\_coordinates\]). When $H$ is Hermitean, the matrix $\|A^k\,_j\|$ is skew-Hermitean, i.e. the infinitesimal generator of a unitary transformation if $H$ defines a self-adjoint operator on $\Hil$.
If we associate a vector field $\Gamma$ with our linear equation, Eq. (\[linear\]), as follows: $$\Gamma =A^k\,_j\,z^j\pd{}{z_k} ,$$ we find that ${\mathcal{L}}_\Gamma \, \braket{\cdot}{\cdot}=0$, then $\Gamma$ preserves the Hermitean product. The vector field $\Gamma$ is at the same time Hamiltonian and Killing, i.e., it preserves both the symmetric and the skew-symmetric part separately.
We observe that for any Hermitean operator $A$ we may define an evaluation function which is real valued: $$f_A(\psi)=\bra\psi A\ket{\psi} ,$$ and an expectation value function: $$e_A(\psi)=\frac{\bra\psi A\ket{\psi}}{\braket\psi\psi} .$$ We can compute the symmetric bracket defined before, Eq. (\[symm\]), for pairs of evaluation functions and we find that: $$(f_A,f_B)=G(df_A,df_B)=f_{AB+BA} .$$ Similarly we can compute the Poisson bracket of two evaluation functions, Eq. (\[poiss\]), and we get: $$\{f_A,f_B\}=\Lambda(df_A,df_B)=f_{\ii(AB-BA)}\,.$$ The function $f_A$ defines a Hamiltonian vector field that with the natural identification of $T\Hil$ with $\Hil\oplus \Hil$, can be seen to be given by $X_A(\ket{\psi})=-\ii A\ket{\psi}$, and whose integral curves are the solutions of the equation: $$\frac d{dt}\ket{\psi}=-\ii A\ket{\psi}\,,$$ therefore the dynamical evolution corresponding to a given Hamiltonian $H$ is given by Schrödinger equation Eq. (\[schrodinger\]). Moreover, the expectation value function is such that: $$(e_A,e_A)=\frac{\bra\psi A^2\ket{\psi}}{\braket\psi\psi}-\left(\frac{\bra\psi
A\ket{\psi}}{\braket\psi\psi}\right)^2\,,$$ i.e. the physical interpretation of such a function is clear: $(e_A,e_A)$ is the square of the standard deviation.
**Remark:** More precisely, the space of pure states of a quantum system is not associated with a Hilbert space $\Hil$ but to the manifolds of rays of the Hilbert space $\Hil$, this is to the projective space $\PH$. For instance, if $\Hil={\mathbb{C}}^2$, we have that ${\mathbb{C}}^2$ is a principal bundle with structural group $\mathbb{C}^*$ and base the projective space $\mathbb{CP}^1$ that can be identified with the two-dimensional sphere $S^2$, that is the space of pure states the system. The bivector field $\Lambda$ on $\mathbb{C}^2$ given by Eq. (\[poiss\]) is not projectable on $S^2$, but at each $\ket\psi$ we define two new tensor fields: $$\widetilde \Lambda_{\ket{\psi}}=\braket\psi\psi\ \Lambda_{\ket{\psi}}\,,\qquad \widetilde
G_{\ket{\psi}}=\braket\psi\psi\ G_{\ket{\psi}}\,.$$
These two tensor fields are now projectable. They define corresponding bi-differential operators on $S^2$. Note that neither the function $f_A$ is projectable onto the quotient, however $e_A$ is projectable. Furthermore, even if the symplectic structure $\omega$ is projectable, the corresponding potential function $\theta$ is not projectable and then the projected symplectic form is not exact anymore [@bcgb].
Now we are in the position of defining observables from elements of ${\mathcal{F}}(S^2,\mathbb{C})$ by requiring that $f\in
{\mathcal{F}}(S^2,\mathbb{R})$ and moreover the Hamiltonian vector field associated with $f$ preserves the projected symmetric tensor on $S^2$. Thus, observables are intrinsically defined without any reference to the original Hilbert space.
One can check that $S^2$ is a Lie-Jordan manifold, i.e., both the Lie and the Jordan product on observables are mutually compatible and $$f*g=(f,g)+\ii \{f,g\}-fg$$ defines a ${\mathbb{C}}^*$-algebra structure when the brackets are extended to complex-valued functions whose real and imaginary parts are observables. We will come back to this point in the following section.
Geometrical description of algebraic structures
===============================================
In the previous section we have seen an explicit example where algebraic structures (Hermitian products) are promoted to geometric objects (Riemannian and symplectic structures) and their properties analyzed from that perspective. Such procedure is only an instance of a general procedure. Let us consider a few more examples of this mechanism of interest not only for quantum theories.
Bilinear maps and Frobenius manifolds
-------------------------------------
To convey the general ideas we consider real vector spaces. Let us consider bilinear (or multilinear) maps like: $$B : V \times V\to V, \quad {\rm or}, \quad b: V\to V^*\,.$$ It is clear that $B\in V^*\otimes V^*\otimes V$ and $b\in
V^*\otimes V^*$. The linear space $V$ itself can be immersed into ${\mathcal{F}}(V^*)$, by means of the canonical map: $$\label{bidual}
v\mapsto \widehat v\,, \quad \widehat v(\alpha)=\alpha(v)\,,$$ and each vector $v\in V$ can be regarded as a linear map in $V^*$; hence we may define polynomial functions out of them.
By introducing a basis for $V$, $\{e_j\}$, and the dual basis $\{ \alpha^k \}$ for $V^*$, we will have: $$B = b^l_{jk}\alpha^k\otimes\alpha^j\otimes e_l\,$$ and now we can promote $B$ to define a tensor field $c_B$ on $V$ by replacing the basis vectors $e_j$ and $\alpha^k$ by $dx_j$ and $\partial/\partial x_k$ respectively, this is: $$\label{general}
c_B = b^l_{jk} \, \frac{\partial}{\partial x_j} \otimes \frac{\partial}{\partial x_k} \otimes dx_l ,$$ (clearly $x_k$ denote linear coordinates on $V^*$ with respect to the basis $\alpha^k$). A first application of this observation lies in considering the structure constants $b^l_{jk}$, or the tensor $B$, as the components of an affinely constant connection $\nabla_B$. We can also imagine that the tensor $B$ defines a composition law $\circ$ on the algebra of differential operators by means of: $$\partial_j\circ \partial_k=b_{jk}\,^l\partial_l .$$ Then the associativity condition for the product is equivalent to the vanishing of the curvature. Finally, if we allow $b_{jk}\,^l$ to depend on the point we get the notion of a Frobenius manifold, as introduced by Dubrovin [@D1; @D2; @D3; @D4]. Even more, substituting the differential operators $\partial_k$ by their corresponding symbols $p_k$ on the cotangent bundle $T^*V$, we can define the quadratic functions: $$F_{jk}=p_jp_k-\Gamma_{jk}\,^lp_l\,,$$ and if $\mathcal{J}$ denotes the ideal generated by them, the associativity condition for the product $\circ$ defined by $B$, becomes $$\{\mathcal{J}, \mathcal{J} \}\subset \mathcal{J}\,.$$ This result constitutes the key observation of Magri and Konopelchenko [@Ko; @KM] to relate Frobenius manifolds and important hierarchies of completely integrable systems.
The Jordan–Lie manifold structure of the space of endomorphisms
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Apart from the association of linear functions on $V^*$ to vectors in $V$ discussed above, Eq. (\[bidual\]), to any vector $v\in V$ we can associate the constant vector field $X_v \colon V\to TV $ on $V$ defined by $X_v(w) = (w,v)$. The Liouville vector field $\Delta\colon v\mapsto (v,v)$ generating infinitesimal dilations, induces a linear structure on the base manifold from the one on the fiber.
Moreover, by using the identification above of vectors on $V$ with tangent vectors to $TV$, any linear transformation $A\colon V\to V$ induces a transformation $T_A$ on tangent vectors: $$T_A : TV\to TV, \quad (w,v)\mapsto (w,Av).$$ For instance, $T_I=\Delta$. In this way the algebra ${\rm End\, }(V)$ is mapped into the algebra of linear endomorphisms on $TV$ preserving the base. The map $T_A\mapsto X_A=T_A(\Delta) $ is injective and it therefore allows us to induce a composition law on vector fields as: $$X_A\cdot X_C=T_{AC}(\Delta)=X_{AC }\,.$$ If we consider now the Jordan product $\circ$ defined in the space of endomorphisms of the linear space $V$ by $A\circ C=\frac 12 (AC+CA)$, a similar product is induced on the corresponding vector fields. $$X_A\circ X_C=T_{A\circ C}(\Delta)=X_{A\circ C }\,.$$ Now, to any bilinear map $B:V\times V\to V$ we can associate in addition to the tensor field $c_B$ given by Eq. (\[general\]) a 2-contravariant tensor field $\tau_B$ on $V^*$ given by $i_\Delta c_B$, or more explicitly: $$\tau_B(df_1,df_2)(\alpha)=\alpha(B(df_1(\alpha), df_2(\alpha)))\,.$$ Hence, if we consider the symmetric bilinear composition $B(A,C) = A\circ C$, defined by the Jordan bracket above, the dual space $\mathcal{E}^*$ of the space of endomorphisms $\mathcal{E}$ of the linear space $V$ inherits a symmetric contravariant 2-tensor $\tau_B$ and the corresponding (Jordan) symmetric bracket $(\cdot, \cdot )$ on the algebra of functions $\mathcal{F}(\mathcal{E}^*)$.
The space of endomorphisms $\mathcal{E}$ also carries the Lie algebra bracket: $$L(A,C) = \frac 12 [A, C] = \frac 12 (AC - CA) ,$$ inducing the corresponding skew-symmetric contravariant 2-tensor $\tau_L$ on $\mathcal{E}^*$ that defines a Poisson bracket $\{ \cdot, \cdot \} $ on $\mathcal{F}(\mathcal{E}^*)$. Thus the space $\mathcal{E}^*$ has the structure of a Jordan-Lie manifold. The two brackets above are compatible in a trivial way because the two tensors add to the canonical tensor induced by the obvious bilinear map $B_0(A,C) = AC$ on $\mathcal{E}$.
Let us consider the simple example of a complex linear space $V$ of dimension 2. The space $\mathcal{E}$ of endomorphisms of $V$ has complex dimension 4. A basis for $\mathcal{E}$ can be chosen as the set of the $2\times 2$ (Hermitean) matrices: $$\sigma_0=\matriz{cc}{ 1&0\\0&1}\,,\quad \sigma_1=\matriz{cc}{ 0&1\\1&0}\,,\quad \sigma_2=\matriz{cc}{ 0&-\ii\\\ii&0}\,,\quad \sigma_3=\matriz{cc}{ 1&0\\0&-1}\,.$$ The corresponding (complex) coordinate functions on matrices are given by: $$z_\mu(A)=\frac 12 {\rm Tr\,}(\sigma_\mu A)\,,$$ and for any $2\times 2$ matrix $A$ we have: $ A = z_\mu \sigma_\mu$. Clearly now the skew-symmetric tensor $\tau_L$ becomes: $$I=\epsilon_{jkl}\,z_j\,\pd{}{z_k}\wedge\pd{}{z_l},$$ while the symmetric tensor $\tau_B$ has the form: $$R=\pd{}{z_0} \stackrel{\otimes}{_s}
\left(z_j\pd{}{z_j}\right)+z_0\left(\pd{}{z_0}\otimes\pd{}{z_0}+\pd{}{z_1}\otimes\pd{}{z_1}+\pd{}{z_2}\otimes\pd{}{z_2} \right)\,.$$ Both tensors define a $(1,1)$ tensor field $J$ such that $J^3=-J$, which is a generalisation of the complex structure.
The ${\mathbb{C}}^*$-algebra approach to Quantum mechanics
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The connection of the Schrödinger picture with the Heisenberg picture is provided by the momentum map associated with the symplectic action of the unitary group on the Hilbert space or the complex projective Hilbert space of the Schrödinger picture. The inverse connection is provided by the GNS construction which generalizes to Quantum Mechanics the concept of symplectic realization of a Poisson manifold.
The linear transformations preserving both $g$ and $\omega$ as defined in Eq. (\[thetwostr\]) constitute the unitary group $\mathcal{U}(\Hil)$. If we denote by $\mathfrak{u}(\Hil)$ its Lie algebra, the set of skew-Hermitean operators acting on $\Hil$, and identify the set of all Hermitean operators with the dual $\mathfrak{u}^*(\Hil)$ via the pairing (in the infinite dimensional case we should restrict to Hilbert-Schmidt operators): $$\langle A,T\rangle=\frac 12\, {\rm Tr\, }(AT)\,,\qquad A\in \mathfrak{u}^*(\Hil)\,,
T\in \mathfrak{u}(\Hil)\,,$$ we can consider $\widehat T$ as the linear map associated with $T$, $\widehat
T(A)=\langle A,T\rangle$. A bracket can then be defined as before by: $$\{\widehat T_1,\widehat T_2\}=[T_1,T_2]\,\,\widehat{}\ \,,$$ and similarly a Jordan bracket is introduced by means of: $$(\widehat{T}_1,\widehat{T}_2)=(T_1T_2+T_2T_1)\,\,\widehat{}\ .$$ These two brackets are compatible in the sense that they define a Lie–Jordan algebra in $\mathfrak{u}^*(\Hil)$. If we consider the inner product on $\mathfrak{u}^*(\Hil)$: $$\langle A,B\rangle_{\mathfrak{u}^*}=\frac 12\, {\rm Tr\, }(AB)\,,$$we find that this inner product is preserved by the Hamiltonian vector fields associated with $\widehat T$ for any $T\in\mathfrak{u}(\Hil)$. These vector fields are related with corresponding vector fields on $\Hil$, namely, $$\frac d{dt}(e^{-\ii tA}\ket{\psi})_{t=0}=-\ii\,A\ket{\psi}=X_A(\ket{\psi})\,,$$ with $\ii\, A\in \mathfrak{u}(\Hil)$. The vector field $X_A$ on $\Hil$ is Hamiltonian with Hamiltonian function $f_A(\ket{\psi})=\frac 12\, \bra{\psi}A\ket{\psi}$. The momentum map which relates $X_A$ with the Hamiltonian vector field on $\mathfrak{u}^*(\Hil)$ associated with $(\ii\, A)\ \widehat{}$ is given by $$\mu : \Hil \to \mathfrak{u}^*(\Hil), \quad \mu(\ket{\psi}) = \ket\psi\bra\psi .$$ The symmetric tensor associated with the Jordan bracket: $$R(d\widehat T_1,d\widehat T_2)=(\widehat T_1,\widehat
T_2)=(T_1T_2+T_2T_1)\,\,\widehat{} \ \,,$$ is a contravariant symmetric 2-tensor as we discussed earlier. Similarly, the skew-symmetric tensor defined by: $$I (d\widehat T_1,d\widehat T_2))=\{\widehat T_1,\widehat T_2\} ,$$ is the Poissson tensor associated with the Lie algebra $\mathfrak{u}(\Hil)$. These two tensor fields are $\mu$-related with the tensors $G$ and $\Lambda$ defined on $\Hil$, respectively, by Eq. (\[GLambda\]).
By considering the complex contravariant tensor $R+\ii\, I$ we obtain a tensor field which allows us to consider the algebra of linear functions on $\mathfrak{u}^*(\Hil)$ of the form $\widehat
T+\ii\, \widehat S$, $T,S\in \mathfrak{u}(\Hil)$, as a $\mathbb{C}^*$-algebra of complex valued functions. In this setting the momentum map relates the Schrödinger picture with the Heisenberg picture. To go from the Heisenberg picture to the Schrödinger picture we consider an Hermitean realization of the Lie–Jordan algebra on $\mathfrak{u}^*(\Hil)$. This is a generalisation of the symplectic realisation of the Poisson structure on $\mathfrak{u}^*(\Hil)$. The existence of these Hermitean realizations for the Lie–Jordan algebra structure on $\mathfrak{u}^*(\Hil)$, the real part of the $\mathbb{C}^*$-algebra we are considering, is the essential content of the so called Gelfand–Naimark–Segal (GNS) construction. We refer to [@CM] for further details on these interesting aspects.
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---
abstract: 'Neural architecture search (NAS) has dramatically advanced the development of neural network design. We revisit the search space design in most previous NAS methods and find the number and widths of blocks are set manually. However, block counts and block widths determine the network scale (depth and width) and make a great influence on both the accuracy and the model cost (FLOPs/latency). In this paper, we propose to search block counts and block widths by designing a densely connected search space, , DenseNAS. The new search space is represented as a dense super network, which is built upon our designed routing blocks. In the super network, routing blocks are densely connected and we search for the best path between them to derive the final architecture. We further propose a chained cost estimation algorithm to approximate the model cost during the search. Both the accuracy and model cost are optimized in DenseNAS. For experiments on the MobileNetV2-based search space, DenseNAS achieves 75.3% top-1 accuracy on ImageNet with only 361MB FLOPs and 17.9ms latency on a single TITAN-XP. The larger model searched by DenseNAS achieves 76.1% accuracy with only 479M FLOPs. DenseNAS further promotes the ImageNet classification accuracies of ResNet-18, -34 and -50-B by 1.5%, 0.5% and 0.3% with 200M, 600M and 680M FLOPs reduction respectively. The related code is available at <https://github.com/JaminFong/DenseNAS>.'
author:
- |
Jiemin Fang$^{1}$, Yuzhu Sun$^{1}$, Qian Zhang$^{2}$, Yuan Li$^{2}$, Wenyu Liu$^{1}$, Xinggang Wang$^{1}$\
$^1$School of EIC, Huazhong University of Science and Technology $\; ^2$Horizon Robotics\
`{jaminfong, yzsun, liuwy, xgwang}@hust.edu.cn`\
`{qian01.zhang, yuan.li}@horizon.ai`
bibliography:
- 'egbib.bib'
title: Densely Connected Search Space for More Flexible Neural Architecture Search
---
Introduction {#sec: intro}
============
In recent years, neural architecture search (NAS) [@zoph2016neural; @zoph2017learning; @pham2018efficient; @Real2018Regularized] has demonstrated great successes in designing neural architectures automatically and achieved remarkable performance gains in various tasks such as image classification [@zoph2017learning; @pham2018efficient], semantic segmentation [@DBLP:conf/nips/ChenCZPZSAS18; @liu2019auto] and object detection [@ghiasi2019fpn; @Xu_2019_ICCV]. NAS has been a critically important topic for architecture designing.
![Search space comparison between conventional methods and DenseNAS. *Upper*: Conventional search spaces manually set a fixed number of blocks in each stage. The block widths are set manually as well. *Bottom*: The search space in DenseNAS allows more blocks with various widths in each stage. Each block is densely connected to its subsequent ones. We search for the best path (the red line) to derive the final architecture, in which the number of blocks in each stage and the widths of blocks are allocated automatically. []{data-label="fig: comp_ss"}](comp_ss.pdf){width="1.0\linewidth"}
In NAS research, the search space plays a crucial role that constrains the architectures in a prior-based set. The performance of architectures produced by NAS methods is strongly associated with the search space definition. A more flexible search space has the potential to bring in architectures with more novel structures and promoted performance. We revisit and analyze the search space design in most previous works [@zoph2017learning; @MnasNet; @cai2018proxylessnas; @fbnet]. For a clear illustration, we review the following definitions. *Block* denotes a set of layers/operations in the network which output feature maps with the same spatial resolution and the same width (number of channels). *Stage* denotes a set of sequential *block*s whose outputs are under the same spatial resolution settings. Different *block*s in the same stage are allowed to have various widths. Many recent works [@cai2018proxylessnas; @fbnet; @chu2019fairnas] stack the inverted residual convolution modules (MBConv) defined in MobileNetV2 [@sandler2018mobilenetv2] to construct the search space. They search for different kernel sizes and expansion ratios in each MBConv. The depth is searched in terms of layer numbers in each block. The searched networks with MBConvs show high performance with low latency or few FLOPs.
In this paper, we aim to perform NAS in a more flexible search space. Our motivation and core idea are illustrated in Fig. \[fig: comp\_ss\]. As the upper part of Fig. \[fig: comp\_ss\] shows, the number of blocks in each stage and the width of each block are set manually and fixed during the search process. It means that the depth search is constrained within the block and the width search cannot be performed. It is worth noting that the scale (depth and width) setting is closely related to the performance of a network, which has been demonstrated in many previous theoretical studies [@raghu2017expressive; @lu2017expressive] and empirical results [@gordon2018morphnet; @tan2019efficientnet]. Inappropriate width or depth choices usually cause drastic accuracy degradation, significant computation cost, or unsatisfactory model latency. Moreover, we find that recent works [@sandler2018mobilenetv2; @cai2018proxylessnas; @fbnet; @chu2019fairnas] manually tune width settings to obtain better performance, which indicates the design of network width demands much prior-based knowledge and trial-and-error.
We propose a densely connected search space to tackle the above obstacles and name our method as *DenseNAS*. We show our novelly designed search space schematically in the bottom part of Fig. \[fig: comp\_ss\]. Different from the search space design principles in the previous works [@cai2018proxylessnas; @fbnet], we allow more blocks with various widths in one stage. Specifically, we design the *routing block*s to construct the densely connected super network which is the representation of the search space. From the beginning to the end of the search space, the width of the routing block increases gradually to cover more width options. Every routing block is connected to several subsequent ones. This formulation brings in various paths in the search space and we search for the best path to derive the final architecture. As a consequence, the block widths and counts in each stage are allocated automatically. Our method extends the depth search into a more flexible space. Not only the number of layers within one block but also the number of blocks within one stage can be searched. The block width search is enabled as well. Moreover, the positions to conduct spatial down-sampling operations are determined along with the block counts search.
We integrate our search space into the differentiable NAS framework by relaxing the search space. We assign a probability parameter to each output path of the routing block. During the search process, the distribution of probabilities is optimized. The final block connection paths in the super network are derived based on the probability distribution. To optimize the cost (FLOPs/latency) of the network, we design a *chained estimation algorithm* targeted at approximating the cost of the model during the search.
Our contributions can be summarized as follows.
- We propose a densely connected search space that enables network/block widths search and block counts search. It provides more room for searching better networks and further reduces expert designing efforts.
- We propose a chained cost estimation algorithm to precisely approximate the computation cost of the model during search, which makes the DenseNAS networks achieve high performance with low computation cost.
- In experiments, we demonstrate the effectiveness of our method by achieving SOTA performance on the MobileNetV2 [@sandler2018mobilenetv2]-based search space. Our searched network achieves 75.3% accuracy on ImageNet [@imagenet] with only 361MB FLOPs and 17.9ms latency on a single TITAN-XP.
- DenseNAS can further promote the ImageNet classification accuracies of ResNet-18, -34 and -50-B [@he2016deep] by 1.5%, 0.5% and 0.3% with 200M, 600M, 680M FLOPs and 1.5ms, 2.4ms, 6.1ms latency reduction respectively.
{width="1.0\linewidth"}
Related Work
============
#### Search Space Design
NASNet [@zoph2017learning] is the first work to propose a cell-based search space, where the cell is represented as a directed acyclic graph with several nodes inside. NASNet searches for the operation types and the topological connections in the cell and repeat the searched cell to form the whole network architecture. The depth of the architecture (, the number of repetitions of the cell), the widths and the occurrences of down-sampling operations are all manually set. Afterwards, many works [@liu2017progressive; @pham2018efficient; @Real2018Regularized; @liu2018darts] adopt a similar cell-based search space. However, architectures generated by cell-based search spaces are not friendly in terms of latency or FLOPs. Then MnasNet [@MnasNet] stacks MBConvs defined in MobileNetV2 [@sandler2018mobilenetv2] to construct a search space for searching efficient architectures. Some works [@cai2018proxylessnas; @eatnas; @fbnet; @ChamNet] simplify the search space by searching for the expansion ratios and kernel sizes of MBConv layers.
Some works study more about the search space. Liu [@liu2018hierarchical] proposes a hierarchical search space that allows flexible network topologies (directed acyclic graphs) at each level of the hierarchies. Auto-DeepLab [@liu2019auto] creatively designs a two-level hierarchical search space for semantic segmentation networks. CAS [@zhang2019customizable] customizes the search space design for real-time segmentation networks. RandWire [@Xie_2019_ICCV] explores randomly wired architectures by designing network generators that produce new families of models for searching. Our proposed method designs a densely connected search space beyond conventional search constrains to generate the architecture with a better trade-off between accuracy and model cost.
#### NAS Method
Some early works [@zoph2016neural; @zoph2017learning; @zhong2018practical] propose to search architectures based on reinforcement learning (RL) methods. Then evolutionary algorithm (EA) based methods [@dong2018dpp; @liu2018hierarchical; @Real2018Regularized] achieve great performance. However, RL and EA based methods bear huge computation cost. As a result, ENAS [@pham2018efficient] proposes to use weight sharing for reducing the search cost.
Recently, the emergence of differentiable NAS methods [@liu2018darts; @cai2018proxylessnas; @fbnet] and one-shot methods [@brock2017smash; @Understanding] greatly reduces the search cost and achieves superior results. DARTS [@liu2018darts] is the first work to utilize the gradient-based method to search neural architectures. They relax the architecture representation as a super network by assigning continuous weights to the candidate operations. They first search on a small dataset, , CIFAR-10 [@krizhevsky2009learning], and then apply the architecture to a large dataset, , ImageNet [@DBLP:conf/cvpr/DengDSLL009], with some manual adjustments. ProxylessNAS [@cai2018proxylessnas] reduces the memory consumption by adopting a dropping path strategy and conducts search directly on the large scale dataset, , ImageNet. FBNet [@fbnet] searches on the subset of ImageNet and uses the Gumbel Softmax function [@JangGP17; @MaddisonMT17] to better optimize the distribution of architecture probabilities. TAS [@dong2019network] utilizes a differentiable NAS scheme to search and prune the width and depth of the network and uses knowledge distillation (KD) [@hinton2015distilling] to promote the performance of the pruned network. FNA [@Fang*2020Fast] proposes to adapt the neural network to new tasks with low cost by a parameter remapping mechanism and differentiable NAS. It is challenging for differentiable/one-shot NAS methods to search for more flexible architectures as they need to integrate all sub-architectures into the super network. The proposed DenseNAS tends to solve this problem by integrating a densely connected search space into the differentiable paradigm and explores more flexible search schemes in the network.
Method
======
In this section, we first introduce how to design the search space targeted at a more flexible search. A *routing block* is proposed to construct the densely connected super network. Secondly, we describe the method of relaxing the search space into a continuous representation. Then, we propose a *chained cost estimation* algorithm to approximate the model cost during the search. Finally, we describe the whole search procedure.
Densely Connected Search Space
------------------------------
As shown in Fig. \[fig:searchspace\], we define our search space using the following three terms, , (*basic layer*, *routing block* and *dense super Network*). Firstly, a *basic layer* is defined as a set of all the candidate operations. Then we propose a novel *routing block* which can aggregate tensors from different routing blocks and transmit tensors to multiple other routing blocks. Finally, the search space is constructed as a *dense super network* with many routing blocks where there are various paths to transmit tensors.
### Basic Layer {#sssec:layer}
We define the *basic layer* to be the elementary structure in our search space. One basic layer represents a set of candidate operations which include MBConvs and the skip connection. MBConvs are with kernel sizes of $\{3, 5, 7\}$ and expansion ratios of $\{3, 6\}$. The skip connection is for the depth search. If the skip connection is chosen, the corresponding layer is removed from the resulting architecture.
### Routing Block
For the purpose of establishing various paths in the super network, we propose the *routing block* with the ability of aggregating tensors from preceding routing blocks and transmit tensors to subsequent ones. We divide the routing block into two parts, *shape-alignment layers* and *basic layers*.
Shape-alignment layers exist in the form of several parallel branches, while every branch is a set of candidate operations. They take input tensors with different shapes (including widths and spatial resolutions) which come from multiple preceding routing blocks and transform them into tensors with the same shape. As shape-alignment layers are required for all routing blocks, we exclude the skip connection in candidate operations of them. Then tensors processed by shape-alignment layers are aggregated and sent to several basic layers. The subsequent basic layers are used for feature extraction whose depth can also be searched.
### Dense Super Network {#sssec: network}
Many previous works [@MnasNet; @cai2018proxylessnas; @fbnet] manually set a fixed number of blocks, and retain all the blocks for the final architecture. Benefiting from the aforementioned structures of routing blocks, we introduce more routing blocks with various widths to construct the *dense super network* which is the representation of the search space. The final searched architecture is allowed to select a subset of the routing blocks and discard the others, giving the search algorithm more room.
We define the super network as $\mathcal{N}_{sup}$ and assume it to consist of $N$ routing blocks, $\mathcal{N}_{sup} = \{B_1, B_2, ..., B_N\}$. The network structure is shown in Fig. \[fig:searchspace\]. We partition the entire network into several stages. As Sec. \[sec: intro\] defines, each stage contains routing blocks with various widths and the same spatial resolution. From the beginning to the end of the super network, the widths of routing blocks grow gradually. In the early stage of the network, we set a small growing stride for the width because large width settings in the early network stage will cause huge computational cost. The growing stride becomes larger in the later stages. This design principle of the super network allows more possibilities of block counts and block widths.
We assume that each routing block in the super network connects to $M$ subsequent ones. We define the connection between the routing block $B_i$ and its subsequent routing block $B_j$ ($j > i$) as $C_{ij}$. The spatial resolutions of $B_i$ and $B_j$ are $H_i \times W_i$ and $H_j \times W_j$ respectively (normally $H_i = W_i$ and $H_j = W_j$). We set some constraints on the connections to avoid the stride of the spatial down-sampling exceeding $2$. Specifically, $C_{ij}$ only exists when $j - i \leq M$ and $H_i / H_j \leq 2$. Following the above paradigms, the search space is constructed as a dense super network based on the connected routing blocks.
Relaxation of Search Space {#ssec: relax}
--------------------------
We integrate our search space by relaxing the architectures into continuous representations. The relaxation is implemented on both the basic layer and the routing block. We can search for architectures via back-propagation in the relaxed search space.
### Relaxation in the Basic Layer
Let $\mathcal{O}$ be the set of candidate operations described in Sec. \[sssec:layer\]. We assign an architecture parameter $\alpha_o^\ell$ to the candidate operation $o \in \mathcal{O}$ in basic layer $\ell$. We relax the basic layer by defining it as a weighted sum of outputs from all candidate operations. The architecture weight of the operation is computed as a *softmax* of architecture parameters over all operations in the basic layer: $$w_o^\ell = \frac{\exp(\alpha_o^\ell)}{\sum_{o' \in \mathcal{O}} \exp(\alpha_{o'}^\ell)}
\label{eq: op_weight}.$$ The output of basic layer $\ell$ can be expressed as $$x_{\ell+1} = \sum_{o \in \mathcal{O}} w_o^\ell \cdot o(x_\ell),$$ where $x_\ell$ denotes the input tensor of basic layer $\ell$.
### Relaxation in the Routing Block
We assume that the routing block $B_i$ outputs the tensor $b_i$ and connects to $m$ subsequent blocks. To relax the block connections as a continuous representation, we assign each output path of the block an architecture parameter. Namely the path from $B_i$ to $B_j$ has a parameter $\beta_{ij}$. Similar to how we compute the architecture weight of each operation above, we compute the probability of each path using a *softmax* function over all paths between the two routing blocks: $$p_{ij} = \frac{\exp(\beta_{ij})}{\sum_{k=1}^m \exp(\beta_{ik})}.$$ For routing block $B_i$, we assume it takes input tensors from its $m'$ preceding routing blocks ($B_{i-m'}$, $B_{i-m'+1}$, $B_{i-m'+2}$ ... $B_{i-1}$). As shown in Fig. \[fig:searchspace\], the input tensors from these routing blocks differ in terms of width and spatial resolution. Each input tensor is transformed to a same size by the corresponding branch of shape-alignment layers in $B_i$. Let $H_{ik}$ denotes the $k$th transformation branch in $B_i$ which is applied to the input tensor from $B_{i-k}$, where $k=1 \dots m'$. Then the input tensors processed by shape-alignment layers are aggregated by a weighted-sum using the path probabilities, $$x_i = \sum_{k=1}^{m'} p_{i-k, i} \cdot H_{ik}(x_{i-k}).$$ It is worth noting that the path probabilities are normalized on the output dimension but applied on the input dimension (more specifically on the branches of shape-alignment layers). One of the shape-alignment layers is essentially a weighted-sum mixture of the candidate operations. The layer-level parameters $\alpha$ control which operation to be selected, while the outer block-level parameters $\beta$ determine how blocks connect.
Chained Cost Estimation Algorithm {#sssec: chain est}
---------------------------------
We propose to optimize both the accuracy and the cost (latency/FLOPs) of the model. To this end, the model cost needs to be estimated during the search. In conventional cascaded search spaces, the total cost of the whole network can be computed as a sum of all the blocks. Instead, the global effects of connections on the predicted cost need to be taken into consideration in our densely connected search space. We propose a *chained cost estimation* algorithm to better approximate the model cost.
We create a lookup table which records the cost of each operation in the search space. The cost of every operation is measured separately. During the search, the cost of one basic layer is estimated as follows, $$\mathtt{cost}^\ell = \sum_{o \in \mathcal{O}} w_o^\ell \cdot \mathtt{cost}_o^\ell,$$ where $\mathtt{cost}_o^\ell$ refers to the pre-measured cost of operation $o \in \mathcal{O}$ in layer $\ell$. We assume there are $N$ routing blocks in total ($B_1, \dots, B_N$). To estimate the total cost of the whole network in the densely connected search space, we define the chained cost estimation algorithm as follows. $$\label{eq: chain cost}
\begin{aligned}
\tilde{\mathtt{cost}}^N =&\ \mathtt{cost}^N_{b} \\
\tilde{\mathtt{cost}}^i =&\ \mathtt{cost}^i_{b} + \sum_{j=i+1}^{i+m} p_{ij} \cdot (\mathtt{cost}^{ij}_{align} + \mathtt{cost}^j_{b}),
\end{aligned}$$ where $\mathtt{cost}^i_{b}$ denotes the total cost of all the basic layers of $B_i$ which can be computed as a sum $\mathtt{cost}_b^i = \sum_\ell \mathtt{cost}_b^{i, \ell}$, $m$ denotes the number of subsequent routing blocks to which $B_i$ connects, $p_{ij}$ denotes the path probability between $B_i$ and $B_j$, and $\mathtt{cost}_{align}^{ij}$ denotes the cost of the shape-alignment layer in block $B_j$ which processes the data from block $B_i$.
The cost of the whole architecture can thus be obtained by computing $\tilde{\mathtt{cost}}^1$ with a recursion mechanism, $$\mathtt{cost} = \tilde{\mathtt{cost}}^1.$$ We design a loss function with the cost-based regularization to achieve the multi-objective optimization: $$\label{eq: loss}
\mathcal{L}(w, \alpha, \beta) = \mathcal{L}_{CE} + \lambda \log_\tau\mathtt{cost},$$ where $\lambda$ and $\tau$ are the hyper-parameters to control the magnitude of the model cost term.
Search Procedure {#sec: search_pro}
----------------
Benefiting from the continuously relaxed representation of the search space, we can search for the architecture by updating the architecture parameters (introduced in Sec. \[ssec: relax\]) using stochastic gradient descent. We find that at the beginning of the search process, all the weights of the operations are under-trained. The operations or architectures which converge faster are more likely to be strengthened, which leads to shallow architectures. To tackle this, we split our search procedure into two stages. In the first stage, we only optimize the weights for enough epochs to get operations sufficiently trained until the accuracy of the model is not too low. In the second stage, we activate the architecture optimization. We alternatively optimize the operation weights by descending $\nabla_w \mathcal{L}_{train}(w, \alpha, \beta)$ on the training set, and optimize the architecture parameters by descending $\nabla_{\alpha, \beta} \mathcal{L}_{val}(w, \alpha, \beta)$ on the validation set. Moreover, a dropping-path training strategy [@Understanding; @cai2018proxylessnas] is adopted to decrease memory consumption and decouple different architectures in the super network.
When the search procedure terminates, we derive the final architecture based on the architecture parameters $\alpha, \beta$. At the layer level, we select the candidate operation with the maximum architecture weight, , $\operatorname*{arg\,max}_{o \in \mathcal{O}} \alpha_o^\ell$. At the network level, we use the Viterbi algorithm [@forney1973viterbi] to derive the paths connecting the blocks with the highest total transition probability based on the output path probabilities. Every block in the final architecture only connects to the next one.
Experiments
===========
In this section, we first show the performance with the MobileNetV2 [@sandler2018mobilenetv2]-based search space on ImageNet [@imagenet] classification. Then we apply the architectures searched on ImageNet to object detection on COCO [@COCO]. We further extend our DenseNAS to the ResNet [@he2016deep]-based search space. Finally, we conduct some ablation studies and analysis. The implementation details are provided in the appendix.
Performance on MobileNetV2-based Search Space
---------------------------------------------
We implement DenseNAS on the MobileNetV2 [@sandler2018mobilenetv2]-based search space, set the GPU latency as our secondary optimization objective, and search models with different sizes under multiple latency optimization magnitudes (defined in Eq. \[eq: loss\]). The ImageNet results are shown in Tab. \[tab: mb\_results\]. We divide Tab. \[tab: mb\_results\] into several parts and compare DenseNAS models with both manually designed models [@howard2017mobilenets; @sandler2018mobilenetv2] and NAS models. DenseNAS achieves higher accuracies with both fewer FLOPs and lower latencies. Note that for FBNet-A, the group convolution in the 1$\times$1 conv and the channel shuffle operation are used, which do not exist in FBNet-B, -C and Proxyless. In the compared NAS methods [@MnasNet; @cai2018proxylessnas; @fbnet], the block counts and block widths in the search space are set and adjusted manually. DenseNAS allocates block counts and block widths automatically. We further visualize the results in Fig. \[fig: comp\_mobile\], which clearly demonstrates that DenseNAS achieves a better trade-off between accuracy and latency. The searched architectures are shown in Fig. \[fig: architecture\].
![The comparison of model performance on ImageNet under the MobileNetV2-based search spaces.[]{data-label="fig: comp_mobile"}](comp_mobile.pdf){width="0.8\linewidth"}
![Visualization of the searched architectures. We use rectangles with different colors and widths to denote the layer operations. $KxEy$ denotes the MBConv with kerner size $x \times x$ and expansion ratio $y$. We label the width in each layer. Layers in the same block are contained in the dashed box. We separate the stages with the green lines.[]{data-label="fig: architecture"}](architectures.pdf){width="1\columnwidth"}
Generalization Ability on COCO Object Detection
-----------------------------------------------
We apply the searched DenseNAS networks on the COCO [@COCO] object detection task to evaluate the generalization ability of DenseNAS networks and show the results in Tab. \[tab: det\]. We choose two commonly used object detection frameworks RetinaNet [@lin2017focal] and SSDLite [@liu2016ssd; @sandler2018mobilenetv2] to conduct our experiments. All the architectures shown in Tab. \[tab: det\] are utilized as the backbone networks in the detection frameworks. The experiments are performed based on the MMDetection [@chen2019mmdetection] framework.
We compare our results with both manually designed and NAS models. Results of MobileNetV2 [@sandler2018mobilenetv2], FBNet [@fbnet] and ProxylessNAS [@cai2018proxylessnas] are obtained by our re-implementation and all models are trained under the same settings and hyper-parameters for fair comparisons. DetNAS [@DBLP:journals/corr/abs-1903-10979] is a recent work that aims at searching the backbone architectures directly on object detection. Though DenseNAS searches on the ImageNet classification task and applies the searched architectures on detection tasks, our DenseNAS models still obtain superior detection performance in terms of both accuracy and FLOPs. The superiority over the compared methods demonstrates the great generalization ability of DenseNAS networks.
Performance on ResNet-based Search Space
----------------------------------------
We apply our DenseNAS framework on the ResNet [@he2016deep]-based search space to further evaluate the generalization ability of our method. It is convenient to implement DenseNAS on ResNet [@he2016deep] as we set the candidate operations in the basic layer as the basic block defined in ResNet [@he2016deep] and the skip connection. The ResNet-based search space is also constructed as a densely connected super network.
We search for several architectures with different FLOPs and compare them with the original ResNet models on the ImageNet [@imagenet] classification task in Tab. \[tab: resnet\_results\]. We further replace all the basic blocks in DenseNAS-R2 with the bottleneck blocks and obtain DenseNAS-R3 to compare with ResNet-50-B and the NAS model RandWire-WS, C=109 [@Xie_2019_ICCV] (WS, C=109). Though WS, C=109 achieves a higher accuracy, the FLOPs increases 600M, which is a great number, 17.6% of DenseNAS-R3. Besides, WS C=109 uses separable convolutions which greatly decrease the FLOPs while DenseNAS-R3 only contains plain convolutions. Moreover, RandWire networks are unfriendly to inference on existing hardware for the complicated connection patterns. Our proposed DenseNAS promotes the accuracy of ResNet-18, -34 and -50-B by 1.5%, 0.5% and 0.3% with 200M, 600M, 680M fewer FLOPs and 1.5ms, 2.4ms, 6.1ms lower latency respectively. We visualize the comparison results in Fig. \[fig: comp\_res\] and the performance on the ResNet-based search space further demonstrates the great generalization ability and effectiveness of DenseNAS.
![Graphical comparisons between ResNets and DenseNAS networks on the ResNet-based search space.[]{data-label="fig: comp_res"}](comp_resnet.pdf){width="0.8\linewidth"}
Ablation Study and Analysis
---------------------------
#### Comparison with Other Search Spaces {#sssec: fixed-block}
To further demonstrate the effectiveness of our proposed densely connected search space, we conduct the same search algorithm used in DenseNAS on the search spaces of FBNet and ProxylessNAS as well as a new search space which is constructed following the settings of block counts and block widths in MobileNetV2. The three search spaces are denoted as FBNet-SS, Proxyless-SS and MBV2-SS respectively. All the search/training settings and hyper-parameters are the same as that we use for DenseNAS. The results are shown in Tab. \[tab: fixed-block\] and DenseNAS achieves the highest accuracy with the lowest latency.
\[tab: fixed-block\]
#### Comparison with Random Search
As random search [@li2019random; @sciuto2019evaluating] is treated as an important baseline to validate NAS methods. We conduct random search experiments and show the results in Tab. \[tab: mb\_results\]. We randomly sample 15 models in our search space whose FLOPs are similar to DenseNAS-C. Then we train every model for 5 epochs on ImageNet. Finally, we select the one with the highest validation accuracy and train it under the same settings as DenseNAS. The total search cost of the random search is the same as DenseNAS. We observe that DenseNAS-C is 1% accuracy higher compared with the randomly searched model, which proves the effectiveness of DenseNAS.
\[tab: connect\]
#### The Number of Block Connections
We explore the effect of the maximum number of connections between routing blocks in the search space. We set the maximum connection number as 4 in DenseNAS. Then we try more options and show the results in Tab. \[tab: connect\]. When we set the connection number to 3, the searched model gets worse performance. We attribute this to the search space shrinkage which causes the loss of many possible architectures with good performance. As we set the number to 5 and the search process takes the same number of epochs as DenseNAS, 150 epochs. The performance of the searched model is not good, even worse than that of the connection number 3. Then we increase the search epochs to 200 and the search process achieves a comparable result with DenseNAS. This phenomenon indicates that larger search spaces need more search cost to achieve comparable/better results with/than smaller search spaces with some added constraints.
#### Cost Estimation Method {#sec: cost-est-exp}
As the super network is densely connected and the final architecture is derived based on the total transition probability, the model cost estimation needs to take the effects of all the path probabilities on the whole network into consideration. We try a *local cost estimation* strategy that does not involve the global connection effects on the whole super network. Specifically, we compute the cost of the whole network by summing the cost of every routing block during the search as follows, while the transition probability $p_{ji}$ is only used for computing the cost of each individual block rather than the whole network. $$\mathtt{cost} = \sum_i^B (\sum_{j=i-m}^{j=i-1} p_{ji} \cdot \mathtt{cost}_{align}^{ji} + \mathtt{cost}_b^i),$$ where all definitions in the equation are the same as that in Eq. \[eq: chain cost\]. We randomly generate the architecture parameters ($\alpha$ and $\beta$) to derive the architectures. Then we draw the approximated cost values computed by *local cost estimation* and our proposed *chained cost estimation* respectively, and compare with the real cost values in Fig. \[fig: comp\_est\]. In this experiment, we take FLOPs as the model cost because the FLOPs is easier to measure than latency. 1,500 models are sampled in total. The results show that the predicted cost values computed by our *chained cost estimation* algorithm has a much stronger correlation with the real values and approximate more to the real ones. As the predicted values are computed based on the randomly generated architecture parameters which are not binary parameters, there are still differences between the predicted and real values.
![Predicted values of FLOPs computed by chained cost estimation and local cost estimation algorithm.[]{data-label="fig: comp_est"}](comp_est.pdf){width="1\columnwidth"}
#### Architecture Analysis
We visualize the searched architectures in Fig. \[fig: architecture\]. It shows that DenseNAS-B and -C have one more block in the last stage than other architectures, which indicates enlarging the depth in the last stage of the network tends to obtain a better accuracy. Moreover, the smallest architecture DenseNAS-A whose FLOPs is only 251M has one fewer block than DenseNAS-B and -C to decrease the model cost. The structures of the final searched architectures show the great flexibility of DenseNAS.
Conclusion
==========
We propose a densely connected search space for more flexible architecture search, DenseNAS. We tackle the limitations in previous search space design in terms of the block counts and widths. The novelly designed routing blocks are utilized to construct the search space. The proposed chained cost estimation algorithm aims at optimizing both accuracy and model cost. The effectiveness of DenseNAS is demonstrated on both MobileNetV2- and ResNet- based search spaces. We leave more applications, semantic segmentation, face detection, pose estimation, and more network-based search space implementations, MobileNetV3 [@howard2019searching], ShuffleNet [@zhang2017shufflenet] and VarGNet [@zhang2019vargnet], for future work.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported by National Key R&D Program of China (No. 2018YFB1402600), National Natural Science Foundation of China (NSFC) (No. 61876212, No. 61733007 and No. 61572207), and HUST-Horizon Computer Vision Research Center. We thank Liangchen Song, Kangjian Peng and Yingqing Rao for the discussion and assistance.
Appendix
========
Implementation Details
----------------------
Before the search process, we build a lookup table for every operation latency of the super network as described in Sec. . We set the input shape as $(3, 224, 224)$ with the batch size of $32$ and measure each operation latency on one TITAN-XP GPU. All models and experiments are implemented using PyTorch [@paszke2017automatic].
For the search process, we randomly choose $100$ classes from the original 1K-class ImageNet training set. We sample $20\%$ data of each class from the above subset as the validation set. The original validation set of ImageNet is only used for evaluating our final searched architecture. The search process takes 150 epochs in total. We first train the operation weights for 50 epochs on the divided training set. For the last 100 epochs, the updating of architecture parameters ($\alpha, \beta$) and operation weights ($w$) alternates in each epoch. We use the standard GoogleNet [@DBLP:conf/cvpr/SzegedyLJSRAEVR15] data augmentation for the training data preprocessing. We set the batch size to $352$ on $4$ Tesla V100 GPUs. The SGD optimizer is used with $0.9$ momentum and $4 \times 10^{-5}$ weight decay to update the operation weights. The learning rate decays from $0.2$ to $1 \times 10^{-4}$ with the cosine annealing schedule [@DBLP:conf/iclr/LoshchilovH17]. We use the Adam optimizer [@DBLP:conf/iclr/2015] with $10^{-3}$ weight decay, $\beta = (0.5, 0.999)$ and a fixed learning rate of $3 \times 10^{-4}$ to update the architecture parameters.
For retraining the final derived architecture, we use the same data augmentation strategy as the search process on the whole ImageNet dataset. We train the model for $240$ epochs with a batch size of $1024$ on $8$ TITAN-XP GPUs. The optimizer is SGD with $0.9$ momentum and $4 \times 10^{-5}$ weight decay. The learning rate decays from 0.5 to $1 \times 10^{-4}$ with the cosine annealing schedule.
[c|c|c|c|c]{} **Stage** & **Output Size** & **DenseNAS-R1** & **DenseNAS-R2** & **DenseNAS-R3**\
1 & 112 $\times$ 112 &\
2 & 56 $\times$ 56 & [$ \left[
\begin{tabular}{c}
3 $$ 3, 64 \\
3 $$ 3, 64
\end{tabular}
\right] \times 1
$]{} & [$ \left[
\begin{tabular}{c}
3 $$ 3, 48 \\
3 $$ 3, 48
\end{tabular}
\right] \times 1
$]{} &
[c]{} [$ \left[
\begin{tabular}{c}
1 $$ 1, 48 \\
3 $$ 3, 48 \\
1 $$ 1, 192
\end{tabular}
\right] \times 1
$]{}\
\
3 & 28 $\times$ 28 & [$ \left[
\begin{tabular}{c}
3 $$ 3, 72 \\
3 $$ 3, 72
\end{tabular}
\right] \times 2
$]{} & [$ \left[
\begin{tabular}{c}
3 $$ 3, 72 \\
3 $$ 3, 72
\end{tabular}
\right] \times 4
$]{} &
[c]{} [$ \left[
\begin{tabular}{c}
1 $$ 1, 72 \\
3 $$ 3, 72 \\
1 $$ 1, 288
\end{tabular}
\right] \times 4
$]{}\
\
4 & 14 $\times$ 14 &
----------------------------
[$ \left[
\begin{tabular}{c}
3 $$ 3, 176 \\
3 $$ 3, 176
\end{tabular}
\right] \times 6
$]{}
[$ \left[
\begin{tabular}{c}
3 $$ 3, 192 \\
3 $$ 3, 192
\end{tabular}
\right] \times 3
$]{}
----------------------------
&
----------------------------
[$ \left[
\begin{tabular}{c}
3 $$ 3, 176 \\
3 $$ 3, 176
\end{tabular}
\right] \times 16
$]{}
[$ \left[
\begin{tabular}{c}
3 $$ 3, 208 \\
3 $$ 3, 208
\end{tabular}
\right] \times 4
$]{}
----------------------------
&
[c]{} [$ \left[
\begin{tabular}{c}
1 $$ 1, 176 \\
3 $$ 3, 176 \\
1 $$ 1, 704
\end{tabular}
\right] \times 16
$]{}\
[$ \left[
\begin{tabular}{c}
1 $$ 1, 208 \\
3 $$ 3, 208 \\
1 $$ 1, 832
\end{tabular}
\right] \times 4
$]{}\
\
5 & 7 $\times$ 7 &
----------------------------
[$ \left[
\begin{tabular}{c}
3 $$ 3, 288 \\
3 $$ 3, 288
\end{tabular}
\right] \times 1
$]{}
[$ \left[
\begin{tabular}{c}
3 $$ 3, 512 \\
3 $$ 3, 512
\end{tabular}
\right] \times 1
$]{}
----------------------------
&
----------------------------
[$ \left[
\begin{tabular}{c}
3 $$ 3, 288 \\
3 $$ 3, 288
\end{tabular}
\right] \times 2
$]{}
[$ \left[
\begin{tabular}{c}
3 $$ 3, 512 \\
3 $$ 3, 512
\end{tabular}
\right] \times 1
$]{}
----------------------------
&
[c]{} [$ \left[
\begin{tabular}{c}
1 $$ 1, 288 \\
3 $$ 3, 288 \\
1 $$ 1, 1152
\end{tabular}
\right] \times 2
$]{}\
[$ \left[
\begin{tabular}{c}
1 $$ 1, 512 \\
3 $$ 3, 512 \\
1 $$ 1, 2048
\end{tabular}
\right] \times 1
$]{}\
\
6 & 1 $\times$ 1 &\
\[tab: res\_arch\]
Viterbi Algorithm for Block Deriving
------------------------------------
The Viterbi Algorithm [@forney1973viterbi] is widely used in dynamic programming which targets at finding the most likely path between hidden states. In DenseNAS, only a part of routing blocks in the super network are retained to construct the final architecture. As described in Sec. , we implement the Viterbi algorithm to derive the final sequence of blocks. We treat the routing block in the super network as each hidden state in the Viterbi algorithm. The path probability $p_{ij}$ serves as the transition probability from routing block $B_i$ to $B_j$. The total algorithm is described in Algo. \[algo: viterbi\]. The derived block sequence holds the maximum transition probability.
$P[0] \gets 1$ $S[0] \gets 0$ $X[0] \gets B_{N+1}$ $idx \gets N+1$ $count \gets 1$ $revers X$
Dropping-path Search Strategy
-----------------------------
The super network includes all the possible architectures defined in the search space. To decrease the memory consumption and accelerate the search process, we adopt the dropping-path search strategy [@Understanding; @cai2018proxylessnas] (which is mentioned in Sec. ). When training the weights of operations, we sample one path of the candidate operations according to the architecture weight distribution $\{w_o^\ell | o \in \mathcal{O}\}$ in every basic layer. The dropping-path strategy not only accelerates the search but also weakens the coupling effect between operation weights shared by different sub-architectures in the search space. To update the architecture parameters, we sample two operations in each basic layer according to the architecture weight distribution. To keep the architecture weights of the unsampled operations unchanged, we compute a re-balancing bias to adjust the sampled and newly updated parameters. $$\mathtt{bias}_s = \ln \frac{\sum_{o \in \mathcal{O}_s} \exp(\alpha_o^\ell)}{\sum_{o \in \mathcal{O}_s} \exp({\alpha'}_o^\ell)},$$ where $\mathcal{O}_s$ refers to the set of sampled operations, $\alpha_o^\ell$ denotes the original value of the sampled architecture parameter in layer $\ell$ and ${\alpha'}_o^\ell$ denotes the updated value of the architecture parameter. The computed bias is finally added to the updated architecture parameters.
Implementation Details of ResNet Search
---------------------------------------
We design the ResNet-based search space as follows. As enlarging the kernel size of the ResNet block causes a huge computation cost increase, the candidate operations in the basic layer only include the basic block [@he2016deep] and the skip connection. That means we aim at width and depth search for ResNet networks. During the search, the batch size is set as 512 on 4 Tesla V100 GPUs. The search process takes 70 epochs in total and we start to update the architecture parameters from epoch 10. We set all the other search settings and hyper-parameters the same as that in the MobileNetV2 [@sandler2018mobilenetv2] search. For the architecture retraining, the same training settings and hyper-parameters are used as that for architectures searched in the MobileNetV2-based search space. The architectures searched by DenseNAS are shown in Tab. \[tab: res\_arch\].
\[tab: cost-est\]
Experimental Comparison of Cost Estimation Method
-------------------------------------------------
We study the design of the model cost estimation algorithm in Sec. . $1,500$ models are derived based on the randomly generated architecture parameters. Cost values predicted by our proposed chained cost estimation algorithm demonstrate a stronger correlation with the real values and more accurate prediction results than the compared local cost estimation strategy. We further perform the same search process as DenseNAS on the MobileNetV2 [@sandler2018mobilenetv2]-based search space with the local estimation strategy and show the searched results in Tab. \[tab: cost-est\]. DenseNAS with the chained cost estimation algorithm shows a higher accuracy with lower latency and fewer FLOPs. It proves the effectiveness of the chained cost estimation algorithm on achieving a good trade-off between accuracy and model cost.
|
---
abstract: 'A mapping from the vertex set of a graph $G=(V,E)$ into an interval of integers $\{0, \dots ,k\}$ is an $L(2,1)$-labelling of $G$ of span $k$ if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices at distance 2 are mapped onto distinct integers. It is known that for any fixed $k\ge 4$, deciding the existence of such a labelling is an NP-complete problem while it is polynomial for $k\leq 3$. For even $k\geq 8$, it remains NP-complete when restricted to planar graphs. In this paper, we show that it remains NP-complete for any $k \ge 4$ by reduction from Planar Cubic Two-Colourable Perfect Matching. Schaefer stated without proof that Planar Cubic Two-Colourable Perfect Matching is NP-complete. In this paper we give a proof of this.'
author:
- 'Nicole Eggemann[^1] , Frédéric Havet[^2] and Steven D. Noble[^3]'
title: '$k$-$L(2,1)$-Labelling for Planar Graphs is NP-Complete for $k\geq 4$'
---
Introduction
============
The Frequency Assignment Problem requires the assignment of frequencies to radio transmitters in a broadcasting network with the aim of avoiding undesired interference and minimising bandwidth. One of the longstanding graph theoretical models of this problem is the notion of distance constrained labelling of graphs. An [*$L(2,1)$-labelling*]{} of a graph $G$ is a mapping from the vertex set of $G$ into the nonnegative integers such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices at distance 2 are different. The [*span*]{} of such a labelling is the maximum label used. In this model, the vertices of $G$ represent the transmitters and the edges of $G$ express which pairs of transmitters are too close to each other so that undesired interference may occur, even if the frequencies assigned to them differ by 1. This model was introduced by Roberts [@Rob] and since then the concept has been intensively studied (see the survey articles [@calamoneri:06; @yeh:06]). Much of the early research involved determining the optimal labelling of grids and is either folklore or buried in the engineering literature.
The minimum span of an $L(2,1)$-labelling of a graph $G$ is denoted by $\lambda_{2,1}(G)$. Considerable effort has been spent on trying to resolve a conjecture of Griggs and Yeh [@griggs+yeh] stating that $\lambda_{2,1}(G) \leq \Delta^2$ for graphs with maximum degree $\Delta$. For general graphs, the result has been established for large $\Delta$ by Havet, Reed and Sereni [@havet+reed+sereni]. For the special case of planar graphs with $\Delta\geq 7$, the conjecture follows from a result of van den Heuvel and McGuiness [@heuvel+mcguiness] and has since been established for planar graphs with $\Delta \ne 3$ by Bella *et al.* [@bella]. Furthermore, generalising Wegner’s conjecture [@wegner:77] on the chromatic number of squares of planar graphs, it is conjectured that if $G$ is planar then $\lambda_{2,1}(G) \leq \frac{3}{2}\Delta +C$ for some absolute constant $C$. Havet et al. [@havet+heuvel+mcdiarmid+reed:list-colouring-conference-version; @havet+heuvel+mcdiarmid+reed:list-colouring-full-version] showed that this conjecture holds asymptotically: $\lambda_{2,1}(G) \leq \frac{3}{2}\Delta +o(\Delta)$. For outerplanar graphs with $\Delta \geq 8$, it has been shown that $\lambda_{2,1}(G) \leq \Delta+2$ [@calamoneri+petreschi:planar; @koller:05; @koller:09].
In their seminal paper, Griggs and Yeh [@griggs+yeh] proved that determining $\lambda_{2,1}(G)$ is an NP-hard problem. Fiala, Kloks and Kratochvíl [@fiala+kloks+kratochvil:fixed-parameter-lambda] proved that deciding $\lambda_{2,1}(G)\le k$ is NP-complete for every fixed $k\ge 4$.
Since then this problem has been shown to be NP-complete for some very restricted classes of graphs. For instance, Bodlaender *et al.* [@bodlaender:04] showed that this problem is NP-complete when restricted to bipartite planar graphs if we require $k\ge 8$ and $k$ even. Later Havet and Thomassé [@havet] proved that for any $k\geq 4$, it remains NP-complete when restricted to a different subclass of bipartite graphs, namely *incidence graphs*, that is, those graphs which may be obtained by making a single subdivision of each edge of a graph.
When the span $k$ is part of the input, the problem is nontrivial for trees but a polynomial time algorithm based on bipartite matching was presented by Chang and Kuo in [@chang+kuo]. Since then faster algorithms have been introduced by Hasunuma *et al.* running in time $O(n^{1.75})$ [@hasunuma+ishii+ono+uni:fast] and more recently $O(n)$ [@hasunuma+ishii+ono+uni:linear] on trees with $n$ vertices. The problem is still solvable in polynomial time if the input graph is outerplanar [@koller:05; @koller:09].
Moreover, somewhat surprisingly, Fiala, Golovach and Kratochvíl have shown that the problem becomes NP-complete for series-parallel graphs [@fiala+golovach+kratochvil:05], and thus the $L(2,1)$-labelling problem belongs to a handful of problems known to separate graphs of tree-width 1 and 2 by P/NP-completeness dichotomy.
In this paper we consider the following problem.
\[pro:main\]\
Let $k\ge 4$ be fixed. Instance: A planar graph $G$.Question: Is there an $L(2,1)$-labelling with span $k$?
As we mentioned above, Bodlaender *et al.* [@bodlaender:04] showed that this problem is NP-complete if we require $k\ge 8$ and $k$ even. We have read a suggestion in the literature that the problem is proved to be NP-complete for all $k\ge8$ in [@fotakis+niko+papa:00]. However this does not seem to be the case. In [@fotakis+niko+papa:00] there is a proof showing that the corresponding problem where $k$ is specified as part of the input is NP-complete. This proof shows that the problem is NP-complete for certain fixed values of $k$. However it is far from clear for which values of $k$ this is true. The same authors also show in [@fotakis+niko+papa:05] that the problem is NP-complete for $k=8$.
In this paper we first prove that Planar Cubic Two-Colourable Perfect Matching, which we define in the next section, is NP-Complete. This result was first stated by Schaefer [@schaefer:78] but without proof. In the second part of this paper we use this result in order to show that Problem \[pro:main\] is NP-complete.
Preliminary results {#seccol}
===================
The starting problem for our reductions is Not-All-Equal 3SAT, which is defined as follows [@schaefer:78].
\
Instance: A set of clauses each having three literals.Question: Can the literals be assigned value true or false so that each clause has at least one true and at least one false literal?
In [@schaefer:78], it is shown that this problem is NP-complete.
Our reduction involves an intermediate problem concerning a special form of two-colouring. In this section we define the intermediate problem and show that it is NP-complete. When $k=4$ or $k=5$, the final stage of our reduction is similar to the reduction in [@fiala+kloks+kratochvil:fixed-parameter-lambda]. However we cannot use induction for higher values of $k$ in contrast with the situation in [@fiala+kloks+kratochvil:fixed-parameter-lambda] and the problem from which the reduction starts in [@fiala+kloks+kratochvil:fixed-parameter-lambda] is not known to be NP-complete for planar graphs. So considerably more work is required.
The following problem is also discussed in [@schaefer:78].
\[pro:matching\]\
Instance: A graph $G$.Question: Is there a colouring of the vertices of $G$ with colours black and white in which every vertex has exactly one neighbour of the same colour?
In [@schaefer:78] it was shown that Two-Colourable Perfect Matching is NP-complete. We are more interested in the case where the input is restricted to being a planar cubic graph. We call this variant, Planar Cubic Two-Colourable Perfect Matching defined formally as follows [@schaefer:78].
\[pro:planarmatching\]\
Instance: A planar cubic graph $G$.Question: Is there a colouring of the vertices of $G$ with colours black and white in which every vertex has exactly one neighbour of the same colour?
Schaefer [@schaefer:78] states that this problem is NP-complete but does not give the details of the proof. We call a colouring as required in Problem \[pro:planarmatching\] a *two-coloured perfect matching*. This section is devoted to the proof of this result, using a reduction from Not-All-Equal 3SAT [@schaefer:78]. As far as we know, no proof of this has ever been published.
We say that a colouring of the vertices of a graph with colours black and white is an *almost two-coloured perfect matching* if every vertex of degree at least two is adjacent to exactly one vertex of the same colour. We say an edge is *monochromatic* if both end-vertices have the same colour and *dichromatic* if its end-vertices have different colours.
(10,7) (5,7)[0.15]{}[a]{} (5.375,7.175)[$a$]{} (5,6)[0.15]{}[b]{} (5.375,6.275)[$b$]{} (3,5.3)[0.15]{}[c]{} (3.1,5.7)[$c$]{} (7,5.3)[0.15]{}[d]{} (7.1,5.8)[$d$]{} (2.5,4.5)[0.15]{}[e]{} (2.1,4.5)[$e$]{} (7.5,4.5)[0.15]{}[h]{} (7.9,4.5)[$h$]{} (3.5,4.5)[0.15]{}[f]{} (3.9,4.5)[$f$]{} (6.5,4.5)[0.15]{}[g]{} (6.1,4.5)[$g$]{} (2.5,2.5)[0.15]{}[i]{} (2.1,2.9)[$i$]{} (7.5,2.5)[0.15]{}[l]{} (7.9,2.9)[$l$]{} (3.5,2.5)[0.15]{}[j]{} (3.9,2.9)[$j$]{} (6.5,2.5)[0.15]{}[k]{} (6.1,2.9)[$k$]{} (3,1.5)[0.15]{}[o]{} (2.55,1.5)[$o$]{} (7,1.5)[0.15]{}[p]{} (6.55,1.5)[$p$]{} (3,0.5)[0.15]{}[q]{} (2.55,0.5)[$q$]{} (7,0.5)[0.15]{}[r]{} (6.55,0.5)[$r$]{} (1.5,2.5)[0.15]{}[m]{} (1.4,2.9)[$m$]{} (8.5,2.5)[0.15]{}[n]{} (8.6,2.9)[$n$]{}
Let $H$ be the planar graph depicted in Fig. \[fig1\]. $H$ plays a key role in showing that Problem \[pro:planarmatching\] is NP-complete. We need the following lemma.
\[le:proph\] Any almost two-coloured perfect matching of $H$ has the following properties.
- Exactly one of the edges $ab,mi,ln$ is monochromatic.
- Vertices $b,i,l$ receive the same colour.
- Vertices $o,p,q,r$ receive the other colour to $b,i,l$.
Consider the triangles on the vertices $c,e,f$ and $g,h,d$. In order to obtain an almost two-coloured perfect matching exactly one of the edges $ce, ef$ and $cf$ must be monochromatic. The same is true for the triangle on the vertices $g,h,d$. Now consider the subgraph of $H$ induced by the vertices $c,e,f,i,j,m,k,o,q$. In Fig. \[fig3\] three of the six almost two-coloured perfect matchings of this subgraph are depicted with monochromatic edges shown by heavy lines. The other three two-coloured perfect matchings are obtained by interchanging the colours. It follows that $oq$ and $pr$ must be monochromatic.
(24,6.5)
(2,5.3)[0.22]{}[c]{}
(1.5,4.5)[0.22]{}[e]{}
(2.5,4.5)[0.22]{}[f]{}
(1.5,2.5)[0.22]{}[i]{}
(2.5,2.5)[0.22]{}[j]{} (5.5,2.5)[0.22]{}[k]{} (2,1.5)[0.22]{}[o]{} (2,0.5)[0.22]{}[q]{} (0.5,2.5)[0.22]{}[m]{} (11,5.3)[0.22]{}[c]{} (10.5,4.5)[0.22]{}[e]{} (11.5,4.5)[0.22]{}[f]{} (10.5,2.5)[0.22]{}[i]{} (11.5,2.5)[0.22]{}[j]{} (14.5,2.5)[0.22]{}[k]{} (11,1.5)[0.22]{}[o]{} (11,0.5)[0.22]{}[q]{} (9.5,2.5)[0.22]{}[m]{}
(20,5.3)[0.22]{}[c]{} (19.5,4.5)[0.22]{}[e]{} (20.5,4.5)[0.22]{}[f]{} (19.5,2.5)[0.22]{}[i]{} (20.5,2.5)[0.22]{}[j]{} (23.5,2.5)[0.22]{}[k]{} (20,1.5)[0.22]{}[o]{} (20,0.5)[0.22]{}[q]{} (18.5,2.5)[0.22]{}[m]{}
By symmetry the same applies to the subgraph of $H$ induced by the vertices $d,g,h$, $j,k,l,n,p,r$. Considering which pairs of these almost two-coloured perfect matchings are compatible and extend to an almost two-coloured perfect matching of $H$ shows that there are only six possibilities. In Fig. \[fig2\] three possible almost two-coloured perfect matchings are depicted. The only other possible almost two-coloured perfect matchings are obtained by interchanging the two colours. Clearly these all have the properties described in the lemma.
(1,0)(25,7) (5,7)[0.22]{}[a]{} (5,6)[0.22]{}[b]{} (3,5.3)[0.22]{}[c]{} (7,5.3)[0.22]{}[d]{} (2.5,4.5)[0.22]{}[e]{} (7.5,4.5)[0.22]{}[h]{} (3.5,4.5)[0.22]{}[f]{} (6.5,4.5)[0.22]{}[g]{} (2.5,2.5)[0.22]{}[i]{} (7.5,2.5)[0.22]{}[l]{} (3.5,2.5)[0.22]{}[j]{} (6.5,2.5)[0.22]{}[k]{} (3,1.5)[0.22]{}[o]{} (7,1.5)[0.22]{}[p]{} (3,0.5)[0.22]{}[q]{} (7,0.5)[0.22]{}[r]{} (1.5,2.5)[0.22]{}[m]{} (8.5,2.5)[0.22]{}[n]{}
(13,7)[0.22]{}[a]{} (13,6)[0.22]{}[b]{} (11,5.3)[0.22]{}[c]{} (15,5.3)[0.22]{}[d]{} (10.5,4.5)[0.22]{}[e]{} (15.5,4.5)[0.22]{}[h]{} (11.5,4.5)[0.22]{}[f]{} (14.5,4.5)[0.22]{}[g]{} (10.5,2.5)[0.22]{}[i]{} (15.5,2.5)[0.22]{}[l]{} (11.5,2.5)[0.22]{}[j]{} (14.5,2.5)[0.22]{}[k]{} (11,1.5)[0.22]{}[o]{} (15,1.5)[0.22]{}[p]{} (11,0.5)[0.22]{}[q]{} (15,0.5)[0.22]{}[r]{} (9.5,2.5)[0.22]{}[m]{} (16.5,2.5)[0.22]{}[n]{}
(21,7)[0.22]{}[a]{} (21,6)[0.22]{}[b]{} (19,5.3)[0.22]{}[c]{} (23,5.3)[0.22]{}[d]{} (18.5,4.5)[0.22]{}[e]{} (23.5,4.5)[0.22]{}[h]{} (19.5,4.5)[0.22]{}[f]{} (22.5,4.5)[0.22]{}[g]{} (18.5,2.5)[0.22]{}[i]{} (23.5,2.5)[0.22]{}[l]{} (19.5,2.5)[0.22]{}[j]{} (22.5,2.5)[0.22]{}[k]{} (19,1.5)[0.22]{}[o]{} (23,1.5)[0.22]{}[p]{} (19,0.5)[0.22]{}[q]{} (23,0.5)[0.22]{}[r]{} (17.5,2.5)[0.22]{}[m]{} (24.5,2.5)[0.22]{}[n]{}
We define what we call the *clause gadget graph* $K$ as follows, see Fig. \[fig13\]. Take three copies of $H$, namely $H_1$, $H_2$ and $H_3$. We label the vertices by adding the subscript $i \in \{1,2,3\}$ to the corresponding label of $H$. Now identify $a_1,a_2,a_3$ into a single vertex $a$, remove vertices $m_1,m_2,m_3,n_1,n_2,n_3$ and their incident edges and replace them with edges $l_1i_2,l_2i_3,l_3i_1$. Notice that $K$ is planar and every vertex has degree three, except for $q_1,q_2,q_3$ and $r_1,r_2,r_3$.
(7,7.5) (3.5,3.5)[0.075]{}[a]{} (3.7,3.6)[a]{} (3.5,4.5)[0.075]{}[b]{} (3.775,4.25)[$b_1$]{} (2.793,2.793)[0.075]{}[c]{} (2.65,3.15)[$b_2$]{} (4.207,2.793)[0.075]{}[d]{} (4.3,3.1)[$b_3$]{} (3,5)[0.075]{}[e]{} (4,5)[0.075]{}[f]{} (2.75,5.5)[0.075]{}[g]{} (3.25,5.5)[0.075]{}[h]{} (3.75,5.5)[0.075]{}[i]{} (4.25,5.5)[0.075]{}[j]{} (2.75,6)[0.075]{}[k]{} (2.5,6.1)[$l_1$]{} (3.25,6)[0.075]{}[l]{} (3.75,6)[0.075]{}[m]{} (4.25,6)[0.075]{}[n]{} (4.55,6.1)[$i_1$]{} (3,6.5)[0.075]{}[o]{} (4,6.5)[0.075]{}[p]{} (3,7)[0.075]{}[q]{} (4,7)[0.075]{}[r]{} (4.91,2.793)[0.075]{}[d1]{} (4.207,2.08)[0.075]{}[d2]{} (5.4,2.693)[0.075]{}[d3]{} (5,2.293)[0.075]{}[d4]{} (4.7,2)[0.075]{}[d5]{} (4.31,1.58)[0.075]{}[d6]{} (5.75,2.343)[0.075]{}[d7]{} (5.95,2.6)[$l_3$]{} (5.35,1.943)[0.075]{}[d8]{} (5.85,1.85)[0.075]{}[d9]{} (5.05,1.65)[0.075]{}[d10]{} (4.66,1.23)[0.075]{}[d11]{} (4.52,0.93)[$i_3$]{} (5.15,1.14)[0.075]{}[d12]{} (5.5,0.85)[0.075]{}[d13]{} (6.2,1.5)[0.075]{}[d14]{} (2.08,2.793)[0.075]{}[c1]{} (2.793,2.08)[0.075]{}[c2]{} (1.6,2.693)[0.075]{}[c11]{} (2,2.293)[0.075]{}[c12]{} (1.25,2.343)[0.075]{}[c13]{} (1,2.5)[$i_2$]{} (1.65,1.943)[0.075]{}[c14]{} (1.15,1.8)[0.075]{}[c15]{} (0.8,1.45)[0.075]{}[c16]{} (2.3,2)[0.075]{}[c21]{} (2.693,1.6)[0.075]{}[c22]{} (1.95,1.65)[0.075]{}[c23]{} (2.343,1.25)[0.075]{}[c24]{} (2.35,0.93)[$l_2$]{} (1.85,1.15)[0.075]{}[c25]{} (1.5,0.8)[0.075]{}[c26]{}
\[le:colouringc\] A two-colouring of $\bigcup^3_{t=1}\{o_t,q_t,p_t,r_t\}\cup \{a\}$ may be extended to an almost two-coloured perfect matching of $K$ if and only if
- For each $t=1,2,3$, $o_tq_t$, $p_tr_t$ are monochromatic and $o_t,p_t,q_t,r_t$ all receive the same colour.
- For exactly two values of $t=1,2,3$, the vertices $o_t,p_t,q_t,r_t$ receive the same colour as $a$.
We first show that any almost two-coloured perfect matching of $K$ must have the two properties in the lemma.
The first property is an immediate consequence of Lemma \[le:proph\].
To show that the second property holds, recall that exactly one neighbour of $a$ must receive the same colour as $a$. Let $b_{t_1}$ for $1 \le t_1 \le 3$ be this neighbour. Then from Lemma \[le:proph\] we know that $b_{t_1}$ must have the opposite colour to $o_{t_1},p_{t_1},q_{t_1},r_{t_1}$. Since the other neighbours of $a$, namely $b_{t_2}$ and $b_{t_3}$ for $t_2,t_3 \in \{1,2,3\} \backslash \{t_1 \}$, receive the opposite colour to $a$, the vertices $o_{t_2},p_{t_2},q_{t_2},r_{t_2},o_{t_3},p_{t_3},q_{t_3},r_{t_3}$ must receive the same colour as $a$.
Now we show that any two-colouring of $\bigcup^3_{t=1}\{o_t,q_t,p_t,r_t,b_t \}\cup \{a\}$ satisfying the conditions of the lemma may be extended to an almost two-coloured perfect matching of $K$. Suppose without loss of generality that $a$ is coloured black and $o_1,p_1,q_1,r_1$ are coloured white. Then colour $l_1,i_1$ black and $l_2,i_2,l_3,i_3$ white. This colouring may be extended to an almost two-coloured perfect matching using the colourings of Fig. \[fig2\] and the colourings obtained from those in Fig. \[fig2\] by interchanging the colours.
We now move a step towards the main result of this section with the following proposition.
\[prop:1\] Problem \[pro:matching\] is NP-complete if the input is restricted to cubic graphs.
Given an instance of Not-All-Equal 3SAT with clauses $C_1,...,C_m$, construct a graph as follows. For every clause $C$ take a copy of the clause gadget graph $K(C)$ and do the following. Suppose without loss of generality that $C$ has literals $x_1,x_2,x_3$. Label the two vertices of degree one of the subgraph $H_i$ of $K(C)$ and their neighbours in $H_i$ with $x_i$.
Now for each literal $x$ do the following. Suppose that literal $x$ appears in clauses $C_{i_1},...,C_{i_k}$. (If $x$ appears twice or three times in a clause $C_r$ then add $C_r$ twice or three times to this list.) For every $j=1,...,k-1$ remove either one of the vertices of degree one labelled $x$ from $K(C_{i_j})$ and from $K(C_{i_{j+1}})$ leaving two half-edges. Now identify these two half edges to form an edge joining $K(C_{i_j})$ and $K(C_{i_{j+1}})$. Finally do the same thing with the remaining two edges labelled $x$ in $C_{i_k}$ and $C_{i_1}$. We call the graph obtained $G$.
Now suppose that there is a solution $S$ of the instance of Not-All-Equal 3SAT given. For all literals $x$ in Not-All-Equal 3SAT, colour all vertices in $G$ that are labelled $x$ with colour white if $S(x)$ is true and black if $S(x)$ is false. We now show that this colouring can be extended to a two-coloured perfect matching of $G$. First note that all edges joining copies of $K$ are monochromatic since their end-vertices are labelled with the same literal. In the next step colour the vertex $a$ in each copy of $K$ so that it has the same colour as the vertices $o_t,p_t$ for exactly two values of $t=1,2,3$. This is possible because for $t=1,2,3$, $o_t,p_t$ are labelled with literals $x_1,x_2,x_3$ which cannot have all the same value as they belong to one clause. By Lemma \[le:colouringc\] we can extend the colouring of each copy of $K$ to an almost two-coloured perfect matching of $K$ which yields a two-coloured perfect matching of $G$.
Now suppose there is a two-coloured perfect matching of $G$. By Lemma \[le:colouringc\] the edges joining the copies of $K$ must be monochromatic. All vertices in $G$ labelled with the same literal therefore must have the same colour and in each copy of $K$, for exactly two values of $t=1,2,3$, the vertices $o_t,p_t$ receive the same colour. It follows that if we assign to each literal $x$ the value true if it is the label of white vertices and false it is the label of black vertices then we obtain a solution to Not-All-Equal 3SAT.
We call the edges joining copies of $K$ *identifying edges*.
In order to prove that Problem \[pro:planarmatching\] is NP-complete we still need to deal with edges that cross. For this reason we define the *uncrossing gadget* $U$ to be the graph depicted in Fig. \[fig80\].
(18,9) (4,0.5)[0.1]{}[z4]{} (4,0.5)[$z_4$]{} (4,1.5)[0.1]{}[z3]{} (4,1.5)[$z_3$]{} (3,2.5)[0.1]{}[u]{} (3,2.5)[$u$]{} (5,2.5)[0.1]{}[v]{} (5,2.5)[$x$]{} (3,3.5)[0.1]{}[n]{} (3,3.5)[$n$]{} (5,3.5)[0.1]{}[q]{} (5,3.5)[$q$]{} (2.5,4)[0.1]{}[l]{} (2.5,4)[$l$]{} (3.5,4)[0.1]{}[m]{} (3.5,4)[$m$]{} (5.5,4)[0.1]{}[r]{} (5.5,4)[$r$]{} (4.5,4)[0.1]{}[p]{} (4.5,4)[$p$]{} (1.5,5)[0.1]{}[v]{} (1.5,5)[$v$]{} (0.5,5)[0.1]{}[w]{} (0.5,5)[$w$]{} (3,4.5)[0.1]{}[k]{} (3,4.5)[$k$]{} (5,4.5)[0.1]{}[o]{} (5,4.5)[$o$]{} (3,5.5)[0.1]{}[f]{} (3,5.5)[$f$]{} (5,5.5)[0.1]{}[i]{} (5,5.5)[$i$]{} (2.5,6)[0.1]{}[e]{} (2.5,6)[$e$]{} (3.5,6)[0.1]{}[d]{} (3.5,6)[$d$]{} (5.5,6)[0.1]{}[j]{} (5.5,6)[$j$]{} (4.5,6)[0.1]{}[h]{} (4.5,6)[$h$]{} (3,6.5)[0.1]{}[c]{} (3,6.5)[$c$]{} (5,6.5)[0.1]{}[g]{} (5,6.5)[$g$]{} (4,7.5)[0.1]{}[b]{} (4,7.5)[$b$]{} (4,8.5)[0.1]{}[a]{} (4,8.5)[$a$]{} (6.5,6)[0.1]{}[s]{} (6.5,6)[$s$]{} (6.5,4)[0.1]{}[t]{} (6.5,4)[$t$]{} (7.5,5)[0.1]{}[z1]{} (7.5,5)[$z_1$]{} (8.5,5)[0.1]{}[z2]{} (8.5,5)[$z_2$]{}
(13,0.5)[0.1]{}[z4]{} (13,0.5)[$\alpha$]{} (13,1.5)[0.1]{}[z3]{} (13,1.5)[$\alpha$]{} (12,2.5)[0.1]{}[u]{} (12,2.5)[$\alpha'$]{} (14,2.5)[0.1]{}[v]{} (14,2.5)[$\alpha$]{} (12,3.5)[0.1]{}[n]{} (12,3.5)[$\alpha$]{} (14,3.5)[0.1]{}[q]{} (14,3.5)[$\alpha$]{} (11.5,4)[0.1]{}[l]{} (11.5,4)[$\beta'$]{} (12.5,4)[0.1]{}[m]{} (12.5,4)[$\beta$]{} (14.5,4)[0.1]{}[r]{} (14.5,4)[$\beta$]{} (13.5,4)[0.1]{}[p]{} (13.5,4)[$\beta'$]{} (10.5,5)[0.1]{}[v]{} (10.5,5)[$\beta$]{} (9.5,5)[0.1]{}[w]{} (9.5,5)[$\beta$]{} (12,4.5)[0.1]{}[k]{} (12,4.5)[$\alpha'$]{} (14,4.5)[0.1]{}[o]{} (14,4.5)[$\alpha'$]{} (12,5.5)[0.1]{}[f]{} (12,5.5)[$\alpha$]{} (14,5.5)[0.1]{}[i]{} (14,5.5)[$\alpha$]{} (11.5,6)[0.1]{}[e]{} (11.5,6)[$\beta'$]{} (12.5,6)[0.1]{}[d]{} (12.5,6)[$\beta$]{} (14.5,6)[0.1]{}[j]{} (14.5,6)[$\beta'$]{} (13.5,6)[0.1]{}[h]{} (13.5,6)[$\beta$]{} (12,6.5)[0.1]{}[c]{} (12,6.5)[$\alpha'$]{} (14,6.5)[0.1]{}[g]{} (14,6.5)[$\alpha'$]{}
(13,7.5)[0.1]{}[b]{} (13,7.5)[$\alpha$]{} (13,8.5)[0.1]{}[a]{} (13,8.5)[$\alpha$]{}
(15.5,6)[0.1]{}[s]{} (15.5,6)[$\beta'$]{} (15.5,4)[0.1]{}[t]{} (15.5,4)[$\beta'$]{} (16.5,5)[0.1]{}[z1]{} (16.5,5)[$\beta$]{} (17.5,5)[0.1]{}[z2]{} (17.5,5)[$\beta$]{}
\[le:uncrossgadget\] There exists an almost two-coloured perfect matching of $U$ if and only if $w,v,z_1,z_2$ have the same colour and $a,b,z_3,z_4$ have the same colour.
Consider the four-cycle on the vertices $c,e,f,d$ in $U$. In a two-coloured perfect matching exactly two of the vertices $c,e,f,d$ must receive the colour black and the other two must receive the colour white. There are two different ways of colouring them. The first way is that exactly two of the edges in the four-cycle are monochromatic, namely $ce$ and $df$, or $cd$ and $ef$. Then none of the edges $cb,dh,ev,fk$ can be monochromatic. The other way is that none of the edges in the four-cycle is monochromatic and all of the edges $cb,dh,ev,fk$ are monochromatic. The analogous thing is true for any four-cycle in $U$.
We now prove that in any almost two-coloured perfect matching the edges $ab,wv,z_1z_2$ and $z_3z_4$ are monochromatic. Suppose $ab$ is dichromatic. Then precisely one of $bc$ and $bg$ must be monochromatic. Without loss of generality assume $bc$ is monochromatic. Then the four-cycle on $c,e,f,d$ cannot have a monochromatic edge. It follows that $dh$ must be monochromatic. But then $bg$ must be monochromatic which is a contradiction. Thus $ab$ and by symmetry $wv$ must be monochromatic. Now suppose $z_1z_2$ is dichromatic. It follows that precisely one of $sz_1$ and $tz_1$ is monochromatic. Without loss of generality assume $tz_1$ is monochromatic. Then $js$ must be monochromatic. It follows that $io$ must be monochromatic and so must $rt$. This is not possible. Thus $z_1z_2$ and due to symmetry $z_3z_4$ must be monochromatic. Hence each of the four-cycles $(c,d,f,e),(g,h,i,j),(k,l,n,m),(o,p,q,r)$ contains exactly two monochromatic edges. So vertices that are opposite of each other in these four-cycles receive opposite colours. It is now easy to see that the only possible colourings are as shown in Fig. \[fig80\] where $\alpha,\beta \in \{\mbox{black},\mbox{white}\}$ and $\alpha'$ denotes the opposite colour to $\alpha$ and $\beta'$ denotes the opposite colour to $\beta$. The result then follows.
We are now able to prove that Problem \[pro:planarmatching\] is NP-complete.
\[th:cubplanp\] Problem \[pro:planarmatching\] is NP-complete.
Given an instance of Not-All-Equal 3SAT, construct the graph $G$ as in the proof of Proposition \[prop:1\]. This graph can be drawn in the plane so that the only edges that cross are the identifying edges, each pair of identifying edges crosses at most once and at most two edges cross at any point.
Now we replace the crossings one by one by replacing a pair of crossing edges by the uncrossing gadget. Suppose $\gamma$ and $\delta$ are two edges that cross. We delete $\gamma$ and $\delta$ and replace them with a copy of the uncrossing gadget attaching $w$ and $z_2$ to the end-vertices of $\gamma$, and $a$ and $z_4$ to the end-vertices of $\delta$. We will also call the four pendant edges $wv,ab,z_1z_2,z_3z_4$ in the uncrossing graph identifying edges. After each replacement we can draw the graph so that only identifying edges cross and such that there is one fewer crossing. We continue until there are no more crossing edges. The final graph can be constructed in polynomial time and is planar and cubic. Each original identifying edge in $G$ now corresponds to one or more identifying edges with each consecutive pair being on opposite sides of a copy of the uncrossing gadget. Lemma \[le:uncrossgadget\] shows that in a two-coloured perfect matching all of these edges must be monochromatic and all the end-vertices of these edges have the same colour.
Now the argument in Proposition \[prop:1\] shows that the final graph has a two-coloured perfect matching if and only if the instance of Not-All-Equal 3SAT is satisfiable.
$k$-$L(2,1)$-labelling for $k\ge4$ fixed {#sec:constgraph}
========================================
Let $G$ be a planar cubic graph. In order to establish our main result we will reduce Planar Cubic Two-Colourable Perfect Matching to Planar $k$-$L(2,1)$-Labelling for planar graphs for each $k \ge 4$. From any planar cubic graph $G$ forming an instance of Planar Cubic Two-Colourable Perfect Matching, we construct a graph $K$. As we see in the next section, the basic form of $K$ does not depend on $k$ but is formed by constructing an auxiliary graph $H$ and then replacing each edge of $H$ by a gadget which does depend on $k$. In this section we define these gadgets and analyse certain $L(2,1)$-labellings of them. Each gadget has two distinguished vertices, which will always be labelled $u$ and $v$, corresponding to the end-vertices of the edge that is replaced in the auxiliary graph defined in the next section. These two vertices have degree $k-1$ in $K$. Any vertex of degree $k-1$ must receive either label $0$ or $k$ in a $k$-$L(2,1)$-labelling because these are the only possible labels for which there are $k-1$ labels remaining to label the neighbours of that vertex, so we will analyse the $k$-$L(2,1)$-labellings of these gadgets in which $u,v$ receive label $0$ or $k$.
$\lambda_{2,1}(G) = 4$
----------------------
In this subsection the gadget $G_{4}$ is simply a path of length three. More precisely the gadget $G_{4}$ is given by $V(G_{4})=\{ u,a_u,a_v,v\}$ and $E(G_{4})=\{ua_u,a_ua_v,a_vv\}$. This gadget is used in [@fiala+kloks+kratochvil:fixed-parameter-lambda], from where we get the following lemma.
\[le:possib2\] There is a $4$-$L(2,1)$-labelling $L$ of $G_4$ with $L(u),L(v) \in \{0,4\}$ if and only if the following conditions are satisfied.
1. If $(L(u),L(v))=(0,0)$, then $(L(a_u),L(a_v)) \in \{(2,4),(4,2)\}$.
2. If $(L(u),L(v))=(4,4)$, then $(L(a_u),L(a_v)) \in \{(0,2),(2,0)\}$.
3. If $(L(u),L(v))=(4,0)$, then $(L(a_u),L(a_v)) =(1,3)$.
4. If $(L(u),L(v))=(0,4)$, then $(L(a_u),L(a_v)) =(3,1)$.
$\lambda_{2,1}(G) = 5$
----------------------
(4,4) (0.5,3.5)[0.075]{}[u]{} (0.5,3.5)[$u$]{} (3.5,3.5)[0.075]{}[v]{} (3.5,3.5)[$v$]{} (1.5,3.5)[0.075]{}[av]{} (1.5,3.5)[$a_u$]{} (2.5,3.5)[0.075]{}[au]{} (2.5,3.5)[$a_v$]{} (2.5,3)[0.075]{}[b1]{} (2.5,3)[$b_v$]{} (1.5,3)[0.075]{}[b2]{} (1.5,3)[$b_u$]{}
(2,2.5)[0.075]{}[b3]{} (2,2.5)[$c$]{} (2,2)[0.075]{}[c3]{} (2,2)[$d$]{} (2.5,2)[0.075]{}[c4]{} (2.5,2)[$e_3$]{} (2.5,1.5)[0.075]{}[d6]{} (2.5,1.5)[$e_1$]{} (1.5,1.5)[0.075]{}[d4]{} (1.5,1.5)[$e_2$]{} (2,1)[0.075]{}[e2]{} (2,1)[$f$]{} (1.5,0.5)[0.075]{}[f3]{} (1.5,0.5)[$g_1$]{} (2.5,0.5)[0.075]{}[f4]{} (2.5,0.5)[$g_2$]{}
Let $G_{5}$ be the graph depicted in Fig. \[fig16\].
\[possib\] There is a $5$-$L(2,1)$-labelling $L$ of $G_5$ with $L(u),L(v) \in \{0,5\}$ if and only if the following conditions are satisfied.
1. If $(L(u),L(v))=(0,0)$, then $(L(a_u), L(a_v))\in\{(2,5), (5,2), (3,5), (5,3)\}$.
2. If $(L(u),L(v))=(5,5)$, then $(L(a_u), L(a_v))\in\{(3,0), (0,3), (2,0), (0,2)\}$.
3. If $(L(u),L(v))=(0,5)$, then $(L(a_u), L(a_v))=(4,1)$.
4. If $(L(u),L(v))=(5,0)$, then $(L(a_u), L(a_v))=(1,4)$.
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1. By the definition of $L(2,1)$-labelling, both $L(a_u)$ and $L(a_v)$ belong to the set $\{2,3,4,5\}$. As $|L(a_u)-L(a_v)|\geq 2$, $$(L(a_u), L(a_v))\in\{(2,4), (4,2), (2,5), (5,2), (3,5), (5,3)\}.$$
Suppose for a contradiction that $(L(a_u), L(a_v))\in\{(2,4), (4,2)\}$. By symmetry, we may assume that $(L(a_u), L(a_v))=(2,4)$. Then $L(b_u)=5$ and $L(b_v)=1$. The vertices $d$ and $f$ have degree four and thus must receive labels from $\{0,5\}$. Because $\operatorname{dist}(b_u,d)=2$ we must have $L(d)=0$ and because $\operatorname{dist}(d,f)=2$ we must have $L(f)=5$. This implies that $\{L(e_1), L(e_2)\}=\{2,3\}$. But then vertex $c$ cannot be labelled, giving a contradiction. An $L(2,1)$-labelling is obtained if $(L(a_u),L(a_v))=(5,2)$ and $L(b_u)=3$, $L(b_v)=4$, $L(c)=0$, $L(d)=5$, $L(e_3)=1$, $L(e_2)=2$, $L(e_1)=3$, $L(f)=0$ $L(g_1)=5$ and $L(g_2)=4$ or if $(L(a_u),L(a_v))=(5,3)$ and $L(b_u)=2$, $L(b_v)=1$, $L(c)=4$, $L(d)=0$, $L(e_3)=5$, $L(e_2)=2$, $L(e_1)=3$, $L(f)=5$ $L(g_1)=0$ and $L(g_2)=1$. The other cases follow by symmetry.
2. Analogous to (i).
3. By the definition of $L(2,1)$-labelling, $L(a_u)\in \{2,3,4\}$ and $L(a_v)\in \{1,2,3\}$. As $|L(a_u)-L(a_v)|\geq 2$, $(L(a_u), L(a_v))\in\{(4,1), (4,2), (3,1)\}$. Suppose for a contradiction that $(L(a_u), L(a_v))\neq (4,1)$. By the label symmetry $x\mapsto 5-x$, we may assume that $(L(a_u), L(a_v))= (4,2)$. Hence $L(b_u)=1$ and $L(b_v)=0$. Now the vertices $d$ and $f$ have degree four and thus $L(d)=0,L(f)=5$ or $L(d)=5,L(f)=0$. As $\operatorname{dist}(b_v,d)=2$, $L(d)=5$ and as $\operatorname{dist}(d,f)=2$, $L(f)=0$. Now $\{L(e_1), L(e_2)\}=\{2,3\}$. But then vertex $c$ cannot be labelled, a contradiction. An $L(2,1)$-labelling is obtained if $(L(a_u),L(a_v))=(4,1)$ and $L(b_u)=2$, $L(b_v)=3$, $L(c)=0$, $L(d)=5$, $L(e_3)=1$, $L(e_2)=2$, $L(e_1)=3$, $L(f)=0$ $L(g_1)=5$ and $L(g_2)=4$.
4. Analogous to (iii).
$\lambda_{2,1}(G) \geq 6$
-------------------------
In this subsection we introduce for any $k \ge 6$ the gadget $G_k$ and consider some of its $L(2,1)$-labellings. The gadgets $G_6$ and $G_7$ are depicted in Fig. \[fig21\] and in Fig. \[fig50\], respectively.
(4,4) (0.5,3.5)[0.075]{}[u]{} (0.5,3.5)[$u$]{} (3.5,3.5)[0.075]{}[v]{} (3.5,3.5)[$v$]{} (1.5,3.5)[0.075]{}[av]{} (1.5,3.5)[$a_u$]{} (2.5,3.5)[0.075]{}[au]{} (2.5,3.5)[$a_v$]{} (2,2.5)[0.075]{}[b1]{} (2,2.5)[$b_1$]{} (1,2.5)[0.075]{}[b2]{} (1,2)[0.075]{}[c2]{} (0.5,2)[0.075]{}[c1]{} (0.5,1.5)[0.075]{}[d1]{} (1,1.5)[0.075]{}[d2]{} (1.5,1.5)[0.075]{}[d3]{} (1,1)[0.075]{}[e1]{} (0.5,0.5)[0.075]{}[f1]{} (1.5,0.5)[0.075]{}[f1]{}
(3,2.5)[0.075]{}[b3]{} (3,2)[0.075]{}[c3]{} (3.5,2)[0.075]{}[c4]{} (3.5,1.5)[0.075]{}[d6]{} (3,1.5)[0.075]{}[d5]{} (2.5,1.5)[0.075]{}[d4]{} (3,1)[0.075]{}[e2]{} (2.5,0.5)[0.075]{}[f3]{} (3.5,0.5)[0.075]{}[f4]{}
(4,3) (0.5,2.5)[0.025]{}[u]{} (0.5,2.5)[ $u$]{} (3.5,2.5)[0.025]{}[v]{} (3.5,2.5)[ $v$]{} (1,2.5)[0.025]{}[av]{} (1,2.5)[$a_u$]{} (3,2.5)[0.025]{}[au]{} (3,2.5)[$a_v$]{} (2,1)[0.025]{}[b1]{} (2,1)[$b_2$]{} (1.5,1)[0.025]{}[b2]{} (1.5,0.75)[0.025]{}[c2]{} (1.25,0.75)[0.025]{}[c1]{} (1.35,0.5)[0.025]{}[d1]{} (1.46,0.5)[0.025]{}[d2]{} (1.54,0.5)[0.025]{}[d2x]{} (1.65,0.5)[0.025]{}[d3]{} (1.5,0.25)[0.025]{}[e1]{} (1.4,0.15)[0.025]{}[f1]{} (1.6,0.15)[0.025]{}[f1]{}
(2.5,1)[0.025]{}[b3]{} (2.5,0.75)[0.025]{}[c3]{} (2.75,0.75)[0.025]{}[c4]{} (2.35,0.5)[0.025]{}[d6]{} (2.46,0.5)[0.025]{}[d5]{} (2.54,0.5)[0.025]{}[d5x]{} (2.65,0.5)[0.025]{}[d4]{} (2.5,0.25)[0.025]{}[e2]{} (2.4,0.15)[0.025]{}[f3]{} (2.6,0.15)[0.025]{}[f4]{} (2,2.25)[0.025]{}[b11]{} (2,2.25)[$b_1$]{} (1.75,2.25)[0.025]{}[b21]{} (1.75,2.15)[0.025]{}[c21]{} (1.65,2.15)[0.025]{}[c11]{} (1.65,2)[0.025]{}[d11]{} (1.72,2)[0.025]{}[d21]{} (1.78,2)[0.025]{}[d212]{} (1.85,2)[0.025]{}[d31]{} (1.75,1.85)[0.025]{}[e11]{} (1.65,1.75)[0.025]{}[f11]{} (1.85,1.75)[0.025]{}[f11]{}
(2.25,2.25)[0.025]{}[b31]{} (2.25,2.15)[0.025]{}[c31]{} (2.35,2.15)[0.025]{}[c41]{} (2.15,2)[0.025]{}[d61]{} (2.22,2)[0.025]{}[d51]{} (2.28,2)[0.025]{}[d512]{} (2.35,2)[0.025]{}[d41]{} (2.25,1.85)[0.025]{}[e21]{} (2.15,1.75)[0.025]{}[f31]{} (2.35,1.75)[0.025]{}[f41]{}
Let $H'$ be defined as follows, see Fig. \[fig22\].
(0,1)(4,4.75) (2,4.5)[0.075]{}[c]{} (2,4.5)[$c$]{} (2,3.75)[0.075]{}[d]{} (2,3.75)[$d$]{} (2.75,3.75)[0.075]{}[e]{} (2.75,3.75)[$e$]{} (2.75,3)[0.075]{}[f1]{} (2.75,3)[$f_{1}$]{} (2,3)[0.075]{}[f2]{} (2,3)[$f_{2}$]{} (1.25,3)[0.075]{}[f3]{} (1.25,3)[$f_{3}$]{} (2,2.25)[0.075]{}[g]{} (2,2.25)[$g$]{} (1.25,1.5)[0.075]{}[h]{} (1.25,1.5)[$h$]{} (2.75,1.5)[0.075]{}[i]{} (2.75,1.5)[$i$]{}
$$\begin{aligned}
V(H')=\{ c,d,e,f_1,...,f_{k-3},g,h,i \}\end{aligned}$$
and $$\begin{aligned}
E(H')= \{ &cd,de,hg,gi,gf_1,...,gf_{k-3},df_{1},...,df_{k-3} \}.\end{aligned}$$
\[le:labelhn\] For any $k \ge 6$ the graph $H'$ is planar and there exists an $L(2,1)$-labelling $L$ of $H'$ with span $k$ if and only if $$\begin{aligned}
(L(c),L(d)) \in \{(0,k),(1,k),(k-1,0),(k,0) \}.\end{aligned}$$
As seen from Fig. \[fig22\], $H'$ is planar for $k=6$. For any higher $k$ we need to connect $k-6$ paths of length two at $d$ and $g$ to the graph $H'$ in Fig. \[fig22\]. So $H'$ is planar for any $k \ge 6$.
If $L$ is a $k$-$L(2,1)$-labelling of $H'$ then $\{L(g),L(d)\} = \{0,k\}$ because $g$ and $d$ have degree $k-1$ and $\operatorname{dist}(g,d)=2$. It follows that $\{f_{1},...,f_{k-3}\} = \{ 2,...,k-2 \}$. If $L(d)=0$ and $L(g)=k$, then $\{ L(h),L(i) \}= \{ 1,0 \}$ and $\{L(c),L(e)\}=\{k-1,k\}$. Similarly if $L(d)=k$ and $L(g)=0$, then $\{ L(h),L(i) \}= \{ k-1,k \}$ and $\{L(c),L(e)\}=\{0,1\}$.
We now define the graph $G_k$ for $k \ge 6$. Take a path of length three with vertices $u,u_a,a_v,v$ and edges $ua_u, a_ua_v,a_vv$. Add vertices $b_1,...,b_{k-5}$, with each joined to $a_u$ and $a_v$. Now for each $i=1,...,k-5$, add two copies of $H'$ with the vertex labelled $c$ in each copy joined to $b_i$. So each of $b_1,...,b_{k-5}$ has degree four. To refer to the vertices in the two copies of $H'$ attached to $b_i$, we add a subscript of $(l,i)$ to the name of the vertices in one copy of $H'$ and $(r,i)$ to the name of the vertices in the other copy of $H'$. Notice that $G_k$ is planar.
\[th:col1\] There exists a $k$-$L(2,1)$-labelling $L$ of $G_k$ with $L(u)=L(v)=0$ if and only if $(L(a_v),L(a_u)) \in \{ (2,k),(k,2),(k-2,k),(k,k-2) \}$.
First notice that we need $k-5$ different colours to colour the vertices $b_1,...,b_{k-5}$ as they are all at distance two from each other. We first show that there exists a $k$-$L(2,1)$-labelling $L$ of $G_k$ with $L(u)=L(v)=0$ and $(L(a_u),L(a_v)) \in \{ (2,k),(k,2),(k-2,k),(k,k-2) \}$. The first case is when $L(a_u)=2$ and $L(a_v)=k$. Take $L(b_j)=j+3$ for $j=1,...,k-5$. A $k$-$L(2,1)$-labelling is obtained by setting $L(d_{l,j})=L(d_{r,j})=k$ and $L(c_{l,j})=0$ and $L(c_{r,j})=1$ for $1 \le j \le k-5$ and then using Lemma \[le:labelhn\] to give a valid labelling of the rest of the graph.
A similar argument shows that we may take $(L(a_u),L(a_v))=(k,2)$.
The second case is $L(a_u)=k-2$ and $L(a_v)=k$. Take $L(b_j)=j+1$ for $j=1,...,k-5$. A $k$-$L(2,1)$-labelling is obtained by setting $L(d_{l,j})=0$, $L(d_{r,j})=k$ and $L(c_{l,j})=k-1$ and $L(c_{r,j})=0$ for $1 \le j \le k-5$ and then using Lemma \[le:labelhn\] to give a valid labelling of the rest of the graph.
A similar argument shows that we may take $(L(a_u),L(a_v))=(k,k-2)$.
Next we show that there is no $k$-$L(2,1)$-labelling $L$ of $G_k$ with $L(u)=L(v)=0$ and $(L(a_v),L(a_u)) \not\in \{ (2,k),(k,2),(k-2,k),(k,k-2) \}$. Assume without loss of generality that $L(a_u) < L(a_v)$. Suppose first that $3 \le L(a_u) \le k-3$ and $L(a_v)=k$. Note that for any $j$ by considering the proximity of $b_j$ to $u$ and $a_v$, we see that $b_j$ cannot be labelled with $0,k-1$ or $k$. Furthermore we cannot have $b_j=1$ because then $\{L(c_{l,j}),L(c_{r,j})\}=\{k,k-1\}$. But they are both at distance two from $a_v$ which is labelled $k$, so this is not possible. So $b_1,...,b_{k-5}$ must receive distinct labels from $\{2,...,k-2\} \backslash \{ L(a_u)-1, L(a_u), L(a_u)+1 \}$, but this only gives $k-6$ labels which is not enough.
Now suppose $L(a_v) \not= k$ and $2 \le L(a_u) \le L(a_v)-2 \le k-3$. Note that for any $j$, $b_i$ cannot be labelled with $0$. Furthermore $b_j$ cannot be labelled $k$ as then $\{L(c_{l,j}),L(c_{r,j})\}=\{0,1\}$ and $L(d_{l,j})=L(d_{r,j})=k$ but this is invalid. So $b_1,...,b_{k-5}$ must receive distinct labels from $\{ 1,...,k-1\} \backslash \{ L(a_u)-1,L(a_u),L(a_u)+1,L(a_v)-1,L(a_v),L(a_v)+1\}$. This is only possible if $L(a_u)=k-3$ and $L(a_v)=k-1$. Then $\{L(b_1),...,L(b_{k-5})\} = \{ 1,...,k-5\}$. However if $L(b_j)=1$ then $\{L(c_{l,j}),L(c_{r,j})\}=\{k-1,k\}$. But this is invalid as $L(a_v)=k-1$.
Analogously we obtain the following lemma.
\[th:col2\] There exists a $k$-$L(2,1)$-labelling $L$ of $G_k$ with $L(u)=L(v)=k$ if and only if $(L(a_v),L(a_u)) \in \{ (2,0),(0,2),(k-2,0),(0,k-2) \}$.
\[th:col3\] There exists a $k$-$L(2,1)$-labelling $L$ of $G_k$ with $L(u)=k$ and $L(v)=0$ if and only if $(L(a_v),L(a_u)) =(k-1,1)$.
Let $l_1=\min\{L(a_u),L(a_v)\}$ and $l_2=\max\{ L(a_u),L(a_v)\}$. The vertices $b_1,...,b_{k-5}$ must be labelled with distinct labels from $S= \{ 1,...,k-1 \}\backslash \{ l_1-1,l_1,l_1+1,l_2-1,l_2,l_2+1\}$. The only way that this set can contain $k-5$ elements is when $(l_1,l_2)=(1,k-1)$, $(l_1,l_2)=(1,3)$ or $(l_1,l_2)=(k-3,k-1)$.
We first show that there exists a $k$-$L(2,1)$-labelling $L$ of $G_k$ with $L(u)=k,L(v)=0$ and $(L(a_v),L(a_u)) =(k-1,1)$.
We need $\{L(b_1),...,L(b_{k-5})\}=\{3,...,k-3\}$. Then let $L(c_{l,j})=0$, $L(d_{l,j})=k$, $L(c_{r,j})=k$ and $L(d_{l,j})=0$ for $1\le j \le k-5$. So by Lemma \[le:labelhn\] we obtain a valid labelling.
Next we show that there is no $k$-$L(2,1)$-labelling of $G_k$ with $L(u)=k$ and $L(v)=0$ and $(L(a_v),L(a_u))\not=(k-1,1)$.
Assume that $l_1=1$ and $l_2=3$ so $L(a_v)=3$ and $L(a_u)=1$. Then the vertices $b_1,...,b_{k-5}$ must take distinct labels from $ \{ 5,...,k-1\}$. However if $b_j$ is labelled $k-1$ the only label $c_{l,j}$ and $c_{r,j}$ can be labelled with is $0$ but $c_{l,j}$ and $c_{r,j}$ must have distinct labels. Therefore this labelling is not possible.
Now suppose $l_1=k-3$ and $l_2=k-1$ then $L(a_v)=k-1$ and $L(a_u)=k-3$. Then the vertices $b_1,...,b_{k-5}$ must take distinct labels from $ \{ 1,...,k-5\}$. However if $b_j$ is labelled $1$ the only label $c_{l,j}$ and $c_{r,j}$ can be labelled with is $k$ but $c_{l,j}$ and $c_{r,j}$ must have distinct labels. Therefore this labelling is not possible.
Summary
-------
The following theorem summarises the results of this section and follows immediately from Lemmas \[le:possib2\], \[possib\], \[th:col1\], \[th:col2\] and \[th:col3\].
\[th:colgadget\] Let $k\ge 4$ be fixed. There is a $k$-$L(2,1)$-labelling $L$ of $G_k$ with $L(u),L(v) \in \{0,k\}$ if and only if the following conditions are satisfied.
1. If $(L(u),L(v))=(0,0)$ then $$(L(a_u),L(a_v))\in \{ (2,k),(k,2),(k-2,k),(k,k-2) \}.$$
2. If $(L(u),L(v))=(k,k)$ then $$(L(a_u),L(a_v))\in \{ (2,0),(0,2),(k-2,0),(0,k-2) \}.$$
3. If $(L(u),L(v))=(k,0)$ then $(L(a_u),L(a_v))=(1,k-1)$.
4. If $(L(u),L(v))=(0,k)$ then $(L(a_u),L(a_v))=(k-1,1)$.
Planar $k$-$L(2,1)$-labelling is NP-complete for $k \ge 4$ {#sec:npcom4}
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We reduce Planar Cubic Two-Colourable Perfect Matching to Planar $k$-$L(2,1)$-Labelling. Suppose we are given a cubic planar graph $G$ corresponding to an instance of Planar Cubic Two-Colourable Perfect Matching. From $G$ we construct a graph $K$ which has the property that $K$ has a $k$-$L(2,1)$-labelling if and only if $G$ has a two-coloured perfect matching.
In order to show this we also construct an auxiliary graph $H$ and define what we call a *coloured orientation*. Then we show that $G$ has a two-coloured perfect matching if and only if $H$ has a coloured orientation and finally that $H$ has a coloured orientation if and only if $K$ has a $k$-$L(2,1)$-labelling.
$H$ is obtained by replacing every edge of $G$ with the gadget as depicted in Fig. \[fig610\], where the end-vertices of the edge being replaced are $u,v$.
(13,15) (0.5,7.5)[0.6]{}[a]{} (0.5,7.5)[$u$]{} (3.5,7.5)[0.6]{}[d]{} (6.5,10.5)[0.6]{}[i]{} (6.5,4.5)[0.6]{}[j]{} (9.5,7.5)[0.6]{}[o]{} (12.5,7.5)[0.6]{}[r]{} (12.5,7.5)[$v$]{} (6.5,13.5)[0.6]{}[u]{} (6.5,13.5)[in]{} (6.5,1.5)[0.6]{}[z]{} (6.5,1.5)[out]{}
The gadget has two special vertices labelled *in* and *out*, which we call the *invertex* and the *outvertex* and we explain in a moment. The edges incident with them are called the *inedge* and *outedge*, respectively. We use the phrase *auxiliary edge* to refer to a subgraph of $H$ that has replaced an edge of $G$, that is, any of the copies of the gadget from Fig. \[fig610\]. A coloured orientation of an auxiliary graph $H$ is a colouring of the vertices of $H$ with black and white and an orientation of some of the edges satisfying certain properties. The indegree and outdegree of a vertex $v$ are the number of edges oriented towards $v$ and the number of edges oriented away from $v$, respectively. Unoriented edges are not counted towards indegree and outdegree. A coloured orientation must satisfy the following properties.
- Every vertex is adjacent to at most one vertex of the opposite colour.
- An edge is oriented if and only if it is monochromatic.
- Every vertex except those labelled *out* has outdegree at most two and indegree at most one.
- Every vertex labelled *out* has indegree zero.
We say a coloured orientation is *good* if every vertex of degree three is adjacent to precisely one vertex of the opposite colour and has indegree and outdegree one.
\[le:main1\] Let $G$ be a cubic planar graph and let $H$ be the corresponding auxiliary graph. If $G$ has a two-coloured perfect matching then $H$ has a good coloured orientation.
First colour the vertices of $H$ that were present in $G$ with the same colour that they receive in $G$.
We next colour the vertices of auxiliary edges where both end-vertices of the corresponding edge in $G$ receive the same colour. The in- and outvertex receive the same colour as the end-vertices of the corresponding edge in $G$ and the vertices on the four-cycle receive the opposite colour. We orient this cycle to form a directed cycle, see Fig. \[fig61\].
(13,15) (0.5,7.5)[0.6]{}[a]{} (0.5,7.5)[$u$]{} (3.5,7.5)[0.6]{}[d]{} (6.5,10.5)[0.6]{}[i]{} (6.5,4.5)[0.6]{}[j]{} (9.5,7.5)[0.6]{}[o]{} (12.5,7.5)[0.6]{}[r]{} (12.5,7.5)[$v$]{} (6.5,13.5)[0.6]{}[u]{} (6.5,13.5)[in]{} (6.5,1.5)[0.6]{}[z]{} (6.5,1.5)[out]{}
Now we colour all the other vertices and orient edges as follows. Vertices remaining uncoloured all belong to auxiliary edges for which the end-vertices of the corresponding edge in $G$ are coloured differently in $G$. We start by choosing an auxiliary edge $e$ between a black vertex $v$ and a white vertex $w_1$ of $G$. In $G$, $v$ has two white neighbours and one black neighbour. Call the other white neighbour $w_2$. Colour the outvertex of $e$ black and orient the edge incident with it away from the outvertex. Now follow the shortest path from the outvertex, through $v$ and to the invertex of the auxiliary edge $vw_2$. Colour every uncoloured vertex on this path black and orient every edge consistently with the path. At each stage of this colouring/orientation process we will colour and orient a path like this from an outvertex of an auxiliary edge, through a vertex present in $G$ to an invertex of a neighbouring auxiliary edge, see Fig. \[fig70\]. It only remains to describe how to choose the outvertex and invertex pair forming the end-vertices of each path. The first pair is chosen as above. Otherwise, if at some stage, we colour an invertex of an auxiliary edge $f$ with colour $c$ and the outvertex $f$ is still uncoloured then at the next stage we form a path from the outvertex of $f$ colouring the vertices on it with the opposite colour to $c$. If the outvertex of $f$ has already been coloured then we choose another auxiliary edge for which the outvertex is uncoloured. This method ensures that at each stage there is at most one auxiliary edge with the outvertex coloured and the invertex uncoloured and at most one auxiliary edge with the outvertex not coloured but the invertex coloured. Such an uncoloured outvertex is always the next one to be coloured.
(4,4) (2,2)[0.075]{}[a]{} (2,2.5)[0.075]{}[b]{} (2.15,2.1)[$v$]{} (2,3)[0.075]{}[c]{} (2,3.5)[0.075]{}[d]{} (2,3.655)[$w_3$]{} (1.75,2.75)[0.075]{}[e]{} (1.25,2.75)[0.075]{}[g]{} (2.25,2.75)[0.075]{}[f]{} (2.75,2.75)[0.075]{}[j]{} (2.53,1.47)[0.075]{}[bb]{} (2.88,1.47)[0.075]{}[ee]{} (3.18,1.77)[0.075]{}[gg]{} (2.88,1.12)[0.075]{}[cc]{} (3.23,0.77)[0.075]{}[dd]{} (3.4,0.6)[$w_1$]{} (2.53,1.12)[0.075]{}[ff]{} (2.23,0.82)[0.075]{}[hh]{} (1.47,1.47)[0.075]{}[bbb]{} (1.47,1.12)[0.075]{}[eee]{} (1.77,0.82)[0.075]{}[ggg]{} (1.12,1.47)[0.075]{}[fff]{} (1.12,1.12)[0.075]{}[ccc]{} (0.77,0.77)[0.075]{}[ddd]{} (0.65,0.6)[$w_2$]{} (0.82,1.77)[0.075]{}[hhh]{}
Due to the construction process, every vertex of $H$ which is also present in $G$ is adjacent to exactly one vertex of the opposite colour and has indegree and outdegree one. Clearly the same is true for all vertices of degree three of auxiliary edges where both end-vertices of the corresponding edge in $G$ receive the same colour. Now consider an auxiliary edge which corresponds to a dichromatic edge $e$ in $G$. Due to the colouring/orientation process the invertex and outvertex must receive opposite colours and the shortest path from each of them to the end-vertices of $e$ with the same colour is monochromatic. It follows that all vertices of degree three on the auxiliary edge must be adjacent to exactly one vertex of the opposite colour and have indegree and outdegree one, see Fig. \[fig70\]. Therefore the method yields a good coloured orientation.
\[le:goodorient\] Let $G$ be a cubic planar graph and let $H$ be the corresponding auxiliary graph. If $H$ has a coloured orientation, then it has a good coloured orientation.
Consider the possible coloured orientations of an auxiliary edge. Because each vertex is adjacent to at most one of the opposite colour the only ways in which the four-cycle of an auxiliary edge may be coloured are with all four vertices receiving the same colour or with a pair of adjacent vertices receiving one colour and the other pair receiving the opposite colour. In the first case we may change the colours of the invertex and outvertex (if necessary) to be the opposite colour to that of the vertices in the four-cycle. (We also remove the orientation of the inedge and outedge if necessary.) In this way both the inedge and the outedge are dichromatic. In the second case the fact that each vertex is adjacent to at most one vertex of the opposite colour forces both the invertex and the outvertex to have the same colour as their neighbour.
From now on we will assume we have a coloured orientation with each auxiliary edge being coloured in this way. We will show that if $H$ has a coloured orientation then it has a good coloured orientation. Let $H'$ be formed from $H$ by deleting all the dichromatic, or equivalently unoriented edges, and consider a connected component $C$ of $H'$. In $H'$ every vertex has outdegree at most two and indegree at most one. So $C$ is either an isolated vertex, a directed circuit with a number of trees rooted on the circuit and directed away from the circuit or a directed rooted tree in which all edges are oriented away from the root. Bearing in mind the constraints on the in- and outdegree of the vertices, we see that every vertex of degree three in the auxiliary graph has total degree at least two in $H'$. Leaves of $H'$ correspond to invertices and roots with degree one correspond to either invertices or outvertices. Notice that the isolated vertices of $H'$ can only be invertices or outvertices and by the remarks at the beginning of the proof exactly half of the isolated vertices are invertices. Hence the numbers of invertices and outvertices appearing in $H'$ that are not isolated are equal. So the number of leaves of $H'$ is at most the number of roots of tree components. Consequently each connected component of $H'$, that is not just an isolated vertex, is either a path beginning at an ouvertex and ending at an invertex or a directed circuit. So every vertex of degree three in the auxiliary graph has one out-neighbour, one in-neighbour and one incident unoriented edge. Therefore the coloured orientation is good.
\[le:main2\] Let $G$ be a cubic planar graph and let $H$ be the corresponding auxiliary graph. If $H$ has a good coloured orientation then $G$ has a two-coloured perfect matching.
Consider a vertex $v$ of $G$ and let $w_1,w_2$ and $w_3$ be its neighbours in $G$. Suppose without loss of generality that $v$ is coloured black in the good coloured orientation of $H$. We will show that in $H$, two of the vertices $w_1,w_2,w_3$ are coloured white and one is coloured black. Then we only need to assign to any vertex in $G$ the colour it receives in the good coloured orientation in $H$ to obtain a two-coloured perfect matching of $G$.
Vertex $v$ has two black neighbours and one white neighbour in $H$. In the proof of Lemma \[le:goodorient\] we showed that in a good coloured orientation a terminal vertex of an auxiliary edge receives the opposite colour to the unique neighbour in the auxiliary edge of the other terminal vertex of the auxiliary edge. Thus two of the vertices $w_1,w_2,w_3$ must be coloured white and the other one black.
Now given an instance $G$ of Planar Cubic Two-Colourable Perfect Matching, we define an instance $K$ of $k$-$L(2,1)$-labelling. First form the auxiliary graph $H$. For every vertex $v$ of $H$ add sufficient vertices of degree one with edges joining them to $v$ to ensure that $v$ has degree $k-1$. Now replace each edge that was originally present in $H$ by the gadget $G_k$ identifying the vertices $u,v$ of $G_k$ with the two end-vertices of edges of $H$ being replaced. Finally for each outvertex $v$ choose a neighbour $w$ of $v$ with degree one and add $k-2$ vertices of degree one joined to $w$. To illustrate this, suppose that $k=4$ and $v$ is adjacent to $w_1,w_2,w_3$ in $G$. In Fig. \[fig300\] we show how the neighbourhood of $v$ is modified in $K$. Note that $K$ can be constructed from $G$ in time $O(n)$.
(1.1,0)(9,4) (7,2)[0.05]{}[a]{} (7,1.85)[$v$]{} (7,2.17)[0.03]{}[r]{} (7,2.34)[0.03]{}[r]{} (7,2.5)[0.05]{}[b]{} (6.92,2.58)[0.03]{}[r]{} (6.83,2.64)[0.03]{}[r]{} (7.08,2.58)[0.03]{}[r]{} (7.16,2.64)[0.03]{}[r]{} (7,3)[0.05]{}[c]{} (7.08,2.91)[0.03]{}[r]{} (7.16,2.83)[0.03]{}[r]{} (7,3.17)[0.03]{}[r]{} (7,3.34)[0.03]{}[r]{} (7,3.5)[0.05]{}[d]{} (7.0,3.7)[$w_1$]{} (3.0,3.7)[$w_1$]{} (6.75,2.75)[0.05]{}[e]{} (6.83,2.83)[0.03]{}[r]{} (6.91,2.91)[0.03]{}[r]{} (6.58,2.75)[0.03]{}[r]{} (6.41,2.75)[0.03]{}[r]{} (6.25,2.75)[0.05]{}[g]{} (6.15,2.65)[0.03]{}[h]{} (6.15,2.85)[0.03]{}[i]{} (6.15,2.85)[0.03]{}[i]{} (6.15,3)[0.03]{}[i1]{} (6.0,2.85)[0.03]{}[i2]{} (7.25,2.75)[0.05]{}[f]{} (7.42,2.75)[0.03]{}[r]{} (7.59,2.75)[0.03]{}[r]{} (7.75,2.75)[0.05]{}[j]{} (7.85,2.85)[0.03]{}[k]{} (7.85,2.65)[0.03]{}[l]{} (7.53,1.47)[0.05]{}[bb]{} (7.18,1.82)[0.03]{}[r]{} (7.36,1.66)[0.03]{}[r]{} (7.88,1.47)[0.05]{}[ee]{} (8.18,1.77)[0.05]{}[gg]{} (8.18,1.92)[0.03]{}[ii]{} (8.33,1.77)[0.03]{}[jj]{} (8.08,2.02)[0.03]{}[ii1]{} (8.28,2.02)[0.03]{}[ii2]{} (7.98,1.57)[0.03]{}[r]{} (8.08,1.67)[0.03]{}[r]{}
(7.64,1.47)[0.03]{}[r]{} (7.76,1.47)[0.03]{}[r]{} (7.88,1.12)[0.05]{}[cc]{} (7.99,1.01)[0.03]{}[r]{} (8.11,0.9)[0.03]{}[r]{}
(8.23,0.77)[0.05]{}[dd]{} (8.25,0.55)[$w_3$]{} (4.25,0.55)[$w_3$]{} (7.88,1.36)[0.03]{}[r]{} (7.88,1.24)[0.03]{}[r]{}
(7.53,1.12)[0.05]{}[ff]{} (7.43,1.02)[0.03]{}[r]{} (7.33,0.92)[0.03]{}[r]{} (7.23,0.82)[0.05]{}[hh]{} (7.08,0.82)[0.03]{}[kk]{} (7.23,0.67)[0.03]{}[ll]{} (7.53,1.36)[0.03]{}[r]{} (7.53,1.24)[0.03]{}[r]{} (7.64,1.12)[0.03]{}[r]{} (7.76,1.12)[0.03]{}[r]{}
(6.47,1.47)[0.05]{}[bbb]{} (6.65,1.65)[0.03]{}[r]{} (6.83,1.83)[0.03]{}[r]{} (6.47,1.12)[0.05]{}[eee]{} (6.57,1.02)[0.03]{}[r]{} (6.67,0.92)[0.03]{}[r]{} (6.77,0.82)[0.05]{}[ggg]{} (6.92,0.82)[0.03]{}[kkk]{} (6.77,0.67)[0.03]{}[lll]{} (6.67,0.57)[0.03]{}[lll1]{} (6.87,0.57)[0.03]{}[lll2]{} (6.47,1.23)[0.03]{}[r]{} (6.47,1.35)[0.03]{}[r]{} (6.12,1.47)[0.05]{}[fff]{} (6.02,1.57)[0.03]{}[r]{} (5.92,1.67)[0.03]{}[r]{} (6.12,1.12)[0.05]{}[ccc]{} (6.01,1.01)[0.03]{}[r]{} (5.89,0.89)[0.03]{}[r]{} (5.77,0.77)[0.05]{}[ddd]{} (5.6,0.55)[$w_2$]{} (1.6,0.55)[$w_2$]{} (6.12,1.23)[0.03]{}[r]{} (6.12,1.35)[0.03]{}[r]{} (6.23,1.12)[0.03]{}[r]{} (6.35,1.12)[0.03]{}[r]{}
(6.23,1.47)[0.03]{}[r]{} (6.35,1.47)[0.03]{}[r]{}
(5.82,1.77)[0.05]{}[hhh]{} (5.67,1.77)[0.03]{}[iii]{} (5.82,1.92)[0.03]{}[jjj]{} (6.9,0.1)[$v \in K$]{} (3,2)[0.05]{}[a]{} (3,1.85)[$v$]{} (3,3.5)[0.05]{}[d]{} (4.23,0.77)[0.05]{}[dd]{} (2.9,0.1)[$v \in G$]{} (1.77,0.77)[0.05]{}[ddd]{}
\[le:main3\] Let $G$ be a cubic planar graph and let $H$ be the corresponding auxiliary graph. Let $K$ be the instance of $k$-$L(2,1)$-labelling constructed from $G$ as described above. Then $H$ has a good coloured orientation if and only if $K$ has a $k$-$L(2,1)$-labelling.
Suppose that $K$ has a $k$-$L(2,1)$-labelling $L$. We now describe how to obtain a coloured orientation of $H$ from $L$. Any vertex in $H$ corresponds to a vertex of degree $k-1$ in $K$ and so must be coloured $0$ or $k$. Colour a vertex of $H$ white if it corresponds to a vertex labelled $0$ in $K$ and black if it corresponds to vertex labelled $k$ in $K$.
We orient some of the edges of $H$ as follows. If $uv$ is an edge of $H$ then there is a path $u,a_u,a_v,v$ between $u,v$ in $K$ where $u$ is adjacent to $a_u$ and $v$ is adjacent to $a_v$. Orient the edge $uv$ from $u$ to $v$ if and only if $a_v \in \{0,k\}$ and orient it from $v$ to $u$ if and only if $a_u \in \{0,k\}$. From Theorem \[th:colgadget\] it follows that in our colouring of $H$, each vertex of $H$ is adjacent to at most one vertex of the opposite colour, and an edge is oriented if and only if it joins two vertices of the same colour. Consider a vertex $v\in H$. All neighbours of $v$ in $K$ must receive different colours, so in $H$, $v$ has at most one incoming edge, and at most two outgoing edges. Finally let $u$ be an outvertex of $H$. Then $u$ is part of exactly one copy of the gadget $G_k$ and has a neighbour $w$ of degree $k-1$ that is not part of this copy of $G_k$. We have $\{L(u),L(w)\}=\{0,k\}$ which means that no other neighbour of $u$ is labelled $0$ or $k$ and hence $u$ has indegree $0$. Therefore $H$ has a coloured orientation and by Lemma \[le:goodorient\], $H$ has a good coloured orientation.
Now suppose that $H$ has a good coloured orientation. We will show how to construct a $k$-$L(2,1)$-labelling $L$ of $K$. First label all vertices $v$ in $K$ that appear in $H$, so that $L(v)=0$ if $v$ is coloured white in $H$ and otherwise $L(v)=k$. Next give labels to all the remaining vertices that appear in a copy of the gadget $G_k$. Let $uv$ be an edge of $H$ and suppose without loss of generality that $L(u)=0$. Let $u,a_u,a_v,v$ be the path of length three from $u$ to $v$ in $K$. If $uv$ is not oriented, let $L(a_u)=k-1$ and $L(a_v)=1$. If $uv$ is oriented from $u$ to $v$ then let $L(a_u)=2,L(a_v)=k$ and if $uv$ is oriented from $v$ to $u$ then let $L(a_u)=k,L(a_v)=2$. Furthermore if $w \in V(H)$ then, because we start from a good coloured orientation of $H$, its three neighbours in $K$ receive different labels. Then Theorem \[th:colgadget\] shows that $L$ may be extended so that any vertex appearing in a copy of $G_k$ receives a label. For each outvertex, its neighbour of degree $k-1$ must be labelled. This can be done because each edge in $H$ adjacent to an outvertex $x$ is oriented away from $x$, so one of the labels $0,k$ is always available. Finally the vertices of degree one form an independent set and are all adjacent to vertices of degree $k-1$ that have received label $0$ or $k$. So they may be labelled. Hence $K$ has a $k$-$L(2,1)$-labelling.
We now return to the main problem of this paper. The following theorem which is the main statement of this paper follows immediately from Theorem \[th:cubplanp\] and Lemmas \[le:main1\], \[le:goodorient\], \[le:main2\] and \[le:main3\].
Problem \[pro:main\] is NP-complete.
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[^1]: Brunel University, Kingston Lane, Uxbridge, UB8 3PH, UK. Supported by the EC Marie Curie programme NET-ACE (MEST-CT-2004-6724). [Nicole.Eggemann@brunel.ac.uk]{}
[^2]: projet Mascotte, I3S(CNRS and University of Nice-Sophia Antipolis) and INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia-Antipolis Cedex, France. Partially supported by the european project [FET - Aeolus]{}. [Frederic.Havet@sophia.inria.fr.]{}
[^3]: Brunel University, Kingston Lane, Uxbridge, UB8 3PH, UK. Partially supported by the Heilbronn Institute for Mathematical Research, Bristol, UK. [Steven.Noble@brunel.ac.uk]{}
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